The Search and Matching Model: Challenges and Solutions

Home , Macroeconomic model

Demetris Koursaros

Submitted in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

in the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2011

Demetris Koursaros

Abstract

This dissertation focuses on explaining the cyclicality of unemployment, job vacancies, job creation and market tightness in the US economy. The framework used to model unemployment and job creation throughout this work, is the search and matching model, created by Mortensen and Pissarides (1994). This dissertation proposes three different mechanisms to improve the performance of a dynamic stochastic general equilibrium model (DSGE) with search unemployment, to align the model’s predictions with the quarterly US data from 1955-2005.

The first chapter proposes a New Keynesian model with search and matching frictions in the labor market that can account for the cyclicality and persistence of vacancies, unemployment, job creation, inflation and the real wage, after a monetary shock. Motivated by evidence from psychology, unemployment is modeled as a social norm. The norm is the belief that individuals should exert effort to earn their living and free riders are a burden to society. Households pressure the unemployed to find jobs: the less unemployed workers there are, the more supporters the norm has and therefore the greater the pressure and psychological cost experienced by each unemployed searcher. By altering the value of being unemployed, this procyclical psychological cost hinders the wage from crowding out vacancy creation after a monetary shock. Thus, the model is able to capture the high volatility of vacancies and unemployment observed in the data, accounting for the Shimer puzzle. The paper also departs from the literature by introducing price rigidity in the labor market, inducing additional inertia and persistence in the response of inflation and the real wage after a monetary shock. The model's responses after a monetary shock are in line with the responses obtained from a VAR on US data.

In the second chapter I attempt to solve the amplification puzzle, the inability of the standard search and matching model to account for the volatility in vacancies and unemployment, by exploring the connection between R&D and employment. R&D affects product creation and product creation affects employment. An improvement in technology benefits the economy in two ways. Same products can be produced more efficiently and also new products are created. Empirical evidence suggests that the increase in production for already existing goods does not imply increases in employment, while new products are associated with increases in employment. The search and matching model implies that changes in technology do not imply large changes in employment for already existing goods which is in line with what the evidence suggest. However, when the search and matching model applies for sectors that innovate and produce new products, changes in employment significantly increase. Therefore, in this model I assume all agents need to innovate first before they create a job opening, because firms that invent new products are the ones that contribute more to the volatility of employment according to the evidence. Since ideas are cheaper to implement after a technological expansion, the cost of vacancies becomes countercyclical which boosts job creation and vacancies. The model can amplify the volatilities of vacancies, unemployment and market tightness approximately by up to 300 percent.

The third chapter investigates the macroeconomic implications from introducing perpetual learning in a simple search and matching model. When the agents with rational expectations are replaced with agents that are boundedly rational, the volatilities of vacancies, unemployment and market tightness are increased significantly. Job creation is connected to the present discounted value of future cash flows, which means that if agents do not form rational expectations, their forecasts of future cash flows are subject to periods of either excess optimism or excess pessimism. Those extra distortions of the agents' forecasts amplify the volatility of job creation. Therefore, the amplification puzzle arising from the search and matching model might be due to the strong assumption that agents are rational; thus, when agents need to form multi-period forecasts using past data as in Preston (2005) and Eusepi and Preston (2010), the search and matching model's amplification potential is enhanced. The model can replicate moments from quarterly US data from 1955Q1 to 2007Q4. However the more amplification added to the model through higher gain parameter, the further the correlations generated by the model are from the ones obtained from US data. That is because higher amplification induces vacancies to fluctuate around the rational expectations equilibrium less smoothly. Moreover, higher amplification through learning worsens the prediction of the model for the slope of the Beveridge curve.

Contents: Chapter 1 - Labor market dynamics when (un)employment is a social norm 1. Introduction ...... 2 2. The puzzles and the social norm mechanism ...... 5 2.1 The amplification puzzle...... 6 2.2 Social norms and the norm of unemployment ...... 7 2.3 Defining unemployment as a social norm ...... 10 2.4 A job market framework and unemployment as a social norm ...... 12 2.5 Unemployment as a social norm and the amplification puzzle...... 14 3. The Model ...... 15

3.1 The agents ...... 15 3.2 Households...... 16 3.2.1 Household Problem ...... 19 3.3 Intermediate firms ...... 20 3.3.1 The value of job and unemployment ...... 21 3.3.2 The value of a job to the firm ...... 22 3.3.3 The value of employment and unemployment ...... 22 3.3.4 The job creation condition ...... 23 3.3.5 The bargaining process ...... 24 3.3.6 The wage equation ...... 16 3.4 Retail firms ...... 28 3.5 Government and Monetary authority ...... 30 3.5.1 Government ...... 30 3.5.2 Monetary authority ...... 30 3.6 Equilibrium ...... 9 4. Calibration, estimation and results ...... 33

4.1 Calibration ...... 33 4.2 The estimation procedure ...... 34 4.3 The estimates ...... 35 4.4 The model’s ability to match the data ...... 37 4.5 The main ingredients ...... 37

4.5.1 The introduction of unemployment as a social norm ...... 37 4.5.2 The introduction of price stickiness in the intermediate sector...... 39 4.5.3 Other ingredients ...... 42 5. Conclusion ...... 43 6. Bibliography ...... 44 7. Appendix A ...... 49 8. Appendix B ...... 52 9. Appendix C ...... 54 10. Appendix D ...... 57 11. Appendix E ...... 60 12. Appendix F ...... 63 13. Appendix G ...... 64

Contents: Chapter 2 - R&D, Product Innovation and Employment 1. Introduction ...... 71 2. The impact of R&D on employment ...... 73 3. R&D firms ...... 76 2.1 The vacancy cost ...... 78 2.2 Puzzle and solution ...... 79 4. The Model ...... 82 4.1 Household problem ...... 82 4.2 The Bellman equations ...... 84 4.2.1 The firms ...... 84 4.2.2 The workers...... 85 4.2.3 Bargaining and wage ...... 85 4.2.4 The solution...... 89 4.2.5 The price of ideas and vacancy dynamics ...... 90 5. Calibration, estimation and results ...... 91

5.1 Calibration ...... 92 5.2 Result ...... 93 6. Conclusion ...... 94 7. Bibliography ...... 95

Contents: Chapter 3 - The Search and Matching Model when Agents are Learning 1. Introduction ...... 101 2. The model ...... 104 2.1 Households...... 105 2.1.1 The households problem ...... 106 2.1.2 The Bellman equations ...... 108 2.2 Firms ...... 109 2.2.1 Nash bargaining ...... 112 2.3 Household problem revisited ...... 113 2.3.1 The intertemporal budget constraint ...... 114 2.3.2 The consumption equation ...... 116 2.4 Equilibrium and aggregate dynamics ...... 116 2.5 Expectations ...... 118 2.5.1 Timing ...... 119 2.5.2 Perpetual learning ...... 120 2.6 The solution ...... 121 3. Simulation and results ...... 123

3.1 Calibration ...... 123 3.2 Data ...... 124 3.3 Results ...... 125 3.4 Alternative model specification ...... 131 3.5 The distribution of beliefs ...... 133 4. Conclusion ...... 134 5. Bibliography ...... 135 6. Appendix A ...... 137 7. Appendix B ...... 138 8. Appendix C ...... 139 9. Appendix D ...... 141 10. Appendix E ...... 145 11. Appendix F ...... 147

List of Figures & Tables: Chapter 1 1. Figure 1: Empirical impulse responses ...... 6

iii

2. Figure 2: Evidence from Clark (2003) ...... 9 3. Figure 3: Model outline ...... 17 4. Figure 4: Main Result ...... 65 5. Figure 5: Result without social norm ...... 66 6. Figure 6: The effect of the social norm ...... 66 7. Figure 7: Get rid of the job market specification ...... 67 8. Figure 8: Result without the job market ...... 67 9. Figure 9: Impulse responses with flexible prices ...... 68 10. Figure 10: Impulse responses and flexible prices in the intermediate sector ...... 69 11. Figure 11: Responses without vacancy adjustment costs ...... 69 12. Table 1: The calibration ...... 35 13. Table 2: The estimated parameters ...... 36

List of Figures & Tables: Chapter 2 1. Figure 1: Model outline ...... 81 2. Figure 2: Impulse responses of vacancies ...... 98 3. Figure 3: Impulse responses of unemployment ...... 98 4. Figure 4: Impulse responses of market tightness ...... 99 5. Table 1: The calibration ...... 92 6. Table 2: The table of second moments ...... 93

List of Figures & Tables: Chapter 3 1. Figure 1: Impulse responses with gain 0.0055 ...... 156 2. Figure 2: Impulse responses with gain 0.006 ...... 157 3. Figure 3: Responses of vacancies ...... 158 4. Figure 4: Convergence of the belief parameters with decreasing gain ...... 160 5. Figure 5: The distribution of beliefs ...... 161 6. Table 1: The calibration ...... 124 7. Table 2: US data moments...... 153 8. Table 3: RE moments ...... 153

9. Table 4: Moments with gain 0.0045 ...... 153 10. Table 5: Moments with gain 0.005 ...... 154 11. Table 6: Moments with gain 0.0055 ...... 154 12. Table 7: Amplification ...... 154 13. Table 8: Autocorrelations ...... 155 14. Table 9: The Beveridge curve ...... 155 15. Table 10: Moments for alternative model specification ...... 155 16. Table 11: Means and medians of the belief distributions ...... 159

Acknowledgements

I would like to express a deep debt of gratitude to my advisers, Professors Bruce Preston, Michael Woodford and Ricardo Reis for providing me with their advice through all the years I spent at Columbia University. Their guidance and encouragement has contributed a lot to be able to finish this dissertation and accumulate the knowledge to follow my career path. I am also grateful to Jon Steinsson, Jaromir Nosal, Emi Nakamura, Marc Giannoni and John Donaldson for their useful comments and advice. Moreover, I would like to thank all the participants of the Macro-colloquium at Columbia University for their participations in those useful presentations. Last but not least, I would like to thank Jody Johnson for helping me through all the bureaucracy during all my years as a student at Columbia University.

Labor market dynamics when (un)employment is a social

norm

12 December 2009

Abstract

This paper proposes a New Keynesian model with search and matching frictions in the

labor market that can account for the cyclicality and persistence of vacancies, unemployment,

job creation, in‡ation and the real wage, after a monetary shock. Motivated by evidence

from psychology, unemployment is modeled as a social norm. The norm is the belief that

individuals should exert e¤ort to earn their living and free riders are a burden to society.

Households pressure the unemployed to …nd jobs: the less unemployed workers there are,

the more supporters the norm has and therefore the greater the pressure and psychological

cost experienced by each unemployed searcher. By altering the value of being unemployed,

this procyclical psychological cost hinders the wage from crowding out vacancy creation after

a monetary shock. Thus, the model is able to capture the high volatility of vacancies and

unemployment observed in the data, accounting for the Shimer puzzle. The paper also departs

from the literature by introducing price rigidity in the labor market, inducing additional inertia

and persistence in the response of in‡ation and the real wage after a monetary shock. The

model’sresponses after a monetary shock are in line with the responses obtained from a VAR

on US data.

1 1 Introduction

The search and matching model introduced by Mortensen and Pissarides (1994), (MP model), has been a workhorse for economists in the last couple of decades. Although the MP model outperforms the standard RBC model with no labor market frictions (Merz (1994), Andolfatto (1996)), it fails in explaining the strong cyclicality and persistence of key labor market variables such as vacancies, unemployment and job creation (Shimer (2005a), Fujita (2003)). Moreover, it fails to account for the inertial and persistent response of in‡ation after a monetary shock, (Trigari (2006), Krause and

Lubik (2006) and Christo¤el and Linzert (2005)).

The ampli…cation puzzle, the di¢ culty of the standard MP model to match the volatility of unemployment and vacancies according to Shimer (2005a), lies in the particular wage formation method imposed by the standard search and matching framework, which makes the wage depend on workers’outside options. During an expansion, vacancy posting increases, boosting hiring and forcing unemployment duration to decrease. Lower unemployment duration increases the value in the unemployment state for a worker, strengthening their bargaining power which entitles them a higher wage contract. Thus, the wage becomes so responsive to workers’outside opportunities that ends up absorbing most of the e¤ect of the shock, leaving pro…ts barely a¤ected and the incentive to post vacancies too low. The ampli…cation puzzle exists not only for productivity shocks, but also for monetary and other exogenous forces as well.

In addition, Shimer (2005a) argues that the standard MP model fails to create enough propagation of the shocks, forcing the responses to die out immediately after the shocks. The contempo- raneous correlations between the market tightness and the productivity shock is equal to -1 instead of -0.4 as the data suggest. The propagation problem (as well as the ampli…cation problem) for the case of monetary shocks, is shown by Christiano, Trabandt and Walentin (2009), where the authors …nd that enough propagation can be created by the search and matching framework only by assuming high degrees of price rigidity for individual …nal-good …rms.

This paper presents a New Keynesian model with nominal rigidities and search and matching frictions. My new modi…cations are the following:

2 1. Unemployment as a social norm. Households support the norm that everyone should

put e¤ort to earn their living and free riders are frowned upon. Thus, there’s pressure on

unemployed workers to …nd jobs. The fewer the unemployed in a group of people, the more

supporters the norm has and the greater the loss or reputation or psychological cost su¤ered

by each unemployed person within the group. There are two unemployment rates a¤ecting

workers’bargaining power, each having an opposite e¤ect on the underlying wage. The …rst

is the economy-wide unemployment rate and the second is the unemployment rate of relevant

others (closest people in someone’s immediate environment). In an expansion, the economy-

wide unemployment rate decreases, decreasing unemployment duration and strengthening

workers’bargaining power as argued by Shimer (2005a). Moreover, during an expansion, the

decreasing unemployment rate of relevant others implies, according to the norm, a greater

loss of reputation for the unemployed within a group, weakening the workers’ bargaining

power. The two opposite e¤ects counterbalance each other preventing the wage from being too

responsive to outside opportunities and hindering it from absorbing most of the e¤ect of the

shock. The importance of social norms in restoring the missing motivation in macroeconomic

models is stressed in Akerlof (2006).

2. Price rigidity in the labor market. I assume every worker is a …rm, producing an interme-

diate di¤erentiated good, facing monopolistic competition and price rigidity. Price rigidity in

the intermediate goods’sector not only adds in‡ation inertia ala Christiano, Eichenbaum and

Evans (2005) but also adds persistence in the model without the need to impose high degrees

of price rigidity. Moreover, price rigidity in the labor market smooths out the response of the

real wage after a monetary shock while also signi…cantly reduces the real wage’s volatility,

which tends to be excessively volatile when monetary shocks are introduced in a standard MP

framework. It is important to note that this model assumes price rigidity, while assuming no

rigidities in the wage bargained by …rms and workers.

I identify a structural VAR on U.S. data and compare the impulse responses implied by my model with the ones obtained by the VAR. My primary goal is to explain the persistent and

3 volatile responses of the labor market variables and at the same time explain the smooth and low responses of in‡ation and hourly wages after a monetary shock. The contribution of this paper is twofold. On one hand, it aims to enhance the performance of the standard MP model in order to be in-line with evidence on vacancies, unemployment and job creation. On the other hand it contributes to the literature of augmenting the standard New Keynesian model with search and matching frictions, in order to explain among others, in‡ation and wage behavior after a monetary shock. I propose new mechanisms to bring the model closer to the data, while borrowing others from the literature.

There are a few attempts to modify a New Keynesian model with the search and matching framework in order to explain the above. The …rst attempt was made by Trigari (2004). Even though the model has no predictions for the responses of the wage, unemployment and vacancies, it shows how the inclusion of search frictions improves the performance of the basic New Keynesian model, to account for in‡ation and output responses to a monetary shock. After Trigari (2004), more New Keynesian models with search frictions appeared, such as Braun (2005) and Kuester

(2007). The main element that ampli…es the responses of vacancies and unemployment in both models is wage rigidity. However, the empirical validity of wage rigidity for new …rms has been rejected recently by Pissarides (2007) and Haefke, Sonntag and van Rens (2008). In addition, both models assume a value for unemployment bene…ts following the calibration of Hagedorn and

Manovskii (2007) who claim that high unemployment bene…ts can amplify shocks. The critique on this approach is that those models lead to the counterfactual prediction that unemployment bene…t policy is extremely e¤ective as argued by Costain and Reiter (2007).

Section 2 presents the facts, the empirical responses from a structural VAR on U.S. data and the empirical evidence supporting the existence of unemployment as a social norm. In the same section, I present the theoretical framework to incorporate the unemployment as a social norm in the job searching environment. In section 3, I present the model in detail, analyzing the behavior of each of the agents. Section 4 presents the calibration along with the estimation1 procedure and

1 I calibrate some parameters to commonly used values from independent studies, while I estimate the rest to match the impulse responses implied by the model, with the ones obtained by a structural VAR on U.S. data after a monetary shock.

4 results.

2 The puzzles and the social norm mechanism

Figure 1 shows the responses to a unit monetary shock estimated by a structural VAR. The U.S. data2 are from 1955:1 to 2005:1 and the identi…cation assumption is that the only variable contemporaneously correlated with the monetary shock is the nominal interest rate, while the remaining variables respond with a lag.

The variables3 are: the log of quarterly real GDP, the annualized quarterly rate of change in the

CPI, the log of Help-Wanted advertising, the civilian unemployment rate, the log of job creation for continuing establishments in the manufacturing sector, the log of average hourly earnings in the business sector, the log of total hours in the business sector and the e¤ective federal funds rate4 .

The solid lines in …gure 1 are the empirical impulse responses from the SVAR and the grey regions the associated 90% con…dence intervals.

The propagation puzzle is evident from …gure 1. All the variables are very persistent, especially output, in‡ation, vacancies, unemployment and the real wage, as the responses die out between the

…fteenth and the twentieth quarter while achieving the maximum e¤ect around the tenth quarter.

Especially, in‡ation persists even after the twentieth quarter. Therefore, a need for a model that can provide enough propagation is necessary to explain the above facts.

Moreover, the ampli…cation challenge is evident by examining the magnitude of the responses of vacancies, unemployment and job creation. The response of vacancies is close to eight times larger than the response of output, while the response of unemployment is around eight times the response of output. Even though job creation is not persistent, it is very volatile with magnitude around …ve times that of output. Thus, a model equipped with a strong ampli…cation mechanism is eminent.

The real wage responses are very low, nearly half the response of the real GDP and quite inertial.

2 I consider the data set up to 2005:1 because of the availability of data for job creation up to that date. 3 All data series come from the St. Louis economic database except for the job creation series. 4 The job creation variable is coming from the job creation and destruction database by Davis, Haltiwanger and Shuh (1996)

5 0.4 0.6 2

0.4 1 0.2 0.2 0 0

0 •1

Inflation

Real GDP V acancies •0.2 •0.2 •2

•0.4 •0.4 •3 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 time time time

3 0.2 1.5

2 0.1 1

1 0 0.5 Wage

0 •0.1 fundsFed rate 0 Unemployment

•1 •0.2 •0.5 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 time time time

1 0.4

0.2 0

0 Hours •1

Job Creation •0.2

•2 •0.4 0 5 10 15 20 0 5 10 15 20 time time

Figure 1: Impulse responses to 1% monetary shock for U.S. data 1955:1 to 2005:1. Shaded areas are 90% con…dence intervals.

It is particularly challenging for a model that combines the search and matching speci…cation along with monetary shocks, to explain the low response of the real wage, as shown by Christiano,

Trabandt and Walentin (2009), as the model tends to overshoot the response of wages after a monetary shock.

2.1 The ampli…cation puzzle

The puzzle arises from the particular bargaining problem which generates a speci…c wage equation.

Wages not only include match-speci…c characteristics but also re‡ect outside opportunities. The outside opportunity for workers is the value of unemployment. Because in any stage of the bargain, the worker threatens to quit and look elsewhere for a job, the value of unemployment (the outside option) becomes a part of the wage equation.

The standard MP model assumes than in an expansion, vacancy creation increases and unemployment decreases. This will decrease the unemployed workers’ unemployment duration and consequently the value of unemployment. Thus, when the workers bargain and threaten to quit and

6 look for a job elsewhere, their threat becomes more credible when the outside conditions are bene…- cial for the unemployed. This increases the bargaining power of the workers too much and enables them to negotiate a higher wage contract. The higher wage decreases …rm pro…ts and decreases the incentive to post vacancies. In other words, in an expansion, the increase in vacancies triggers a signi…cant improvement in workers’outside options (value in unemployment state), which through a wage increase reduces the initial increase in job vacancies.

2.2 Social norms and the norm of unemployment

The way I solve the ampli…cation puzzle is through the introduction of unemployment as a social norm. Norms are rules of behavior members of a social group are expected to follow and distinguish appropriate from inappropriate behavior. Failure to follow the rule or code might result in loss of reputation within the group, isolation, guilt, psychological pressure or even discharge from the group. The consequences from disobeying the code depend on how strong is the support of the norm within the group members. The stronger the support of a norm, or a sudden increase in the supporters of the norm, exacerbates the loss of reputation or the psychological cost of those who deviate from the code. Akerlof (1980) investigates the importance of norms in economic models and

Akerlof (2006) emphasizes that macro models have important missing motivation in the preferences assumed simply by overlooking the existence of norms.

There are many examples of norms that de…ne the behavior of a group of individuals interacting with each other. A norm can be a simple handshake, gift exchanging on birthdays or respecting and supporting the elder and the disabled within a group. Even the idea of external habit formation

(catching up with the jonesses), can be thought as an embodiment of a social norm in macroeconomic models. The norm in the case of habit would be the belief that an individual’s consumption capabilities are evident for the individual’s success within the group. Failure to match or surpass the others’ consumption will result in loss of reputation within the group. The greater is the consumption of the rest in the group from one’sown, the greater the loss of reputation experienced by that individual.

In this paper, the social norm is that every individual should be a productive and useful member

7 of the society by earning their living with their own e¤ort, or in other words, the unemployed are free riders and free riders are nothing more than a burden to society. According to the norm, the unemployed are clearly the code breakers, consuming part of the economy’s goods and services without contributing at all in the production process. Since the unemployed break the rule, they su¤er disutility from loss of reputation within the group. The loss of reputation can be accompanied by feelings of uselessness, isolation, depression and constant feeling of being pressured by others

(friends, family, society) to …nd a job.

The larger the unemployment within the group is, the fewer the supporters of the norm, therefore, the less severe is the loss of reputation or psychological cost su¤ered by each unemployed individual. In other words, the more unemployed workers there are in one’s immediate environment, the less the feeling of embarrassment, uselessness or inadequacy encountered by the particular unemployed individual. Moreover, since households put pressure on the unemployed to …nd jobs, the more unemployed there are, the less is the pressure experienced by the individual unemployed person. The blame feels always less intense if it can be shared among others.

There is empirical evidence supporting unemployment as a social norm. Clark (2003) uses as a proxy for utility the General Health questionnaire5 (GHQ) which is a measure of mental well-being

(see Goldberg, 1972), and Clark’s goal among others, is to examine the e¤ect of relevant others’ unemployment, on ones own unemployment experience. The unemployment of relevant others is de…ned in three alternative ways: the regional unemployment rate; the unemployment status of the partner of the individual; the unemployment status of all other adults within the same household.

Irrespective of which one of the three de…nitions is used to represent relevant others, the author reports that increases in the unemployment rate of relevant others increases the psychological well- being of the individual unemployed worker.

Figure 2, coming from Clark (2003), presents some preliminary evidence of the positive impact of others’unemployment on the well-being of the unemployed, where the measure of others’unemployment is the regional unemployment rate. In the …gure, the psychological cost of unemployment is the di¤erence between GHQ values between employed and unemployed and is larger the larger the

5 The GHQ is an indicator of psychological health state and is used extensively in medical, psychological and socioeconomic research.

8 Figure 2: From Clark (2003), a graph of the negative correlation between unemployment psychological cost (CHQ, General Health Questionnaire) and the unemployment rate for various British regions. The regression line has a slope of -0.127.

GHQ di¤erence between employed and unemployed is. The …gure plots average psychological cost of unemployment against the unemployment rate in di¤erent geographical regions in Britain. It is prominent from the …gure that when an unemployed person moves to a region with a higher unemployment rate, experiences an improvement in psychological well-being. The results are unchanged when the regression includes other individual characteristics.

The e¤ect of relevant others on one’s own unemployment becomes more exaggerated when the author alternatively de…nes others’unemployment as the unemployment rate of other household members instead of regional unemployment, since the former de…nition captures the idea of a person’simmediate environment more accurately. Therefore one of the key …ndings of Clark’spaper is: the psychological cost of individual unemployment is signi…cantly lower when more members of the household become unemployed which veri…es my intuition that the loss of reputation or blame hurts less if can be shared among others.

In a similar exercise, Clark (2009) reports the same evidence for OECD data and Clark, Knabe

9 and Ratzel (2009) con…rm the result in the case of German regions. Powdthavee (2007) also comes to the same conclusion in an experiment with South African regions. Unemployment as a social norm is also supported by Stutzer and Lalive (2004), where the intensity of the social norm is constructed by votes in favour of lowering unemployment bene…ts from a referendum in Switzerland in 1997. In the regions where votes against unemployment bene…ts have been the most, the unemployed reported signi…cantly lower psychological well-being. Also similar results supporting unemployment as a social norm are reported by Shields and Price (2005) for United Kingdom and Shields, Price and

Wooden (2008).

Those results are also in-line with other studies in unemployment psychology that take a di¤erent route in de…ning well-being, other than GHQ. For example Jackson and Warr, (1987) report better mental health for the unemployed in higher unemployment rate regions. Similar claims are made by other authors using suicide and para-suicide rates who report that suicide and para-suicide attempts occur the most in low unemployment regions e.g. Platt, Micciolo and Tansella (1992), Platt and

Kreitman (1990), or Neeleman (1998).

2.3 De…ning unemployment as a social norm

I follow Akerlof (1980) and speci…cally his social norm formulation, to incorporate in my model the e¤ect of the unemployment of relevant others on the individual unemployed, as the empirical evidence suggest. Akerlof de…nes a social norm by augmenting a usual utility function with the reputation function:

R = R(A; ) (1) where A is a dummy variable that determines an agent’s obedience or disobedience to the com- munity’s rules of behavior and is the portion of the group’s population that support the rule.

Akerlof’ssocial norm speci…cation suggests that an individual who disobeys the rule, has to su¤er disutility from loosing reputation in the group, whilst the more group members there are disobeying the rule (low ), the less is the loss of reputation su¤ered by each individual that goes against the rule.

10 Following Akerlof, I introduce in a standard utility function the following:

1+` u U u Gu = ht (2) u 1 +` Uht ! u where Uht is the unemployment rate of relevant others, the unemployment rate in an individual’s immediate environment, which comprises friends, other household members, family, etc. The parameter controls the relative importance of the social norm in the utility of the individual, relative to the utility from consumption and leisure. In addition, the parameter `u governs the curvature of this disutility function. The parameter u, determines the extent the reputation e¤ect depends on the supporters of the norm. The greater u is, the greater the impact of the unemployment of relevant others’on the psychological well-being of the respective unemployed person. I call the parameter u: the Relevant Others’Psychological E¤ect (ROPE). Every unemployed worker in my model su¤ers disutility from loss of reputation within their own household according to equation (2). The larger the unemployment of relevant others Uht within the group, the less is the loss of reputation experienced by the individual unemployed. The unemployment rate Uht corresponds to in Akerlof’ssocial norm speci…cation, equation (1). This speci…cation captures the relation between unemployment psychological cost and the unemployment of relevant others’observed in the empirical evidence.

The speci…c functional form for the social norm in equation (2), I derive with a simple model for the job market, where the above function arises in the utility function as disutility from job searching. There is no evidence as to what functional form an unemployment norm should have, thus I derive the functional form (2), from other widely used functional forms in macroeconomic models. Those functional forms arise naturally from the way the rest of the model is speci…ed.

I simply assume, the same functional forms that determine the employed’s production function and disutility from work, are also the same functional forms that determine the unemployed’s

"searching" production function and disutility from search. The next section derives my reputation function more formally, giving a meaning to the parameters that appear in equation (2).

11 2.4 A job market framework and unemployment as a social norm

As in the standard MP model, potential employers are posting vacancies every period. Each unemployed worker provides application units to employers, to claim a position in the job openings6 .

