(a) Output only takes on two values, so the wage will also take on two values: z(0) = 0 0 z(0) 0. max s(d)z { d. n { z 1 0 (n + d) 2.

Transcription

1 Steve Pischke/Jin Li Labor Economics II Problem Set Answers. An Agency Problem (a) Output only takes on two values, so the wage will also take on two values: z( ) z 0 z The worker s problem: z(0) 0 0 z(0) 0 max s(d)z d d0 F.O.C. Solution n z 0 (n + d) For z An: d s nz n if z An d 0 otherwise r n s(d) z ³ s(d)z d z s s s, nz + n z n This will be positive when z An, so there is no problem with the participation constraint in this case. Thre is no problem when z n either, since then d s(d) 0. Now turn to the firm s problem. If z n, d 0, s(d) 0, 0for sure. Therefore, as long as A n it is worth setting Az Anto induce at least some eort, so that there is at least some probability of making profit ( z). So the firm s problem is à r! n max ( z) z z F.O.C. r z z z + ( z) n This gives the implicit sharing rule. Note that the two constraints z and z An take care of themselves: provided that An, there exists a solution that satisfies both constraints. Cite as: Steve Pischke and Jin Li, course materials for 4.66 Labor Economics II, Spring 007.

2 (b) Only the firms with n? will produce.. Optimal Linear Contract Consider an environment with one agent and one principal. The agent s reservation utility is K If the agent works for the principal, he chooses an action d at cost F (d) The output is d + % where % Q 0> The principal chooses a linear contract, of the form v + (thus no limited liability). In addition, the principal is risk neutral, while the utility function of the agent is X (v> d) exp ( u (v F (d))) (a) Show that the program for solving the optimal linear contract can be written as max H (( )(d + %) ) > d f μ + u k f where k ln K u Sol: When the principal chooses a linear contract in the form of v + > the agent s utility by choosing action d is H[exp u + (d + %) ] exp( u( + d u%)] () exp( u( + d exp( u ) () exp( u( + d fd )) (3) This implies that the maximization of the agent s problem can be written as max + d fd (4) d Therefore, the FOC of the agent implies that When the agent chooses d f > his utility is d (5) f u exp( u( + )) (6) f exp( u( + ( u )) (7) f Cite as: Steve Pischke and Jin Li, course materials for 4.66 Labor Economics II, Spring 007. on [DD Month YYYY].

3 This leads to the participation constraint that μ + u k f (8) Finally, when the contract is in the form of v + and the agent choosesanactionof d>the principal s utility is H (( )(d + %) ) (9) Therefore, the principal s program can be written as max H (( )(d + %) ) > d ; f μ + u k f (b) Show that the solution to this problem is + uf and uf k f ( + uf ) > and the equilibrium level of eort is d f ( + uf ) Sol: It is clear that the participation constraint must be satisfied with equality, i.e., + μ f u k (0) Also, we know that d f Therefore, we can write the substitute into the maximization prob lem of the principal: max H (( )(d + %) ) > μ max H ( )(d + %)+ μ f u k () () μ ( + u k (3) f 3 Cite as: Steve Pischke and Jin Li, course materials for 4.66 Labor Economics II, Spring 007.

4 This is a maximization problem of single variable, and taking the FOC gives the optimal solution as (4) + uf The expressions of and the equilibrium eort level d canthenbe easily backed out from using the participation constraint and the incentive compatibility constraint. (c) Suppose that there is another signal of the eort } d + where is Q 0> and is independent of %. Now restricting attention to linear contracts of the form v + + } Show that this contract can } be written as v + > where } What s the interpretation of here? Sol: Follow the same procedure as in (a) and (b), we can write the program of the principal as max ( } )d (5) > > } + f ( + } ) u( + } ) d ( + } )@f (6) k (7) You can directly solve for the optimal contract by taking FOCs of the program above (if you like manipulating with the equations in particular). Alternatively, we can define that z > (8) df The incentive compatiblity constraint of the agent then implies that z } df then we can rewrite the program as max ( df)d (9) >d>z + f (df) uf (z +( z ) ) k (0) In this program, z only appears in the participation constraint, so the principal should choose z to minimize z +( z ) 4 Cite as: Steve Pischke and Jin Li, course materials for 4.66 Labor Economics II, Spring 007.

