DOCSLIB.ORG

Lecture Notes for Mathematical Economics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Lecture Notes1 Mathematical Ecnomics Guoqiang TIAN Department of Economics Texas A&M University College Station, Texas 77843 ([email protected]) This version: August 2018 1The lecture notes are only for the purpose of my teaching and convenience of my students in class, please not put them online or pass to any others. Contents 1 The Nature of Mathematical Economics 1 1.1 Economics and Mathematical Economics . 1 1.2 Advantages of Mathematical Approach . 2 2 Economic Models 4 2.1 Ingredients of a Mathematical Model . 4 2.2 The Real-Number System . 4 2.3 The Concept of Sets . 5 2.4 Relations and Functions . 7 2.5 Types of Function . 8 2.6 Functions of Two or More Independent Variables . 10 2.7 Levels of Generality . 10 3 Equilibrium Analysis in Economics 12 3.1 The Meaning of Equilibrium . 12 3.2 Partial Market Equilibrium - A Linear Model . 12 3.3 Partial Market Equilibrium - A Nonlinear Model . 14 3.4 General Market Equilibrium . 16 3.5 Equilibrium in National-Income Analysis . 19 4 Linear Models and Matrix Algebra 20 4.1 Matrix and Vectors . 20 4.2 Matrix Operations . 23 4.3 Linear Dependance of Vectors . 25 4.4 Commutative, Associative, and Distributive Laws . 26 i 4.5 Identity Matrices and Null Matrices . 27 4.6 Transposes and Inverses . 29 5 Linear Models and Matrix Algebra (Continued) 32 5.1 Conditions for Nonsingularity of a Matrix . 32 5.2 Test of Nonsingularity by Use of Determinant . 34 5.3 Basic Properties of Determinants . 38 5.4 Finding the Inverse Matrix . 42 5.5 Cramer's Rule . 45 5.6 Application to Market and National-Income Models . 49 5.7 Quadratic Forms . 51 5.8 Eigenvalues and Eigenvectors . 54 5.9 Vector Spaces . 56 6 Comparative Statics and the Concept of Derivative 62 6.1 The Nature of Comparative Statics . 62 6.2 Rate of Change and the Derivative . 63 6.3 The Derivative and the Slope of a Curve . 64 6.4 The Concept of Limit . 64 6.5 Inequality and Absolute Values . 68 6.6 Limit Theorems . 69 6.7 Continuity and Differentiability of a Function . 70 7 Rules of Differentiation and Their Use in Comparative Statics 73 7.1 Rules of Differentiation for a Function of One Variable . 73 7.2 Rules of Differentiation Involving Two or More Functions of the Same Vari- able . 76 7.3 Rules of Differentiation Involving Functions of Different Variables . 79 7.4 Integration (The Case of One Variable) . 82 7.5 Partial Differentiation . 84 7.6 Applications to Comparative-Static Analysis . 87 7.7 Note on Jacobian Determinants . 89 ii 8 Comparative-static Analysis of General-Functions 92 8.1 Differentials . 93 8.2 Total Differentials . 95 8.3 Rule of Differentials . 96 8.4 Total Derivatives . 97 8.5 Implicit Function Theorem . 100 8.6 Comparative Statics of General-Function Models . 104 8.7 Matrix Derivatives . 105 9 Derivatives of Exponential and Logarithmic Functions 107 9.1 The Nature of Exponential Functions . 107 9.2 Logarithmic Functions . 108 9.3 Derivatives of Exponential and Logarithmic Functions . 109 10 Optimization: Maxima and Minima of a Function of One Variable 111 10.1 Optimal Values and Extreme Values . 111 10.2 General Result on Maximum and Minimum . 112 10.3 First-Derivative Test for Relative Maximum and Minimum . 113 10.4 Second and Higher Derivatives . 115 10.5 Second-Derivative Test . 116 10.6 Taylor Series . 118 10.7 Nth-Derivative Test . 120 11 Optimization: Maxima and Minima of a Function of Two or More Vari- ables 122 11.1 The Differential Version of Optimization Condition . 122 11.2 Extreme Values of a Function of Two Variables . 123 11.3 Objective Functions with More than Two Variables . 128 11.4 Second-Order Conditions in Relation to Concavity and Convexity . 130 11.5 Economic Applications . 133 12 Optimization with Equality Constraints 136 12.1 Effects of a Constraint . 136 iii 12.2 Finding the Stationary Values . 137 12.3 Second-Order Condition . 141 12.4 General Setup of the Problem . 143 12.5 Quasiconcavity and Quasiconvexity . 145 12.6 Utility Maximization and Consumer Demand . 149 13 Optimization with Inequality Constraints 152 13.1 Non-Linear Programming . 152 13.2 Kuhn-Tucker Conditions . 155 13.3 Economic Applications . 160 14 Linear Programming 164 14.1 The Setup of the Problem . 164 14.2 The Simplex Method . 166 14.3 Duality . 174 15 Continuous Dynamics: Differential Equations 177 15.1 Differential Equations of the First Order . 177 15.2 Linear Differential Equations of a Higher Order with Constant Coefficients 180 15.3 Systems of the First Order Linear Differential Equations . 184 15.4 Economic Application: General Equilibrium . 190 15.5 Simultaneous Differential Equations. Types of Equilibria . 194 16 Discrete Dynamics: Difference Equations 197 16.1 First-order Linear Difference Equations . 199 16.2 Second-Order Linear Difference Equations . 201 16.3 The General Case of Order n . 202 16.4 Economic Application:A dynamic model of economic growth . 203 17 Introduction to Dynamic Optimization 205 17.1 The First-Order Conditions . 