Impacte Do Subsidio De Desemprego Na Oferta De Trabalho E No Bem Estar Dos Trabalhadores

Unemployment Subsidy and Labour Supply Construction, calibration and analysis of a dynamic model

Pedro Cosme da Costa Vieira ([email protected]) Faculdade de Economia do Porto R. Dr. Roberto Frias, s/n 4200-464 PORTO PORTUGAL

Abstract: In this work I study the impact of the unemployment subsidy in the supply labour function, considering that worker maximises an inter-temporal utility function separable in time. Computed the model by simulation methods, results that the existence of unemployment subsidy moves the supply labour function toward the diminution of the supply because be unemployed becomes an appealing "occupation". As this conclusion is compatible with stylised economic facts, it is possible to use my model as an "experimental environment" in the discovery of how to "encourage" the search for a new job.

Key Words: Unemployment subsidy; Inter-temporal utility; Calibration; Dynamic programming.

1. Introduction In this work, I intend to study the influence of the existence of unemployment subsidy in the supply labour function. In this vein, I formalise a partial equilibrium model of the labour market.

1 In the model, the workers have rational expectations a la Muth, they are price takers and adopt a supply of labour that maximise an inter-temporal utility function separable in the time.

2. Model of partial equilibrium with rational expectations In a certain period, the worker, if employed, he receives a real wage W per unit of labour supplied and, if unemployed, he receives an unemployment subsidy. In the beginning of the present period, the worker has the wealth r in real terms. If he is employed, during the present period he earns the wage W.s, being s the labour supplied and he pays an insurance employment premium in percentage terms, . At the end of the present period, the worker consumes c in real terms. In this way, at the beginning of next period, the worker has the wealth (r + s.W. – c) and may continue employed or become unemployed. If the worker is unemployed, instead of wage, he earns the unemployment subsidy, Sub, search for a job and in the beginning of the next period he may find a job or continue unemployed. The worker's utility function considers all the periods from the current one to the time horizon (to see, for example, Varian, 1999). The utility function has as variables of decision, the consumption and the amount of work, and five variables of state: the employment, E, the wealth, r, the unitary wage, W, the unemployment subsidy, Sub, and the contribution percentage for the unemployment fund, . All variables are continuos positive except the variable E that is a Dummy (1 if the worker employed and 0 if he is unemployed).

U (ct ,st |Et ,rt ,W,Sub,)  u(ct ,st )   U (ct 1 ,st1|Et1 ,rt 1 ,W,Sub,), (0) s.a rt 1  rt  st .W.1 .Et  Sub.1 Et   ct

This model is known in the literature as Model of Bellman, Dynamic Programming or Mathematical Programming. The worker sets the variables of decision in a way that maximises this utility function.

V Et ,rt ,W,Sub,  maxUct ,st |Et ,rt ,W,Sub, (0) ct ,st

It results from this maximisation problem the worker's labour supply function and goods demand function. The worker knows that may loose his job in the end of the present period, but he doesn't know if for sure. As worker has rational expectations a la Muth (1961), he assumes a probability of loosing there job that uses in the model to forecast the impact of the present decisions in the future. This inter-temporal recursive model is known in the literature as Stochastic Dynamic Programming (about recursive models see Stokey, Lucas Jr. and Prescott, 1989). Being that the worker is employed, naming PLJ as the worker's probability of losing the employment that is a decreasing function of the supply of work, the worker's utility quantifies by:

V 1,rt ,W ,Sub,  maxu(ct ,st ) ct ,st  .V 1,r ,W ,Sub,.1 PLJ (s )  stays employed t1 t (0)

 .V 0,rt1 ,W ,Sub,.PLJ (st )  looses the job

rt1  rt  st .W.1  ct

Being that the worker is unemployed, naming PFJ as the worker's probability of finding a job that is a increasing function of the intensity that he of search a job, the worker's utility quantifies by:

V 0,rt ,W ,Sub,  maxu(ct ,st ) ct ,st  V 1,r ,W ,Sub,.PFJ (s )  finds a job t1 t (0)

 V 0,rt1 ,W ,Sub,.1 PFJ (st ) stays unemployed

rt1  rt  Sub  ct

Now, I can join expression (3) and (4) using the state dummy variable E.

3 V Et ,rt ,W , Sub,  maxu(ct , st ) ct ,st

 .V 1,rt1 ,W , Sub,.Et .1 PLJ (st )  stays employed  .V 0,r ,W ,Sub,.E .PLJ (s )  losses the job t1 t t (0)

 .V 1,rt1 ,W , Sub,.1 Et .PFJ (st )  finds a job

 .V 0,rt1 ,W ,Sub,.1 Et .1 PFJ (st )  stays unemployed

rt+1  rt  st .W.1 .Et  Sub.1 Et  ct

Rearranging and simplifying this expression, it results the recursive model in its final form:

V Et ,rt ,W , Sub,  maxu(ct , st ) ct ,st  .V 1,r ,W , Sub,.E .1 PLJ (s ) 1 E .PFJ (s ) t1 t t t t (0)

 .V 0,rt1 ,W ,Sub,.Et .PLJ (st )  1 Et .1 PFJ (st )

rt+1= rt  st .W.1 .Et  Sub.1 Et  ct

In aggregated terms and in a situation of stationary equilibrium, the market labour supply function results from the sum of the individual supply functions of the entire N worker. In an equivalent way, this market supply function results from the sum of the supply functions of one worker for all the periods (see, e.g., Blanchard and Fischer, 1989, p. 92):

N T SW , Sub,   s(Et,i , rt,i ,W , Sub, ).Et,i  s(Et,i , rt,i ,W , Sub, ).Et,i  (0) i1 tt0

The unemployment subsidy is financed by the employed workers' contributions. The deficit of the unemployment fund, calculated as the sum of all subsidies paid subtracted from all contributions collected, must be zero if we don’t want that there is a wealth effect.