5 This implies that z z () Since z z } > so + } + () } + } + (3) Finally, we see that v + + } } (4) }} +( + } )( + ) + } + } (5) }} +( + } )( + ) + } + } (6) +( + } )( + + }) + (7) + + The variable + } is a su!cient statistic that is the weighted average of the two observables (> }) >where the weights are determined according to the variance(precision) of the observables. In particular, the weights are chosen to minimize the "combined risk" from the two obserables imposed on the agent. In this example, the noises associated with the two observables are independent. You should also try how the argument works when the two signals are correlated. 3. Limited liability A worker can choose h 0 or h The worker s utility is x(z) Nh> where z is the wage, NA 0 and x(z) is concave. The worker cannot be paid less than zero, so z 0; he also has a reservation utility equal to 0 If h 0> the project has a probability of success probability equal to s> in which case it produces a revenue of If it s unsuccessful, it produces a revenue of zero. If h > the project has a probability of success equal to ta s (a) Write the optimization problem of the principal assuming that she wants to implement h 5 Cite as: Steve Pischke and Jin Li, course materials for 4.66 Labor Economics II, Spring 007.

6 Sol: Suppose the agent is paid z v in case of success and z i in case of failure. Then the principal wants to max t( ) ( (8) z v >z i z v t)z i (t s)(x(z v ) x(z i )) N Incentive Compatibility Constraint tx( z v )+( t)x(z i ) N x(0) Participation Constraint z v 0 Limited Liability Constraint z i 0 Limited Liability Constraint (9) (b) Determine the wage contract and indicate the conditions under which theworker needstobepaidan"e!ciency wage." Sol: It is easy to see that the IC constraint in (a) must bind with equality (you should check this using the concavity of x). Therefore, for any level of z i > let z v (z i ) be the wage paid out by the principal in case of success so that the IC condition binds exactly, i.e. z v (z i ) x (x(z i )+ N@(t s)) (30) It is clear that z v is an increasing function. Moreover, we have z v (z i ) A z i Since the principal wants to minimize his payments, we can write the program as Plq z i (3) such that tx(z v (z i ))+( t)x(z i ) N x(0) (3) z i 0 (33) In general, only one of the two inequalities above will bind, so the optimal contract is determined by which constraint binds first. When tx(z v (0)) + ( t)x(0) N x(0)> (34) then the optimal contract has z i 0 and z v z v (0) When tx(z v (0)) + ( t)x(0) N?x(0)> (35) the optimal contract specifies a z i such that and z v z v (z i ) tx(z v (z i ))+( t)x(z i ) N x(0) (36) 6 Cite as: Steve Pischke and Jin Li, course materials for 4.66 Labor Economics II, Spring 007.

7 There is e!ciency wage if and only if tx(z v (0)) + ( t)x(0) NAx(0) (37) so the IR constraint is slack because of the incentive constraint and limited liability constraint. (c) Determine whether the principal prefers h to h 0 Sol: If the principal chooses to implement h 0> then the optimal contract must pay out a wage of zero regardless of the outcome. In this case, the principal s payo is s When the principal chooses to implement h >there are two cases to consider. First, suppose, tx(z v (0)) + ( t)x(0) N x(0) In this case, zi 0 and zv z v (0) The principal prefers to implement h if and only if t tz v (0) As (38) Second, suppose tx(z v (0)) + ( t)x(0) N? x(0)in this case, zi A 0 and satisfy tx(z v (z i )) + ( t)x(z i ) N x(0); z v z v (z i ) The principal prefers to implement h if and only if t tz v (z i ) ( t)z i As (39) 7 Cite as: Steve Pischke and Jin Li, course materials for 4.66 Labor Economics II, Spring 007.