205 17.2 Present-Value and Current-Value Hamiltonians . 207 17.3 Dynamic Problems with Inequality Constraints . 208 17.4 Economics Application:The Ramsey Model . 208 iv Chapter 1 The Nature of Mathematical Economics The purpose of this course is to introduce the most fundamental aspects of the mathe- matical methods such as those matrix algebra, mathematical analysis, and optimization theory. 1.1 Economics and Mathematical Economics Economics is a social science that studies how to make decisions in face of scarce resources. Specifically, it studies individuals' economic behavior and phenomena as well as how individual agents, such as consumers, households, firms, organizations and government agencies, make trade-off choices that allocate limited resources among competing uses. Mathematical economics is an approach to economic analysis, in which the e- conomists make use of mathematical symbols in the statement of the problem and also draw upon known mathematical theorems to aid in reasoning. Since mathematical economics is merely an approach to economic analysis, it should not and does not differ from the nonmathematical approach to economic analysis in any fundamental way. The difference between these two approaches is that in the former, the assumptions and conclusions are stated in mathematical symbols rather than words and in the equations rather than sentences so that the interdependent relationship among economic variables and resulting conclusions are more rigorous and concise by using math- 1 ematical models and mathematical statistics/econometric methods. The study of economic social issues cannot simply involve real world in its experiment, so it not only requires theoretical analysis based on inherent logical inference and, more often than not, vertical and horizontal comparisons from the larger perspective of history so as to draw experience and lessons, but also needs tools of statistics and econometrics to do empirical quantitative analysis or test, the three of which are all indispensable. When conducting economic analysis or giving policy suggestions in the realm of modern eco- nomics, the theoretical analysis often combines theory, history, and statistics, presenting not only theoretical analysis of inherent logic and comparative analysis from the historical perspective but also empirical and quantitative analysis with statistical tools for exami- nation and investigation. Indeed, in the final analysis, all knowledge is history, all science is logics, and all judgment is statistics. As such, it is not surprising that mathematics and mathematical statistics/econometrics are used as the basic analytical tools, and also become the most important analytical tools in every field of economics. For those who study economics and conduct research, it is necessary to grasp enough knowledge of mathematics and mathematical statistics. There- fore, it is of great necessity to master sufficient mathematical knowledge if you want to learn economics well, conduct economic research and become a good economist. 1.2 Advantages of Mathematical Approach Mathematical approach has the following advantages: (1) It makes the language more precise and the statement of assumptions more clear, which can deduce many unnecessary debates resulting from inaccurate definitions. (2) It makes the analytical logic more rigorous and clearly states the boundary, application scope and conditions for a conclusion to hold. Otherwise, the abuse of a theory may occur. (3) Mathematics can help obtain the results that cannot be easily attained through intuition. (4) It helps improve and extend the existing economic theories. 2 It is, however, noteworthy a good master of mathematics cannot guarantee to be a good economist. It also requires fully understanding the analytical framework and research methodologies of economics, and having a good intuition and insight of real economic environments and economic issues. The study of economics not only calls for the understanding of some terms, concepts and results from the perspective of mathematics (including geometry), but more importantly, even when those are given by mathematical language or geometric figure, we need to get to their economic meaning and the underlying profound economic thoughts and ideals. Thus we should avoid being confused by the mathematical formulas or symbols in the study of economics. All in all, to become a good economist, you need to be of original, creative and academic way of thinking. 3 Chapter 2 Economic Models 2.1 Ingredients of a Mathematical Model A economic model is merely a theoretical framework, and there is no inherent reason why it must mathematical. If the model is mathematical, however, it will usually consist of a set of equations designed to describe the structure of the model. By relating a number of variables to one another in certain ways, these equations give mathematical form to the set of analytical assumptions adopted.
Recommended publications
  • Aggregate Comparative Statics