The …xed application units per unemployed worker is in line with Shimer (2005a) …ndings that search intensity is acyclical, according to the Current Population Survey data. The matching function each period is: ~ 1 Ht = hFt Vt

where Ft is the total application units provided every period by households. In total, the application units demanded by a given household is:

D Fht = Uht (3)

th where Uht = 1 Nht is the unemployment rate within the h household. D Workers can produce the Fht = Uht units of aggregate application units within the household, S u 7 by putting e¤ort according to the following production function Fht = Lht. The aggregate e¤ort u supplied, Lht is produced by combining each unemployed worker’sindividual e¤ort by the following function: u u 1 u 1 Lu = Au lu u dj (4) ht t 2 hjt 3 UZht 4 5 u th u u where lhjt is the e¤ort provided by the j unemployed worker that induces G (lt ) units of disutility per unemployed worker that gives rise to the reputation function (2). This disutility su¤ered by the unemployed is going to be an important part of the match surplus and also of the wage equation

6 I assume the application units are …xed, but in principle, the more application units transferred by a worker relative to the rest, the more likely it is become employed relative to the others, thus the application units could be a choice variable for the household. 7 I implicitly assume there is a …rm selling "aggregation" services, which takes the individual unemployed’ssearch e¤ort units as inputs and produces the aggregate application units via equation (4), which then sells back to the household. For simplicity I leave the full de…nition of such framework in the appendix A.

12 that will eventually enhance the model’sampli…cation. In addition,

1 u u u 1 A U t ht

The parameter u enters equation (4) as a parameter governing the complementarity in the production of aggregate e¤ort. It’sthe elasticity with respect to Uht of the following function:

1+u qUht u M = = Uht qUht

The numerator is the aggregate search e¤ort from having Uht individuals put q units of e¤ort each. The denominator is the aggregate search e¤ort produced, by having only a single worker putting qUht units of e¤ort. The elasticity of M with respect to Uht determines how much more e¢ cient it is for the household to produce the aggregate e¤ort units from many di¤erent unemployed workers, than producing all the aggregate e¤ort units from a single individual.

u u Since every worker puts the same amount of e¤ort, in equilibrium lhjt = lht and the equation (4) simpli…es to:

S u 1+u u Fht = Lht = Uht lht (5)

D S In equilibrium the aggregate application units supplied, equal the ones demanded Fht = Fht = F , by equating (3) and (5) the individual e¤ort demand for each unemployed worker in this economy is: lu = (6) ht u Uht

u The above e¤ort implies G u units of disutility. The above framework for the job market will Ut impose a utility "penalty" (reputation e¤ect) for unemployed workers in the household that takes the same form as equation (2). The parameter corresponds to the application units to participate in the job market and the complementarity parameter u measures the degree of ROPE (Relative Others’Psychological E¤ect). The e¤ort demanded by each unemployed worker to participate in the job market is inversely related to the size of the unemployment pool. This is exactly what

13 the empirical evidence about social norms suggested. Higher unemployment rate of relevant others implies higher psychological well-being for the individual unemployed which is captured here by lower perceived search e¤ort. The relevant others in the model are the other household members.

The unemployed’s psychological well-being is a¤ected by increases in the household’s unemployment rate, since higher unemployment within the household makes the reputation loss from being unemployed more severe as there are less supporters of the norm.

Note here the special case where u = 0. The unemployed searcher has to simply put a …xed e¤ort as part of her job market experience. Therefore, the e¤ect of the unemployment of relevant others does not exist in this case. Under such assumption the model will behave similar to the standard

MP model. The interesting case is when u is positive. Since relative others’unemployment matters for one’s psychological well-being, according to (6), the larger the unemployment rate within the household is, the less perceived e¤ort is needed by the individual job seeker in order to provide this

application units to the employers.

2.5 Unemployment as a social norm and the ampli…cation puzzle

Recall that the ampli…cation puzzle arises because the wage responds too much after a shock, absorbing most of the e¤ect of the shock, eating away pro…ts and leaving vacancy postings barely volatile. What the social norm does, is to add an extra psychological cost term in workers’ unemployment state, which during an expansion increases the perceived e¤ort to participate in the job market, reducing the value of outside opportunities for workers, thus hindering the wage from absorbing most of the e¤ect of the shock, which results into a signi…cant boost in vacancy creation.

Another way to interpret my mechanism in action is the following. The ampli…cation puzzle arises because in expansions unemployment duration decreases dramatically and gives employed workers the right to bargain a high wage, which eventually crowds out employers’pro…ts and prevents them from posting a lot of vacancies. Thus, the vacancy creation and the congestion externality of the falling economy-wide unemployment rate, increase the probability of the unemployed to …nd a job so much, that the state of unemployment becomes much less severe, giving bargaining power in the hands of workers to negotiate a high wage. On the contrary, falling unemployment

14 rate among relevant others induces greater psychological cost for the individual unemployed through increasing the support of the norm, resulting in lower value in unemployment state for the workers and lower negotiated wage.

To sum up, in an expansion there are two opposite e¤ects that tend to a¤ect the wage bargained.

On the one hand is the congestion externality which states that the decreasing economy-wide unemployment makes it easier for a worker to …nd a job, increasing the value in the state of unemployment and on the other hand, the social norm which states that decreasing relevant others’(household) unemployment rate decreases the psychological well-being of the unemployed, decreasing the value in the state of unemployment for the workers. The latter e¤ect counterbalances to some extent the former, hindering the wage from being too responsive to outside opportunities and thus boosting vacancy creation. In a sense, in an expansion, unemployment is not going to last long, but it is psychologically painful even for the short period it is expected to persist8 . On the contrary, in a recession, unemployment lasts longer but the individual shares the unemployment experience and also the pressure to …nd a job with many others, making the whole unemployment experience less severe than otherwise.

3 The Model

3.1 The agents

There are four agents in my model:

1. Households

2. Intermediate …rms (Labor market)

8 The idea of the search e¤ort to be used to lower wage volatility can be also seen in the standard model under variable search intensity. The di¤erence there, is that the cost of search is a function of search intensity and not unemployment. It is also tied to the future expected wages of the searcher upon a match. This means that in order to have high search cost you need high wage. However high search cost according to the wage equation implied by those models means low wage. In a sense you cannot have too high search cost and too low wages at the same time because the way those two are dynamically related it is impossible. This limits the ampli…cation power of the standard variable search intensity formation as in Pissarides (2000).

15 3. Final good …rms or retail …rms

4. Monetary Authority/Government

Figure 3 provides an outline of the model with all the agents and their basic characteristics.

There is a large number of identical households in the economy. Households are composed of workers that may be either employed or unemployed. Households create job vacancies and also have unemployed workers searching for vacant jobs. If an unemployed worker …nds a vacant job, they perform a match and create a di¤erentiated intermediate …rm. A worker-vacancy match (job) can only be broken at an exogenous rate. Unemployed workers need to put e¤ort in order to …nd a job each period. Upon a match, the output of any intermediate …rm is the labor service of a worker of a household. Retail …rms produce using as inputs the aggregate labor service of the intermediate

…rms. Those …rms are producing the …nal consumption good. There is a continuum of retail …rms and since there is no entry or exit, the retail sector has a unit measure. The monetary authority and government are responsible for monetary and …scal policy respectively.

3.2 Households

There is a large number of identical households in this economy, indexed by h. Each household has the same fraction of members employed and unemployed as any other and each household has a measure of I. Households maximize utility which is separable in three arguments:

1 L1+` u1+`u (Cht Ct 1) hjt lhkt U (Cht;Lhjt;Lhjt) = $ dj $u dk (7) 1 1 + ` 1 + `u NZw I ZN w ht ht where Cht is the aggregate consumption good chosen by the household and Ct 1 the economy-wide consumption of the aggregate good of the previous period, G (Lhjt) is the disutility from the Lhjt

th th w labor hours of the j worker of the h household at time t. The variable Nht is the measure of members of this household currently employed. The third argument of the utility function (7), is

u the disutility from devoting lhkt units of e¤ort per unemployed worker for participating in the job market. The household maximizes the above objective function, subject to the following budget

16 Figure 3: The agents and their characteristics.

constraint:

PtCht + Bht + PtKtVht

w = (I N ) Ptbt + WhjtLhjtdj + Tht ht w NZht 1 w + Ptsht itdi + Pt hstds + (1 + it 1) Bht 1 Z0 ZF Nht

Vt where Kt = kv . The variable Bht represents the nominal bond holdings of the household, Vt 1 KtVht the real cost of having Vht job vacancies open. The variable Kt is an adjustment cost in vacancies and it is a function of the economy-wide number of vacancies and not in the household’s control. The expression I N w is the measure of unemployed workers in the given household, ht th Whjt is the nominal hourly wage the j worker earns, bht is the unemployment bene…t, Tht are the

17 transfers from the government to the household, sht is the fraction of the total shares of retail …rms

th the household possesses and it is the dividend the i retail …rm is o¤ering to its shareholders.

F The variable Nht is the number of intermediate …rms the household owns by the vacancies it posted w th in the past and hst is the dividend the s intermediate …rm owned by the household pays its shareholders, while it is the nominal interest. Note that households and intermediate …rms are di¤erent agents, even though the household posts the vacancies. Each job is one …rm, therefore a vacancy is a …rm waiting to be created.

Since any vacancy does not provide cash ‡ows yet, someone has to pay the cost until it becomes a job. Households provide the fee for the vacancy in exchange of the …rm’s shares in the event of a match. Thus, a household accumulates shares of the intermediate …rms by posting vacancies that successfully match.

Since households post the vacancies, in order to avoid the same household hiring their own workers, I assume that there are many households of the same type. A household is too small compared to the economy, so the probability to bene…t their unemployed members from posting

F w F vacancies is zero. This will make clear what is the di¤erence between Nht and Nht. The …rst, (Nht), w is the number of intermediate …rms/jobs the household owns. The latter, (Nht), is the number of workers of the household that are currently employed. The distinction between the two, guarantees that the employment level in the household is not a¤ected by the individual household’s vacancy postings.

F The law of motion for Nht is:

F F N = (1 ) N + qtVht (8) ht+1 ht

There is a probability of a job to be exogenously destroyed so the number of …rms owned by the household next period is equal to those that survived from last period (1 ) N F , plus the new ht matches that took place in the current period and are ready to be productive the next. Those new matches are depending on the number of vacancies opened by the household Vht and the probability

18 of a vacancy to become a match,

Ht qt = = ht Vt where t is the market tightness.

w The law of motion for Nht is similarly de…ned as

w w h w N = (1 ) N + t L N (9) ht+1 ht ht The di¤erence is that in this case, the number of household members exiting the unemployment pool, depends on the number of unemployed workers in household I N w and the probability ht

Ht 1 t = = ht Ut which is the probability, at a given period, an unemployed worker to enter the labor force. The

w evolution of Nht is out of the household’scontrol.

3.2.1 Household Problem

The household’sproblem is the maximization of the household’sutility function (7) subject to the constraints, by choosing Cht consumption, Bht bond holdings and Vht vacancies. The …rst-order conditions give the usual Euler equation

1 ht+1 Pt = Et 1 + it ht Pt+1 where

ht = (Cht Ct 1) along with the household budget constraint and the evolution of the intermediate …rms owned by the household, equation (8). The vacancy posting condition obtained from maximizing the household’s objective with respect to Vht, is presented in the intermediate …rm section below.

19 3.3 Intermediate …rms

A measure Nt of the whole population is employed, leaving Nt intermediate …rms producing the labor service each period. Each employed worker in this model is a di¤erentiated product since workers are unique individuals. Call xjt the di¤erentiated labor service denoting with j [0; 1] the 2 identity of the worker and with t the current period. The worker produces the di¤erentiated labor service xjt, via the following production function

xjt = ZtLjt (10)

where Ljt is the labor hours the worker devotes in production and Zt is productivity.

All the currently employed workers together produce the aggregate intermediate good Xt which will be used as input by the retail …rms. The aggregator is

w w 1 w 1 w Xt = At x dj (11) 2 jt 3 NZt 4 5 where 1 w 1 At N t and Nt is the number of employed workers, w is the elasticity of substitution between the di¤erent good varieties. The variable corresponds to the love of variety (LOV) for intermediate goods.

The price charged to the retail …rms for the composite good Xt is:

1 1 w 1 w w 1 w Pt = pjt dj (12) At 2 3 NZt 4 5

The demand for each intermediate good xjt from the retail …rms is determined by the usual expenditure minimization problem and is equal to:

w w p w 1 jt x = A X (13) jt t P w t t

20 In my model the labor market (intermediate good sector) will be characterized by nominal rigidities. This means that existing …rms will update their price in the current period with a probability 1 w, and the remaining fraction will set the previous period’s price indexing it to last in‡ation rate (as Christiano et al. (2005)). The new …rms created in this economy are going to be productive next period. This means that if a match occurs in period t, the newly created …rm, produces for the …rst time in t + 1. The pricing decision of the new …rms is particularly interesting, since vacancy posting and job creation is a statement about the new …rms. Assumptions on the ongoing …rms has little to do with job creation. I assume that newly created …rms are price adjusters. Since I argue that new …rms must be wage adjusters according to evidence, it would be legitimate, if not necessary, to assume that newly created …rms are also price adjusters.

3.3.1 The value of job and unemployment

A matched job generates a surplus not only for the worker but also for the …rm. This economic rent created by the search and matching process has to be shared by workers and …rms through a bargaining process. To analyze the bargaining problem I need to …nd expressions for the surpluses of workers and …rms. By adequately de…ning the surplus I will be able to describe the pricing decision of …rms, de…ne the wage and the job creation condition. Keep in mind that the following expressions can be derived from the …rst-order conditions and envelope conditions in the household’s maximization problem.

Vacancies can be created simply by su¤ering a sunk cost each period, payable in units of the

…nal good. The value of a vacant job to the household is characterized by the following Bellman equation:

V J w V F = Kt + EtQt;t+1 qtF p + (1 qt) F (14) t t+1 t+1 t+1 n o V where Ft is the value of a vacant job and Kt is the cost of keeping a vacancy open. The remaining terms in the expression above are simply the present discounted value of the vacancy next period.

One period ahead the job opening remains vacant with probability 1 qt, or can be transformed J w to a match with probability qt. The variable Ft+1 pt+1 is the value of a job to the …rm in period

21 w t + 1, given it sets the optimal price pt+1 next period. I assume that if a …rm is matched today, then it will be productive next period where it will be a price adjuster.

V V In equilibrium, vacancies are freely posted until the value of a vacancy is zero, so Ft = Ft+1 = 0, which transforms equation (14) to:

Kt J w = EtQt;t+1 Ft+1 pt+1 (15) qt n o

3.3.2 The value of a job to the …rm

The value of a job to the owner of the …rm that is optimizing it’s price in the current period, is characterized by the following Bellman equation:

J w w w Ft pt = t pt + (16) J w (1 w) Ft+1 pt+1 (1 ) E Q t t;t+1 8 w 9 J w Pt > + wFt+1 pt w > < Pt 1 = :> ;> w w where t pt is the pro…t of an intermediate …rm currently adjusting its price. The value of a job to the …rm adjusting its price at t is equal to the current pro…ts, plus the present discounted value of the asset value this job is going to have next period. The next period, the …rm readjusts it’sprice with probability 1 w and with probability w sets the same price as the previous period, indexing it to the latest intermediate good in‡ation. The equation for the pro…t of an intermediate

…rm adjusting its price today is:

w w w pt w Wt w t pt = xt pt Lt pt Pt Pt

3.3.3 The value of employment and unemployment

The value of a job to the worker employed in a currently adjusting …rm is represented by the following Bellman equation:

22 E w G (Lt) Wt pt = wtLt (17) t E w (1 ) (1 w) Wt+1 pt+1 + E Q t t;t+1 8 w 9 E w Pt U > + w (1 ) Wt+1 pt w + W t+1 > < Pt 1 = :> ;> Wt where wt = is the real wage. This Bellman equation states that the asset value of a job at Pt period t to the worker employed at a …rm currently adjusting its price, is the wage payment, minus the disutility of work in terms of the real good, plus the present discounted value of the job next period. Next period, if the …rm is destroyed, the worker will enter the unemployment pool, which

U is worth to the worker Wt+1 units of the real good. The value of unemployment to the worker is:

u u U G (lt ) E w U Wt = bt + EtQt;t+1 tWt+1 pt+1 + (1 t) Wt+1 (18) t n o where bt is the unemployment bene…t and t is the probability an unemployed worker to join the labor force. To participate in the current-period’s job market, the worker has to su¤er disutility

u u u u G (lt ) from lt units of e¤ort, where the functional form G (:) is de…ned in (2). The value of

E w unemployment to the worker next period is Wt+1 pt+1 since a new job is always a price adjusting

…rm. Also there is a probability 1 t the worker to be still unemployed in the next period with U value Wt+1. The di¤erence between (17) and (18) can be derived from an envelope condition with w respect to Nt from the household’sproblem.

3.3.4 The job creation condition

By using the vacancy posting condition (14), along with the value of a job to the …rm, equation

(16), I get the job creation condition:

K w pw + (1 ) t+1 Kt t+1 t+1 qt+1 = EtQt;t+1 (19) 8 P w 9 qt J w t+1 J w > + w (1 ) Et+1Qt+1;t+2 Ft+2 pt+1 P w Ft+2 pt+2 > < t = :> ;> 23 The proof of the above is in Appendix F . Intuitively the above condition states, the optimal number of vacancies should be at the point where the cost of the extra vacancy (left hand side of (19)), equals the present discounted value of future cash‡ows, when the vacancy …nds a match,

(right hand side of (19)).

3.3.5 The Bargaining Process

The search and matching process creates economic rent/surplus that needs to be distributed among workers and …rms every period. The surplus of a match, St, is divided by Nash Bargaining between workers and …rms and it is a function of the Bellman equations presented earlier:

J V E U St = F F + W W t t t t

The total surplus is the sum of the surplus of a job to the …rm F J F V and the surplus of a job t t to the worker W E W U . Workers and …rms bargain for the wage every period such that of the t t surplus goes to workers and 1 to …rms. Intermediate …rms and workers that are allowed to adjust the price of their labor service, choose the price that maximizes the total surplus, and then split this maximized surplus according to the …xed ratio as explained above. This means that …rms currently adjusting their price solve the following problem:

1 S pw = max F J (pw) F V W E (pw) W U (20) t t w t t t t t t pt ;wt n o and …rms not currently setting their price, solve the same problem, maximizing the surplus only with respect to wt

1 w J w V E w U St pjt = max Ft pjt Ft Wt pjt Wt wt n o

24 Maximizing the above with respect to wt gives the following …rst-order condition for every …rm j [0;Nt], price adjuster or not: 2

(1 ) W E pw W U = F J pw F V (21) t jt t t jt t w Next I present the problem of maximizing (20) with respect to pt , the problem of maximizing the total surplus by choosing the appropriate price for the intermediate good (labor service).

Proposition 1 Under the speci…cations of the model, the solution to the surplus maximization problem 1 max F J pw F V W E pw W U w t t t t t t pt is equivalent to the solution of:

w w w pt Pt+k 1 w Pt+k 1 1 w xt+k pt w k k Pt+k Pt 1 Pt 1 max (1 ) EtQt;t+k w w 8 w 9 pt w Pt+k 1 k=0 > MRSt+kLt+k pt w > < Pt 1 = X :> ;> The proof of the above is in the Appendix B

From Proposition 1, in order to optimally set the price, an intermediate …rm should maximize the following objective:

w w w pt Pt+k 1 w Pt+k 1 1 w xt+k pt w k k Pt+k Pt 1 Pt 1 max (1 ) EtQt;t+k (22) w w 8 w 9 pt w Pt+k 1 k=0 > MRSt+kLt+k pt w > < Pt 1 = X subject to the production function :> ;>

xt = ZtLt and the demand w w p w 1 jt x = A X jt t P w t t Firms maximize the surplus and not the pro…t, so the real cost of labor becomes the marginal rate of substitution. If one wishes to deviate from the classic assumption that the cost of labor for the

25 …rm is the MRS, then one needs to either change the bargain problem or assume wage rigidity as in Kuester (2007) , Christo¤el and Linzert (2005) and others. The bargaining problem does not a¤ect the marginal cost of the intermediate …rms in my speci…cation.

Under Flexible prices the F.O.C from the surplus maximization is:

w pt w = t Pt w 1 w w pt Pt w w = t (23) P Pt w 1 t

w MRS pt where t = and after using (12), and considering all intermediate …rms charge the same Zt price, w x Pt w Pt = = Nt t (24) Pt w 1 P w where P x = t is the relative price of the aggregate intermediate good. t Pt The …rst-order condition of the intermediate …rm with Calvo contracts is:

w w w w 1 w 1 P k k w 1 pt pt t+k 1 Et w (1 ) Qt;t+kAt+k w w Xt+k (25) Pt+k P P k=0 t+k t 1 X w w w w 1 w 1 P 1 w k k pt t+k 1 = Et w (1 ) Qt;t+kAt+k t+k w w Xt+k (w 1) P P k=0 t+k t 1 X 3.3.6 The Wage equation

The wage in this Economy is perfectly ‡exible. This means that every period workers and …rms share the surplus created by a job match, no matter if this surplus is maximized or not. Keep in mind that the surplus is maximized when …rms are currently adjusting their prices. Also, price and wage rigidity in this model are di¤erent things. Price rigidity (Calvo contracts) might force the

…rm not to maximize it’ssurplus, however, every …rm splits the surplus between workers and …rms, therefore the wage is not rigid. The Wage equation for any intermediate …rm in this model is the following:

26 u u w G (Ljt) G (lt ) pjt wjtLjt = (1 ) + bt + Kt#t + xjt (26) P t t t The above expression is derived combining (21) along with (16), (17) and (18). The details are in

Appendix D.

The wage equation splits the surplus generated by the match. It includes terms that are match- speci…c and others that depend on variables outside of the match, variables that re‡ect the state of the labor market. The …rst and last terms in the wage equation are the only ones that depend on the match. The …rst match-speci…c term G(Ljt) , is the disutility of labor in terms of the real good. More t hours of work implies more disutility from working therefore the worker needs to be compensated more. This term is the wage that would prevail in the standard New Keynesian model without labor w pjt market frictions. The last term xjt is the revenue of the …rm. Higher revenue creates higher Pt surplus from a match leading to higher wage. Those two terms are based on the characteristics of the speci…c …rm and worker and they are match-speci…c. The remaining terms arise due to the nature of the bargain. Workers theoretically have the option to break the negotiations with the

…rm and enter the unemployment pool if the wage bargain seems unfavorable. Hence, those terms arise in the wage equation as outside opportunities that a¤ect the relative bargaining power of the two sides. The less severe is the outside opportunity (unemployment value) the more bargaining power is earned by the workers. The variable bt is the constant unemployment bene…t. Higher unemployment bene…t makes the outside opportunity (value of unemployment to worker) higher, which increases the bargaining power of the worker and entitles her to higher wage compensation.

t Vt As I already mentioned, #t = = is the market tightness. The tightness #t, is high when qt 1 Nt there are relatively more job openings than unemployed workers. In such case, for a worker to …nd a job is more likely than a vacant job to …nd a match. High #t then, increases the bargain power of workers again by improving outside opportunities (unemployment duration 1 is low), earning a t Gu(lu ) higher wage, since workers are in relatively high demand. The extra term jt is the disutility of t search e¤ort in terms of the real good su¤ered by each unemployed worker in order to participate in the job market. This term also captures the e¤ect of relevant others on the value of a worker’s

27 unemployment state, which is the psychological cost of being unemployed and searching. Again, this term is associated with the outside opportunities of breaking the bargain.

u Moreover, the appearance of marginal utility t serves the same purpose as lt , since t is also a countercyclical variable. In an expansion, marginal utility is decreasing, implying that a utility unit in a boom, corresponds to more units of the real good than one utility unit in a recession. If the variety coe¢ cient was set to zero, u = 0, every worker would put …xed e¤ort every period. A constant unit of e¤ort will require constant disutility Gu () per worker to participate in the

u job market. However, this constant disutility, when transformed to units of the real good G () t becomes a procyclical variable that can, to some extend, counterbalance part of the change in market tightness, enhancing the model’sampli…cations.

The above wage expression is the wage equation for any …rm, price adjusting or not. Job creation is a statement about the new …rms, therefore the wage relevant for vacancy posting is the wage of a price adjusting …rm. However the wage response in the implied impulse responses is the average wage of the whole economy. The aggregation of all possible wages in the economy is derived in

Appendix E.

3.4 Retail …rms

Retail …rms use the output of wholesale …rms to produce the …nal good. There is a continuum of retail …rms with measure of one. Those …rms use as inputs the output of the wholesale …rms, and produce according to the following production function

r yit = Zt Xit

th where Xit is the quantity of the composite intermediate good employed by the i retail …rm. Each

h of those retail …rms is producing a di¤erentiated product. Let cit be the demand for consumption

28 good i from the hth household. The aggregate consumption good for the hth household is:

1 1 1 Ch = ch di (27) t 2 it 3 Z0 4 5 The underlying demand for consumption derived from the usual cost minimization problem is:

p ch = it Ch it P t t

The price of the composite good is the usual

1 1 1 1 Pt = p di (28) 2 it 3 Z0 4 5 There’snominal rigidity in the market for the …nal goods. A fraction of the …rms are allowed to adjust their price each period. The ones that fail to adjust their price simply index their price to the most recent in‡ation rate.

A retail …rm currently adjusting it’sprice is solving the usual Pro…t maximization problem:

1 1 k pt Pt+k 1 pt Pt+k 1 max EtQt;t+k MCt+k Yt+k pt Pt+k Pt 1 Pt+k Pt 1 (k=0 ! ) X where MC is the retail …rm’smarginal cost, which is simply:

w x Pt 1 Pt MCt = r = r Pt Zt Zt

x The marginal cost for the retail …rms is the price Pt the intermediate …rms charge for the aggregate 9 labor service Xt divided by the marginal product of labor . Under fully ‡exible prices, the above necessary condition becomes:

pt = MCt Pt 1 9 r In this case the marginal product is simply Zt a common to all retail …rms productivity shock.

29 which means that the relative price each individual retail producer sets is a markup over marginal cost.

From the aggregate price index (28) I get:

1 1 1 1 Pt 1 Pt = (1 ) pt + Pt 1 (29) Pt 2

3.5 Government and Monetary authority

The monetary authority conducts monetary policy following a Taylor rule. There in no government spending in this model and the Government is following a Ricardian policy regime.

3.5.1 Government

The Government redistributes any amount raised from the issue of government bonds and revenue from vacancy posting with the use of lump sum transfers and unemployment bene…ts. Every period the government keeps a balanced budget. The government’sbudget constraint is:

(1 Nt) Ptbt + Bt 1 (1 + it 1) = Bt + KtPtVt + Tt

Solving the government budget constraint for Tt and plugging it into the household’s budget constraint I get the aggregate resource constraint.

Ct = Yt

It is worth noting that the cost of vacancies KtVt appears in the household budget constraint but not the resource constraint since the cost of vacancies in modeled as a tax cost.

3.5.2 Monetary authority

The monetary authority controls the nominal interest rate via a Taylor rule:

30 (1 i) P i t y (1 i) ut 1 + it = (1 + it 1) Yt e Pt 1

The parameters and y are the coe¢ cients determining the response of the Central Bank to in‡ation and output respectively. The coe¢ cient i is measuring the degree of interest rate smoothing, included in the Taylor rule following the empirical work of Clarida, Gali and Gertler

(2000). The variable ut is a monetary policy shock.

3.6 Equilibrium

Since all households are the same, in equilibrium I can ignore the index h. Also in equilibrium

w F Nht = Nht = Nt, sht = 1 and I = 1. The …rst-order conditions give the usual Euler equation

1 t+1 Pt = Et (30) 1 + it t Pt+1 where

t = (Ct Ct 1) is the marginal utility. In equilibrium the household’s intermediate …rm ownership becomes the usual law of motion for employment:

Nt+1 = (1 ) Nt + Ht (31)

where Ht is the matching function. The resource constraint10 is:

Ct = Yt (32)

10 Here we assume that the costs of vacancies is subsitized by the government, therefore those expenditure do not appear in the resource constraint. Trigari (2004), Walsh (2005) and Kuester (2007) make the exact same assumption that simpli…es the model especially the evaluation of its steady state.

