SOLUTION PROBLEM SET 3 LABOR ECONOMICS

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1 SOLUTION PROBLEM SET 3 LABOR ECONOMICS Question : Answers should recognize that this result does not hold when there are search frictions in the labour market. The proof should follow a simple matching model in which the worker bargaining power parameter is set at zero, and the rm appropriates the whole rent from job matches. In this set-up equilibrium unemployment is given by + () Hence, the only way to eliminate unemployment when > 0 is to have!, which makes it in nitely easy to nd a job. Labour market tightness and wages are jointly determined by the following job creation and wage setting conditions: c w P (r + ) (2) w P + ( )z (3) z when 0 Therefore, the equilibrium tightness parameter is such that r + c P z Hence is nite and, although very low, unemployment is not eliminated. The opposite case, where implies that w P > P Then the Job Creation condition reads c c (r + ) However, it can never be satis ed since the term on the left-hand side is positive and the one on the right-hand side is negative. The intuition is quite simple: since the worker appropriates all the rent, rms cannot recover their searching costs. Consequently no rm will enter the market and we have full unemployment. In other words, the equilibrium conditions for an interior solution cannot be satis ed when : Question 2: The reservation wage of a worker facing a non-degenerate wage o er distribution is given by Z + w z + (w w ) df (w) (4) r + w where is the rate at which workers contact rms. The term z is the ow income of unemployed workers. It aggregates unemployment income, alternative revenues and searching costs. Hence, the rst obvious answer to the question is that governments do not have full control over the worker outside option and so cannot directly determine the reservation wage.

2 The second and more interesting point is that, even when z is equal to zero, workers might turn down some job o ers (as seen in PS.5). The rationale behind such a behavior is that, by accepting a given job o er, the worker abandons the possibility to search for better opportunities. This is why he will sometimes refuse job o ers although they provides him with a higher income than what he gets while being unemployed. In other words, an acceptable job o er must also compensate the worker for abandoning his option to search. The discussion of the provisions of reasonableness could led to di erent answers. It is related to the rst part of the question, since we have seen that unemployment income do not allow direct control over the reservation wage. This is why governments have turned to di erent policies in order to modify the behavior of job seekers. By excluding workers from the services provided to job seekers, they can reduce the rate of contact and to some extent stigmatize the unemployed workers for not complying with the rule. Therefore, the provisions of reasonableness is likely to increase the rate at which workers leave the unemployment pool. However, such a policy also has important drawbacks. The principal one is that workers choose the reservation wage which maximizes their utility. By interfering, the government is likely to generate ine ciencies from the worker s point of view. It is also di cult to imagine that workers will be motivated for jobs that they would have refused in the rst place. Of course, many other answers are possible and were given points. Question 3: The common mistake was to discuss shocks that increase whereas the question was about shocks that shift the Beveridge curve. The expression of the Beveridge curve is derived from + (5). The rst type of shocks directly a ect the matching function. Such shocks re ect an increase in the degree of mismatch in the economy. They might be due, for example, to a higher proportion of long term unemployed workers whose skills have become obsolete. Consider for example a Cobb-Douglas matching function, so that A with 2 (0; ) A decrease in the scale factor A leads to an increase in unemployment for a given value of. Since the matching function does not enter the JC and WC, the equilibrium value of and w do not change. Hence the new equilibrium can be illustrated considering solely the (u; v) plane. V U* 2. The second type of shocks increase the separation rate. They might be due, for example, to a reduction in employment protection. A change in also a ects the JC curve, so the new equilibrium looks as follows. 2

3 V W* θ U* Exercise:. This question asks to redo the calculations explained during the lectures. We brie y repeat them here. Job Creation Curve: The asset equations for Vacancies and Filled Jobs are rv C + (J V ) rj P w + (V J) Free-Entry implies that in steady-state V 0, therefore J P w r + C (JC) Wage Curve: The asset equations for Unemployed and Employed workers are ru Z + (E U) re w + (U E) The second asset equation can be rewritten as follows w (r + )(E U) + ru (6) To obtain the value of (E U) in terms of the model s parameters, we use the Nash-bargaining solution P w C ( )(E U) (J V ) r + where the last equality follows from the Free-Entry condition. Now replace into the asset equation for U the value of (E U) to obtain ru Z + (E U) Z + C Finally reinsert this solution into (6) to obtain w (r + )(E U) ru (P w) + Z + C 3

4 Simplifying this expression yields w P + C + ( ) Z (W) 2. If the minimum wage is below the market wage, then the regulation is irrelevant. Thus the lower bound is w w P + C + ( ) Z: If the minimum wage is above the job s output, rms have no incentives to post vacancies since they cannot recover their searching costs. Thus the upper bound is w P: For any minimum wage between these two bounds, the equilibrium is well de ned and di ers from the free-market equilibrium. Since the wage is xed exogenously, we do not need to consider the wage curve. The equilibrium tightness is given by the JC condition, so that P w m (r + ) C When w m is above the market wage, it must be the case that increases. Since q 0 () < 0, this implies that decreases. In other words, there is less vacancies per unemployed worker. To obtain the impact on the equilibrium rate of unemployment, we use the Beveridge curve + We know that decreases and since (q()) > 0, we can deduce that the unemployment rate increases. 3. The hazard rate of leaving unemployment is equal to. Thus the aggregate ow of workers is given by u () + Di erentiating this expression with respect to yields (u) (q()) ( + ) ( + ) 2 (q()) 2 ( + ) 2 > 0 (q()) Since is a decreasing function of the minimum wage, the size of the ows out of the unemployment pool also decreases. In other words, the unemployment rate increases proportionally less than the hazard rate decreases. 4. We replace the new expressions in the WC to obtain w P P + ( ) zw ) w P > 0 since z < : ( ) z Therefore the wage is given by a constant factor times the job s output. 5. Replace the new expression into the JC condition P w P (r + ) cp r + ( ) z q() ) ( ) z (r + ) c q() : The equilibrium tightness and so unemployment are independent of the job s output. This is because the wage and the cost of posting a vacancy increase proportionally to output. 4

5 6. P is a measure of job s productivity. Thus one can think of changes in P as measuring technological progress. If the minimum wage grows at a slower rate than technological progress, it is obvious that after a su cient length of time w (t) P (t) ( + ( ) z) > w m (t) so that the market wage will eventually exceeds the minimum wage. On the contrary, if the minimum wage grows at a faster rate than technological progress, it will always be true that lim t! w m (t) > lim t! P (t). Then the equilibrium will collapse and there will be full-unemployment. To conclude, for the minimum wage to be consistent with steady state and growth, it must increase at the same rate than technological progress. 5