    Aggregate Comparative Statics

    This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/authorsrights Author's personal copy Games and Economic Behavior 81 (2013) 27–49 Contents lists available at SciVerse ScienceDirect Games and Economic Behavior www.elsevier.com/locate/geb Aggregate comparative statics ✩ ∗ Daron Acemoglu a, , Martin Kaae Jensen b a Department of Economics, Massachusetts Institute of Technology, United States b Department of Economics, University of Birmingham, United Kingdom article info abstract Article history: In aggregative games, each player’s payoff depends on her own actions and an aggregate of Received 31 August 2011 the actions of all the players. Many common games in industrial organization, political Available online 23 April 2013 economy, public economics, and macroeconomics can be cast as aggregative games. This paper provides a general and tractable framework for comparative static results in JEL classification: aggregative games. We focus on two classes of games: (1) aggregative games with strategic C60 C72 substitutes and (2) nice aggregative games, where payoff functions are continuous and concave in own strategies.
  • 2B Tax Incidence : Partial Equilibrium the Most Simple Way of Analyzing

    2B Tax Incidence : Partial Equilibrium the Most Simple Way of Analyzing

    2b Tax Incidence : Partial Equilibrium The most simple way of analyzing tax incidence is to use demand curves and supply curves. This analysis is an example of a partial equilibrium model. Partial equilibrium models look at individual markets in isolation. They are appropriate when the effects of the tax change are | for the most part | confined to one market. They are inappropriate when there are changes in other markets. Why might partial equilibrium analysis be misleading in many cases? Recall that a demand curve for some commodity expresses the quantity consumers wish to buy of a good as a function of the price they pay for that good | holding constant a bunch of stuff. In particular, when drawing an ordinary [as opposed to a compensated] demand curve for socks, we are holding constant the prices buyers pay for all other goods (other than socks), and the incomes of all the buyers. Very often, a tax will change not only the price that consumers pay for socks, but the prices they pay for other goods. For example, an 8 percent sales tax should have some effects on the market for socks. But since a sales tax also applies to many more goods than socks, the sales tax will also probably affect the equilibrium price of shoes, pants, and many other commodities. All these changes will shift the demand curve for socks, if there are any important substitutabilities or complementarities in the demand for socks with the prices of other goods. So partial equilibrium analysis of the market for some commodity, or group of commodities, will be useful, and reasonably accurate, only when looking at at tax which applies only to that group of commodities.
  • PARTIAL EQUILIBRIUM Positive Analysis