31 From the …rst-order condition for vacancies arises the job creation condition

Kt Kt+1 w w = (1 ) EtQt;t+1 + EtQt;t+1ht+1 pt (33) qt qt+1

Vt The cost of posting a vacancy is Kt = kv . The reason of assuming adjustment costs Vt 1 in vacancies is because the search and match framework does not explain the inertial response of vacancies observed in the data. Vacancies and wages are jump variables in the standard MP setting.

Similar assumption for vacancy costs is made by Braun (2005).

Using the F.O.C of the intermediate …rm’sproblem, equation (25) and the price of the aggregate service (12) in log-linear form while …xing the elasticity of output to labor to = 1 (my benchmark calibration), I get an expression for intermediate good’sin‡ation equation

w 1 w w ^x t = t 1 + t+1 + mrs^ t Pt nn^t + n^t 1 + n^t+1 (34) 1 + 1 + 1 + 1 +

2 2 1 1 w(1 ) w(1 ) 1+ w(1 ) where = , and n = , 1+ 1 w(1 ) 1+`w 1+ w(1 ) mrs^ t = `X^t ^t + ` (w 1) A^t+k. All the steps for deriving the intermediate good Philips curve are in Appendix B.

From the retail …rm’s…rst-order condition combined with (29), I get the usual Philips curve for the evolution of …nal good in‡ation:

1 (1 ) (1 ) ^x r t = t 1 + Ett+1 + Pt zt (35) 1 + 1 + (1 + )

P w By de…nition the relative price of the intermediate good …rms is P x = t . Log-linearizing this, I t Pt can express the price of the composite labor service, as a function of the intermediate and retail good in‡ation as follows ^x ^x w Pt = Pt 1 + t t 1

32 4 Calibration, estimation and results

I partition the parameters of the model into two sets. The …rst set of parameters is:

1 = ; ; ; ; w; `; ; ; b; q; ; ; f g while the second set of parameters is:

2 = ; `u; u; ; ; ; ; w; i; y f g

I calibrate the …rst set of parameters 1 and I estimate the parameters in the second set 2 to match the model implied impulse responses after a monetary shock, with the empirical responses obtained from a structural VAR on U.S. data.

The …rst partition 1, contains parameters for which either microevidence exists, or those parameters are not too important for the models dynamics. The second partition 2, contains variables that are either important for the model’s dynamics, or parameters for which microevidence is not available.

4.1 Calibration

The performance criterion upon which I evaluate my model is whether it can match, up to an acceptable extent, the impulse responses to a monetary shock, identi…ed in an SVAR on US data.

A summary of the calibrated parameters is presented in table 1. I calibrate the parameters of the model to values from independent studies. I set the steady state probability a vacancy to be matched, q, to 0.7 like Cooley and Quadrini (1999) and den Haan, Ramey and Watson (2000).

I set the probability of a worker to …nd a job, , to 0.6, as Cole and Rogerson (1999) implying an unemployment duration of 1.67 quarters. By picking those values for the two probabilities I immediate pin down the value of the e¢ ciency of the match, h, to 0.6511 . I set the unemployment

11 The variable h = h~ according to our de…nition of the matching function

33 bene…t b to zero12 , the lowest possible value, to make sure it is not driving my result13 .

The separation rate, , to 0.035. This implies a 6% steady state unemployment level. Hall (1999) estimates a value of 0.08 and usually values vary around 0.08-0.1. My choice of the separation rate is not important in achieving my target. I use a discount rate, , to 0.989, which implies a 1% real interest rate per quarter. The elasticity of substitution for …nal goods, , is set to 11, which entails a markup of 1.1. The elasticity of substitution for the intermediate goods, w, is set to 25. Boivin and Giannoni (2005) propose a value of 11 and Altig, Christiano, Eichenbaum and Linde (2005) set the own price elasticity of demand to 101. My value of w = 25 lies in between the values proposed in the literature. I set the risk aversion coe¢ cient, , to 3. The inverse of labor supply elasticity, `, is set to 10, which corresponds to a rather inelastic labor supply. It implies a labor supply elasticity of 1 = 0:1. Trigari (2005) and Kuester (2005) use the same value. Card (1994) estimates that 1 ` ` 2 (0; 0:5). My parameter value lies in the interval.

I set the value for the …nal good price rigidity, , to 0.5, along with evidence by Bils and Klenow

(2004) suggesting prices adjust on average every two quarters. I set the nominal rate response of the central bank to in‡ation, , to 1.1, as in Bernanke, Gertler and Gilchchrist (1999).

4.2 The estimation procedure

I use the techniques introduced by Rotemberg and Woodford (1999) to bring the model to the data.

I estimate a set of parameters in order to minimize the distance between model implied impulse responses and the responses obtained from a structural VAR on US data. More weight is placed on the most accurately estimated responses. The exact methodology is presented in detail below.

De…ne with M ( ) the vector valued function of the impulse responses from my model. It is a function of the parameter vector . I denote with V the vector of the impulse responses from the

12 Setting b = 0:4 as in Shimer (2005) has a very small impact on the responses and estimated parameter and this is one of my goals, to create a model that’snot too sensitive to unemployment bene…t changes. 13 A high value for the unemployment bene…t increases the steady state value of the wage while decreasing the steady state of pro…ts. This decreases the deviation of wage from it’ssteady state thus increasing vacancy volatility. This mechanism creates ampli…cation in Hagedorn and Manovskii (2007).

34 Type Par Name/Explanation Value Source Preferences time discount factor 0:989 real rate 1% per quarter elasticity of subs …nal goods 11 corresponds to 1.1 markup risk aversion coe¢ cient 3 number usually used 2 LOV intermediate goods 0 no Love Of Variety w elast of substitution int goods 25 101 Christiano (2005), Kuester 23 ` inverse of labor supply elast 10 1 (0,0.5) Card (1994) ` 2 Labor market s.s unemployment probability 0:6 matches un. duration of 1.67 separation rate 0:035 Implies 6% s.s. unemployment b unemployment bene…t 0 match data with lowest un. bene…t q steady state vacancy probability 0:7 den Haan et al (2000) elasticity of int. output to labor 1 Price & policy …nal good price rigidity 0:5 Bils & Klenow (2004) response to in‡ation 1:1 Bernanke et al estimate1.1 (1999)

Table 1: This is a summary of the baseline calibration. structural VAR on US data (1955:1 to 2005:1) . The vector of estimated parameters is the vector

^ , the solution to the following minimization problem:

^ = min [M ( ) V ]0 V [M ( ) V ] 2 subject to the possible constraints imposed by the parameter space coming from theory. The diagonal matrix V is the matrix with diagonal elements the inverses of the VAR impulse response variances.

4.3 The estimates

The estimates associated with preferences, labor market, price rigidity and policy parameters appear in table 2. Starting from the household preferences related parameters, the habit persistence parameter is estimated to be 0.96. The estimate is close to the 0.97 obtained by Kuester (2005) and the 0.91 estimated by Boivin and Giannoni (2005). The disutility of the search e¤ort, `u, is estimated to be 7.1, the ROPE parameter, u, is set to 0.23 and the application units, , to 1.97. My estimates stress that unemployment for a person who is actually actively trying to …nd a job is quite painful. Speci…c estimates for this may not be available, but the idea that the job searching

35 Type Par Name/explanation Value st.dev. Preferences habit persistence 0:96 (0:002) `u disutility of search e¤ort param. 7:1 (0:073) u ROPE coe¢ cient 0:23 (0:003) application units per person 1:97 (0:026) Labor Market elast. matching function 0:46 (0:019) workers bargaining power 0:81 (0:009) vac adjustment cost parameter 4:1 (0:413)

Price & policy w intermediate good price rigidity 0:55 (0:031) i policy inertia 0:83 (0:005) y policy response to output 0:2 (0:064)

Table 2: The table presents the estimated parameters from the minimum distance procedure to match the model implied impulse responses with the empirical. The numbers in parenthesis are standard deviations. experience to involve frustration, stress and inevitably high disutility seems reasonable.

For the labor market parameters, the elasticity of the matching function, , is set to 0.45, close to the value of 0.4 estimated by Blanchard and Diamond (1989). Petrongolo and Pissarides (2001) estimate the elasticity of the matching function with respect to unemployment to be between 0.5-

0.7. My estimate lies a little lower than their bound. The share of the surplus that goes to workers,

, is estimated in my model to 0.8. Although there are no microevidence for this parameter it is close to 0.77 obtained by Braun (2005). Trigari (2004) estimates = 0:81, while Kuester (2005)

…nds = 0:21 and Shimer (2005a) calibrates it to 0.7. The vacancy adjustment cost, , is estimated to be 4.1 and Braun estimates it to be around 2.

My estimate of price rigidity, w, is 0.55 close to 0.5 suggested by Bils & Klenow (2004). For the intermediate goods Christiano et al. (2005) …nd an estimate for w = 0:64. For the policy parameters, I …nd the policy inertia parameter, i, to be 0.83 as in Kuester (2005) and close to Trigari’s (2004) estimate of 0.85. The policy inertia parameter is important for the size and persistence of the responses. An extended discussion about this fact can be found in Walsh (2005).

The monetary authority’s policy response to output y is estimated to be 0.2. Both policy values are mostly in line with the …ndings of Clarida, Gali and Gertler (2000).

36 4.4 The model’sability to match the data

I present the model implied along with the empirical impulse responses of the variables of interest after a unit monetary shock in …gure 4. The solid lines represent the responses obtained from running a VAR on quarterly US data from 1955:1 to 2005:1, which are the same as the ones in

…gure 1. The solid lines with bullet marks represent the model implied responses, and the gray areas are 90% con…dence intervals. The model’s responses are consistent with the persistence and cyclicality of various key macroeconomic variables in the data.

The model-predicted response of in‡ation is as low and inertial as it’s empirical counterpart.

The model implied responses of vacancies, unemployment and job creation can match the level of magnitude, as well as the persistence of the responses observed in the data.

The low and inertial response of the real wage is matched fairly well, despite the tendency of search and matching models augmented with monetary shocks to overshoot the responses of the real wage. Below I provide details as to how my di¤erent assumptions contribute to the picture in

…gure 4.

4.5 The main ingredients

I refer to as main ingredients, the two key departures from the literature that I’ve already stated in the introduction, which are the introduction of unemployment as a social norm and nominal rigidities in the intermediate sector. The e¤ect of each of my key assumptions on the …t of the model in …gure 4 is discussed below, as well as the performance of the model when those mechanisms are excluded.

4.5.1 The introduction of unemployment as a social norm

Figure 4 shows that the model can create enough ampli…cation for vacancies, unemployment and job creation. The volatilities of those variables are ampli…ed by the introduction of unemployment as a social norm. As I’ve already covered, after an expansionary monetary shock, the unemployment of relevant others is declining, while marginal utility is decreasing. According to the norm, and the

37 decreasing marginal utility, the searching experience by each individual job seeker becomes more unpleasant, which lowers the value of unemployment for a worker who eventually looses bargaining power. The loss of bargaining power, hinders the wage from responding too much to the increased likelihood of …nding a job elsewhere, boosting this way vacancy creation.

Figure 5 shows the optimal …t14 of the model without the social norm assumption ( = 0). I do not present responses other than vacancies, unemployment and job creation because the model performs fairly well in explaining those15 . Clearly the model cannot create the ampli…cation needed without the social norm speci…cation which means that a non-zero and the ROPE parameter u play an important role in the labor market variables’volatilities. However, some ampli…cation is present in the model without the social norm, since the addition of a monetary side can supply the model with some extra volatility for vacancies and unemployment as also stressed by Barnichon

(2007). The addition of a monetary side can boost the present value of …rm pro…ts via interest rate changes that directly a¤ect the stochastic discount factor. In other words, in an expansion the interest rate drops and the stochastic discount factor increases, inducing vacancy creation through higher present value of future cash ‡ows.

To further motivate the importance of my model’s job market framework, I present some responses to a unit monetary shock for di¤erent degrees of ROPE, u, that governs the importance of the measure of the supporters of the norm to the disutility of the code breakers. Figure 6 examines the e¤ect on the model implied responses using my benchmark calibration for di¤erent values of

ROPE. It is distinct that higher degrees of the ROPE parameter u, can amplify the responses of vacancies, unemployment and job creation in the model.

I proceed by setting u = 0 and simply vary the value of application units per person , to observe how the countercyclicality of marginal utility can also create a procyclical response for the perceived search e¤ort and therefore enhance the ampli…cation of the shocks in my model. Figure

7 shows the response of vacancies, unemployment and job creation after a unit monetary shock, by varying the application units , while keeping ROPE equal to zero. It is evident that by simply

14 By optimal …t I mean running the algorithm in order to match the empirical responses subject to parameter constraints coming from micro evidence. The …t of the model compare to the empirical responses is my measure of success. 15 This is expected since the ampli…cation puzzle a¤ects primarily those three variables in this model.

38 assuming the search cost is denominated in e¤ort units instead of units of the …nal good (as usually assumed in the literature), ampli…cation can be improved in the model.

Despite the fact that the countercyclicality of both the psychological cost of unemployment and the marginal utility contribute in ampli…cation of the shocks in my model, the cyclicality of the marginal utility alone cannot create the ampli…cation needed in the data. This can be motivated by looking at …gure 8 which shows the optimal …t of the model by keeping u = 0 and letting the cyclicality of the marginal utility alone to provide ampli…cation. I only present the labor market variables since the remaining responses can match the empirical ones fairly well. The marginal utility alone cannot provide enough ampli…cation especially for vacancies and job creation. Also a large number is needed for the vacancy adjustment cost parameter. This result speci…es that both e¤ects are important, even though the social norm assumption is more vital in boosting ampli…cation.

4.5.2 The introduction of price stickiness in the intermediate sector

The assumption of price stickiness of the form of Calvo contracts is very important in this model.

The purpose of this assumption is twofold. On the …rst hand it aims in enhancing the model’s persistence and on the other hand to lower the volatility of the wage response after a monetary shock.

Many authors like Gertler and Trigari (2006), Shimer (2005b) and others have used wage rigidity for new …rms, to obtain ampli…cation, an assumption that is rejected for it’s empirical validity by

Pissarides (2007) and Haefke, Sonntag and van Rens (2008). Wage rigidity for new matches might be rejected by the evidence, but what about wage rigidity for continuing …rms? Wage rigidity, at least for continuing …rms, seems to be essential to guarantee the low and inertial response of the real wage observed in the data. The only problem is that, according to Krause and Lubik (2005), the wage rigidity assumption could aid me in matching the wage response, but the assumption will not contribute at all in obtaining in‡ation inertia. As the authors report, this is because the partition of the production activity in two sectors, makes the retail …rm’smarginal cost una¤ected by wage rigidity in intermediate sector, leaving the in‡ation response barely a¤ected.

39 Instead, the assumption of price rigidity for intermediate …rms, not only enables the model to match the real wage response, but also creates additional in‡ation inertia. Calvo pricing in the intermediate sector, enables me to take advantage of the assumption that every worker is an intermediate …rm, to boost the in‡ation inertia and persistence implied by the model. Since an intermediate …rm currently adjusting it’s price uses a single worker’s labor hours in production, the …rm takes into consideration how the price of labor is directly a¤ected by the …rm’s pricing decision. In an expansion, when a price adjusting …rm considers increasing it’s price, the demand for it’sproduct must fall. The fall in demand induces a fall in demand for labor hours, which implies a lower disutility of labor for the worker and therefore a lower marginal cost for the …rm, which applies pressure on the price initially set by the …rm to decrease. Stated otherwise, an attempt of an intermediate …rm to increase it’s price, triggers a mechanism that tends to decrease the initial price change. Thus, the changes in prices are achieved in smaller increments, increasing the degree of strategic complementarity between intermediate …rms and eventually induce smaller adjustments in marginal costs faced by retail …rms. This is an e¤ective way of increasing in‡ation inertia in the model. Treating each worker as a single …rm will add the term 1 + `w in the denominator of the parameter in the Philips curve for the intermediate goods (34), that tends to decrease the volatility of intermediate good in‡ation and justi…es my choice of a fairly inelastic labor supply

(inverse of labor supply elasticity ` calibrated to 10). A discussion about the di¤erence between

…rms employing a single worker against employing partly all workers in the economy can be found in chapter 3 of Woodford (2003).

Therefore, price rigidity in intermediate …rms allows me to increase the real rigidity in the model.

The important question now is: can the model match the data by simply assuming price rigidity only in the retail sector? To answer this question, I set the parameter governing the degree of price stickiness in the intermediate sector, w, to zero, while letting the degree of price rigidity for retail …rms, , to vary. In addition, I remove the social norm assumption temporarily for this speci…cation.

In other words, in this exercise, there’s only rigidity in the retail sector, while intermediate …rms have ‡exible prices. Figure 9 shows the optimal …t of that model for three di¤erent speci…cations.

The lines with dot marks correspond to the optimal …t when is estimated with no restriction

40 imposed on the parameter. The lines with circle marks and star marks, correspond to the same exercise with calibrated to 0.85 and 0.55 respectively. The estimated degree of price rigidity for the …rst speci…cation is: = 0:97, an implausible value according to evidence by Bils and Klenow

(2004).

What might strike you as unusual is that the responses with the dot mark can actually create enough ampli…cation without the need of the social norm assumption. According to Andres,

Domenech, and Ferri (2006), in a model with two sectors, increases in the degree of price rigidity can create ampli…cation. This is due to the fact that the intermediate …rm’srevenue depends on the

w pt w relative price , where p is the price of the intermediate …rm and Pt the price of the consumption Pt t good. In an expansion, additional rigidity in intermediate …rms, lowers the volatility of the price of the consumption good Pt, therefore, the relative price of intermediate …rms’signi…cantly increases, boosting …rm pro…ts and vacancy creation16 . However, when the degree of price rigidity assumed for retail …rms is reduced to more acceptable values, the lines with the circle and star marks in

…gure 9 show that not only the persistence in the model disappears, but also the ampli…cation, which can be only sustained by values of greater than 0.9.

In addition, …gure 9 shows that whichever value for is used, the model with ‡exible intermediate …rms overshoots the wage responses, an observation also shown in Christiano, Trabandt and

Walentin (2009). Thus, it is evident from the above exercise that the assumption of price rigidity in the intermediate sector is vital for preventing the wage from being too volatile after a monetary shock.

Other than the wage, the remaining responses with star and circle marks, can match nearly none of the empirical responses. This means that when more reasonable values for retail price rigidity

are assumed, the model’s performance is not even close to satisfactory. A model with ‡exible intermediate …rms and no social norm framework cannot account for the empirical responses. The question now is: can a model with ‡exible intermediate …rms augmented with the social norm assumption, account for the data without the assumption of price rigidity in intermediate …rms?

The answer to this question is provided in …gure 10. Clearly the model cannot match the responses,

16 In my model this e¤ect doesn’texist since the rigidity is in both sectors, therefore there is no signi…cant change in relative price.

41 even though the social norm assumption seems to create at least the appropriate ampli…cation for vacancies, unemployment and job creation. The response of in‡ation is not persistent enough, stressing the necessity of price rigidity in the intermediate sector that can induce greater in‡ation inertia and persistence in the model.

Therefore, both the assumptions of price rigidity in intermediate …rms and unemployment as a social norm are indispensable ingredients for my model’sperformance.

4.5.3 Other ingredients

The output response in …gure 4 is inertial and persistent. This is achieved through the employment of habit persistence in preferences. The real hourly wage is also in-line with the data. The response of the real wage is about half the response of output, along with the estimates of Giannoni and

Woodford (2005). The small and inertial response of the real wage comes from the nominal rigidities in the intermediate sector. First of all, because the new …rms are price adjusters in the …rst period, they face a decline in demand when increasing their price and employ less labor hours in production.

Thus, in the wage equation, the revenue of the new …rm, which is a part of the wage (last term in

(26)), is negative during an expansion. The disutility of labor (…rst term in (26)) is also negative due to the declining demand the price adjusting …rm is facing. On aggregate, this lowers the average wage in the economy. In the standard MP model with monetary shocks, the wage tends to be way too volatile. Similar discussion as to how to lower the wage implied by a search and matching framework with market power, can be found in Rotemberg (2006).

An important role towards my goal is the assumption of adjustment costs in vacancies, Kt =

Vt kv . The necessity of this assumption stems from the forward looking nature of the job Vt 1 creation condition, equation (19). Since …rm exit in the model is quite low (as it should be according to evidence), there is no restriction in the model as to when the intermediate …rms should be created.

There is a tendency for the vacancies to be posted all at once and employment to reach it’s peak fairly early. However, the adjustment costs account for the inertial response of vacancies observed in the data. This kind of vacancy cost can be obtained by assuming time to build in vacancy creation as assumed for jobs in this model. Lucca (2005) creates a framework where a vacancy cost of the

42 type I already assumed rises. In …gure 11, I examine the …t of the model by assuming no vacancy adjustment costs. It is apparent that vacancies respond immediately and unemployment reaches it’s pick fast. Thus, the assumption of vacancy adjustment costs is crucial for vacancy inertia and persistence. However, assuming vacancy adjustment costs, sacri…ces ampli…cation for persistence in the model, which implies that the virtues of the vacancy adjustment cost assumption can only be prominent in a model employed with a good ampli…cation mechanism.

5 Conclusion

I have presented a fairly simple new Keynesian model with search unemployment to explain the joint response of key macroeconomic variables after a monetary shock. My main departure is the introduction of unemployment as a social norm that incorporates the belief that the unemployed are a burden to society, resulting in additional pressure from households upon the unemployed to

…nd a job. The lower the unemployment within the household raises the support of the norm and the greater the pressure is upon the individual unemployed. This extra psychological cost imposed by the norm counterbalances the bargaining power gained by decreasing unemployment duration during an expansion, preventing the wage from crowding out vacancy creation, boosting the model’s ampli…cation mechanism.

I also assume nominal rigidities (Calvo contracts) in the intermediate good sector (labor market) in order to create enough persistence in the model. Price rigidity in the intermediate sector not only contributes in creating in‡ation inertia but also aids me in keeping the wage response low and smooth. In a monetary model with no nominal rigidities in the labor market, the wage implied is too volatile.

The model can match the impulse responses implied by a structural VAR on quarterly US data from 1955:1 to 2005:1 and can be used for optimal monetary policy analysis. My goal was to match the data by assuming moderate degrees of price rigidity and without the aid of wage rigidity or unreasonable calibration, mechanisms that have been heavily criticized recently. In other words, I tried to address the Shimer puzzle by employing monetary shocks and at the same time avoiding

43 stickiness in wages and high unemployment bene…ts.

In addition, there is an interesting interaction between the in‡ation process and the search and match framework created by the appearance of the extensive margin in the labor market, which can be important for policy analysis. As a matter of fact, the ultimate goal of my paper is to show that the model can perform well according to the facts, in order to apply the model for optimal monetary policy design and welfare analysis. I intend to consider those extensions in the future.

Moreover, there is an interaction between the model’s dynamics and the degree of Love Of

Variety (LOV), , that I wish to further investigate, as the degree of in‡ation inertia can be linked to this parameter and also the evolution of intermediate …rms.

At last, I’minterested in investigating the case where the number of application units per period

is variable and constitutes a choice variable for the household, in an attempt to address puzzles associated with search intensity.

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A Appendix

This part derives more formally my labor market speci…cation by de…ning the …rm that aggregates the individual search e¤orts more properly. There is a …rm that combines household’sunemployed workers’individual e¤ort and sells it to households. The household uses this aggregate e¤ort to provide the needed application units to send it’sunemployed workers to the job market.

hD w I assumed that each household demands Ft = Uht application units in order to have their unemployed workers participate in the job market, where U w = I N w is the number of unemployed ht ht u workers in the household. The household needs Lht aggregate e¤ort units to produce the aggregate hS u application units using the following production function Ft = Lht. There is a …rm that takes unemployed workers’individual e¤ort and transforms it to aggregate e¤ort units according to the following production function:

u u 1 u 1 u Lu = Au l u dj ht t 2 hjt 3 UZt 4 5 where 1 u u u 1 A U t t

The pro…t maximization problem of the …rm is:

max P l Lu pl lu dj u ht ht hjt hjt lhjt UZt

49 u u 1 u 1 s:t: Lu = Au lu u dj ht t 2 hjt 3 UZt 4 5 The pro…t maximization problem derives the demand for individual e¤ort units:

u l u 1 p u u hjt u lhjt = At l Lht Pt ! and 1 1 u 1 u l 1 l Pht = u phjt dj At 2 3 UZt 4 5 Since all …rms are the same the above conditions become:

l u l phjt = Ut Pt (36) and

u (1+u) u lhjt = Ut Lht (37)

I return to the household problem to de…ne our job market framework more formally. The utility function of the household is the following:

u u U (Cht Ct 1) G (Lhjt) dj G (lhkt) dk H N w I N w W ( t) = max 8 ht ht 9 Cht;Bht;Vht R R f g > H > <> + EtW ( t+1) => > > :> ;>

50 The budget constraint is:

l h PtCht + Bht + KtPtVht + Pt Ft =

l u WhjtLhjtdj + phktLhktdk NZw I ZN w ht ht 1 w + (I N ) Ptb + Tht + Ptsht itdi ht Z0

w + Pt hstds + (1 + it 1) Bht 1 ZF Nht where F h = U h = (I N w ) t t ht and

w w w N = (1 ) N + t (I N ) ht+1 ht ht

w The envelope condition with respect to Nht determines the value of the job to the household, important ingredient for the wage equation.

H u u l l u WN w ( t) = G (Lhjt) + G lhjt + tPt tphktLhjt + t (WhjtLhjt Ptb) ht H + (1 t) EtWN w ( t+1) ht+1

Let’ssimply focus on what is new in the above expression and keep in mind than in equilibrium household speci…c variables are equal to the economy wide variables. What’s new in the above equation is the following sum (after dividing everything by marginal utility):

u1+`u l l 1 u u l l u 1 lt Pt pt u NT = G (lt ) + tPt tptlt = + lt (38) t 1 + `u t Pt Pt where t = tPt

51 The cost of e¤ort for intermediate e¤ort good j is:

` pl lu u jt = t Pt t l l u`u pjt Pt lt l = Pt Pt t

l u`u Pt u lt = Ut (39) Pt t

u (1+u) u u Plugging (39) into (38) and using lt = Ut Lt = Ut I get the following for the new terms:

u1+ù uù uù 1 lt u lt u lt NT = + Ut Ut 1 + ù t t t 1+` 1 lu u = t 1 + ù t

u u which is the G (lt ) term introduced in the wage equation due to my job market speci…cation.

