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166 "For simplicity, neglect the constraint that L cannot exceed L Accounting for this con straint, one would hnd that for A, above some critical level, L, would equal I rather than being determined by the condition derived in part (b), below In the United States, a hrms unemployment insurance taxes only partly account for the extent to which the hrms workers obtain unemployment insurance; that is, the taxes are only partially experience rated Thus f is between and one. productivity and lowproducUvity workers the hrm hires. Assume that the fair wage for typei workers is given by w* = (wi + W2)/2  bu,, where w, is the average wage paid to workers of type / and u, is their unemployment rate. Finally, assume there are L workers of each type. (i) In equihbrium, is there unemployment among highproductivity workers Explain. (¹nt: if U] is positive, hrms are unconstrained in their choice of wi.) (h) In equilibrium, is there unemployment among lowproductivity workers Explain. 10.6 Implicit contracts without variable hours. Suppose that each worker must either work a fixed number of hours or be unemployed. Let C, denote the consumption of employed workers m state /, and C, the consumption of unemployed workers The hrms prohts in state i are therefore A, F{L,)  [C,L, iC,{L  L,)], where L is the number of workers. Similarly, workers expected utility m state i is (L, /L)[U(C,) K] + {(L  L,)lLUKC,), where > 0 is the disutility of working. ia) Set up the Lagrangian for the hrms problem of choosing the L,s, C,s, and C,"s to maximize expected prohts subject to the constraint that the representative workers expected utility is ? ib) Find the hrstorder condiUons for L,, C,, and C,". How, if at all, do C and C depend on the state What is the relation between Cf and C, (c) After A is reahzed and some workers are chosen to work and others are chosen to be unemployed, which workers are better off? 10.7 Unemployment insurance. (This follows Feldstein, 1976.) Consider a hrm with revenues AF(L). A has two possible values, and Ac {Ab < Ac), each of which occurs half the nme. Workers who are employed when A = Ac and unemployed when A = Ab receive an unemployment insurance beneht of > 0 when A = Afi. Workers are riskneutral; thus the representative workers expected utility is = {w~K)/2+ [(Lb/Lc.)(wK) + [{Lo Lb)/Ig]B}/2, where w is the wage (which is assumed without loss of generahty to be independent of the state), is the disutihty of working, and Lb and Lc are employment in the two states. The hrms expected prohts are [AgF(Lc) wL(,]/2 + IAbF(Lb)wLb  fB(Lc  Lb)]/2, where f is the fraction of unemployment benehts that are paid by the hrm. Assume 0 < f < I?* (a) Set up the Lagrangian for the hrms problem of choosing w, Lq , and Lb to maximize expected prohts subject to the constraint that workers expected utility is uo (b) Fmd the hrstorder conditions for w,Lc, and Lb.
(c) Show that a fall in f (or a rise in if < 1) reduces Lb (d) Show that a fall in f (or a rise in if < 1) raises . 10.8 Implicit contracts imder asymmetric information. (Azariadis and Stiglitz, 1983.) Consider the model of Section 10.5. Suppose, however, that only the firm observes A In addition, suppose there are only two possible values of A, Ab and Ac (Ab < Ac), each occurring with probabiUty one half. We can think of the contract as specifying w and L as functions of the firms announcement of the state, and as being subject to the restriction thai it is never in the firms interest to announce a state other than the actual one; formally, the contract must be incentivecompatible. (a) Is the efficient contract under symmetric information derived in Section 10.5 incentivecompatible under asymmetric information? Specifically, if A is Ab, is the firm better off claiming that A is Ac (so that and L are given by Cc and Lc) rather than that it is Ab? And if A is Ac, is the firm better off claiming It is Ag rather than Ac? (b) One can show that the constraint that the firm not prefer to claim that the state is bad when it is good is not binding, but that the constraint that it not prefer to claim that the state is good when it is bad is binding. Set up the Lagrangian for the firms problem of choosing Cc.Cb.Lg, and Lb subject to the constraints that workers expected utility is Uo and that the firm is indifferent about which state to announce when A is Ab  Find the firstorder conditions for Cc,Cb,Lc, and Lb (c) Show that the marginal product and the marginal disutility of labor are equated in the bad statethat is, that AbF(Lb) = V{Lb)/U(Cb). id) Show that there is "overemployment" in the good statethat is, that AcF{Lc)<V(Lc)/U{Cc). (e) Is this model helpful in understanding the high level of average unemployment? Is it helpful in understanding the large size of employment fluctuations? 10.9 Does worker influence on the wage after shocks to labor demand are realized affect the cyclical characteristics of the labor market? (a) (This follows McDonald and Solow, 1981.) Consider a union with the objective function [[/(w)l]Li[/(Wu)(NL), [/(•) > 0, where N is the number of union members, I is the number who are employed, > Ois the disutility of working, w is the wage, and is unemployment compensation. The firms profits are AL"la  wL,A > 0,0 < a < 1. The union sets w after A is known, and the firm then chooses L given w and A. (Assume throughout the problem that the constraint that L cannot exceed N is not binding.) (/) What is the firms choice of L given w and A? (ii) Given its knowledge of how the firm will behave, what Is the unions choice of w given A? Given this, how does L vary with A? (b) Given the unions objective function, what is labor supply under spot marketsthat is, if the union takes w as given and chooses L to
maximize its objective function? How do w and L vary with A under spot markets? (c) Suppose the unions objective function is wL  [crKa + 1)]!*+*, o > 0, instead of the expression in part (a). (i) How do w and L vary with A under spot markets? (ii) Redo part ( )( ) using the modified union objective function. Does assuming that the wage is determined by the union rather than by spot markets affect the elasticities of w and I with respect to A? 10.10 Does worker influence on the wage and employment after shocks to labor demand are realized affect the cyclical characteristics of the labor market? (a) (This is based on McDonald and Solow, 1981.) Consider a union and a firm with the objective functions assumed in part (a) of Problem 10.9. The union chooses w and L given A, subject to the constraint that the firms profits must be at least some level, ttq. (i) Set up the Lagrangian for the unions maximization problem. (ii) Find the firstorder conditions for w and L. (ill) What role does w play in this model? (iv) How does L vary with A? Compare your answer with your finding in part (b) of Problem 10.9 concerning how L varies with A under spot markets. (b) Suppose the unions objective function is given instead by the expression in part (c) of Problem 10.9. Redo parts (i) and (ii) of part (a) using the modified union objective function. Compare how L varies with A with your finding in part (c)( i) of Problem 10.9 concerning how it varies under spot markets. 10.11 The HarrisTodaro modeL (Harris and Todaro, 1970.) Suppose there are two sectors. Jobs in the primary sector pay Wp\ jobs in the secondary sector pay Ws. Each worker decides which sector to be in. All workers who choose the secondary sector obtain a job. But there is a fixed number, Np, of primarysector jobs. These jobs are allocated at random among workers who choose the primary sector. Primarysector workers who do not get a job are unemployed, and receive an unemployment benefit of b. Workers are riskneutral, and there is no disutility of working. Thus the expected utility of a primarysector worker is qwp + (l~q)b, where q is the probability of a primarysector worker getting a job. (a) What is equilibrium unemployment as a function of Wp,Ws,Np,b, and the size of the labor force, N? (b) How does an increase in Np affect unemployment? Explain intuitively why, even though unemployment takes the form of workers waiting for primarysector jobs, increasing the number of these jobs can increase unemployment. (c) What are the effects of an increase in the level of unemployment benefits?
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