    PARTIAL EQUILIBRIUM Positive Analysis

    PARTIAL EQUILIBRIUM Positive Analysis [See Chap 12 ] 1 Equilibrium • How are prices determined? • Partial equilibrium – Look at one market. – In equilibrium, supply equals demand. – Prices in all other markets are fixed. • General Equilibrium – Look at all markets at once. – Consider interactions. 2 Example: Car Market • What happens if the Chinese Govt builds more roads? • Partial equilibrium – Demand for cars rises. – Quantity of cars rises. – Price of cars rises. • General equilibrium – Price of inputs rises. Increases car costs. – Value of car firms rises. Shareholders richer and buy more cars. 3 1 Model • We are interested in market 1. – Price is denoted by p 1, or p. – Firms/Consumers face same price (law of one price). – Firms/Consumers are price takers. 4 Model • There are J agents who demand good 1. – Agent j has income m j – Utility u j(x 1,…,x N) – Prices {p 1,…,p N}, with {p 2,…,p N} exogenous. • There are K firms who supply good 1. – Firm k has technology f k(z 1,…,z M) – Input prices {r 1,…,r M} exogenous. 5 Competitive Market • To understand how the market functions we consider consumers’ and firms’ decisions. • The consumers’ decisions are summarized by the market demand function. • The firms’ decisions are summarized by the market supply function. 6 2 MARKET DEMAND 7 Market Demand • Market demand is the quantity demanded by all consumers as a function of the price of the good. – Hold constant price of the other goods. – Hold constant agents’ incomes. 8 Market Demand • Assume there are two goods, x1 and x2.
  • Estimating Impact in Partial Vs. General Equilibrium: a Cautionary Tale from a Natural Experiment in Uganda∗

    Estimating Impact in Partial Vs. General Equilibrium: a Cautionary Tale from a Natural Experiment in Uganda∗

    Estimating Impact in Partial vs. General Equilibrium: A Cautionary Tale from a Natural Experiment in Uganda∗ Jakob Svensson† David Yanagizawa-Drott‡ Preliminary. August 2012 Abstract This paper provides an example where sensible conclusions made in partial equilibrium are offset by general equilibrium effects. We study the impact of an intervention that distributed information on urban market prices of food crops through rural radio stations in Uganda. Using a differences-in-differences approach and a partial equilibrium assumption of unaffected urban market prices, the conclusion is the intervention lead to a substantial increase in average crop revenue for farmers with access to the radio broadcasts, due to higher farm-gate prices and a higher share of output sold to traders. This result is consistent with a simple model of the agricultural market, where a small-scale policy intervention affects the willingness to sell by reducing information frictions between farmers and rural-urban traders. However, as the radio broadcasts were received by millions of farmers, the intervention had an aggregate effect on urban market prices, thereby falsifying the partial equilibrium assumption and conclusion. Instead, and consistent with the model when the policy intervention is large-scale, market prices fell in response to the positive supply response by farmers with access to the broadcasts, while crop revenues for farmers without access decreased as they responded to the lower price level by decreasing market participation. When taking the general equilibrium effect on prices and farmers without access to the broadcasts into account, the conclusion is the intervention had no impact on average crop revenue, but large distributional consequences.
  • Introduction

    Introduction

    APPLIED GENERAL EQUILIBRIUM MODELS: AN ASSESSMENT OF THEIR USEFULNESS FOR POLICY ANALYSIS Antonio M. Borges CONTENTS Introduction . ..... .. 8 I. History and brief survey . ..... .. 9 II. Strengths and weaknesses of the general equilibrium approach .. 15 A. Strengths . .... .. 15 B. Weaknesses . .... .. 19 111. The "state of the art" - directions of current research .. .. 23 IV. Operational aspects and constraints . .... .. 26 Conclusion: Assessment of general equilibrium models for policy analysis ............................ .. 30 Bibliography ............................ .. 33 Antonio M. Borges is Professor of Economics at INSEAD; he has prepared this paper under contract with the OECD Economics and Statistics Department. 7 INTRODUCTION Applied general equilibrium modelling is nowadays one of the most active areas of research in economics. It has generated much interest among policy makers and policy analysts as a new methodology capable of providing coherent answers to complicated questions in a systematic way. The popularity and prestige of this fairly new approach in applied economics have occasionally led to exagerated optimism about its usefulness to handle any issue or answer any question. Applied general equilibrium models are a powerful and informative tool to deal with important practical policy issues; but they should be developed with great care and used with prudence. A careful and comprehensive model requires substantial resources, which certainly does not make it a fast and ready instrument to handle trivial problems. Its usefulness is unquestionable for certain types of issues, while dubious for others. Based on the Walrasian tradition, applied general equilibrium models describe the allocation of resources in a market economy as the result of the interaction of supply and demand, leading to equilibrium prices.
  • Conditional Comparative Statics