B Appendix

Proof of proposition 1

I start with the surplus maximization problem:

1 J w V E w U max Ft (pt ) Ft Wt (pt ) Wt pw f t g n o The F.O.C of this problem is: @ F J (pw) + W E (pw) t t t t = 0 @pw t Let’sdetermine the expression in the brackets. From (16) and (17) is:

w J w w w J w J w Pt F (p ) = (p ) + (1 ) EtQt;t+1 (1 w) F p + wF p t t t t t+1 t+1 t+1 t P w t 1

52 E w $G (L ) (1 ) (1 w) Wt+1 pt+1 W E (pw) = w L t + E Q t t t t t t;t+1 8 w 9 t E w Pt U > + w (1 ) Wt+1 pt w + W t+1 > < Pt 1 = > w> w adding the two together and plugging in the equation: for Pro…ts of the intermediate …rms t; pt =

w pt xt wtLt, the above becomes Pt

w J w E w pt $G (Lt) U Ft (pt ) + Wt pt = xt + EtQt;t+1Wt+1 Pt t J w E w (1 w) Ft+1 pt+1 + Wt+1 pt+1 + (1 ) E Q t t;t+1 8 w w 9 J w Pt E w Pt > + w Ft+1 pt w + Wt+1 pt w > < Pt 1 Pt 1 = :> ;> If I solve the equation forward, eliminating the terms F J (pw) + W E (pw) for k = 1; :::; I get t+k t t+k t 1

J w E w Ft (pt ) + Wt (pt ) = (40)

w w w pt Pt+k 1 w Pt+k 1 P P w xt+k pt P w 1 k k t+k t 1 t 1 P w w (1 ) EtQt;t+k w t+k 1 8 $G Lt+k pt P w 9 t 1 k=0 <> => X t+k > > : F J ;pw 1 k k t+k+1 t+k+1 + (1 ) (1 w) w (1 ) EtQt;t+k+1 8 E w 9 k=0 > +W p > X < t+1 t+k+1 =

1 k k U > > + (1 ) EtQt;t+k+1W : ; w t+k+1 k=0 X The above expression is the expression in the brackets of the F.O.C I found before

@ F J (pw) + W E (pw) t t t t = 0 (41) @pw t

w Since the last two summations in (40) are independent of pt , implying that after plugging (40) in (41) I get:

P w w t+k 1 w w w $G Lt+k pt P w 1 k k pt Pt+k 1 w Pt+k 1 t 1 @ w (1 ) EtQt;t+k w xt+k pt w Pt+k Pt 1 Pt 1 t+k 0k=0 8 91 P < = = 0 @ @pw A : t ;

53 which is equivalent to:

w w w w 1 k k pt Pt+k 1 w Pt+k 1 $GL(Lt+k) w Pt+k 1 @ w (1 ) EtQt;t+k w xt+k pt w Lt+k pt w Pt+k Pt 1 Pt 1 t Pt 1 k=0 n w o = 0 P @pt

$GL(Lt+k) and by de…ning MRSt = , the above is the solution to a maximization problem like the t+k following:

w w w w 1 k k pt Pt+k 1 w Pt+k 1 w Pt+k 1 max (1 ) E Q x p MRS L p w w t t;t+k w t+k t w t+k t+k t w pt Pt+k P P P k=0 t 1 t 1 t 1 X

C Appendix

I’ll derive the Philips curve for the intermediate …rm’sin‡ation. The FOC of the intermediate …rm’s problem is:

w w w w 1 w 1 P k k w 1 pt pt t+k 1 Et w (1 ) Qt;t+kAt+k w w Xt+k Pt+k P P k=0 t+k t 1 X w w w w 1 w 1 P 1 w k k pt t+k 1 = Et w (1 ) Qt;t+kAt+k t+k w w Xt+k (w 1) P P k=0 t+k t 1 X The …rst-order condition can be rewritten as:

w w 1 w w 1 w 1 P k k w 1 x pt Pt t+k 1 (w 1) Et (1 ) Qt;t+kA P Xt+k w t+k t+k P w P w P w k=0 t t+k t 1 X w w w w w 1 w 1 1 P w k k pt Pt t+k 1 = Et (1 ) Qt;t+kA X t+k w t+k t+k P w P w P w k=0 t t+k t 1 X

P w where P x = t t Pt w w pt De…ne Gt = w . Log-linearizing the above and using (23) at the steady state: Pt

1 1 1 x (w 1) w 1+w w (w 1) P A G X = (42) 54 I get

x w (w 1) A^t+k + P^ + (1 w) g t+k t 1 k k k 0 1 E (1 ) w ^w ^w (43) t w + 1 w + Pt Pt 1 k=0 B C X B ^w ^ 1 ^ C B + (1 w) Pt+k 1 Pt+k + 1 Xt+k C B C 1 @ A k k k w 1 ^ w w w ^w ^ = Et w (1 ) '^t+k + At+k gt Pt+k 1 Pt+k k=0 X

Hat variables are deviations from steady state. Now let’s substitute out 't+k. Let’s remind the reader that the real marginal cost is:

MRSt+k t+k = Zt+k

1 P w w t+k 1 ` xt+k pt P w $Lt+k t 1 where MRSt+k = , Lt+k = , t+k 2 Zt+k 3

w w w w w w Pt+k 1 w 1 pt 4Pt Pt+k 1 5 xt+k pt w = A w w w Xt . Pt 1 t+k Pt Pt+k Pt 1 Log-linearizing the above expressions one by one yields:

'^t+k = mrst+k z^t+k

mrs^ t+k = `^lt+k t+k 1 ^lt+k = (^xt+k z^t+k)

^ w ^w ^w ^w ^w x^t+k = (w 1) At+k w gt + Pt Pt+k + Pt+k 1 Pt 1 + Xt+k Combining the above four equations I get:

` w ^w ^w ^w ^w ` ` ^ ` '^t+k = w gt + Pt Pt+k + Pt+k 1 Pt 1 + Xt+k t + (w 1) At+k 1 + zt+k

55 ` ` ` De…ne mrs^ t+k = Xt+k t+k + (w 1) A^t+k 1 + z^t+k so ` w ^w ^w ^w ^w '^t+k = mrs^ t+k w g^t + Pt Pt+k + Pt+k 1 Pt 1

Substitute this into (43) and rearranging I get

w w gt + t

w 1 ^ ^x 1 (1 ) 1 mrs^ t+k w 1 At+k P w k k k t+k = Et w (1 ) w 0 1 1 1 w + (1 + `) w w ^ k=0 + 1 w + (1 + `) t+k 1 Xt+k X B C @ A Writing the above in recursive form and rearranging I get:

w 1 w (1 ) x w 1 1 g = mrs^ t P^ w 1 A^t 1 X^t (44) t w t 1 w + (1 + `) w w w + w (1 ) Et g^ + t+1 t+1 t ` ` ` where mrs^ t = Xt t + (w 1) A^t 1 + z^t Now I’ll log-linearize the price of the aggregate intermediate good (12). Since the price of the aggregator is: 1 1 w 1 w w 1 w Pt = pjt dj At 2 3 NZt 4 5 1 w 1 w 1 w w w 1 w w 1 w Pt = At pjt dj + At pjt dj

(1 Z)Nt 1 Nt (1 Z)Nt 1

56 1 w 1 w w w 1 w P = (1 w) (1 ) A p dj t t t NtZ 1 w 1 w 1 w P At w 1 w t 1 + w (1 ) At 1 pjt 1 w dj At 1 Pt 2 NtZ 1 1 w w 1 w + At pt dj

Nt (1 Z)Nt 1

1 w w 1 w Log-linearizing the above and using the steady state condition A NG = 1

w w (1 ) w w g^t = (^nt w (1 )n ^t 1) + t t 1 1 w (1 ) 1 w (1 ) Combining this last equation with (44) I get the Philips curve for intermediate good in‡ation:

w 1 w w t = t 1 + t+1 1 + 1 +

1 1 w (1 ) w (1 ) + w 1 + 1 w (1 ) 1 w + (1 + `) x w 1 1 mrs^ t P^ w 1 A^t 1 X^t t 2 2 1 + w (1 ) n^t 1 + w (1 ) + n^t 1 1 +

+ n^ 1 + t+1

D Appendix

In this section I’llderive the wage equation for the jth intermediate …rm. For this task I’ll repeat the necessary equations. The …rst is the condition that divides the total surplus between …rms and workers.

(1 ) W E pw W U = F J pw F V (45) t jt t t jt t 57 The next step is simply to derive the two di¤erences in brackets. The …rst can be obtained by subtracting the Bellman equations (17) and (18). This di¤erence is:

Kt J w = EtQt;t+1 Ft+1 pt+1 qt n o

w J w w w J w J w Pt F p = p + (1 ) EtQt;t+1 (1 w) F p + wF p t jt t t t+1 t+1 t+1 jt P w t 1

E w G (L ) (1 ) (1 w) Wt+1 pt+1 W E pw = w L jt + E Q t jt jt jt t t;t+1 8 w 9 t E w Pt U > + w (1 ) Wt+1 pjt w + W t+1 > < Pt 1 = :> ;> Gu Lu U jt E w U Wt = b + EtQt;t+1 tWt+1 pt+1 + (1 t) Wt+1 t n o The left hand side of (45) is:

E w U W p W = wjtLjt Dt (46) t jt t

E w U (1 t) W p W t+1 t+1 t+1 8 P w 9 + E Q E w t U t t;t+1 > Wt+1 pjt P w Wt+1 > > t 1 > > + w (1 ) 0 1 > < E w U = Wt+1 pt+1 Wt+1 > B C > > @ A > :> ;> u u G(Ljt) G (Ljt) where Dt = +b . Equation (45) holds for every period and for every …rm, no matter t t if it is adjusting it’sprice or not. This is due to the fact that all intermediate …rms negotiate their price every period (wage is ‡exible). This means that I can write the following equalities:

E w U J w (1 ) W p W = F p t+1 t+1 t+1 t+1 t+1 and w w E w Pt U J w Pt (1 ) Wt+1 pjt w Wt+1 = Ft+1 pjt w Pt 1 Pt 1

58 Then after using the above fact I write (46) as:

E w U W p W = wjtLjt Dt t jt t

J w (1 t) 1 Ft+1 pt+1 w 8 J w Pt 9 + EtQt;t+1 Ft+1 pjt w > Pt 1 > > + (1 ) > > w 1 0 1 > < J w = Ft+1 pt+1 > B C > > @ A > :> ;> Finally I use the vacancy posting condition, equation (15),

E w U Kt Wt pjt Wt = wjtLjt Dt + (1 t) (47) 1 qt w w w + w (1 ) EtQt;t+1 p ; p (48) 1 t+1 jt t+1

w w w w J w Pt J w where t+1 pjt; pt+1 = Ft+1 pjt w Ft+1 pt+1 Pt 1 The di¤erence in the right hand sight of (45) is just the value of the job to the …rm, where

V Ft = 0 in equilibrium. I repeat below equation (16)

w J w w w J w J w Pt J w F p = p + (1 ) EtQt;t+1 F p + w F p F p t jt t jt t+1 t+1 t+1 jt P w t+1 t+1 t 1 and using the vacancy posting condition (15)

J w w w Kt w w w Ft pjt = t pjt + (1 ) + w (1 ) EtQt;t+1t+1 pjt; pt+1 (49) qt

w w w w w J w Pt J where t+1 pt ; pt+1 = Ft+1 pjt w Ft+1 pt+1 . Pt 1 Now I plug (47) and (49) in (45) after using the following expression for the pro…t equation

59 w w w pjt t p = xjt wjtLjt jt Pt

Kt wjtLjt Dt + (1 t) 1 qt (1 ) 0 w w w 1 + 1 w (1 ) EtQt;t+1t+1 pjt; pt+1 B C @ pw A jt Kt xjt wjtLjt + (1 ) = Pt qt 0 w w w 1 + w (1 ) EtQt;t+1t+1 pjt; pt+1 B C @ A t After cancelling out the delta terms, rearranging and use the fact that = #t (where #t is market qt tightness)

u u w G (Ljt) G Ljt pjt wjtLjt = (1 ) (1 ) + (1 ) bt + Kt#t + xjt t t Pt

E Appendix

Here I derive an expression for aggregate wage. Since some …rms are price adjusters and some are not, I need to …nd a way to …nd an average wage for the whole economy despite the heterogeneity between …rms induced by price rigidity. I start by de…ning the aggregate wage to be the weighted sum of the Nt …rms operating each period

1 ! = w pw dj t N t jt t Z Nt

60 after some manipulations I get

1 w 1 w !t = wt pjt + wt pjt dj Nt Nt (1 Z)Nt 1 Nt (1 Z)Nt 1

(1 w) (1 ) w = wt pt dj Nt NtZ 1 Nt 1 1 w + w (1 ) wt 1 pjt 1 Nt Nt 1 NtZ 1

1 w + wt pt dj Nt Nt (1 Z)Nt 1 w (1 ) w w + wt pjt 1 wt 1 pjt 1 Nt NtZ 1

and …nally the average wage is

(1 w) (1 ) w !t = Nt 1wt pt Nt Nt 1 + w (1 ) !t 1 Nt

Nt (1 ) Nt 1 w + wt pt Nt w (1 ) + w;t;t 1dj Nt r NtZ 1

w w where w;t;t 1 = wt pjt 1 wt 1 pjt 1 which is another delta (inverted delta) variable. The r integral in the last term of the expression is impossible to compute. However when log linearized it can be computed easily. This is the purpose of the delta variables I keep using. As you can see,

w w w log-linearizing wt pjt 1 wt 1 pjt 1 I manage to cancel pjt 1 away which is exactly what gave us a hard time in the above integral.

61 Log-linearizing the average wage equation I get the equation for the average economy-wide wage

!^t = (1 ) (^nt 1 n^t +w ^t) w (1 ) (^nt 1 n^t +w ^t)

+ w (1 ) (^nt 1 n^t +! ^t 1)

+w ^t (1 ) (^nt 1 n^t +w ^t) w (1 ) + ^ t;t 1 w r

The only thing needs to be de…ned is w;t;t 1. I repeat below the wage equation for a …rm charging r w a price pjt 1 on period t 1

(1 ) D pw + (1 ) b w 1 t 1 jt 1 wt 1 p = jt 1 w 2 pw 3 Lt 1 p u jt 1 w jt 1 (1 ) D + Kt 1#t 1+ xt 1 p t 1 Pt 1 jt 1 6 7 4 5 and the wage this same …rm would pay if it failed to adjust its price next period. (Still there is in‡ation indexation in this case)

P w w w t 1 P (1 ) Dt pjt 1 w + (1 ) b w t 1 1 Pt 2 w p = t jt 1 w P w 2 w w w 3 P w t 1 u pjt 1Pt 1 w Pt 1 t 2 L p w t jt 1 P (1 ) Dt + Kt#t + P P w xt pjt 1 P w t 2 t t 2 t 2 6 7 4 5 u u G(Lt) u G (Lt ) where in both equations Dt = is the disutility from labor and D = is the disutility t t t from search e¤ort.

Log linearizing both expressions and subtract I get

^ w 1 ^ ^ w w w 1 ^ ^ w;t;t 1 = w At At 1 + t t 1 + Xt Xt 1 r 1+` L w 1 ^ ^ w w w 1 ^ ^ 1 (1 ) At At 1 + t t 1 + Xt Xt 1 1+` (^t ^t 1) 1 2 3 h (1 ) D u D^ u D^ u + K # K^ K^ + #^ #^ i L t t 1 t t 1 t t 1 6 7 6 x X x x ^ ^ w w ^ ^ 7 6 +P N Pt Pt 1 + (w 1) At At 1 + (w 1) t t 1 + Xt Xt 1 7 6 7 4 h i 5

62 F Appendix

I derive here the job creation condition. I start from the vacancy posting condition

Kt J w = EtQt;t+1 Ft+1 pt+1 (50) qt n o

I substitute in the value of a job to the …rm when it is adjusting it’sprice in the next period

P w J w w w J w J w t+1 Ft+1 pt+1 = t+1 pt+1 + (1 ) Et+1Qt+1;t+2 (1 w) Ft+2 pt+2 + wFt+2 pt+1 w Pt

It is clear from here that the relevant information for vacancy posting is the value of new …rms and the new …rms are always price adjusters next period. Manipulating the above expression I get

J w w w J w F p = p + (1 ) Et+1Qt+1;t+2 F p t+1 t+1 t+1 t+1 t+2 t+2 nP w o J w t+1 J w + w (1 ) Et+1Qt+1;t+2 Ft+2 pt+1 w Ft+2 pt+2 Pt using (50) I get

J w w w Kt+1 w w w Ft+1 pt+1 = t+1 pt+1 + (1 ) + w (1 ) Et+1Qt+1;t+2t+2 pt+1; pt+2 (51) qt+1

P w w w w J w t+1 J w where t+2 pt+1; pt+2 = Ft+2 pt+1 P w Ft+2 pt+2 t Plugging (51) back in (50) I get the job creation condition

Kt w w Kt+1 w w w = EtQt;t+1 t+1 pt+1 + (1 ) + w (1 ) Et+1Qt+1;t+2t+2 pt+1; pt+2 qt qt+1

w w w The de…nition of t+2 pt+1; pt+2 is useful because …rst of all, the steady state is zero and also it becomes zero when log linearized. Since I’m about to log-linearize the model, the reader could as well as ignore this delta term in the above expression, especially because with or without it the log

63 linearized expressions will be exactly the same.

G Appendix

In this section I present the details of the structural VAR in …gure 1. The Central Bank sets it’s instrument it every period given the information available. Therefore the Taylor rule the Central Bank uses is the following:

m it = z (It) + "t where it is the nominal interest rate, It the information set at period t , z (:) is a linear function m and "t is the monetary policy shock. To be able to identify the e¤ect of the monetary policy shock, I de…ne the following structural VAR:

yt = + t+1yt 1 + 2yt 2 + ::: + pyt p + B"t (52)

E ("t) = 0

E ("t"t0 ) =

In my example yt is a (8 1), A is a (8 8) matrix of constants, c and are (8 1) coe¢ cients for the constant and time trend t, p is the (8 8) matrix of VAR coe¢ cients for p = 1; ::; 4 and m "t is the (8 1) vector of shocks with the monetary shock " as the last element of this vector. I t use the usual short run restrictions in order to identify the monetary policy shock. I assume that a policy shock only a¤ects the nominal interest rate contemporaneously. This means that monetary policy shock a¤ects all variables other than the policy instrument after one period lag. In terms of the VAR above, this means that the B matrix is lower triangular with unit diagonal elements.

64 0.4 0.5 2

0.2 0

0 0 Inflation

Real GDP Real •2 Vacancies •0.2

•0.4 •0.5 •4 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 time time time

3 0.2 1.5

2 0.1 1

1 0 0.5 Wage 65

0 •0.1 0

Fed funds rate funds Fed Unemployment

•1 •0.2 •0.5 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 time time time

1 0.4

0.2 0

0 Hours •1

JobCreation •0.2

•2 •0.4 0 5 10 15 20 0 5 10 15 20 time time

Figure 4: The solid lines are the empirical responses after a unit monetary shock, while the red sold lines with bullet marks are the model implied responses. The gray areas are 90% con…dence intervals. 1.5 2.5 0.5

1 2

0 0.5

1.5

•0.5 1 •0.5

Vacancies •1

0.5 Creation Job

Unemployment •1

•1.5

•2 •1.5

•0.5 90% CI •2.5 model empirical

•3 •1 •2 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 ti me ti me ti me

Figure 5: This is the optimal …t of our model without our job market speci…cation. The model adds enough persistence and …ts fairly well all variables other than the job market ones and that’sthe reason I

exclude the rest of the responses from the picture.

0.5 2 0.2

0.1

0 1.5

•0.1 •0.5 1

•0.2

Vacancies Job Creation Job •1 Unemployment 0.5 •0.3

•0.4

•1.5 0

xu = 0.3 •0.5 xu = 0.23 xu = 0

•2 •0.5 •0.6 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 ti me ti me ti me

Figure 6: Impulse responses after a monetary shock for di¤erent values of u under the baseline

calibration.

66 0.4 1 0.3

0.2 0.8 0.2

0 0.6 0.1

•0.2

0.4 0

•0.4 Vacancies

0.2 Creation Job •0.1 Unemployment •0.6

0 •0.2 •0.8

f = 0 •0.2 f = 1.5 •0.3 •1 f = 3

•1.2 •0.4 •0.4 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 ti me ti me ti me

Figure 7: I assume a constant disutility of e¤ort (u = 0) and let the application units vary.

1.5 2.5 0.5

1 2

0 0.5

1.5

•0.5 1 •0.5

Vacancies •1

0.5 Creation Job Unemployment •1

•1.5

•2 •1.5 90% CI •0.5 model •2.5 empirical

•3 •1 •2 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 ti me ti me ti me

Figure 8: This …gure presents the optimal …t of the model when u = 0 with a non-zero . I focus on the

labor market variables again since the rest of the responses perform fairly well.

67 0.4 0.6 2

0.4 1 0.2 0.2 0 0

0 •1

Inflation

Real GDP Vacancies •0.2 •0.2 •2

•0.4 •0.4 •3 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 time time time

3 2 1.5

2 0 1

1 •2 0.5

68 Total Wage Total

0 •4 funds rate Fed 0 Unemployment

•1 •6 •0.5 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 time time time

1 0.4

0.2 0 g = 0.97 0 g = 0.85 •1 g = 0.55

JobCreation •0.2

•2 •0.4 0 5 10 15 20 0 5 10 15 20 time

Figure 9: This is the optimal …t of the model without the social norm and with ‡exible intermediate sector, for di¤erent degrees of price rigidity for retail …rms . 0.4 0.6 2

0.4 1 0.2 0.2 0 0

0 •1

Inflation

Real GDP V acancies •0.2 •0.2 •2

•0.4 •0.4 •3 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 time time time

3 100 1.5

2 50 1

1 0 0.5 Total Wage

0 •50 fundsFed rate 0 Unemployment

•1 •100 •0.5 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 time time time

1 0.4

0.2 0

0 Hours •1

Job Creation •0.2

•2 •0.4 0 5 10 15 20 0 5 10 15 20 time time

Figure 10: This is the …t of the model with the social norm assumption, when there are ‡exible prices in

the intermediate …rms and the degree of price rigidity for retail …rms , is set to 0.85.

1.5 2.5 0.5

1 2

0 0.5

1.5

•0.5 1 •0.5

Vacancies •1

0.5 Creation Job

Unemployment •1

•1.5

•2 •1.5 90% CI •0.5 model •2.5 empirical

•3 •1 •2 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 ti me ti me ti me

Figure 11: Those are the optimal impulse responses after a 1% monetary shock of our model without

vacancy adjustment costs.

69 R&D, Product Innovation and Employment

10 December 2010

Abstract

In this study I attempt to solve the ampli…cation puzzle, the inability of the standard search and matching model to account for the volatility in vacancies and unemployment, by exploring the connection between R&D and employment. R&D a¤ects product creation and product creation a¤ects employment. An improvement in technology bene…ts the economy in two ways.

Same products can be produced more e¢ ciently and also new products are created. Empirical evidence suggest that the increase in production for already existing goods does not imply increases in employment, while new products are associated with increases in employment.

The search and matching model implies that changes in technology do not imply large changes in employment for already existing goods which is in line with what the evidence suggest.

However, when the search and matching model applies for sectors that innovate and produce new products, changes in employment signi…cantly increase. Therefore, in this model I assume all agents need to innovate …rst before they create a job opening, because …rms that invent new products are the ones that contribute more to the volatility of employment according to the evidence. Since ideas are cheaper to implement after a technological expansion, the cost of vacancies becomes countercyclical which boosts job creation and vacancies. The model can amplify the volatilities of vacancies, unemployment and market tightness approximately by up to 300 percent.

70 1 Introduction

The standard search and matching model by Mortensen and Pissarides (1994), despite it’sintuitive framework and predictions, it cannot replicate the volatility of vacancies, unemployment and market tightness observed in empirical exercises (Shimer (2005), Fujita (2003)). There are many attempts to resolve the issue and the solutions proposed insinuate di¤erent implications for policy analysis1 .

I follow a di¤erent route from the literature and attempt to explore how the connection between investments in R&D and the cost of inventing new products (jobs) can increase the volatilities of vacancies, unemployment and market tightness.

The importance of Research And Development (R&D) on technology and productivity has been studied extensively over the past decades especially by growth economists. Although the in‡uence of R&D on productivity is undisputed, the impact of innovation and R&D on employment is controversial. R&D and innovation can a¤ect productivity through two di¤erent channels, process innovation and product innovation. Process innovation involves original ways of producing goods and services while Product innovation involves the creation of new or di¤erentiated products. Even though authors like Greenan and Guellec (2000), Bogliacino and Pianta (2010) and others …nd that the e¤ect of process innovation on employment is arguable, they all agree that the e¤ect of product innovation on employment is positive and statistically signi…cant. That is because process innovation may destroy jobs since newly invented machines or advances in productivity can deem some jobs obsolete while product innovation leads to more job openings, by unveiling new economic channels. Some vacancies can be thought as a by-product of product innovation. Some jobs are created by …rms that create new products. My argument is that those particular …rms are the ones that make vacancies and job creation so volatile in the data. A new job can be considered as a new

…rm or product and a new product is a new idea that needs to be invented which can be put to production only after it …nds a worker to put the idea to life.

I create a framework where every …rm/job is a di¤erentiated product and product innovation needs the services of an R&D …rm for the idea to appear and a worker to implement this idea for

1 Form more information about the issues and recent attempts you can check Cardullo (2008).

71 production purposes. To open a vacancy you need a new idea for a product. The R&D …rm is charging a fee for it’s service and this is the only cost to post a vacancy2 . Moreover, R&D …rms invest in the quality of the ideas they provide and higher quality entails greater e¢ ciency in the production of ideas for new products. This implies that in an expansion, vacancy demand increase and investment in R&D rises which makes the cost of producing vacancies lower and even more vacant positions to be opened.

The reduced cost of ideas a¤ects vacancy creation in two ways. First it lowers vacancy costs directly and second it extends …rm’s pro…ts by diminishing the wage bargained by the workers.

The two main ingredients for innovative …rms to exist are an idea and a worker. In an expansion, although workers are harder to …nd and expensive to keep, ideas are easier to be created and implemented. A job is a combination of an idea and a worker and the degree up to which a …rm values each element determines it’s bargaining power and the wage it pays it’s worker. However, when one element becomes more valuable, the other becomes more cheap, balancing the bargaining power of the …rm, leaving the wage less responsive and pro…ts along with incentives to post vacancies more volatile. In other words, lower vacancy costs tend to decrease the wage from a job match because the outside opportunity of the …rm (the vacancy state) becomes more bene…cial whenever bargain power in the current match slips from the hands of the …rm and vice versa.

The addition of investments in R&D can amplify the volatilities of vacancies, unemployment and market tightness considerably. The model cannot account for the volatility of the real wage, but could be considered as part of a more complex model where some kind of wage rigidity is present to keep the wage from overshooting. The connection of R&D investment and job creation might be one of the missing links in the search and matching theory that may need to be further explored by researchers.

The following sections include the following features: The exploration of the connection between

R&D and employment along with some empirical evidence by other authors. A detailed derivation of the R&D sector that explicitly de…nes the price of a vacancy as a function of aggregate vacancy

2 There could be also a cost to advertise the vacancy which is the standard assumption in the search and matching models. Refraining to assume that on top of the cost to come up with the idea to open a vacancy there are advertizing costs does not change the results of this paper.

72 demand. After the R&D sector is fully de…ned, I present the rest of the model, followed by the calibration section and the results.

2 The Impact of R&D on Employment

Does innovation and new technology create or destroy jobs? This is a rather di¢ cult question that has been bothering researchers from classical economists to today. During the …rst industrial revo- lution, British workers in fear of technological unemployment destroyed the machines in industrial areas as well as the countryside. Technology can actually destroy a lot of jobs rapidly but can also create some. Lots of workers have been replaced by machines during the last few decades where technological progress has become more pervasive. However, technological progress is a rather vague idea as technological innovation could mean di¤erent things. Technological progress could either come in the form of process innovation or in the form of product innovation.

Process innovation involves advances in the way of producing certain goods and services. Process innovation is the form of technological progress that could rapidly shed a large number of jobs in- stantaneously, at least in the short run. However, economists responded to the critics of technological innovation by putting forward a theory that Marx called "compensation theory". The theory includes di¤erent compensation mechanisms in various markets that can actually counterbalance the initial shed of jobs due to process innovation. Empirical analysis by various authors such as

Greenan and Guellec (2000), Pianta (2005), Garcia et. al (2002), Evangelista (2000), Blanch‡ower and Burguess (1998) report controversial evidence about the relation between process innovation and employment.

Product innovation is a form of technological progress that involves the creation of new or di¤erentiated products. Product innovation can create new economic branches along with new job positions. It is the most "labor-friendly" form of technological progress. Despite any ambiguity in the impact of process innovation on employment, there is a consensus that product innovation stimulates employment as proclaimed by Harrison et al (2005), Peters (2005), Bogliacino and Pianta

(2010), Vivarelli and Pianta (2000), Edquist, Hommen and McKelvey (2001), Freeman and Soete

73 (1994) and others. Moreover, R&D expenditures can boost product innovation which consequently increase job creation. The direct impact of R&D on employment is investigated by Bogliacino and

Vivarelli (2010).

In this paper I impose a rather strong assumption. I assume that for a job to be created, an idea of a new product needs to be generated. This is not entirely realistic since some jobs are simply created along premises that already exist. For example, if I have a secretary and I wish to hire another one, then there is no need for an idea or research or R&D of any sort to create this job opening. Not all jobs come from new and innovative products or ideas. Nevertheless, I make this assumption based on the empirical evidence about the e¤ect of product innovation I have already discussed. The ‡uctuations in employment tend to be related to the e¤ect of new product creation.

Therefore, I assume that in the model economy there are only …rms that are innovative. You could think of this model as a simpli…ed version of a relatively more complicated model, where there are two types of …rms. Firms that do not innovate and …rms that innovate and produce new goods. The

…rms that create already existing jobs with no need of R&D are going to have the same structure as the standard search and matching model where the e¤ect of a technology shock on job creation in that sector would be minimal. The other sector, containing the …rms that are creative to the point of designing new products, is going to be the one that is going to generate most of the volatility in employment, which is in line with what the empirical evidence suggest. In my model I discard completely the …rst sector and just assume that all new jobs produce new and innovative products since that particular sector is going to be the one that creates the largest employment ‡uctuations in the …rst place.