4  N Deficit Sub,  Sub. 1 E  s(E ,r ,W,Sub).W.1 .E  0 (0)     t , i  t , i t , i t , i  t 0 i1

3. Formalisation and condensation of parameters The model I presented in 2., has already some functional form: First, I consider that the function of utility is separable in the time and discounted to the factor  per period; Second, I consider that the wage per unit of labour, the unemployment subsidy and the contribution rate for the fund of unemployment are invariant in the time; Third, I consider that the possibility of an employed worker become unemployed and vice-versa is summarised in a probability that is function of the worker's effort. However, I still consider that the utility function is exponential-linear:

 u(ct ,st )  (ct )  st .Ks, ct  0, st  0 , (0)

That the probabilities of loosing or finding a job are hyperbolic functions:

1 PLJ (st )  1 st KL (0)

1 PFJ (st )  1 1 st KF  (0)

4. Calibration and simulation of the model The model I present, being recursive on a non-linear function of utility, doesn't have a general analytic solution, being necessary to solve it by numeric simulation (see, e.g., Amman, Kendrick and Rust, 1996). As I intend to study the effect of the existence of unemployment subsidy with a balanced unemployment fund, I simulated first the situation with no unemployment subsidy and then I introduced the unemployment subsidy.

5 From a situation with W = 2.5,  = 1/3, KL = 10, KF = 1, Ks = 10, initial wealth is 3.2 units and the utility is discounted at the rate of 10% by period, first I present the short- term and long-term labour supply function.

0,75 s Short term 0,5 Long term 0,25

0 1 2 3 4 W5

Fig. 1 - Short-term vs. long-term labour supply function

In the next figure, I present the function "search for a new job" by an unemployed worker in the period immediate to the lost of the job and after a long period of unemployment (after he depleted his precautionary savings).

s Long-term 1

0,5 Short-term 0 1 2 3 4 W5

Fig. 2 - Short-term vs. long-term search of job

In this simulation, the probability an employed worker looses his job in each period is 22.6% and the probability an unemployed worker finds a new job in each period after he depleted his precautionary savings is 44.4%. In the next figure, I present the evolution of the probability of an unemployed worker finds a new job with de increase of unemployment length.

6 PFJ 0,4

0 0 5 10 15 20 25t

Fig. 3 - Probability of finding a new job with increase of unemployment length

Now, I introduce an unemployment subsidy of 0,25 M.U. and I compare the new situation with the one without unemployment subsidy. First, I present the labour supply function in the short-term, and I observe that the subsidy turns the labour supply function more elastic.

0,75s With subsidy

0,5 Without subsidy

0 1 2 3 4 W5

Fig. 4 - Short-term labour supply function with and without unemployment subsidy

In the next picture, Now, I present in the long-term the shift toward the diminution of labour supply function induced by the existence of an unemployment subsidy.

7 0,75s Without Subsidy 0,5

With Subsidy 0,25

0 1 2 3 4 W5

Fig. 5 - Long-term labour supply function with and without unemployment subsidy

4. Conclusion Using an inter-temporal utility optimisation model, I present a study about the influence of the existence of unemployment subsidy in the labour supply function. On the model calibrated in a way I assume convenient, I observe that the existence of unemployment subsidy moves the supply labour function toward the diminution of the supply. This diminution of labour supply results from the fact that the unemployment subsidy turns the situation of unemployment more attractive. In this way, by one side, the worker employed don't care so much if he loose his job, and, by other side, the worker unemployed don’t search so intensely for a new job. Being that my conclusions are compatible with the stylised economic facts, my model may be used as an "experimental environment" to check the ability of alternative mechanisms in the "encouragement" of an unemployed worker in the search for a new job.

Bibliography Amman, Hans M., David A. Kendrick e John Rust (ed.), 1996, Handbook of Computational Economics, Elsevier, Amsterdam Bellman, R. (1957), Dynamic Programming, Princeton University Press, Princeton. Blanchard, Olivier e Stanley Fischer (1989), Lectures on Macroeconomics, MIT Press, Cambridge, Massachusetts.

8 Muth, John (1961), “Rational Expectations and the Theory of Price Movements”, Econometrica, vol. 29, pp.315-335.

Romer, David (2001), Advanced Macroeconomics, MacGraw-Hill, Boston Stokey, Nancy L., Robert E. Lucas Jr., e Edward C. Prescott (1989), Recursive Methods in Economic Dynamics, Harvard University Press.

Varian, Hal R., 5ª ed. (1999), Intermediate Microeconomics - A Modern Approach, W. W. Norton, London.