    Conditional Comparative Statics

    Conditional Comparative Statics James A. Robinsony Ragnar Torvikz August 15, 2010 Abstract Why was the Black Death followed by the decline of serfdom in Western Europe but its’ intensi…cation in Eastern Europe? What explains why involvement in Atlantic trade in the Early Modern pe- riod was positively correlated with economic growth in Britain but negatively correlated in Spain? Why did frontier expansion in the 19th Century Americas go along with economic growth in the United States and economic decline in Latin America? Why do natural re- source booms seem to stimulate growth in some countries, but lead to a ‘curse’in others, and why does foreign aid sometimes seem to encour- age, other times impede economic growth? In this paper we argue that the response of economies to shocks or innovations in economic oppor- tunities depends on the nature of institutions. When institutions are strong, new opportunities or windfalls can have positive e¤ects. But when institutions are weak they can have negative e¤ects. We present a simple model to illustrate how comparative statics are conditional on the nature of institutions and show how this perspective helps to unify a large number of historical episodes and empirical studies. Keywords: JEL: Paper prepared for the 2010 World Congress of the Econometic Society. We are grateful to Marco Battaglini for his incisive comments. yHarvard University, Department of Government and Institute for Quanti- tative Social Science, 1737 Cambridge St., Cambridge MA02138. E-mail: [email protected]. zNorwegian University of Science and Technology, Department of Economics, Dragvoll, N-7491 Trondheim, Norway.
  • Robust Comparative Statics  Susan Athey, Paul Milgrom, and John Roberts

    Robust Comparative Statics  Susan Athey, Paul Milgrom, and John Roberts

    Robust Comparative Statics Susan Athey, Paul Milgrom, and John Roberts DRAFT ONLY - DO NOT CITE OR CIRCULATE Comments Welcome This Draft: October, 1998 OUTLINE Chapter 0: Introduction. Part I: Univariate Optimization Problems Chapter 1: An Introduction to the Theory. Chapter 2: Single Variable Problems with Separable Contexts. Chapter 3: Single Variable Problems with non-Separable Contexts. Part II: Multivariate Optimization Problems. Chapter 4: Multivariate Comparative Statics on Product Sets. Chapter 5: Elementary Lattice Theory and Comparative Statics on Lattices. Part III: Stochastic Optimization Problems. Chapter 6: Supermodularity and monotonicity. Chapter 7: Log-Supermodularity. Chapter 8: Single Crossing Properties. Part IV: Equilibrium Models. Chapter 9: One dimensional models. Chapter 10: Multidimensional models. Chapter 11: Conclusion. 2 0 Introduction This monograph describes a new approach to comparative statics analysis that has developed rapidly in the past several years. Comparative statics – the study of how the solutions of an economic model change as the model parameters and specification are changed – is important because (1) most of the testable predictions of economic theory are comparative statics predictions and (2) many economic equilibrium analyses are built from comparative statics analyses of the model’s components. The results of comparative statics analyses thus form the basis for much of our understanding of the behavior of the economy. Comparative statics predictions can be qualitative or quantitative, but the focus of the new methods is on the former. Typical examples of qualitative predictions include assertions that increasing employment taxes or a minimum wage will increase unemployment, or that relaxing trade restrictions will cause industrial pollution to fall in developed countries and rise in less developed ones.
  • Capitalist Spirit and the Markets: Why Income Inequality Matters