The above authors have a lot of explanations as to why process innovation might not necessarily be translated to more jobs. My own explanation lies exactly along the lines of the standard search and matching model’s woes. The search and matching model predicts that after a productivity shock, job creation and employment are barely responsive. This should not be a problem at all since process innovation actually does not seem to create new jobs. Even though the e¤ect of a technology shock raises productivity and output, the e¤ect on employment is rather mild. Product innovation is the one that is associated with employment ‡uctuations. The standard search and

74 matching model assumes that increases in productivity lead to a very mild increase in the creation of more of those same …rms, which is what the empirical evidence suggest. If one is interested to see after a appositive productivity shock,whether many more of the same jobs/…rms are going to appear, the search and matching model gives actually the right answer. Not many of the same jobs will suddenly appear. If we allow new products to be created, the search and matching model

-as I tend to show in this paper- implies that a positive technology shock can amplify job creation signi…cantly, exactly what those authors predict the connection between product innovation and technology should be.

Speci…cally, in this paper I assume that an increase in employment can only be achieved through product innovation. Every …rm is a single job producing a di¤erentiated product. Therefore, for a

…rm to be created, a di¤erentiated product needs to be created and a worker needs to be employed.

I create the necessity for R&D services for agents seeking to create vacancies. I assume vacancies are potential di¤erentiated products that need to be developed and their development needs ideas.

R&D …rms are in charge of providing ideas for a fee and also invest in the quality of their services.

In a sense, R&D …rms are in charge of product innovation. The higher the quality investment in

R&D, the more e¢ cient those research …rms are in producing ideas or vacancies. Higher quality research makes ideas ‡oating from the R&D sector more e¤ective and the fee they charge for each vacancy lower. In an expansion, vacancies increase and R&D …rms invest more in the quality of their service. Improved quality enhances R&D …rms’productivity and higher productivity in R&D makes the cost of vacancies to drop, creating another wave of potential job positions, boosting overall vacancies and job creation. In other words I create a theoretical framework where vacancies are considered as ideas of new product along with an opening for a worker to put this idea into production. I demonstrate that the additional impact of improvements in R&D on employment is important for job creation and this is something that might have been ignored in the search and matching literature so far. Product innovation is important for employment and the level of

R&D can boost the capacity of ideas produced. Thus, as suggested by the empirical evidence, the cyclicality of R&D might be one of the driving forces of the large cyclical variations in employment that is missing from the standard search and matching model and this is what the purpose of this

75 paper is all about.

3 R&D Firms

Jobs in my model are di¤erentiated products, therefore vacancies are ideas about di¤erentiated products and the invention of a vacant position needs research. Consumers in the economy are capable of producing ideas and ideas are the main ingredient for vacancy creation. Consumers are basically the R&D …rms even though the speci…cation of who provides the ideas is of minor importance. Therefore there’sa market for ideas and agents attempting to create a …rm through vacancy creation, need those services. Individual ideas vst can be produced by the following production function:

1 v vst = (Lst) (1)

v th where Lst is part of the …nal good and vst the number of ideas the s individual provides. In some sense this is the amount of food needed to produce ideas. It’sbasically food for thought.

The aggregate research service or vacancies produced Vt in this sector is a combination of all

th the ideas in the economy, where qst is the quality of the s researcher

v 1 v 1 v 1 v Vt = q vst ds 2 st 3 Z 0 4 5 The cost minimization problem of the agent that purchases this aggregate research good in an attempt to create vacancies implies that the demand for ideas for the individual R&D …rm is:

v v (v 1) pst v = q V (2) st st P v t t

v where pst is the price the individual R&D …rm charges for it’s services. Also from the same cost

76 minimization problem the price of the aggregate vacancies posted is:

1 1 1 v v 1 v v pst Pt = ds 2 qst 3 Z0 4 5 which states that the aggregate price is the aggregate of the individual quality adjusted prices.

Since all agents provide the same amount of ideas the above relative price becomes:

v pst v = qst (3) Pt and the demand (2) can be written alternatively as

( 1) Vt vst = qst (qst) Vt = (4) qst

The pro…t maximization problem of the individual R&D …rm which decides the price and investment in research quality is:

v pt Pt v max E0 Q0;t vt L c (qt) v t p ;qt Pt Pt f t g t=0 X where c (qt) is the cost of employing quality qt and Q0;t stochastic discount factor. s.t.

the demand for v v (v 1) pt v = q V t t P v t t the idea production function

v vt = Lt

I drop the s subscript because every individual is identical and the decision problem is the same

77 for all R&D …rms. Substitute the constraints into the objective to get

v v v v v pt (v 1) pt (v 1) pt max E0 Q0;t q Vt q Vt c (qt) v t v t v p ;qt 8Pt P P 9 f t g t=0 t " t # X < =

: v ; The F.O.C with respect to pt is

1 v v v pt v (v 1) pt = qt v Vt (5) Pt v 1 P " t #

Use the fact that every research …rm charges the same price, equation (3) and the relative price of an individual’sideas is: v 1 pt v Vt = (6) Pt v 1 q st x This means that the relative price of vacancies Pt is

v 1 x Pt v 1 Vt Pt = (7) Pt (v 1) q q t st

The F.O.C with respect to qt is

1 v v v v v pt (v 1) pt (v 1) 1 pt (v 1) q Vt q Vt = c0 (qt) (8) 2P t P v 3 t P v t t ! t 4 5 and after using (3), (5) and (6)

1 Vt (1+) (1 ) q V = c0 (q ) (9) q t t t t

3.1 The Vacancy Cost

Start from the cost of vacancies as a function of both the aggregate vacancy number Vt and quality qt, equation (7), which is repeated below

v 1 x Pt 1 v Vt Pt = Pt qt (v 1) qt

78 If < 1, it means the production of ideas exhibits increasing returns to scale and therefore the cost of a vacancy would be decreasing without the need of quality to play any particular role. The negative relation between vacancies and its cost can be guaranteed by the increasing returns to scale assumption. I do not intent to assume increasing returns to scale thus I set = 1. My goal in this section is to eliminate qt in the above equation to derive an expression between the cost of vacancies and the aggregate demand for vacancies in the economy.

In order to be able to observe what the introduction of quality in R&D can imply for vacancy dynamics, assume c0 (qt) = c. Use the …rst order condition for quality, equation (9). Solve for qt and get 1 1 c 1+ q = V 1+ (10) t t

Then eliminate qt from the expression for the cost of a vacancy, equation (7). That is

1+ v x 1+ Pt = Vt (v 1) c

The cost of vacancies is a constant times a function of vacancies

x 1+ Pt = Vt (11)

where = v 1+ is that constant. Equation (11) summarizes the result of the paper and I (v 1) c am going to refer to it immediately after I restate the puzzle and my goal in the next section.

3.2 Puzzle and Solution

The ampli…cation puzzle arises in the search and matching model due to the wage implied by the speci…c bargaining problem faced by both …rms and workers according to this framework. The problem is that the wage is too dependent on the value of workers’ outside opportunity which is unemployment. After an expansionary productivity shock, vacancies rise and unemployment drops. The few unemployed are facing a large number of available jobs which gives employed workers bargaining power. The employed worker can reach for a higher wage by threatening to

79 break the current match now that the unemployment duration has dropped signi…cantly. The …rm is in a bad position to negotiate because if the match breaks, the average duration of a vacancy is relatively high compared to unemployment duration. Therefore the worker can negotiate such a high wage that basically diminishes …rm’s pro…ts and the incentive to post vacancies to begin with. In the standard search and matching model, the wage absorbs most of the e¤ect of the shock leaving pro…ts and consequently vacancies barely responsive.

However, this is the e¤ect of a technology shock on vacancies and employment if more of the already existing jobs are created. The story is di¤erent for …rms that innovate. Even before I present the whole model, the main result can be observed simply by looking equation (11) that I repeat below x 1+ Pt = Vt

This equation states that the cost of a vacancy is decreasing in the number of vacancies demanded.

The above relationship arises due to quality improvements by the research …rms. The greater the vacancy demand, the larger the pro…t the R&D …rm experiences and thus the larger the capability of investment in the quality of the …rm’sproduct. The quality improvement increases productivity and enables the R&D …rm to further decrease it’scost.

After a positive technology shock, the cost of a vacancy drops according to the equation above which can boost job creation and vacancies through two channels. The …rst channel is the obvious one. Lower cost of vacancies means more vacancies posted. However there is another channel that is not as obvious. Lower cost of a vacancy a¤ects the relative bargaining power between …nal good …rms and workers. In an expansion the duration of a vacancy increases, but the cost of a vacancy drops since R&D …rms can improve their quality. Lower vacancy cost puts …rms back into the negotiations game. It enables the …rm to deny a too high wage to the worker since upon a destruction of a match, the …rm is not in a bad situation anymore. The average vacancy duration may be large in a boom but according to (11) the vacancy cost decreases improving …rms’outside opportunities which is the value of the vacancy state. This prevents the wage from absorbing most of the e¤ect of the shock as in the standard search and matching model, thus boosting pro…ts and

80 Figure 1: The agents in the model, their basic interactions and the ‡ow of goods and services.

the incentive to post even more vacancies.

A job has two elements when only the innovative …rms are considered. The …rst element is an original idea and the second a worker to put this idea into the production process. Along the business cycle the scarcity of those elements varies. Novertheless, it varies in such a way that when workers are more scarce, ideas are more wasy to grasp or impliment and vice versa, meaning that the …rm does is not facing a wage that absorbes most of the e¤ect of the shock anymore.

Therefore, innovative …rms face a quite di¤erent bargaining problem than usual …rms and this is the main reason that the search and matching model fails to explain the ampli…cation puzzle. How- ever, the standard model actually predicts that advances in technology may enhance productivity or output but not employment as much simply because all …rms in the standard model produce the same product. When …rms that create new products are examined, the wage structure implied from the search and matching model does not hinder job creation to expand.

81 4 The Model

The model consists of three agents as can be seen in …gure 1. Households that choose consumption every period and also post vacancies which when successfully matched can become a …rm with the household as the only shareholder. Vacancies are new products or jobs and their creation needs the services of creative minds. R&D …rms provide those ideas to households trying to open job positions in exchange for a fee. Moreover households provide labor to the …nal good …rms which use it as an input. The number of employed workers, hours and wages are determined as in the traditional search and matching model. Final good …rms use labor to produce the …nal good which is partly consumed by the households and the rest is used as an input by R&D …rms for the creation of ideas for product (or job) innovation.

4.1 Household Problem

There is a large number of identical households each containing Ih members. Therefore all house-

w holds are identical supporting market completeness as in Merz (1995). A measure of Nht household w members are employed while the rest Ih N remains unemployed. Households get utility from ht consumption and disutility from labor. In addition, households post vacancies in an attempt to gain ownership of …nal good …rms. Every consumer in the economy employed or not is able to produce ideas that are a necessary ingredient for job creation. There are many households each maximizing the following utility function:

1 1+` (Cht Ct 1) (Lhjt) U (Cht;Lhjt) = dj 1 1 + ` w NZht

82 subject to:

x Cht + Bht + Pt Vht

w = (Ih N ) bt + whjtLhjtdj + Tht ht w NZht 1 v + itdi + sht hstds + (1 + it 1) Bht 1 NZht Z0

Nht+1 = (1 ) Nht + gtVht where Cht is the consumption chosen by the household h, Ct 1 the aggregate consumption, Lhjt w v the labor hours in production from the Nht employed workers and Lhkt the e¤ort chosen in the x production of ideas from each of the Ih members of the household. Real Bond holdings is Bht, Pt the price R&D …rms charge per vacancy and Vht the vacancies the household posts. The unemployment

w bene…t received by the household’sunemployed Ih N is bt. The household receives whjtLhjt wage ht w income from the Nht employed workers. The household earns dividends from consumption good v …rms it and research …rms hst, the …rms providing the "ideas". The number of …rms owned by the household Nht+1 next period is the surviving …rms from the previous period (1 ) Nht (…rms are destroyed with probability ) plus the vacancies the household posts Vht weighted by the probability of a vacancy to match gt. The …rst order conditions for bonds give:

1 ht+1 = Et 1 + it ht

where ht is marginal utility. The …rst order condition for vacancies implies

x x Pt Pt+1 = EtQt;t+1t+1 + (1 ) EtQt;t+1 gt gt+1

83 Moreover, in equilibrium the equation of the evolution of the economy-wide employment is

Nt+1 = (1 ) Nt + Mt (12) where

1 Mt = mV (1 Nt) t is the constant returns to scale matching function.

4.2 The Bellman Equations

4.2.1 The …rms

The value of a vacancy to the …rm is:

V x J V F = P + gtEtQt;t+1F + (1 gt) EtQt;t+1F (13) t t t+1 t+1

x J where Pt is the cost of a vacancy. Next period the vacancy becomes a job with value Ft+1 weighted by the probability of a vacancy to be matched gt or it remains a vacancy with probability

1 gt. Since the value of a vacancy is zero due to free entry, then the above condition in equilibrium becomes x Pt J = EtQt;t+1Ft+1 gt

The value of a job to the …rm is:

J J F = t + (1 ) EtQt;t+1F (14) t t+1

where t is the pro…t of the …nal good …rm which survives the next period with probability 1 . Since the value of a vacancy is zero, then (13) becomes:

x Pt J = EtQt;t+1Ft+1 gt

84 and after using (14) we get the job creation condition which can also arise from the household maximization problem with respect to Vht

x x Pt Pt+1 = EtQt;t+1t+1 + (1 ) EtQt;t+1 (15) gt gt+1

4.2.2 The workers

The value of a job to the worker is:

E G (Lt) E U Wt = wtLt + EtQt;t+1 (1 ) Wt+1 + Wt+1 (16) t where wtLt is the wage received and G (Lt) is the disutility of labor in utility units and t the marginal utility. Next period the job is either continued or terminated with probabilities 1 and respectively.

The value of unemployment to the worker is:

U E U W = bt + EtQt;t+1 tW + (1 t) W (17) t t+1 t+1 where bt is the unemployment bene…t and t the probability corresponding to the event of a worker to …nd a job.

4.2.3 Bargaining and wage

Firms maximize the surplus from a match by choosing the price. The total surplus is the sum of the surpluses a successful match generates.

J V E U St = F F + W W t t t t

Firms and workers engage into Nash bargaining and split the surplus by assigning share to the workers and the rest 1 to the …rms. The above procedure is similar to maximizing the objective

85 below with respect to the wage.

1 J V E U max Ft Ft Wt Wt wt n o The …rst order condition results to:

(1 ) W E W U = F J F V t t t t Substitute in the above the the value of a job to the …rm equation (14), the value of a job to the workers equation (16) and the value of unemployment equation (17). This implies the wage equation is

G (Lt) x pt wtLt = (1 ) + bt + P #t + yt (18) t P t t 1+` 1 Lt x Yt = (1 ) + bt + Pt #t + 1 + ` t ! Nt

There are Nt employed individuals or equivalently, there are Nt …nal good …rms. Each …rm uses

Ljt units of the worker’s e¤ort as input to produce the di¤erentiated good yjt via the following production function

yjt = ZtLjt (19)

where Zt is an AR(1) productivity shock following

Zt = (1 z) + zZt 1 + "t

and "t is iid. The aggregate consumption good is a combination of all individual …rms’outputs. The aggregation is according to the following Dixit-Stiglitz function

1 1 Yt = At y dj 2 jt 3 NZt 4 5

86 where 1 1 At N t and is the Love Of Variety parameter. The aggregate price of the composite good Yt is

1 1 1 1 Pt = pjt dj (20) At 2 3 NZt 4 5 and the demand for each good yjt which is used mainly for consumption is:

1 pjt y = A Y (21) jt t P t t

The …rms maximize the total surplus from a match by choosing every period the price they charge for their good. The total surplus to be maximized is

J V E U St = F F + W W (22) t t t t

From the Bellman equations of the precious section substitute the …rm’sand worker’ssurpluses in the above for the total surplus, to …nd the surplus as a function of pjt the price the …rm charges

pjt for its services. Moreover, use the fact that the pro…t is jt = yjt wtLt. After the above Pt manipulations, maximizing the total surplus is equivalent to maximizing the following objective function:

pjt max yjt tLt pt P t

G(Lt) subject to the demand function (21), and the production function (19). The function t = t is the disutility from providing Lt labor e¤ort in terms of the real good. The …rm maximizes the surplus by treating t as …xed. For further details on this result check Koursaros (2009). The …rst order condition implies the usual monopoly result that the price is a markup over the real marginal cost. Ignore the j subscript since all …rms are identical. The relative price of a single

87 …rm then should be p t = t Pt 1 Zt Final good …rms should set the relative price equal to the marginal cost which is the disutility of labor p t = Lt Pt 1 Zt ` and substitute the expression for the disutility of labor G (Lt) = Lt to get

p G (L ) t = L t Pt 1 tZt and since pt = N I get Pt t

` Lt 1 Nt = 1 t Zt 1+` L 1 Yt t = t Nt

Plug this into the wage equation (18) to get

1 1 Yt x Yt wtLt = (1 ) + bt + P #t + 1 + ` N t N t t

In a similar fashion the pro…t of the …rm after substituting in the wage equation above is:

pt t = yt wtLt Pt 1 + ` Yt x = (1 ) (1 ) bt Pt #t (1 + `) Nt

Keep in mind that the pro…t of the …rm which appears in the job creation condition (15) and the objective of the …rm are not the same. The …rm maximizes the surplus and not the pro…t. The pro…t is part of the surplus. However, investors care about cash ‡ows and not the surplus, therefore, the job creation condition is a statement about cash ‡ows.

88 4.2.4 The solution

The following system characterizes the solution:

The feasibility constraint:

x Yt = Ct + Pt Vt (23) where I assume the vacancy costs are subsidized by the government.

The law of motion of employment:

1 Nt+1 = (1 ) Nt + h (Vt) (1 Nt) (24)

The job creation condition:

x x Pt Pt+1 = EtQt;t+1t+1 + (1 ) EtQt;t+1 (25) gt gt+1

The pro…t of the …rm producing the consumption good:

1 + ` Yt x t = (1 ) (1 ) bt Pt #t (26) (1 + `) Nt

The pro…t maximizing condition of the …rm producing the consumption good

` `(1+)+ 1+` Yt Nt Zt = (27) 1 t

The cost of a vacancy (cost of producing ideas)

1+ v x 1+ Pt = Vt (28) (v 1) c and the condition for the evolution of quality

1 1 c 1+ q = V 1+ (29) t t 89 Equations (23) to (29) characterize the solution.

4.2.5 The price of ideas and vacancy dynamics

It is easy now to show the mechanism that can boost job creation by examining the appropriate equations as I have already presented all the equations governing the dynamics in the model. Solve forward the job creation condition (15)

x Pt 1 j = Et (1 ) Qt;t+j+1t+j+1 (30) g t j=0 X

x The above equation states that in equilibrium the cost of a vacancy is Pt (the per unit of time cost) times the duration of a vacancy 1 . The cost should be equal to the present discounted value gt of all future pro…ts the job is expected to produce for as long it is expected to survive. During an expansion, jobs are created and vacancies rise. However according to (11) the cost of a vacancy diminishes since the cost is decreasing in vacancies due to investments in quality that boosts research productivity. Therefore the …rst channel is through lowering the left hand side of the job creation condition which represents the cost of a vacancy.

x The second channel in which the cost of a vacancy Pt a¤ects job creation is through pro…ts t in the above equation and speci…cally through the wage. Remember the wage is part of the …rm’s pro…t and is determined by:

1 1 Yt x Yt wtLt = (1 ) + bt + P #t + 1 + ` N t N t t

x x The cost of a vacancy Pt a¤ects the bargaining power of …rms, thus lower Pt grants more negoti- ating power to the …rms because the …rm can …nd another worker less costly. Therefore, during an expansion, vacancies are created and the increase in vacancies induces a decrease in prices through improvements in quality. The decrease in the cost of a vacancy boosts vacancies through lower costs

(left hand side of (30)) and through higher pro…ts (right hand side of (30)). The latter comes from the dependence of the wages on outside factors such as the cost of a vacancy. During a bargaining

90 with the worker, the …rm has the option to leave the bargaining stage and keep the job vacant.

The cost of the vacancy is the …rm’soutside option and lower costs give bargaining power to …rms and eventually lead to a lower wage for the worker. The lower wage increases …rm’s pro…ts and investors incentives to open vacancies.

The story so far might have created a logical paradox. I assumed the cost of a vacancy involves the cost of producing ideas. The reader might be wondering how does that enter the wage equation.

In the wage equation, the cost of a vacancy enters as an outside option for the …rm in the case the match is destroyed. If a match is destroyed, the ideas still exists. Why would the …rms need to recreate an idea if a match is destroyed? I assume that upon destruction …rms and ideas disappear.

This might not be entirely realistic but if one thinks about the cost of …rm and job destruction, it certainly involves the destruction of ideas, capital and human capital, stocks that need some di¤erentiation to be able to reused in the production process again.

5 Calibration and results

In the following sections I calibrate the model to common parameter values either from estimates from US data or values obtained by other macroeconomic studies. I simulate the model in order to provide second moments for the volatilities of vacancies, unemployment, tightness, output, wages, hours and quality in R&D. The volatilities implied by the model are compared to the ones from quarterly US data from 1955-2005.

The variables in the US data are basically: the log of quarterly real GDP, the annualized quarterly rate of change in the CPI, the log of Help-Wanted advertising, the civilian unemployment rate, the log of job creation for continuing establishments in the manufacturing sector, the log of average hourly earnings in the business sector, the log of total hours in the business sector and the e¤ective federal funds rate.

91 Par Name/Explanation Value time discount factor 0:989 elasticity of subs …nal goods 25 risk aversion coe¢ cient 3 habit persistence 0:7 elast. matching function 0:6 workers bargaining power 0:5 quality coe¢ cient 2:5 N steady state employment 0:9 ` inverse of labor supply elast 10 z AR coe¢ cient of shock 0:8 s.s unemployment probability 0:63 separation rate 0:07 b unemployment bene…t 0:4 q steady state vacancy probability 0:7

Table 1: This is a summary of the baseline calibration.

5.1 Calibration

The time discount factor is set to 0:989. The elasticity of substitution for …nal goods is set to 25 which lies between Boivin and Giannoni’s(2005) estimate of 11 and Altig, Christiano, Eichenbaum and Linde’s (2005) estimate of 101. The risk aversion coe¢ cient is set to 3 and the habit persistence to 0.7. The elasticity of the matching function with respect to unemployment is set to

0.6 which lies in the interval of acceptable values (0.5-0.7) according to Petrongolo and Pissarides

(2001). The worker’s share of the surplus is set to 0.5. This is the usual value assumed in the literature since empirical work has no predictions for this parameter.

The quality coe¢ cient is set to 2.5. Empirical evidence do not exist for this parameter either.

The inverse labor supply elasticity is set to 10 as in Trigari (2005). The steady state employment rate is set to 0.9 which along with a separation rate of 0.07 (close to 0.08 by Hall (1999)) implies a steady state probability to get a job to 0.63 which is close to unemployment duration of 1.67 observed in US data. The unemployment bene…t is set to 0.4 which implies forty percent of steady state income. The steady state vacancy probability is set to 0.7 as common in the literature. The coe¢ cient of the lag of the technology shock is set to 0.8. The model is unable to create endogenous persistence if this coe¢ cient is set to zero.

92 Stat US M1 M2 M3

Technology z 1:03 0:9 0:73 Output Y 1:58 1:58 1:58 1:58

Vacancies V =Y 8:85 3:44 6:68 10:7 Tightness #=Y 16:33 4:03 8:21 13:3 Unemployment u=Y 7:83 1:42 2:7 4:4 Wage w=Y 0:69 1:72 1:85 2 Hours L=Y 1:15 0:3 0:3 0:33 Quality q=Y 3:38 3:01 Table 2: This is a table of second moments. The …rst column of numbers are second moments from the US data. The moments with the label M1 come from a model without quality boost. The moments under M2 are the moments under the benchmark calibration.

5.2 Result

The result is displayed in table 2. The …rst column of numbers contains statistics from the US data. M1 to M3 are second moments implied by three models for di¤erent values of the quality parameter . The model M1 corresponds to a model with the same parameter values as table 1 except for setting = 0, a model with no quality update and simply constant cost of vacancies. M2 corresponds to a model with = 1 and M3 is a model following the benchmark calibration with

= 2:5. Clearly the volatilities for vacancies, unemployment and tightness implied by M1 are too far from the ones empirically observed in US data.

However, M2 and M3, models that allow R&D …rms to invest in quality appear to be much superior to M1 in terms of the volatilities of the labor market variables. M3 might not be perfect but improves the standard model’sperformance in creating volatilities for vacancies, unemployment and tightness by about 300%. Figures 1 to 3 present impulse responses to a productivity shock for vacancies, unemployment and market tightness for di¤erent values of the quality parameter . It is clear from the responses that the addition of quality improvements can boost the volatilities of those variables.

This might be one of the missing links in the search and matching framework. R&D and improvements in R&D are important for product creation which is also important for job creation.

This model shows how the incentives to improve quality in R&D can boost product innovation

93 which implies increases in employment as the empirical evidence also suggest.

All three models overshoot the wage volatility. Wage rigidity could …x this issue even though wage rigidity should be imposed only on existing …rms to be consistent with evidence by Pissarides

(2007) and Haefke Sonntag and van Rens (2007). In the current paper I do not propose a mechanism to account for the volatility of the real wage, however wage rigidity for ongoing …rms can be the best candidate.

Employment is not as volatile and makes tightness not as volatile as it could be. The reason is the equation for the evolution of employment (12). It behaves the same way as adjustment costs. The speci…cation sacri…ces volatility for persistence. Removing this assumption will remove all persistence in the model. One of the biggest challenges in this literature is to create both enough ampli…cation and persistence at the same time. However, persistence usually comes at a cost of reducing ampli…cation considerably and as I should have mentioned this far, vacancies, unemployment and market tightness are very volatile variables.

6 Conclusion

I developed a model that explores the connection between investment in R&D as a means to boost employment. There is a big literature supporting the idea that product innovation can boost employment and R&D investment can trigger higher product innovation and higher employment. I argue that this might be one of the missing links in the search and matching literature. Vacancies can be considered as ideas that are provided by R&D …rms. Larger investment in R&D can boost the ability to innovate for the economy which creates more di¤erentiated products and jobs.

This framework can be a part of a more sophisticated macroeconomic model that could capture even more stylized facts. For example the model possesses no mechanism to match the very low volatility of wages that is observed in the data. This paper is mainly concerned with the volatilities of vacancies, unemployment and market tightness. For those three variables, the model can boost their volatilities compared to the standard model by 300%.

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97 14

6 Vacancies 4

2 d = 0 0 d = 1 d = 2.5 •2 0 1 2 3 4 5 6 7 8 9 t i me

Figure 2: Impulse responses of vacancies under the benchmark calibration for di¤erent values of the quality paremeter .

0.5 0 •0.5 •1 •1.5 •2

•2.5 Unemployment •3 •3.5 d = 0 •4 d = 1 d = 2.5 •4.5 0 1 2 3 4 5 6 7 8 9 t i me

Figure 3: Impulse responses of unemployment under the benchmark calibration for di¤erent values of the quality paremeter .

98 16

6 Market Tightness Market 4

2 d = 0 0 d = 1 d = 2.5 •2 0 1 2 3 4 5 6 7 8 9 time

Figure 4: Impulse responses of market tightness under the benchmark calibration for di¤erent values of the quality paremeter .

99 The Search and Matching Model when Agents are Learning

20 May 2011

Abstract

This study investigates the macroeconomic implications from introducing perpetual learn-

ing in a simple search and matching model. When the agents with rational expectations are

replaced with agents that are boundedly rational, the volatilities of vacancies, unemployment

and market tightness are increased signi…cantly. Job creation is connected to the present dis-

counted value of future cash ‡ows, which means that if agents do not form rational expectations,

their forecasts of future cash ‡ows are subject to periods of either excess optimism or excess

pessimism. Those extra distortions of the agents’forecasts amplify the volatility of job creation.

Therefore, the ampli…cation puzzle arising from the search and matching model might be due

to the strong assumption that agents are rational; thus, when agents need to form multiperiod

forecasts using past data as in Preston (2005) and Eusepi and Preston (2010), the search and

matching model’sampli…cation potential is enhanced. The model can replicate moments from

quarterly US data from 1955Q1 to 2007Q4. However the more ampli…cation added to the

model through higher gain parameter, the further the correlations generated by the model are

from the ones obtained from US data. That is because higher ampli…cation induces vacancies

to ‡uctuate around the rational expectations equilibrium less smoothly. Moreover, higher am-

pli…cation through learning worsens the prediction of the model for the slope of the Beveridge

curve.

100 1 Introduction

The search and matching model introduced by Mortensen and Pissarides (1994) has been criticized by various authors such as Shimer (2005), Fujita (2004) and others for it’sability to match various statistics from the US data even though as a …rst pass it seems to outperform the standard RBC model (Merz (1995), Adolfatto (1996)). Speci…cally, the standard search and matching model cannot explain the ampli…cation in vacancies, unemployment and market tightness that is observed in the US data, a drawback that is referred to as the ampli…cation puzzle. Moreover, the model fails to provide enough propagation either, as the e¤ects of the shock disappear immediately after the shock dies out. A survey about the problems and possible …xes and critics on the search and matching model can be found in Cardullo (2008).

Various attempts to enhance the performance of the standard model have been reported. The

…rst wave of attempts involved changes in the wage formation such as including wage rigidity. Those attempts are criticized by Pissarides (2007) and Haefke, Sonntag and van Rens (2008) implying wages for new …rms to be too sticky, an assumption rejected by empirical evidence. Another wave of attempts aimed at changing the calibration in the standard model as in Hagedorn and Manovskii

(2007). However, the higher ampli…cation comes at the cost of making the model too sensitive to certain policy variables like unemployment bene…ts. The most promising attempts seem to be those that aim at changing the model’s speci…cation. Although there are numerous attempts to change the model speci…cation to address the issues I already discussed, in my knowledge there is no attempt to deviate from the rational expectations assumption, a channel that might be important in the alignment of the model’spredictions with the data.

The amount of vacancies to be posted every period in a standard search and matching model are determined by the cost of posting a vacancy and the present value of all future cash ‡ows the job is expected to create in the future. The assumption that agents have rational expectations is very crucial in the determination of vacancies and jobs. If agents’forecasts are fully speci…ed, vacancies are barely volatile. However, once deviating from the rational expectations assumption and assume bounded rationality, agents need to provide forecasts of variables outside their control such as future

101 dividends, …rm revenues and labor supply as well as vacancy costs. Possible misspeci…cations in the formation of those beliefs are going to have a big impact on vacancy and job creation.

To formulate the above concept in the search and matching framework, I assume the agents have a fully speci…ed belief system but do not have complete knowledge about equilibrium determination; that is they do not know how shocks and state variables get mapped into prices and policy variables.

Thus, the agents understand that the solution to their optimization problem implies every variable is a linear function of the state variables and shocks. What agents do not know exante is the coe¢ cients that de…ne this linear relation between every variable and the state variables, something known if agents are fully rational. This deviation from rational expectations through learning is discussed extensively in the adaptive learning literature in Evans and Honkapohja (2001). I assume agents form their beliefs for the future through a constant gain learning algorithm. However, I postulate that agents do not make forecasts of future variables only a period in advance which is the common assumption in the learning literature. Agents in this model need to make multi-period forecasts as de…ned in Preston (2005) and Eusepi and Preston (2010). Their …nding is that this deviation from the literature can enhance the volatility of the standard learning model compared to the common speci…cation in Evans and Honkapohja. Since the ampli…cation problem in the search and matching framework is a severe one, I choose to assume agents need to form multi-period forecasts as a way to confront the issue.

Therefore, in this study agents know that the solution to their optimization problem implies that every variable in equilibrium is a linear combination of the model’sstate variables and shocks. The only missing piece for the agents is the determination of the forecasts for all the forward looking variables in their …rst order conditions. Since the agents realize that those future variables are linear combinations of the state variables, forecasts can be constructed as long as estimates for those coe¢ cients of the state variables can be obtained. The coe¢ cients are obtained through a recursive least squares algorithm with constant gain. The parameters are updated period by period and they are a function of past data.

The agents are boundedly rational in the sense that they make forecasts of future variables using a misspeci…ed model assuming that over the forecast horizon forecasts will remain intact

102 when in fact they will be reversed later when new information arrives. Learning distorts the …xed point mapping from perceived to actual law of motion leading to situations of excess optimism or pessimism. In an expansion, labor productivity increases and …rm pro…ts increase as well. Agents about to post vacancies need to forecast future cash ‡ows from …rms and those forecasts are formed from past realizations of dividends. The increase in pro…ts makes agents posting vacancies to be too optimistic about future returns and they end up posting more vacancies than what a rational agent would. The misspeci…ed beliefs boost the volatility of vacancies, unemployment and market tightness. Under bounded rationality, agents behave irrationally in the sense that they form beliefs through a misspeci…ed model, beliefs that agents assume will hold for the periods to come, even though they are eventually going to revise them.

There are a few papers that have demonstrated that the standard adaptive learning model can imply no signi…cant di¤erences from the rational expectations model as claimed by Williams (2003).

In addition, Huang, Liu and Zha (2008) con…rm the claim for high training periods and low gain parameter. However, Eusepi and Preston (2010) found that by forcing the agents to make decisions via multi-period forecasts, the di¤erences between the learning and rational expectations models become very signi…cant. This is the formulation I am going to assume and under it, the model can match standard deviations from quarterly US data from 1955:Q1 to 2007:Q4.

Even though increasing the gain parameter can enhance the model’s ampli…cation potential, learning seems to deteriorate the model’sperformance in matching …rst order autocorrelations and correlations of the variables under scrutiny. This happens for a couple of reasons. First, optimism or pessimism about future returns induces ‡uctuations in consumption around it’s rational expectations response which forces the vacancy schedule to do the same. Moreover, there is no endogenous job destruction. If job creation is very volatile compared to the rational expectations case but output is to remain almost the same between the two models, then job creation and vacancies under learning have to simply ‡uctuate violently around the rational expectations equilibrium. For example, in an expansion, consumption is low compared to a rational agents since consumers form beliefs from past data. their decrease in consumption leads to an big increase in vacancies. Next period consumers start to believe that income is going to be high forever and increase consumption

103 too much. This makes vacancies too low that period and the story continues going from too high to too low vacancy creation. This implies that vacancies are going to ‡uctuate violently around the rational expectations case, a behavior that forces all variables to be less autocorrelated and the correlation of vacancies and unemployment (Beveridge curve) less negative or even positive after a speci…c value for the gain parameter.

The second section presents the model where all the details about the agents and their decisions as well as the bargaining problem of the …rm and worker are determined. Moreover in the same section, I present the formation of beliefs which eventually in‡uences the actual law of motion of the model economy. The third section presents the calibration along with the results.

2 The Model

I develop in the next section a simple search and matching model in the spirit of Mortensen and

Pissarides (1994) in discrete time. There is a large number of identical households containing a continuum of consumers each. This assumption guarantees market completeness as in Merz

(1994) because each household has the same ratio of employed to unemployed people and the same portfolio each period. Households choose every period their consumption basket and vacancies to post. Posting vacancies is the means of job creation as in the standard model. A household that manages to successfully create a …rm/job by posting a vacancy is the rightful owner of the particular …rm and is entitled to any dividend the …rm can pay for as long as it survives. Firms are jobs consisting of a single worker each and they appear and vanish according to the standard search and matching literature.

The main di¤erence is the departure from the rational expectations framework by assuming that agents are learning and can thus have non rational beliefs when forming expectations. The learning framework is of the kind presented in Preston (2005) and Eusepi and Preston (2010) where agents use multi-period forecasts upon making optimal decisions instead of making forecasts only a period in advance as in the standard learning model of Evans and Honkapohja (2001).

104 2.1 Households

There is a large number of households, each one consisting of a continuum of consumers that can be either employed or unemployed. The hth household attempts to maximize the following utility function each period: 1 1+` Ch Lh U Ch;Lh = t N h t t t 1 t 1 + ` h h The household gets utility from consumption Ct and disutility from the labor e¤ort Lt supplied h by each of the Nt employed workers in the household. The ‡ow budget constraint in terms of the real good is:

h v h h h h h h h h h C + P V = Ih N b + N w L + T + N~ d t t t t t t t t t t t v h where Pt is the cost per vacancy posted and Vt the vacancies posted by the particular household. The households can post vacancies. Whenever a vacancy is …lled with a worker, the household obtains the shares of the …rm that is created by the successful match. The right hand side of the budget constraint includes the period t income from various sources. The government pays

h h unemployment bene…t b to the Ih N unemployed workers in the household denoting as Ih the t t h measure of the household. The labor income comes from the Nt employed household members h h providing Lt hours each and getting compensated wt units of the real good per unit of e¤ort. The h transfers from the government are indicated by Tt . The household is entitled to dividend payment h ~ h dt from each of the Nt …rms the household owns. The evolution of the household’sstock portfolio is

h h h N~ = (1 ) N~ + qtV t+1 t t

~ h where the number of …rms the household owns next period Nt+1, equals the surviving …rms from last period (1 ) N~ h, since is the exogenous destruction rate, along with the new …rms created by the t th h h household. The new matches are equal to the Vt vacancies posted by the household weighted by the probability of a vacancy to match qt. The number of employed workers the household has evolves according to

h h h N = (1 ) N + t Ih N t+1 t t 105 where t is the probability of an unemployed worker to leave the unemployment pool. The household has no means of in‡uencing the number of employed workers it has since it cannot bene…t it’sown workers as no control variables appear in the household employment constraint above.

2.1.1 The Household Problem

h The state variable for the household is the number of it’s employed workers Nt and the number ~ h of …rms/jobs it owns Nt . Even though those two numbers are the same in equilibrium, at the stage of the optimization problem the household knows that posting vacancies cannot boost the employment level within its borders. That is because the household realizes that it is too small compared to the whole economy and therefore the probability one of it’sown workers matches with a vacancy the household opens is negligible. Moreover, households neglect the fact that all other households are the same and thus cannot anticipate the identical and coordinated response of every ~ h h other household to the same shocks. Thus the state of the economy is t = Nt ;Nt . De…ne ^h n o with Et the household’sexpectation operator which may be di¤erent from rational. The household problem then is characterized by the following Bellman equation, since agents are maximized present discounted utility with discount factor . That is:

H h h h ^h H W ( t) = max U Ct Nt G Lt + Et W ( t+1) (1) h h Ct ;Vt f g n o subject to the budget constraint,

h v h h h h h h h h h C + P V = Ih N b + N w L + T + N~ d (2) t t t t t t t t t t ht the law of motion for jobs owned by the household,

h h h N~ = (1 ) N~ + qtV (3) t+1 t t

106 and the law of motion for the household employment level

h h h N = (1 ) N + t 1 N (4) t+1 t t The …rst order condition with respect to consumption is:

h h Uc Ct = t (5) h where t is the Lagrangian multiplier. The …rst order condition with respect to vacancies is

v ^h H h Pt Et WN~ h ( t+1) = t (6) t+1 qt

~ h The envelope condition with respect to Nt is

H ^h H h h WN~ h ( t) = (1 ) Et WN~ h ( t+1) + t dt (7) t t+1

Lead (7) one period forward and plug into (6). Then substitute (6) in after leading (6) a period in advance to get the job creation condition

v P v Pt ^h h h ^h h t+1 = Et Qt;t+1dt+1 + (1 ) Et Qt;t+1 (8) qt qt+1 where h h h t+1 Uc Ct+1 Qt;t+1 = h = h (9) t Uc Ct is the household’sstochastic discount factor. This expression is the job creation condition. The left

v hand side of the job creation condition is the cost of a vacancy because Pt is the per unit of time cost of a vacancy and 1 is the vacancy duration which is the inverse of the probability a vacancy qt

…nds a match qt.

107 The transversality condition is

U Ch P v j c t+j t+j ~ h lim Nt+j+1 = 0 (10) j U Ch q !1 c t t+j P v The term t+s N~ h is the opportunity cost for having N~ h jobs in period t + s + 1 since a qt+s t+s+1 t+s+1 P v job next period is worth as much as the cost to post a vacancy the previous period1 t+s . The qt+s ~ h transversality condition states that the present discounted value of the cost to produce Nt+s+1 the …nal period should be zero.

2.1.2 The Bellman Equations

The household problem introduced right above can further give us the surplus of a job to the worker and the surplus of a job to the …rm by simply taking the appropriate envelope condition. I present those expressions here which I am going to use in the next few sections where I present the wage determination. The di¤erence between the value of a job to the employer and the value of a vacancy to the employer2 is the envelope condition (7). However this condition is in utility units. Transform it in units of the real good by dividing with the marginal utility

H H W ( ) h W h ( t+1) N~ h t N~ t ^h t+1 t+1 h h = (1 ) Et h h + dt t t t+1

H W ~ h ( t) J Nt and de…ne Ft h which is the surplus of a job to the employer expressed in terms of the t real good while using (9). The above envelope condition becomes

F J = dh + (1 ) E^hQh F J (11) t t t t;t+1 t+1 The above equation states that the value of a job to the …rm is the dividend plus the future value of the …rm given that it survives next period, an event that occurs with probability 1 . Otherwise, v 1 Pt+s The term is the cost of a vacancy since the numerator is the cost of a vacancy per unit of time and 1=qt+s qt+s the duration of a vacancy. 2 The employers in this model are the households since vacancy posting can give the household new …rm ownership.

108 if the above is solved forward, then the value of the …rm is the present discounted value of all future cash ‡ows for as long as the …rm is expected to survive, which is exactly where our intuition guides us.

The surplus of a job to the worker, which is the di¤erence between the value of a job and the value of unemployment to the worker can be found in a similar fashion by taking an envelope

h condition in the household problem with respect to Nt , which represents the marginal utility gain for the household when an extra member switches from unemployment to employment. That is

H h h h h h h ^ H WN h ( t) = t wt Lt G Lt t bt + (1 t) EtWN h ( t+1) t t+1 Again I divide the expression above by the marginal utility to represent the above in real units and H W h ( t) E U Nt de…ne Wt Wt h which is the value of employment minus the value of unemployment t for the household. That means the surplus of a job to the worker in real units is

G Lh E U h h t h ^ h E U Wt Wt = wt Lt h bt + (1 t) EtQt;t+1 Wt+1 Wt+1 (12) t E U The surplus of a job to the household Wt Wt , according to the above equation, is the total h h h G(Lt ) wage wt Lt the employed worker earns minus the disutility of work h and the unemployment t h bene…t lost from the transition to the unemployment pool bt , plus the surplus the household is E U expecting to keep enjoying if the job continuous to the next period E^t W W , an event t+1 t+1 with associated probability t, the probability a worker …nds a job. Equations (11) and (12) de…ne the pieces of the total surplus from a match, useful ingredients for the determination of the wage which I present in one of the sections to follow.

2.2 Firms

A job is a …rm which employs a single worker. Therefore, there are Nt employed individuals in the j economy and therefore Nt …rms. Each …rm uses Lt units of the worker’se¤ort as input to produce

109 j the di¤erentiated good yt via the following production function

j j yt = AtLt (13)

where At is an AR(1) productivity shock with autoregressive coe¢ cient A and steady state value of one. That is

At = 1 A + AAt 1 + "t and "t is a zero mean iid shock. The aggregate consumption good is a combination of all individual …rms’outputs. The aggregation is according to the following Dixit-Stiglitz function

1 1 j Yt = t y dj 2 t 3 NZt 4 5 where 1 1 t N t and is the Love Of Variety parameter (LOV). The aggregate price of the composite good Yt is

1 1 1 1 j Pt = pt dj (14) t 2 3 NZt 4 5 and the demand for each good yjt which is used for consumption and vacancy creation is:

j j 1 pt yt = t Yt (15) Pt !

As we demonstrated in the previous section, a job creates a surplus for both workers and …rms.

Since this surplus is what workers and …rms are going to split, they have to make sure it is as large as possible. The …rms maximize the total surplus from a match by choosing every period the price they charge for their good. This is optimal for the part of the …rm because the …rm knows the bargaining stage will split the surplus in …xed proportions to workers and …rms. Knowing that, the

110 …rm can do nothing better than maximize that surplus. Even if the …rm is getting a very small part of the surplus, the best strategy is to maximize it picking the appropriate price. The total surplus to be maximized is

J E U St = F + W W (16) t t t

Use equations (11) and (12) to substitute the …rm’sand worker’ssurpluses in the above expression, to relate the total surplus to the choice variable of the …rm, its price pjt. In addition, use the fact j j pt j j j that the dividend is dt = yt wt Lt . After the pre-mentioned substitutions, maximizing the total Pt surplus is equivalent to maximizing the following:

j pt j j j j j max yt pt t Lt pt pt (Pt ) subject to the demand function, equation (15) and the production function (13). The variable j j GL(Lt ) t = j is the marginal disutility from providing Lt labor units in terms of the real good. t

The …rm maximizes the surplus by treating t as …xed. For further details on this result check Appendix A.

The …rst order condition implies the usual monopoly result that the price is a markup over the real marginal cost. Ignore the j subscript since all …rms are identical. The relative price of a single

…rm then should be p G (L ) t = L t Pt 1 At

` pt Given that the disutility of an extra unit of labor is GL (Lt) = L and the relative price is = N t Pt t from equation (14), the …rst order condition of the …rm becomes

` $Lt 1 Nt = (17) 1 t At

Use the production function (13) and the demand function (15) to transform the above to

` `(1+)+ 1+` Yt Nt At = (18) 1 t

111 which determines output as a function of marginal utility and the state variables and summarizes the pricing decisions of the …rms.

2.2.1 Nash Bargaining

The …rm and the worker bargain together to set the wage. The current match creates a surplus,

(16), that needs to be split to the two parties. A Nash bargaining regime suggests that the surplus is split by transferring part of the surplus to workers and 1 to …rms. In other words, the above problem is the same as maximizing the following objective function with respect to the wage

1 max F J W E W U (19) h t t t wt n o The …rst order condition implies

(1 ) W E W U = F J t t t where the individual surpluses are envelope conditions from the household problem, equations (11) and (12). After substituting those expressions in the …rst order condition above, I get the wage equation 1+` 1 Lh whLh = (1 ) t + bh + P v# + phryh (20) t t 1 + ` h t t t t t t ! The derivation of the wage is in the Appendix B. The wage at every period is positively related to

h the disutility of devoting Lt hours of labor. Higher disutility of labor requires higher wage since the h extra e¤ort needs to be compensated. Higher unemployment bene…t bt grants bargaining power v to workers who can negotiate a higher wage. Either an increase in the cost of a vacancy Pt or and increase in market tightness #t gives bargaining power to the worker who can eventually leave the bargaining table with a higher wage. Higher market tightness means workers can …nd a match more easily relative to vacancies thus workers demand a higher wage in such a scenario.

112 2.3 Household Problem Revisited

The rational expectations assumption is a very convenient tool since it imposes a lot of structure on the solution of any model. In my model agents separate control variables from forecasting variables and they provide forecasts for those same variables a rational agent does. Therefore, I force the agents to form expectations of variables that …rst of all are outside of their control3 . In order to …nd the appropriate forms of expectations the agents are facing I use the …rst order conditions of the agents without imposing any equilibrium conditions on them. For example the household doesn’t know that all other households are the same or that the other workers are receiving the same wages or what the pricing formula of a …rm is. In the formation of expectations agents know only about themselves. Any future variables that are outside the agents’control need to be forecasted using the state variables as regressors.

The solution to the household problem involves the combination of the job creation condition

(consumption and vacancies FOCs combined) which I repeat below

v P v Pt ^h h t+1 ^h h h = (1 ) Et Qt;t+1 + Et Qt;t+1dt+1 (21) qt qt+1 and the household budget constraint which I also repeat

h v h h h h h h h h h C + P V = N WtL + I N b + N~ d + T t t t t t t t t t t h As I already mentioned in the household section, the variable Nt is the number of employed people ~ h ~ h within the household while Nt the number of …rms/jobs the household owns. In equilibrium Nt h and Nt are the same and equal to the economy-wide employment but households do not know that simply because they ignore that every other household is identical to their own in every aspect. ~ h The equation for the law of motion of Nt is

h h h N~ = (1 ) N~ + qtV (22) t+1 t t 3 If on the contrary a variable to be forecasted is in the agent’scontrol, using least squares techniques would mean that the agent is not behaving optimally.

113 h where qt is the probability a vacancy to match. The number of employed workers Nt the household has, evolves according to

h h h N = (1 ) N + t 1 N (23) t+1 t t where t is the probability an unemployed worker to …nd a job. Since the government keeps a balanced budget and all households are identical, the transfers to each household are going to be equal to the unemployment bene…t per household. The households at this stage cannot possibly have this information. It doesn’tunderstand that all other households

h h are identical in every aspect. However, I assume agents know bt and transfers Tt are the same for all households and also equal in magnitude every period for simplicity. If I did not assume households possess this knowledge about …scal arrangements, the households would need to provide forecasts for future unemployment bene…ts and transfers from the government. The budget constraint after imposing Ih N h bh = T h becomes t t t h v h h h h ~ h h Ct + Pt Vt = Nt wt Lt + Nt dt (24)

2.3.1 The Intertemporal Budget Constraint

The households derive an optimal decision rule for consumption that is valid for arbitrary expectations which requires to combine the Euler equation and the intertemporal budget constraint. It is a version of permanent income theory in general equilibrium adjusted to a search and matching framework. For the intertemporal budget constraint …rst substitute in the ‡ow budget constraint

(24) the law of motion for the number of …rms the household owns, equation (22) to get

v h Pt ~ h ~ h h h h ~ h h Ct + Nt+1 (1 ) Nt = Nt wt Lt + Nt dt qt h i and rearrange to get a recursive budget constraint

h h h h h h h C + KtN~ = N w L + d + (1 ) Kt N~ (25) t t+1 t t t t t

114 v Pt v where I de…ned Kt = as the cost of vacancies per unit of time. The per unit cost P will be qt t held constant. Log linearize the budget constraint equation (25) to get

h h h h h h h CC^ + K NK^t + K Nn^ = NW L N^ +w ^ + L^ + dNd^ + dN + (1 ) K N n^ t t+1 t t t t t The intertemporal budget constraint should relate consumption which is a choice variable for the household, to forecasts of variables outside of the household’s choice that can be either prices determined by the market or aggregate variables. The wage is determined by the bargaining problem between the household and the …rms, hence it is not outside the household’scontrol. Log-linearize the wage equation (20). The log-linearized wage equation is

1+` 1+` h h (1+) C h 1 (1+) C w^ + L^ = (1 ) N Y L^ + N Y C^t t t W L t 1 + ` W L Pv# Y + #^ + p^hr +y ^h W L t NW L t t which apart from consumption it is comprised of variables outside the household’s control. The

h hr wage involves variables such as the demand of the …rm y^t , the price the …rm sets p^t and technology

A^t which are variables the household has no control over them. Appendix C goes over the details to get to the intertemporal budget constraint

^h 1 j ^h ^h 1 j ^ ^h 1 j ^ h ^h 1 j h cEt vCt+j + kEt vKt+j = wEt vNt+j + LEt vLt+j (26) j=0 j=0 j=0 j=0 X X X X ^h 1 j h ^h 1 j ^h h + yEt vr^t+j + dEt vdt+j +n ^t j=0 j=0 X X

h hr h where r^t =p ^t +y ^t denotes …rms revenue and the epsilon variables de…ned in Appendix C contain structural parameters and steady state values.

115 2.3.2 The Consumption Equation

The intertemporal budget constraint (26) is consisted of expectations of future variables outside of the household’scontrol except for the …rst term which is the weighted sum of all future consumption levels of the household. I need to substitute away future consumption and the way to go is to use the job creation condition. The asset pricing equation for the jobs which is nothing more than a rearrangement of the job creation condition (21), is

h h C (1 ) Kt+1 + d 1 = E^h t t+1 (27) t Ch K t+1 t after assuming log utility for preferences. Appendix D presents the derivation of the consumption equation which basically combines the log-linearized asset pricing equation for jobs, equation

(27) and the intertemporal budget constraint (26) to eliminate future consumption. The resulting equation for consumption is:

^h Ct = (28)

1 v 1 v v 1 h 1 j v + v v (1 ) k + w E^ K^ 1 (1 ) t v t+j+1 c c v j=0 X 1 v h 1 j h v (1 (1 )) d E^ d^ t v t+j+1 c j=0 X 1 v 1 1 v 1 + E^h j Lh + E^h j R^h v L t v t+j+1 v y t v t+j+1 c j=0 c j=0 X X 1 v v v 1 ^ v + c k + w Kt c 1 v 1 v (1 ) 1 v ^h ^h h h 1 ^ h + ddt + yRt + LLt +n ^t + w Nt c 1 v (1 )

2.4 Equilibrium and aggregate dynamics

All households are identical, having the same ratio of unemployed to employed workers and facing the same constraints4 . Firms are also identical facing the same technology and decision problems

4 However, the foreward looking variables that need to be forecasted arised from the fact that agents do not understand that others behave identically.

116 and bargain wages using the same Nash Bargain framework. Since all households are the same, the h subscripts are dropped and after aggregating all consumption equations (28) for all households ^ ^h in the economy and de…ne the average expectations as Et = Et dh the aggregate consumption Z becomes

^ ^ 1 j ^ v ^ 1 j ^ ^h 1 j ^ 1 j ^ Ct = 1Et vKt+j+1 + 2Et vdt+j+1 + 3Et vLt+j+1 + 4Et vRt+j+1 (29) j=0 j=0 j=0 j=0 X X X X h ^h ^h ^ v + 5Lt + 6Rt + 7dt + 8n^t + 9Kt

1 v 1 v v 1 where 1 = v v (1 ) k + w , c c 1 v (1 ) 1 v h 1 v 1 v i 2 = v d (1 (1 )) , 3 = vL, 4 = vy, c c c 1 hv 1 v i 1 v 5 = L, 6 = y, 7 = d, c c c

1 v 1 1 v v v 1 8 = 1 + w , 9 = c k + w c 1 v (1 ) c 1 v 1 v (1 ) The aboveh consumption equationi is the onlyh equation that involves forward lookingi variables that need to be forecasted by agents. The only thing remaining is to present the framework according to which agents form expectations which is the subject of the following section. h ~ h Furthermore, in equilibrium Nt = Nt = Nt since all households are the same. Therefore the equilibrium employment law of motion becomes

Nt+1 = (1 ) Nt + Mt (30) where

1 Mt = mVt Ut is a constant returns to scale matching function. The above law of motion states that employment next period is known a period in advance.

After imposing equilibrium conditions on the wage equation (20) and the fact that every worker

117 and …rm are the same, the wage becomes

1+` 1 Lt v Yt wtLt = (1 ) + bt + Pt #t + (31) 1 + ` t ! Nt

Manipulate equation (17) to …nd 1+` L 1 Yt t = t Nt

Substitute this into (31) and get

1 1 Yt v Yt wtLt = (1 ) + bt + P #t + 1 + ` N t N t t

In a similar manner the dividends of the …rms in equilibrium can be expressed as

pt dt = yt wtLt Pt 1 + ` Yt v = (1 ) (1 ) bt Pt #t (1 + `) Nt

2.5 Expectations

According to (29) the households need to form multi-period forecasts of dividend, vacancy cost, labor demand and the revenue of the …rm. The households keep in mind that every variable at any point in time is a function of the state variables and they use econometric techniques to re‡ect this knowledge. The regression equations to be estimated are

^ d d d ^ d dt = 0 + 1n^t + 2At 1 + "t (32)

^ k k k ^ k Kt = 0 + 1 n^t + 2 At 1 + "t (33)

^ L L L ^ L Lt = 0 + 1 n^t + 2 At 1 + "t (34)

r r r ^ r r^t = 0 + 1n^t + 2At 1 + "t (35)

118 Moreover, as agents need to make multi-period forecasts, they also need to forecast the evolution of the state variables. In a similar manner employment relates to the past state variables according to n n n ^ n nt+1 = 0 + 1 nt + 2 At 1 + "t (36) and the technology shock according to

A^t = AA^t 1 + "t (37)

I assume the law of motion of the technology shock is exogenous and completely known to the

5 agents for simplicity meaning that no estimate for A is necessary.