    Capitalist Spirit and the Markets: Why Income Inequality Matters

    School of Business Economics Division Capitalist Spirit and the Markets: Why Income Inequality Matters Aristotelis Boukouras, University of Leicester Working Paper No. 16/16 Capitalist Spirit and the Markets: Why Income Inequality Matters Aristotelis Boukouras∗ October 30, 2016 Abstract I develop a simple static general equilibrium model with capitalist-spirit prefer- ences and prices set by firm owners (entrepreneurs). The model’s pure symmetric Nash equilibria differ markedly from the canonical model: (i) A positive output gap and unemployment may emerge in equilibrium, despite the absence of price rigidities or information asymmetries. (ii) Income and wealth inequality affect equilibrium prices and employment. (iii) The model generates ambiguous com- parative statics. Specifically, an increase in inequality of either type may reduce employment and increase the output gap of the economy, while productivity re- ductions may have the opposite effect. As a result, minimum wage policies may increase employment. These results provide some justification for a number of arguments used in public debates. Keywords: capitalist spirit, general equilibrium, income distribution, income inequal- ity, minimum wage, output gap, unemployment, wealth distribution, wealth inequality JEL Classification: D31, D63, E24, E25 ∗University of Leicester, AC206, Astely Clarke Building, University Road 1, LE1 7RH, Leicester, UK, email: [email protected] 1 1 Introduction Many western economies, including US, UK and the European Union, suffer from weak growth, low inflation and extremely low, or even negative, interest rates in the aftermath of the Great Recession. Summers (2014a, 2014b) calls this phenomenon “secular stagna- tion” and many economists agree that it is a result of weak aggregate demand (Summers, 2014a).
  • Global Comparative Statics in General Equilibrium: Model Building from Theoretical Foundations

    Global Comparative Statics in General Equilibrium: Model Building from Theoretical Foundations

    Global Comparative Statics in General Equilibrium: model building from theoretical foundations James R. Markusen Department of Economics University of Colorado, Boulder Abstract International trade economists made seminal contributions to general equilibrium theory, moving away from an emphasis on existence of equilibrium to algebraic formulations which enabled us to characterize key relationships between parameters and variables, such as that between tariffs and domestic factor prices and welfare. But the analysis remained limited in value for policy evaluation: the analysis was local, it provided only qualitative results, it was limited to very small models, and strictly interior solutions had to be assumed. The contribution of this paper is pedagogic and methodological, providing a primer for those wishing to do or teach general- equilibrium counterfactuals on computable general-equilibrium (CGE) or structural econometric models. I show how the tools from early local comparative statics analyses can be generalized via the use of Shepard’s lemma, duality, complementarity and the Karush-Kuhn-Tucker theorem into a global, quantitative analysis of large changes in high-dimension models which also allows for regime changes and corner solutions. I then show how the resulting non-linear complementarity problem directly translates into a numerical model using GAMS (general algebraic modeling system). The paper concludes with two examples: (a) comparison of a tax versus a real trade/transactions cost, (b) comparison of a tax versus a quantitative restriction such as a quota or license. Keywords: duality, complementarity, KKT conditions, global comparative statics, corner solutions, GAMS JEJ codes: F23, D50, D58 This paper is a re-orientated and much expanded version of a paper prepared for a special issue of the International Journal of Economic Theory in honor of Ronald W.
  • Partial Equilibrium Sequential Search