2.5.1 Timing

At the beginning of every period t, agents are equipped with estimates of the coe¢ cients that determine beliefs using data from the previous period t 1. For example, in equation for consumption (29) the agents need among other things to form expectations for future dividends dt+j. Thus, the j period ahead forecast of the real dividend is

d d d ^ Etdt+j = 0;t 1 + 1;t 1Etnt+j + 2;t 1EtAt+j 1

d d d where 0;t 1, 1;t 1 and 2;t 1 are belief parameters inherited from the previous period. Agents tend to assume that the belief parameters above are going to remain …xed for the periods to come even though they will eventually alter each one of those in subsequent periods. The way to estimate

Etnt+j, is to iterate on equation (36) and EtA^t+T 1 by iterating on (37). Therefore, the agents d d d n n n need estimates not only for 0, 1 and 2 but also for 0 , 1 and 2 . 5 If alternatively I forced the agents to estimate the law of motion of the technology shock the regression coe¢ cients would converge very fast to the true values especially because the technology process is exogenous and its realization is not in‡uenced from past forecastng mistakes.

119 2.5.2 Perpetual Learning

The next challenge for the agents is to estimate all those parameters that correspond to the regression coe¢ cients in the system above, equations (32) - (35). To achieve this goal the agents use perpetual learning. De…ne

d k L r n 0 0 0 0 0 0 2 3 2 3 d k L r n = 1 1 1 1 1 1 6 7 6 7 6 d k L r n 7 6 7 6 2 2 2 2 2 7 6 2 7 6 7 6 7 4 5 4 5 as the matrix of regression coe¢ cients. The agents need to estimate the matrix of coe¢ cients every period by using as regressors a constant, current and past employment realizations as well as technology realizations6 . To specify the updating of the estimated parameters I de…ne the following vectors, the vector of the regressors 1 2 3 qt 1 = n^t 6 7 6 ^ 7 6 At 1 7 6 7 4 5 and the vector of the dependent variables

^ ^ ^ r zt0 = dt Kt Lt r^t n^t+1

The algorithm that recursively de…nes the estimated parameters is:

1 t = t 1 + gRt qt 1 zt t0 1qt 1 0 (38)

Rt = Rt 1 + g qt 1qt0 1 Rt 1 (39) 6 I include a constant in the estimation process even though the model has a steady state of zero since agents being irrational they might believe the steady state might be changing.

120 7 where g (0; 1) is a constant gain parameter . Agents estimate t at the end of the period t after 2 making consumption and vacancy decisions. This means that the period t decisions are based on the estimated parameters t 1. The derivation of the actual law of motion (ALM) for consumption is presented in Appendix E.

The ALM for consumption is

^ ^ ^h h ^h ^ v Ct = 0 + 1n^t+1 + 2At + 5Lt + 6r^t + 7dt + 8n^t + 9Kt

1 k d L R where 0 = 1 + 2 + 3 + 4 1 v 0;t 1 0;t 1 0;t 1 0;t 1 n v 0;t 1 k d L R + n 11;t 1 + 21;t 1 + 31;t 1 + 41;t 1 , 1 v 1 v 1;t 1 1 k d L R 1 = n 11;t 1 + 21;t 1 + 31;t 1 + 41;t 1 and 1 v 1;t 1 v 1 n k d L R n 2;t 1 11;t 1 + 21;t 1 + 31;t 1 + 41;t 1 1 v A 1 v 1;t 1 2 = 2 1 k d L R 3 + 1 + 2 + 3 + 4 1 v A 2;t 1 2;t 1 2;t 1 2;t 1 6 7 4 5 2.6 The solution

The following log-linear equations de…ne the solution of the model along with the recursive rule for updating beliefs, equations (38) and (39).

The ALM for consumption

^ ^ ^h h ^h ^ v Ct = 0 + 1n^t+1 + 2At + 5Lt + 6r^t + 7dt + 8n^t + 9Kt

The relative price

L^t = Y^t (1 + )n ^t A^t

The demand

h r^ = Y^t n^t t 7 Higher g means the households weight current observations more heavily than the past ones. For example a gain value of 0.005 for example means that observations 30 years ago receive a weight of 0.55 and observations 100 years ago 0.14.

121 Dividends 1 + ` Y Pv# dd^t = (1 ) Y^t n^t K^t (1 + `) N

The foc of the …rm’sproblem

[` (1 + ) + ]n ^t + (1 + `) A^t = `Y^t + C^t

The technology process

A^t = AA^t 1 + "t

The law of motion for employment

N 1 n^t+1 = 1 n^t + K^t 1 N

v The cost of vacancies after assuming Pt is a constant

^ ^v ^ ^ Kt = Pt + #t = #t

The wage equation from bargaining

1 1 Y Pv# Y wL!^t = (1 ) Y^t n^t + K^t + Y^t n^t 1 + ` N N where

!^t =w ^t + L^t

and the resource constraint which after some manipulations becomes

PvV N CC^ + K^ = Y Y^ + PvV n^ t t t 1 N t

122 Appendix F presents the model to be estimated in a form that can be easily simulated using any software.

3 Simulation and results

I calibrate the model by setting the parameters to common values used in the search and matching literature. I simulate the model in an attempt to match statistics from the US data from 1955:Q1 to 2007:Q4, for a total of 212 observations. I run the algorithm 3000 times. Each simulation corresponds to 7000 + T periods. The …rst 7000 periods are ignored just to guarantee convergence of the estimated parameters to the model stationary distribution. Thus, the estimated moments are obtained from the last T periods. I set T = 212, the number of observations in my data set.

The algorithm is initialized according to Carceles-Poveda and Giannitsarou (2007). The following sections present the calibration and the main results.

3.1 Calibration

Table 1 summarizes the benchmark calibration. The time discount factor is set to 0:99, a considerably common value compared to most macroeconomic models. The elasticity of substitution for the consumption good is set to 11 as in Boivin and Giannoni (2005) a value which corresponds to 1.1 markup for the …rms. The inverse of labor supply elasticity `, is set to 2. The elasticity of the matching function with respect to unemployment is set to 0.4 which matches the elasticity of

0.4 estimated by Blanchard and Diamond (1989). According to Petrongolo and Pissarides (2001) acceptable values should lie within the interval (0.5 - 0.7). The worker’sshare of the surplus is set to 0.5 since it is the usual value assumed in the literature even though empirical work has no predictions for this parameter. Moreover I set unemployment bene…t equal to zero since I attempt to create ampli…cation without changing the benchmark calibration in Shimer (2005) by much. Hage- dorn and Manovskii (2005) show that setting the unemployment bene…t to a high value, around

0.9 and the worker’sshare of surplus to a small value, around 0.01, enough ampli…cation can be created in the basic search and matching model. However Costain and Reiter (2007) argue that

123 Par Name/Explanation Value Source time discount factor 0:99 common in macro models elasticity of subs …nal goods 11 corresponds to 1.1 markup elast. matching function 0:4 as Blanchard and Diamond (1989) workers bargaining power 0:5 commonly used. No microevidence ` inverse of labor supply elast 2 1 (0,0.5) Card (1994) ` 2 love of variety (LOV) 0 no LOV s.s unemployment probability 0:6 matches un. duration of 1.67 separation rate 0:04 match 0.06 steady state unempl. rate b unemployment bene…t 0 match data with lowest un. bene…t A technology parameter 0:9 common in RBC models q steady state vacancy probability 0:7 den Haan et al (2000)

Table 1: This is a summary of the baseline calibration. such a calibration makes the model too sensitive to unemployment bene…t variations, reducing the quality of it’spredictions for policy analysis.

The steady state unemployment probability is set to 0.6, as in Cole and Rogerson (1999), which ends up matching the average unemployment duration of 1.67 observed in quarterly US data. The steady state probability a vacancy to match q is set to 0.7 as in Cooley and Quadrini (1999) and den Haan, Ramey and Watson (2000). The separation rate is set to 0.04. Commonly used values lie in the interval of between 0.08-0.10. That particular value of used in this paper is necessary to lock the steady state unemployment rate to 0.06.

The rest that needs to be chosen is the number of periods the model is simulated before starting to compute moments and the gain parameter g (0; 1). The total number of periods I run the 2 model is 7212 periods. The gain parameter g is set to 0.0045, 0.005 and 0.006, since higher values of the gain parameter imply greater deviations from the rational expectations equilibrium. A gain parameter of zero replicates the rational expectations model. Common in the literature are values from 0.007-0.05 as in Branch and Evans (2006), Orphanides and Williams (2005) and others.

3.2 Data

The moments I am after to match come from quarterly US data from 1955:Q1 to 2007:Q1 where all series are HP …ltered and those moments are summarized in table 2. All variables but the

124 vacancies series on that same table come from data series of the St. Louis economic database.

Speci…cally output is the log of quarterly real Gdp, unemployment rate the civilian unemployment rate, vacancies is the log of Help-Want advertising and the wage the hourly earnings in the business sector. I simulate the model to match second moments from quarterly US data. Table 2 presents a few statistics from the US data which are basically the target moments. It is clear from the table that the ampli…cation task is very challenging. Vacancies are nearly nine times the volatility of output, unemployment nearly seven times the volatility of output while tightness is as large as seventeen times more volatile than output. Moreover, the correlation between vacancies and unemployment around -0.9 implies a high slope for the Beveridge curve. All series are highly autocorrelated, pointing out that all those variables are very persistent. The standard deviation of the wage is a bit higher than half the standard deviation of output. The low volatility of the wage compared to output has been an issue in the RBC literature as pointed out by Christiano and

Eichenbaum (1992). As a matter of fact, in the search and matching literature the wage volatility implied by the standard model is exaggerated even more.

3.3 Results

Consider the impulse responses to a unit technology shock presented in …gure 1 (or …gure 2). The blue dashed lines correspond to the perpetual learning model with gain 0.005 (or 0.006 for …gure

2) and each one is the median point-wise response function from 3000 simulations. The red solid lines corresponds to the impulse responses from the rational expectations model. In the rational expectations case, a rise in labor productivity increases the demand for labor hours on impact. In the following periods changes in production are mostly met by changes in employment and that is why labor hours become negative once employment starts increase the period after the shock.

The increase in the demand for hours increases both hours and the real wage. Since the shock is persistent and agents fully rational, the increase in productivity creates a wealth e¤ect since consumers forecast an increase in their future income. This wealth e¤ect increases consumption on impact. The di¤erence between income and consumption is simply vacancies. What consumers don’tconsume they use it to create jobs. The increased consumption shifts the labor supply curve

125 upwards and along with the shifted labor demand, the wage overshoots and the labor hours are barely increased.

Learning changes the picture a bit as the blue dashed lines in …gure 1 suggest. The increase in labor productivity again raises labor demand and increases wages and hours on impact through a substitution e¤ect. However, the di¤erence with rational expectations is that agents learn using past data which dampens the wealth e¤ect. Since consumption is determined from expectations of future income and agents learn using past data, on impact consumption does nor rise as much as the rational expectations case. That is why the response of consumption in …gure 1 is lower on impact from the rational expectations response. The low response of consumption leads to higher vacancy posting since whatever is not consumed it is invested in new …rms. This explains the large response of vacancies under learning on impact. The next period expectations about future income are revised and agents perceive that the big increase in employment is going to persist forever as well as the large wages enjoyed the previous period and thus the consumers increase consumption above the rational expectations response. This implies that once output is realized on the second period, vacancies are going to be too few, because the consumption increase crowds out investment.

In other words, the second period, optimism about future returns increases consumption too much and diminishes vacancy posting. This story continues until the responses return to their long-run values leaving the vacancies response ‡uctuating around the rational expectations response.

Moreover, as Huang, Liu and Zha (2008) …nd, this decrease in consumption on impact (lower wealth e¤ect) is going to have a di¤erent impact compared to rational expectations case. In their model, lower consumption means the labor supply is not going to shift upwards as much, leading to lower wages and higher labor hours. In my model the decrease of the impact of the wealth e¤ect increase hours compared to the rational expectations case, in line with the …ndings of the authors above. However, in my model wages are increased in the learning model besides the decrease of the wealth e¤ect. The reason is because the wage structure is much di¤erent now than in the standard RBC case. The dampening of the wealth e¤ect decreases the wage on the …rst hand but on the other hand the rise in vacancies increases market tightness too much which gives bargaining power to workers that grants them a higher wage contract. Thus in my model, the decrease in

126 consumption leads to a big increase in the wage due to the importance of outside opportunities in the wage formation in the search and matching framework.

The tables from 3 to 6 present the same set of statistics as table 2 and are obtained from simulating the model for di¤erent gain parameters. Table 3 corresponds to gain g=0 or the rational expectations model. Tables 4 to 6 represent the models with perpetual learning and each table summarizes the statistics gathered by simulating the models with constant gains g=0.0045, g=0.005 and g=0.0055 respectively. Let us focus on the ampli…cation e¤ects of the models which are isolated in table 7. Each row presents relative standard deviations of vacancies, unemployment market tightness and the real wage with respect to output, from data and di¤erent model speci…- cations. The …rst row of numbers corresponds to the statistics from US data. The second row is the rational expectations model’smoments. Clearly the rational expectations model cannot generate any ampli…cation for vacancies unemployment and market tightness. Although in this model speci…cation the wage volatility is a bit exaggerated, it is not as high as the models with perpetual learning. From the third row to the end are the statistics obtained from the learning models. As the gain parameter g increases the model’sampli…cation increases dramatically. When the constant gain parameter g is set to 0.0055 the model can amplify the labor market variables as much as the data suggests.

The extra ampli…cation in the model with perpetual learning arises because agents make forecasts using a misspeci…ed model. The agents’beliefs stemming from the misspeci…ed forecasting model are assumed to last forever while in fact the agents are going to revise them the following period. This behavior is what creates extra ampli…cation in the model, boosting the volatilities of vacancies and unemployment signi…cantly.

Although added volatility seems to be bene…cial for most of the labor market variables, I cannot claim the same thing for the wage series. Wage in the data is a little more than half as volatile as output. The rational expectations seems to be closer to the data than the rest of the models, even though it attains a value that is more than double the one observed in the data. Learning does not seem to resolve this issue but it was absolutely expected. Take a look again at the wage implied by the search and matching model, equation (31). When vacancies increase and unemployment shrinks,

127 market tightness becomes very large since tightness #t equals vacancies over unemployment Vt=Ut. This means that boosting the volatilities of vacancies and unemployment immediately increases the volatility of the real wage. This is due to the fact that outside opportunities tend to a¤ect the wage in the search and matching literature due to Nash bargain. More vacancies than unemployed gives bargaining power to workers because they can break the match at any time and …nd a job elsewhere very easily, while for the …rm to …nd another match is relatively more di¢ cult. Since tightness is very volatile, the wage becomes too volatile and absorbs most of the e¤ect of the shock. However, the increased volatility that appears in the learning model does not come from a change in the wage formation. It comes from mistakes in the formation of forecasts about the future dividends of the

…rms. Therefore, the wage is going to be more volatile than in the rational expectation model’scase, since there’snothing to keep tightness from causing the wage to overshoot. Increasing the volatility of the forecasts of future dividends boosts the volatility of vacancies and unemployment but also that of the wage. In a more complex model with sticky wages for continuing …rms, the volatility of the wage could be signi…cantly reduced to the data levels. Moreover, such a modi…cation would not sacri…ce the ampli…cation for the rest of the variables. On the contrary, it would even provide some additional ampli…cation on top.

Once persistence is examined the picture changes a lot. The responses in …gure 1 suggest that the responses from perpetual learning die out more slowly. However, observing the autocorrelations of the variables in tables 3 to 6 which are summarized in table 8, one can infer that the rational expectations model can outperform the learning model. When the gain parameter increases, the …rst order autocorrelation of vacancies becomes negative. The reason lies in the behavior of vacancies which is the main reason for the distortion in autocorrelations. It is easier to visualize what is going on if you take a look at …gure 3 where I present the mean impulse response of vacancies under rational expectations after a unit technology shock, along with the mean impulse responses of vacancies from the learning model for di¤erent gain parameters. The red solid line corresponds to the rational expectations case while the blue dashed line is the learning model with gain g=0.005, the black dotted line is the learning model with gain g=0.0055 and the mixed line corresponds to g=0.006. It is obvious that higher gain leads to higher volatility for vacancies by making vacancies

128 ‡uctuate around the rational expectations response. When vacancies one period is above average, next period it appears below average and so on. This makes the degree of autocorrelation in vacancies to drop as the gain parameter increases and when the gain is su¢ ciently large it even becomes negative. The impulse response for vacancies in the learning model is high on impact and the period following the shock and then goes below the response of the rational expectation model and then on top again. It ‡uctuates around the rational expectations response. This kind of movement is what distorts all correlations in the model.

In this model output does not seem to be much a¤ected by expectations. I don’t mean that income is not a¤ected at all, but the e¤ect of learning on output is rather mild. The behavior of the output response can be understood by examining the foc of the …rm which I repeat below

`n^t + (1 + `) A^t = `Y^t + C^t

Set = 0 which is the benchmark calibration and use the log-linear resource constraint

v Y Y^t = CC t + P V V^t to substitute for consumption. That is

Y V + ` Y^ = `n^ + (1 + `) A^ + Pv V^ C t t t C t

The parameter Y 1 because Y = C + V and V should be as small as possible not only to be able C ' to match US data facts, but also to be able to have any ampli…cation in the model8 . The output response is then equal to ` V Y^ = A^ + n^ + Pv V^ t t 1 + ` t C t

The change in output equals the change in technology A^t plus a fraction of the change in employment

` 1+` n^t plus a fraction of the change in vacancies. Employment steady state is very high and thus

8 Deviations of vacancies from the steady state are larger the smaller the vacancy steady state is. Assuming a high value for Vacancy steady state value diminishes the model’s ampli…cation potential.

129 deviations from it are very small. Moreover, the steady state of vacancies is too small which makes v V ^ the term P C equal to 0.03. Even if Vt is as large as 3 times the response of output, output is barely volatile. Learning can boost employment and vacancies but unemployment response cannot be too high since it’ssteady state is high and also the e¤ect of vacancies on output is small since vacancies are a very small part of aggregate Gdp. Therefore, when expectations determine consumption, output is barely a¤ected and vacancies are whatever part of the output is not consumed. The ability of the learning to a¤ect output is crucial in order to replicate the results as in Eusepi and

Preston (2010).

To summarize, it is important to note that the greater ampli…cation seems to come at the cost of distorting correlations, but this is not solely due to the way beliefs are formed. It is also a¤ected by the fact that the simple search and matching model I have used, implies output is not much in‡uenced by expectations. By employing a di¤erent framework, the way vacancies ‡uctuate can be reduced enhancing the performance of the model against autocorrelations. However it is important to note that the search and matching model needs a very high ampli…cation boost to match the data as can be seen by comparing the …rst two rows of table 7. If this ampli…cation is solely provided by a perpetual learning framework, there is inevitably going to be some trade-o¤ between ampli…cation and correlation. The model with gain g=0.0045 can amplify the vacancies and unemployment processes by more than 50% compared to the rational expectations model.

However, the correlations remain more or less the same between the two models, denoting that learning is not distorting the rest of the model implied moments. Nevertheless, the gap between the volatilities in the data and the model with g=0.0045 is still signi…cant. The ampli…cation that is needed to eliminate the gap is around nine times the ampli…cation in the rational expectations model which forces me to employ higher values for the gain parameter creating a trade-o¤ between success in providing ampli…cation over correlation in the model.

The most important e¤ect of the distortion in the correlations implied by the model with perpetual learning is the reduction in the correlation between vacancies and unemployment. Table

9 presents the correlations between vacancies and unemployment which de…nes the slope of the

Beveridge curve. The correlation of the two variables in the data is high and negative with a value

130 of -0.91. The rational expectations model implies a correlation of -0.63 and it is not even close to match the data. The models with higher gain imply an even lower correlation in absolute value and a positive correlation in the last two model speci…cations. Note that the correlations are low in absolute value even though vacancies are procyclical and unemployment countercyclical. As …gure

1 demonstrates, while new matches start producing a period after the match, on impact, vacancies achieve their highest value while unemployment remains at zero. When unemployment increases later on, vacancies are already fading away inducing a very low correlation between the two variables. For higher gains, vacancies become negative when unemployment is at it’s pick, inducing even positive correlation for vacancies and unemployment. As the gain parameter increases this correlation drops. Keep in mind that the ampli…cation puzzle arises because in the wage equation

(31) the market tightness is too volatile making the wage too depended on outside opportunities.

The increased volatility in the wage decreases the volatility in dividends and leaves vacancies barely a¤ected. Imagine a situation where vacancies and unemployment are highly positively correlated.

That would imply that market tightness #t would not be too volatile simply because vacancies and unemployment cancel each other out. Such a model would support high volatilities for vacancies and unemployment simply because the two variables cancel each other out and wages do not crowd out vacancies. In the current models, vacancies and unemployment are not as highly negatively correlated as the data suggest. This makes the ampli…cation issue in the model a bit less severe simply because market tightness is not as volatile as it should be if vacancies and unemployment didn’t partly o¤set each other. Therefore the models with high gain parameter provide some ampli…cation for the wrong reason. Since tightness is less volatile than it would if the slope of the

Beveridge curve was around -0.9, wage becomes less volatile and it’s not as e¤ective in absorbing most of the e¤ect of the shock.

3.4 Alternative Model Speci…cation

The model I presented in this paper, besides the alternative expectations assumption, it di¤ers from the standard search and matching model in two areas. First of all, in my model future cash

‡ows are discounted by a stochastic discount factor instead of a constant one. The addition of

131 a stochastic discount factor can boost job creation a little bit. The reason is because agents are risk averse and job/…rm creation provides some insurance to the investor. After a productivity shock income increases. Vacancies are posted on the current period where income is high and they will make payments next period if a match is found, where income is lower. This means that vacancy posting becomes a form of insurance and agents would keep posting vacancies even if the dividend they are promised to pay next period is really small. However, when the discount factor is constant, agents would stop posting vacancies when dividends are expected to be too low. Risk neutral agents are only concerned about returns and not the state of the business cycle the cash

‡ows appear. Nonetheless, the ampli…cation created by the inclusion of a stochastic discount factor is not signi…cantly high. Nevertheless, assuming that agents are actually risk averse is a realistic assumption.

Moreover, another di¤erence between my model and the standard case is that I assume there is also labor e¤ort decision for the …rm, while the standard search and matching model allows changes in labor only through the extensive margin. I can replicate the results of a model with just the extensive margin by setting the parameter de…ning disutility for labor ` to zero9 . Table

10 compares the benchmark learning model with a gain of 0.005 for di¤erent parameters for the disutility of labor `. The model with no intensive margin (` = 0) can match the ampli…cation in the US data more easily while the one with ` = 2 (the benchmark model parameterization) cannot. Since the disutility from labor in the model with no extensive margin is constant, the wage is not as responsive and therefore it creates some extra pro…t margin for the …rms compared to the benchmark model, which results in an additional wave of vacancy postings. After a productivity shock, on impact, labor hours/e¤ort increase. Agents expect next period’shours will be as high and infer that the disutility of labor (the cost of labor) will be high for the periods to come. Therefore, agents expect pro…ts to be lower and end up posting less vacancies on impact. This channel is

9 Solve the model I presented above and simply change the following assumptions:Assume the aggregator function 1 1 1 j j j j th is Yt = yt dj and production function is yt = Nt where Nt is the number of workers the j …rm "0 # R h employes. Assume the disutility for labor in the utility function is Nt G (L) where the function G (L) is now constant. Solving the model under this speci…cation and solving my original model with ` = 0 is going to produce the same dynamics.

132 absent if the intensive margin is eliminated and the disutility of labor is constant. This observation can be con…rmed by comparing the last columns of table 10 where it shows that the volatility of the wage in the models without the extensive margin is much lower than the benchmark model.

Decreasing the volatility of the wage signi…cantly is crucial for the search and matching model.

Despite the fact that the model with ` = 0 seems to outperform the benchmark model, the main reason I keep the benchmark model as the one with both the intensive and extensive margin is simply because it is more realistic10 .

3.5 The Distribution of Beliefs

The formation of beliefs is crucial for my result since the substitution or rational beliefs with boundedly rational beliefs is what creates the additional ampli…cation in this model. The way beliefs are updated using perpetual learning through equations (38) and (39), suggests that as time passes, beliefs will not converge to a particular value but to a distribution, since agents tend to forget old data up to a certain threshold that is controlled by the gain parameter.

Figure 5 presents the densities of the belief coe¢ cients after simulating the model for a gain of 0.005. The number of observations in each density are around 2,000,000 data points. Each row corresponds to a variable that is forecasted and each column to the state variable used as a regression for the formation of those forecasts. It is obvious that there is a very high dispersion for each belief coe¢ cient. This is what actually creates the extra volatility in the learning model. This dispersion endogenously creates shifts in agents’expectation that becomes a source of business cycle

‡uctuations and probably the source that seems to be ignored in the standard search and matching literature.

The medians of the densities for di¤erent gain parameters are reported in table 11 along with the means. Although there is a very high scattering around the median, it seems to be very close to the rational expectations case even though the mean states that the distributions are skewed.

Figure (4) shows the evolution of beliefs when instead of a constant gain I use a decreasing gain of the form gt = 1=t. Each row of 3 graphs represent a coe¢ cient for a particular variable that needs

10 Mind that in the standard search and matching model there is no intensive margin.

133 to be forecasted by agents. Each row represents the coe¢ cient of the particular state variable. The blue lines correspond to the beliefs while the red ‡atline marks the rational expectations coe¢ cient.

All coe¢ cients eventually tend to converge to their rational expectations counterpart.

The impulse responses in …gure (1) are di¤erent than the rational expectations responses on impact which might suggest that the distribution of believes is not centered at the rational expectations belief coe¢ cients even though the conditions for E-Stability are satis…ed and the gain coe¢ cient is between zero and one. The reason is because of the non-linear dynamics the belief system is following. Under decreasing gain, the coe¢ cients tend to converge to the rational expectations beliefs solely from the same side. For example, no matter how the belief framework is initialized, the coe¢ cient of the state variable one for the dividend process, always converges from above as can be seen in the second graph on the …rst row of …gure (4) . Therefore,as the constant gain parameter increases, the belief parameters tend to converge to a distribution with means a bit shifted from the rational expectations parameters.

4 Conclusion

In this study I consider a perpetual learning algorithm in an attempt to enhance the standard search and matching model’s performance. The ampli…cation problem of the standard search and matching model is a very severe one and very e¤ective ampli…cation mechanisms are needed for the puzzle to be successfully resolved. Learning algorithms tend to have mild impact in the dynamics of macroeconomic models as studies like Williams (2003) have shown. However, by assuming agents rely on multi-period forecasts as in Eusepi and Preston (2010) rather than forecasts at most a period in advance , the ampli…cation prospective of the learning model is signi…cantly improved.

The learning model can provide ampli…cation to match moments from the US data from 1955:Q1 to 2007:Q4. However, the extra ampli…cation comes at the cost of distorted correlations since the more ampli…cation is added the more vacancies ‡uctuate around the rational expectations equilibrium achieving a low autocorrelation coe¢ cient that a¤ects the autocorrelation of every single variable in the model. Moreover, learning distorts the correlation of vacancies and unemployment

134 known as the Beveridge curve implying even a positive slope for the Beveridge curve as the gain parameter increases.

The model could perform better in a more complex setup where there exists some mechanism that could potentially keep vacancies in a smoother path. For example, endogenous job destruction could open another channel for some corrections in employment to take place without violent shifts in vacancies to be necessary. This is something that I tend to investigate in future research.