    Partial Equilibrium Sequential Search

    Labor Economics, 14.661. Lecture 11: Partial Equilibrium Sequential Search Daron Acemoglu MIT December 6, 2011. Daron Acemoglu (MIT) Sequential Search December6,2011. 1/43 Introduction Introduction Introduction Central question for labor and macro: what determines the level of employment and unemployment in the economy? Textbook answer: labor supply, labor demand, and unemployment as “leisure”. Neither realistic nor a useful framework for analysis. Alternative: labor market frictions Related questions raised by the presence of frictions: is the level of employment e¢ cient/optimal? how is the composition and quality of jobs determined, is it e¢ cient? distribution of earnings across workers. Daron Acemoglu (MIT) Sequential Search December6,2011. 2/43 Introduction Introduction Introduction (continued) Applied questions: why was unemployment around 4-5% in the US economy until the 1970s? why did the increase in the 70s and 80s, and then decline again in the late 90s? why did European unemployment increase in the 1970s and remain persistently high? why is the composition of employment so di¤erent across countries? male versus female, young versus old, high versus low wages Daron Acemoglu (MIT) Sequential Search December6,2011. 3/43 Introduction Introduction Introduction (continued) Challenge: how should labor market frictions be modeled? Alternatives: incentive problems, e¢ ciency wages wage rigidities, bargaining, non-market clearing prices search Search and matching: costly process of workers …nding the “right” jobs. Theoretical interest: how do markets function without the Walrasian auctioneer? Empirically important, But how to develop a tractable and rich model? Daron Acemoglu (MIT) Sequential Search December6,2011. 4/43 McCall Model McCall Sequential Search Model McCall Partial Equilibrium Search Model The simplest model of search frictions.
  • Chapter 1 Introduction

    Chapter 1 Introduction

    Chapter 1 Introduction The art of successful theorizing is to make the inevitable simplifying assumptions in such a way that the final results are not very sensitive. Robert M. Solow (1956, p. 65) 1.1 Macroeconomics 1.1.1 The field Economics is the social science that studies the production and distribution of goods and services in society. Then, what defines the branch of economics named macroeconomics? There are two defining characteristics. First, macroeconomics is the systematic study of the economic interactions between human beings in society as a whole. This could also be said of microeconomic general equilibrium theory, however. The second defining characteristic of macroeconomics is that it aims at understanding the empirical regularities in the behavior of aggregate economic variables such as aggregate production, investment, unemployment, the general price level for goods and services, the inflation rate, the level of interest rates, the level of real wages, the foreign exchange rate, productivity growth etc. The focus is on the short-run fluctuations of these variables as well as their long- run trend. Thus, macroeconomics studies the major lines of the economics of society as a whole. The aspiration of macroeconomics is three-fold: 1. to explain the levels of the aggregate variables as well as their movement over time in the short run and the long run; 2. to make well-founded forecasts possible; 3 4 CHAPTER 1. INTRODUCTION 3. to provide foundations for rational economic policy applicable to macroeco- nomic problems, be they short-run distress in the form of economic recession or problems of a more long-term, structural character.
  • Comparative Statics and Heterogeneity

    Comparative Statics and Heterogeneity

    Towson University Department of Economics Working Paper Series Working Paper No. 2016-01 Comparative Statics and Heterogeneity by Finn Christensen October, 2016 © 2016 by Author. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source. Comparative Statics and Heterogeneity Finn Christensen Towson University October 7, 2016 Abstract This paper elucidates the role played by the heterogeneity of interactions between the endogenous variables of a model in determining the model’sbehav- ior. It is known that comparative statics are well-behaved if these interactions are relatively small, but the formal condition imposed on the Jacobian which typically captures this idea–diagonaldominance–ignores the distribution of the interaction terms. I provide a new condition on the Jacobian–mean positive dominance–which better captures a trade-o¤ between the size and heterogeneity of interaction terms. In accord with Samuelson’s(1947) correspondence princi- ple, I also show that mean positive dominance yields stability and uniqueness results. Applications are provided to optimization problems, di¤erentiable games, and competitive exchange economies. Keywords: comparative statics, heterogeneity, mean positive dominance, correspondence principle, B matrix, stability, uniqueness, optimization, dif- ferentiable games, Cournot oligopoly, general equilibrium JEL Classi…cation: D11 1 Introduction Comparative statics analysis is complicated by interactions between endogenous vari- ables. To see this, consider a game-theoretic context where the total equilibrium e¤ect of a positive shock on a player’s action can be decomposed into a partial e¤ect and an interactions e¤ect. The partial e¤ect is the increase in the player’saction holding constant all other players’actions.