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A Appendix

In this section I introduce the surplus maximization for the …rm that is choosing the appropriate price to set. The …rm is trying to …nd pt either by maximizing

1 J E U pt = arg max Ft (pt) Wt (pt) Wt (pt) pt n o or the total surplus

J E U pt = arg max Ft + Wt (pt) Wt (pt) pt The two objectives give the same answer given that the surplus is split according to (1 ) W E W U = t t J Ft which implies that the surplus is split in …xed portions to …rms and workers. Therefore, maximizing either of the objective functions leads to

J E U @ F + W (pt) W (pt) t t t = 0 @p t

Expand the expression in the brackets using the Bellman equations (11) and (12) in the above …rst order condition to get:

J E U F + W (pt) W (pt) t t t G Lh h ^h h J h h t h = dt + (1 ) Et Qt;t+1 Ft+1 (pt+1) + wt Lt h bt t h E U + (1 t) E^tQ W (pt+1) W (pt+1) t;t+1 t+1 t+1 Use the fact that the dividends process is dh = phryh whLh and the above equation becomes t t t t t

p G Lh F J + W E (p ) W U (p ) = t yh t + t:i:p t t t t t P t h t t

137 where t:i:p stands for terms independent of price. Therefore, maximizing the surplus becomes

G Lh pt h ( t ) J E U @ P yt h @ F + W (pt) W (pt) t t t t t = = 0 @p @p t t

By the chain rule

G Lh ph G Lh hr h h h ( t ( t )) pt h h L( t ) h h @ pt pt yt pt h @ P yt pt h Lt pt t t t = @pt @pt

Thus, the objective of the …rm that is trying to maximize the surplus is equivalent to

h pt h h GL Lt h h max yt pt h Lt pt ph P t ( t t )

h GL(Lt ) h by treating h as independent of the price pt . This form has interesting implications because t it reveals that in a search and matching model, the cost of labor is the marginal rate of substitution, the disutility of labor in real units. The standard New Keynesian model assumes that the …rms solve the exact same problem since the wage in that case equals the marginal rate of substitution, which is assumed …xed when …rms maximize their pro…t.

B Appendix

This part of the appendix presents the wage determination. The wage in a search and matching model splits the surplus to workers and …rms in …xed portions. The Nash bargain solution comes from the following …rst order condition

(1 ) W E W U = F J (40) t t t

138 which comes from the maximization of the surplus with respect to wage, equation (19). Plug in the individual surpluses, equations (11) and (12) to get

G Lh h h t h ^ h E U h ^h h J (1 ) wt Lt h bt + (1 t) EtQt;t+1 Wt+1 Wt+1 = dt + (1 ) Et Qt;t+1 Ft+1 t ! plug in the de…nition for the …rm’sdividends

dh = phryh whLh t t t t t and rearrange to get

G Lh h h t h ^ h E U hr h ^h h J wt Lt +(1 ) h bt + +(1 t) Et (1 ) Qt;t+1 Wt+1 Wt+1 = pt yt + (1 ) Et Qt;t+1 Ft+1 t !

E U First use (40) in the above expression to substitute Wt+1 Wt+1 away. After, use the vacancy H W ~ h ( t) J Nt t posting condition, equation (6), along with the de…nition Ft h and #t = q to get t t

h h h G Lt h hr h v wt Lt + (1 ) h bt = pt yt + Pt #t t !

C Appendix

^h After using the wage equation to substitute away the wage income w^t+Lt from the budget constraint (26) I get

1 1+` (1+) 1 v C 1 N 1+` N Y K P # ^ ^ K h Ct + Kt + n^t+1 h dN + (1 ) K N i d+ (1 ) K d+ (1 ) K 1+` C (1 ) N (1+)Y W L ^ h h Y hr h d ^h h = Nt + Lt + p^t +y ^t + dt +n ^t d + (1 ) K d + (1 ) K d + (1 ) K N d + (1 ) K h th ^ h where n^t are the …rms owned by the h household, Nt the number of the household’s employed workers and N^t the aggregate employment level and evolve according to the laws of motion (which

139 need to be linearized …rst) (22), (23) respectively. The household controls the number of …rms to

h own n^t by posting vacancies as (22) shows. However, it has no control on the evolution of it’sown ^ h ^ employment level Nt or the aggregate employment Nt because …rst of all it is way too small to a¤ect aggregate variables and also does not understand that the rest of the households are actually identical and make decisions in exactly the same way. 1 (1+) 1+` C 1 N N Y C 1 v 1+` ( ) K P # K W L De…ne = , = , = , = , c h [dN+(1 )K N] i k [d+(1 )K ] v [d+(1 )K ] w [d+(1 )K ] (1+) 1+` C(1 )(N Y ) = ; = Y , = d and also de…ne the revenue of the L [d+(1 )K ] y [d+(1 )K ]N d [d+(1 )K ] h hr h …rm as r^t =p ^t +y ^t The log linear budget constraint is

^ ^ h ^ h h h ^h h cCt + kKt + vn^t+1 = wNt + LLt + yr^t + ddt +n ^t (41)

Moving it a period in advance

h h h h h h cC^t+1 + kK^t+1 + vn^ wN^ LL yr^ dd^ =n ^ t+2 t+1 t+1 t+1 t+1 t+1

h h h h r h h cC^ + kK^t+1 + vn^ wN^ yy^ + AA^t+1 pp^ dd^ =n ^ t+1 t+2 t+1 t+1 t+1 t+1 t+1 and substitute it back into (41)

^ ^ ^ ^ h cCt + vcCt+1 + kKt + vkKt+1 + vvn^t+2

^ h ^ h ^h ^h h h ^h ^h h = wNt + vwNt+1 + LLt + vLLt+1 + yr^t + vyr^t+1 + ddt + vddt+1 +n ^t

h Keep doing the same until all n^t+j are eliminated for all j = 1; 2; 3; :::

j h 1 j ^h 1 j ^ 1 j ^ h 1 j ^h 1 j h 1 j ^h h lim vn^t+j+1+c vCt+j+k vKt+j = w vNt+j+L vLt+j+y vr^t+j+d vdt+j+^nt j !1 j=0 j=0 j=0 j=0 j=0 j=0 X X X X X X

140 The linearized trasversality condition, equation (10) states that both

j lim K^t+j = 0 j !1 and

j h lim n^t+j+1 = 0 j !1 Substitute in = K the fact that in the steady state K = d. This implies that v [d+(1 )K ] 1 (1 )

v =

j h j h The limit lim vn^t+j+1 = lim n^t+j+1 = 0 from the transversality condition. After taking j j !1 !1 expectations the intertemporal budget constraint is the following expression

^h 1 j ^h ^h 1 j ^ ^h 1 j ^ h ^h 1 j ^h cEt vCt+j + kEt vKt+j = wEt vNt+j + LEt vLt+j j=0 j=0 j=0 j=0 X X X X ^h 1 j h ^h 1 j ^h h + yEt vr^t+j + dEt vdt+j +n ^t j=0 j=0 X X

D Appendix

Log linearizing the above yields

h h h h h K K^t = E^ (1 ) K K^t+1 + dd^ (1 ) K + d E^ C^ C^ t t+1 t t+1 t h i Keep in mind that at the steady state

K = d 1 (1 )

141 which makes the log linearized asset pricing equation

h h h h h h K^t = (1 ) E^ K^t+1 + (1 (1 )) E^ d E^ C^ C^ t t t+1 t t+1 t

The asset pricing equation at period t + j 1 is

^h ^ ^h ^ ^h h ^h ^h ^h Et Kt+j 1 = (1 ) Et Kt+j + (1 (1 )) Et dt+j Et Ct+j Ct+j 1 (42)

Use the same condition a period back

^h ^h ^h ^ ^h ^ ^h h ^h ^h Et Ct+j 1 = (1 ) Et Kt+j 1 Et Kt+j 2 + (1 (1 )) Et dt+j 1 + Et Ct+j 2 and substitute it in the initial one, equation (42)

^h ^h ^h ^ ^h ^ ^h ^ ^h ^ Et Ct+j = (1 ) Et Kt+j + Et Kt+j 1 Et Kt+j 1 + Et Kt+j 2 ^h h ^h h ^h ^h + (1 (1 )) Et dt+j + Et dt+j 1 + Et Ct+j 2

^h th Keep eliminating Ct+j up to the t period which implies the expected consumption in period t + j is

j 1 j 1 j 1 ^h ^h ^h ^ v ^h ^ v ^h ^h ^h Et Ct+j = (1 ) Et Kt+s+1 Et Kt+s + (1 (1 )) Et dt+s+1 + Ct (43) s=0 s=0 s=0 X X X

142 Substitute the Euler equation (43) to the intertemporal budget constraint (26)

v h 1 j v v h 1 j v (1 ) c E^ K^ c k E^ K^ 1 t v t+j+1 1 t v t+j v j=0 v j=0 X X v h 1 j h c h + c (1 (1 )) E^ d^ + C^ 1 t v t+j+1 1 t v j=0 v X ^h 1 j ^ h ^h 1 j ^h ^h 1 j h = wEt vNt+j + LEt vLt+j + yEt vr^t+j j=0 j=0 j=0 X X X ^h 1 j ^h h + dEt vdt+j +n ^t j=0 X

where I used the fact that

j 1 1 1 j X = v j X v t+s+1 1 v t+j+1 j=0 s=0 v j=0 X X X

Isolate in the above expression the terms that need to be forecasted and the ones that are already known in period t and then solve for consumption

1 v c

^h Ct

1 v h 1 j v = v v (1 ) k E^ K^ t v t+j+1 c j=0 X 1 v h 1 j h v (1 (1 )) d E^ d^ t v t+j+1 c j=0 X 1 v 1 1 v 1 1 v 1 + E^h j N^ h + E^h j L^h + E^h j r^h w t v t+j v L t v t+j+1 v y t v t+j+1 c j=0 c j=0 c j=0 X X X 1 v c ^ v 1 v ^h 1 v h 1 v ^h 1 v h + v k Kt + ddt + yr^t + LLt + n^t c 1 v c c c c

143 ^ h The household knows that the law of motion of the household’semployment level Nt is:

h h N^ = (1 ) N^ + ^t t+1 t

The same law of motion at period j is:

^ h ^ h Nt+j = (1 ) Nt+j 1 + ^t+j 1

Solving it backwards up to period t

j 1 ^ h j ^ h s Nt+j = (1 ) Nt + (1 ) ^t+j s 1 s=0 X since the process depend on the exogenous to the household probability for a worker to …nd a job. j ^ h The expression j1=0 vNt+j then becomes P j 1 1 j ^ h 1 j j ^ h 1 j s vNt+j = v (1 ) Nt + v (1 ) ^t+j s 1 j=0 j=0 j=0 s=0 X X X X 1 1 = N^ h + v j ^ 1 (1 ) t 1 (1 ) v t+j v v j=0 X after using the fact that

j 1 1 j s v 1 j v (1 ) ^t+j s 1 = v^t+j 1 (1 ) j=0 s=0 v j=0 X X X

144 ^ h Plug this fact in the consumption equation above to eliminate Nt+j

^h Ct

1 v 1 v v 1 h 1 j v = v v (1 ) k + w E^ K^ 1 (1 ) t v t+j+1 c c v j=0 X 1 v h 1 j h v (1 (1 )) d E^ d^ t v t+j+1 c j=0 X 1 v 1 1 v 1 + E^h j L^h + E^h j r^h v L t v t+j+1 v y t v t+j+1 c j=0 c j=0 X X 1 v v v 1 ^ v + c k + w Kt c 1 v 1 v (1 ) 1 v ^h h ^h h 1 ^ h + ddt + yr^t + LLt +n ^t + w Nt c 1 v (1 )

E Appendix

Now consider the expectation at period t of the law of motion for employment (36) in the t + j + 1 period and iterate back until the expectation becomes a function of variables known at period t

j 1 j 1 n n s n j n n s j 1 s Etnt+j+1 = 0;t 1 1;t 1 + 1;t 1 nt+1 + 2;t 1 1;t 1 (A) At (44) s=0 s=0 X X Take equations (32) (33), and (34) respectively from period t+j +1 and solve backwards till period t and substitute in equation (44) and equation (37) each solved backwards from period t + j + 1 till period t. Therefore the expectations of the dividends, vacancy costs and aggregate output are

^ d d j ^ Etdt+j+1 = 0;t 1 + 2;t 1 (A) At j 1 j 1 d n n s n j n n s j 1 s ^ + 1;t 1 0;t 1 1;t 1 + 1;t 1 n^t+1 + 2;t 1 1;t 1 (A) At s=0 s=0 ! X X

145 ^ k k j ^ EtKt+j+1 = 0;t 1 + 2;t 1 (A) At j 1 j 1 k n n s n j n n s j 1 s ^ + 1;t 1 0;t 1 1;t 1 + 1;t 1 n^t+1 + 2;t 1 1;t 1 (A) At s=0 s=0 ! X X

^ L L j ^ EtLt+j+1 = 0;t 1 + 2;t 1 (A) At j 1 j 1 L n n s n j n n s j 1 s ^ + 1;t 1 0;t 1 1;t 1 + 1;t 1 n^t+1 + 2;t 1 1;t 1 (A) At s=0 s=0 ! X X

r r j ^ Etr^t+j+1 = 0;t 1 + 2;t 1 (A) At j 1 j 1 r n n s n j n n s j 1 s ^ + 1;t 1 0;t 1 1;t 1 + 1;t 1 n^t+1 + 2;t 1 1;t 1 (A) At s=0 s=0 ! X X which are all known at period t. Consider the following facts

j 1 1 j n s v 1 j n j v 1 v 1;t 1 = v 1;t 1 = n 1 v 1 v 1 v1;t 1 j=0 s=0 j=0 X X X

j 1 1 j n s j 1 s v 1 v 1;t 1 (A) = n 1 vA 1 v1;t 1 j=0 s=0 X X

146 Use the above forecasts and the facts below in the consumption equation (29). The actual law of motion for consumption becomes

^ 1 k d L r Ct = 10;t 1 + 20;t 1 + 30;t 1 + 40;t 1 1 v n v 0;t 1 k d L r + n 11;t 1 + 21;t 1 + 31;t 1 + 41;t 1 1 v 1 v1;t 1 1 k d L r + n 11;t 1 + 21;t 1 + 31;t 1 + 41;t 1 n^t+1 1 v1;t 1 v 1 n k d L r 1 1 n 2;t 1 11;t 1 + 21;t 1 + 31;t 1 + 41;t 1 v A v 1;t 1 ^ At 2 1 k d L r 3 + 1 + 2 + 3 + 4 1 v A 2;t 1 2;t 1 2;t 1 2;t 1 6 7 4 5 ^h h ^h ^ v + 5Lt + 6r^t + 7dt + 8n^t + 9Kt

F Appendix

De…ne

C^t 2 3 Y^t 6 7 6 ^ 7 6 dt 7 6 7 6 7 6 !^t 7 6 7 6 7 Xt = 6 K^ 7 6 t 7 6 7 6 ^ 7 6 Lt 7 6 7 6 7 6 r^t 7 6 7 6 7 6 n^ 7 6 t+1 7 6 7 6 ^ 7 6 At 7 6 7 4 5 The following linear system can be simulated to produce predictions for consumption, output, dividends, total wage, vacancy cost, …rm demand, …rm price, and employment after a shock to the

147 technology "t.

3Xt = 0 + 1Xt 1 + 2"t 1 1 1 Xt = 3 0 + 3 1Xt 1 + 3 2"t where ’s are agent’s estimates and model fundamentals. Starting from X0 and draws from "t s 2 N 0; " the above can give simulations for the model. The parameters in the above expression are related to estimates and model fundamentals according to:

1 (1+) 1+` C 1 N N Y 1+` c = h dN + (1 ) K N i 1 v K P # k = d+ (1 ) K K v = d+ (1 ) K W L w = d+ (1 ) K (1+) 1+` C (1 ) N Y L = d+ (1 ) K Y y = d+ (1 ) K N d d = d+ (1 ) K 1 v v 1 1 v 1 = v v (1 ) k + w c 1 v (1 ) c

1 v 2 = v d (1 (1 )) c

1 v 3 = vL c

148 1 v 4 = vy c

1 v 5 = L c

1 v 6 = y c

1 v 7 = d c

1 v 1 8 = 1 + w c 1 v (1 )

1 v v v 1 9 = c k + w c 1 v 1 v (1 )

1 k d L r 0 = 10;t 1 + 20;t 1 + 30;t 1 + 40;t 1 1 v n v 0;t 1 k d L r + n 11;t 1 + 21;t 1 + 31;t 1 + 41;t 1 1 v 1 v1;t 1

1 k d L r 1 = n 11;t 1 + 21;t 1 + 31;t 1 + 41;t 1 1 v1;t 1

v 1 n k d L r n 2;t 1 11;t 1 + 21;t 1 + 31;t 1 + 41;t 1 1 v A 1 v 1;t 1 2 = 0 1 k d L r 1 + 1 + 2 + 3 + 4 1 v A 2;t 1 2;t 1 2;t 1 2;t 1 B C @ A

149 1 0 7 0 9 5 6 1 2 2 3 1 ` 0 0 0 0 0 0 (1 + `) 6 7 6 1+` Y Pv # 7 6 0 (1 ) (1+`) N d 0 0 0 0 0 7 6 7 6 1 1 Y Y Pv # 7 6 0 (1 ) 0w L 0 0 0 0 7 6 1+` N N 7 6 Pv V 7 3 = 6 C Y 0 0 0 0 0 0 7 6 7 6 7 6 7 6 0 1 0 0 0 1 0 0 1 7 6 7 6 7 6 0 1 0 0 0 0 1 0 0 7 6 7 6 7 6 0 0 0 0 1 0 0 1 0 7 6 7 6 7 6 7 6 0 0 0 0 0 0 0 0 1 7 6 7 4 5

0 2 3 0 6 7 6 7 6 0 7 6 7 6 7 6 0 7 6 7 6 7 0 = 6 0 7 6 7 6 7 6 7 6 0 7 6 7 6 7 6 0 7 6 7 6 7 6 0 7 6 7 6 7 6 7 6 0 7 6 7 4 5

150 0 0 0 0 0 0 0 8 0 2 3 0 0 0 0 0 0 0 ` (1 + ) + 0 6 7 6 1+` Y 7 6 0 0 0 0 0 0 0 (1 ) (1+`) N 0 7 6 7 6 1 1 Y Y 7 6 0 0 0 0 0 0 0 (1 ) 0 7 6 1+` N N 7 6 N v 7 1 = 6 0 0 0 0 0 0 0 P V 0 7 6 1 N 7 6 7 6 7 6 0 0 0 0 0 0 0 (1 + ) 0 7 6 7 6 7 6 0 0 0 0 0 0 0 1 0 7 6 7 6 7 6 0 0 0 0 0 0 0 1 N 0 7 6 1 N 7 6 7 6 h i 7 6 0 0 0 0 0 0 0 0 A 7 6 7 4 5 0 2 3 0 6 7 6 7 6 0 7 6 7 6 7 6 0 7 6 7 6 7 2 = 6 0 7 6 7 6 7 6 7 6 0 7 6 7 6 7 6 0 7 6 7 6 7 6 0 7 6 7 6 7 6 7 6 1 7 6 7 4 5 The parameters are the ones estimated by the households and should be updated according to the recursive algorithm using

1 t = t 1 + gRt qt 1 zt t0 1qt 1 0

Rt = Rt 1 + g qt 1qt0 1 Rt 1

151 where d k L r n 0;t 0;t 0;t 0;t 0;t 2 d k L r n 3 t 1;t 1;t 1;t 1;t 1;t 6 7 6 d k L r n 7 6 2;t 2;t 2;t 2;t 2;t 7 6 7 4 5 1 2 3 qt 1 = n^t 6 7 6 ^ 7 6 At 1 7 6 7 4 5 and

zt0 = d^t K^t L^t r^t n^t+1 Just this part of the appendix and table 1 that summarizes the calibration is enough to replicate the results of this paper.

152 US data Stat Gdp Vac Un tight wage

St. dev relative to output x=Y 1 9:1 7:4 16:1 0:6 Quarterly autocorrelation t;t 1 0:84 0:9 0:89 0:9 0:73 Correlationmatrix Gdp 1 0:87 0:85 0:88 0:25 Vac 1 0:91 0:98 0:17 Un 1 0:97 0:16 tight 1 0:17 wage 1 Table 2: Summary statistics from quarterly U.S Data 1955Q1:2007Q1. All variables are log deviations from an HP trend with smoothing parameter 1600

REE g = 0 Stat Gdp Vac Un tight wage St. dev relative to output x=Y 1 0:68 0:66 1:2 1:05 Quarterly autocorrelation t;t 1 0:91 0:68 0:95 0:92 0:91 Correlationmatrix Gdp 1 0:91 0:89 0:99 1 Vac 1 0:63 0:91 0:92 Un 1 0:90 0:89 tight 1 0:99 wage 1 Table 3: Those are the moments implied by the rational expectations model. The standard deviations in the top row are reported relative to the standard deviation of output

Learning g = 0:0045 Stat Gdp Vac Un tight wage St. dev relative to output x=Y 1 1:97 1:27 2:6 1:44 Quarterly autocorrelation t;t 1 0:9 0:64 0:95 0:9 0:9 Correlationmatrix Gdp 1 0:81 0:83 0:92 0:93 Vac 1 0:49 0:9 0:9 Un 1 0:79 0:79 tight 1 0:99 wage 1 Table 4: Those are the moments implied by the learning model with constant gain parameter g=0.0045. The standard deviations in the top row are reported relative to the standard deviation of output

153 Learning g = 0:005 Stat Gdp Vac Un tight wage St. dev relative to output x=Y 1 8:35 4:77 9:93 3:58 Quarterly autocorrelation t;t 1 0:9 0:1 0:59 0:28 0:39 Correlationmatrix Gdp 1 0:49 0:62 0:68 0:74 Vac 1 0:01 0:91 0:9 Un 1 0:37 0:37 tight 1 0:99 wage 1 Table 5: Those are the moments implied by the learning model with constant gain parameter g=0.005. The standard deviations in the top row are reported relative to the standard deviation of output

Learning g = 0:0055 Stat Gdp Vac Un tight wage St. dev relative to output x=Y 1 18:9 10:3 21:8 7:1 Quarterly autocorrelation t;t 1 0:68 0:05 0:34 0:15 0:17 Correlationmatrix Gdp 1 0:39 0:57 0:63 0:66 Vac 1 0:27 0:92 0:92 Un 1 0:11 0:1 tight 1 0:99 wage 1 Table 6: Those are the moments implied by the learning model with constant gain parameter g=0.0055. The standard deviations in the top row are reported relative to the standard deviation of output

Vac Un tight wage

St. dev relative to output V =Y U =Y #=Y W =Y US data 9:1 7:4 16:1 0:6 REE 0:68 0:66 1:2 1:05 g = 0:0045 1:97 1:27 2:6 1:44 g = 0:005 8:35 4:77 9:93 3:58 g = 0:0055 18:9 10:3 21:8 7:1

Table 7: The table presents the ampli…cation results of the models with di¤erent gain parameters along with the data. All moments are computed relative to output

154 Gdp Vac Un tight wage

First order autocorrelations Yt;Yt 1 Vt;Vt 1 Ut;Ut 1 #t;#t 1 wt;wt 1 US data 0:84 0:9 0:89 0:9 0:73 REE 0:91 0:68 0:95 0:92 0:91 g = 0:0045 0:9 0:64 0:95 0:9 0:9 g = 0:005 0:9 0:1 0:59 0:28 0:39 g = 0:0055 0:68 0:05 0:34 0:15 0:17 Table 8: The table presents the …rst order autocorrelations of the variables under di¤erent speci…- cations of the model along with the same moments from the US data

US data corr(V,U)

St. dev relative to output v;u US data 0:91 REE 0:63 g = 0:0045 0:49 g = 0:005 0:1 g = 0:0055 0:27

Table 9: This is a summary of the implied correlations of vacancies and unemployment in the data and in the models with di¤erent gain parameters

Vac Un tight wage

St. dev relative to output V =Y U =Y #=Y W =Y US data 9:1 7:4 16:1 0:6 REE 0:68 0:66 1:2 1:05 g = 0:005, ` = 2 8:35 4:77 9:93 3:58 g = 0:005, ` = 0 23:4 13:1 27:5 1:53

Table 10: The table presents the ampli…cation results of the models with di¤erent gain parameters along with the data. All moments are computed relative to output

155 •3 •3 x 10 x 10 1 3

0 2

•1 1 V acancies

Unem ploym ent •2 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 ti me ti me •3 •3 x 10 x 10 1.5 4

1 2

Incom e 0.5 T ightness

0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 ti me ti me •3 •5 x 10 x 10 2 5

1 0

Wage Hours

0 •5 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 ti me ti me •3 •3 x 10 x 10 1 1.5

1 0.5

0.5

Productivity Consum ption 0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 ti me ti me

Figure 1: Those are impulse responses after a unit technology shock. The red solid line represents the rational expectations equilibrium responses while the blue dashed lines the impulse responses under perpetual learning. The gain parameter g is set to 0.0055

156 •3 •3 x 10 x 10 0 6

4 •1

2 V acancies

Unem ploym ent •2 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 ti me ti me •3 •3 x 10 x 10 1.5 6

1 4

Incom e 0.5 2 T ightness

0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 ti me ti me •3 •5 x 10 x 10 3 5

0 Wage 1 Hours

0 •5 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 ti me ti me •3 •3 x 10 x 10 1 1

0.5 0.5

Productivity Consum ption 0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 ti me ti me

Figure 2: Those are impulse responses after a unit technology shock. The red solid line represents the rational expectations equilibrium responses while the blue dashed lines the impulse responses under perpetual learning. The gain parameter g is set to 0.006

157 •3 x 10 4.5 g=0 (REE) g=0.005 g=0.0055 4 g=0.006

3.5

2.5

Vacancies 2

1.5

0.5

0 0 1 2 3 4 5 6 7 ti me

Figure 3: Responses of vacancies after a unit productivity shock. The solid red line is the responce in the rational expectations model, the blue dashed line is the model with learning and constant gain parameter 0.005. The black dotted responce corresponds to the learning model with gain parameter 0.0055 and the green dashed and dotted line to gain 0.006

158 Coe¢ cient mean median RE g=0.005 g=0.005 d 0 0 0 0 k 0 0 0 0 L 0 0 0 0 r 0 0 0 0 n 0 0 0 0 d 1 37:9 10:02 13:74 k 1 2:2 0:69 0:64 L 1 0:48 0:53 0:53 r 1 0:48 0:44 0:52 n 1 0:47 0:38 0:37 d 2 2:24 0:11 0:01 k 2 0:27 0:40 0:41 L 2 0:003 0:002 0:002 r 2 0:89 0:88 0:9 n 2 0:02 0:02 0:03

Table 11: Those are the means and medians of the distributions of the belief coe¢ cients induced by perpetual learning along with the rational expectations coe¢ cients.

159 •4 x 10 Intercept State Variable 1 (employment) State Variable 2 (technology) 2 0 1

0 .5 0 •20

0 Dividend

•2 •40 • 0 .5 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000

•5 x 10 2 4 0 .8

0 2 0 .6

•2 0 0 .4 Vacancy cost Vacancy •4 •2 0 .2 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000

•7 •3 x 10 x 10 2 •0.45 4

2 0 • 0 .5

0 Labor Supply Labor •2 •0.55 •2

160 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000

•5 x 10 5 5 1 .5

0 0 1

•5 •5 Firm Demand Firm •10 •10 0 .5 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000

•6 x 10 1 0 .8 0.04

0 0 .6 0.03

•1 0 .4 0.02 Employment •2 0 .2 0.01 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000

Figure 4: Those are the estimates of the belief parameters when the learning model is simulated for 10000 periods with a decreasing gain gt = 1=t. The red ‡at lines are the rational expectations coe¢ cients. The parameters are converging to the rational beliefs in the long run. •3 •4 Intercept x 10 State variable 1 (employment) x 10 State variable 2 (technology) 0.01 2 1

0.005 1 0 .5 Dividend 0 0 0 •60 •40 •20 0 20 40 60 •5000 0 5000 10000 15000 20000 •2 •1 0 1 2 3 4 x 10 •3 x 10 0.03 0.04 1

0.02 0.02 0 .5 0.01

Va ca nco cy st 0 0 0 •4 •3 •2 •1 0 1 2 3 •1200 •1000 •800 •600 •400 •200 0 200 400 •2000 •1500 •1000 •500 0 500 1000 1500

4 1 0.03

0.02 2 0 .5

0.01 Labor Supply Labor

161 0 0 0 •1 • 0 .5 0 0 .5 1 1 .5 •40 •30 •20 •10 0 10 •60 •40 •20 0 20 40

2 1 0 .1

1 0 .5 0.05

Firm Demand Firm 0 0 0 •0.15 • 0 .1 •0.05 0 0.05 0 .1 •40 •30 •20 •10 0 10 20 •60 •40 •20 0 20 40

0 .4 1 0.02

0 .2 0 .5 0.01

Employment 0 0 0 • 0 .2 •0.15 • 0 .1 •0.05 0 0.05 0 .1 0.15 •80 •60 •40 •20 0 20 •150 •100 •50 0 50 100

Figure 5: Distribution of Agent’sbelief coe¢ cients under perpetual learning with gain g=0.15