Economics 452/551 Assignment 3

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Econ 452, assign 3, 2007

Economics 452/551

Assignment 3

Dr. L. Welling March 19, 2007

Due: 4 pm Thursday, March 22 Marks: 45
1.(18) Suppose an agent has a utility function , where w denotes wage income and is effort expended by the agent. Let the agent’s reservation utility by The principal is risk neutral, and receives a value from production of 1000 if output is high, and 0 is output is low (so ). Let payments to the agent by The relation between output and effort is given in the following table:

Prob of output of













1. (3) What would effort level, wages, and agent’s utility be under full information if the principal had all the bargaining power?


1. A’s reservation utility is 4, so that will be expected utility under full info, given P has all bargaining power.

2. Under full info, effort is observable – A can be paid on the basis of effort.

3. if A chooses high effort, expected output is 0.8x1000=800, at cost 49+4, so total surplus is 747; if A chooses low effort, expected output is 0.1x1000=100, minus A’s reservation utility of 4, so social surplus is 100-16=84.. Since social surplus is highest when high effort is chosen, P will choose high effort, and pay

2. (2) If effort is not observable, what are the incentive compatibility, participation, and zero-profit constraints for high effort?
Ans: If effort is not observable, pay must be conditioned on output:

ICC for high effort:

PC for high effort:

Zero profit constraint:

3. (3) What would utility be if the wage were fixed and could not depend on output or effort?

Ans: if the wage did not depend on effort or output, then A’s EU would be ; to max EU, A would choose e=0. Knowing this, P would pay only for low effort, so to allow A to have reservation utility P would set t=16.

4. (4) What is the optimal contract? What is the agent’s utility?


Assume first that P wants high effort: then the optimal contract is a pair of payments, one for each realized output, which max’s the P’s EU subject to the constraints that A is willing to participate, and to expend high effort. That is, the solution to

Suppose both constraints bind. Then (ii) says . Substituting this into (i) and solving for gives ; then A’s EU had better be 4! P’s EU=.
(If P is willing to accept low effort, then the cheapest way to induce low effort is to offer the first best contract for low effort: Then P’s EU is 84. Since P’s EU is higher with high effort, the first contract will be the optimal one.)
5. (3) What is the “agency cost”, the loss in utility due to asymmetric information? Who bears this cost?


A’s EU is 4 with and without full info. Expected output is also the same with and without full info. The only difference between the two settings is then the expected payment to the agent. Under asymmetric information, the expected payment to the agent is . Since under full info, the payment to A is 121, the cost of asymmetric information is 16, paid by P.

6. (3) Suppose now that the agent has all the bargaining power. How does the agency cost in this case compare to that in (5)? Explain briefly.
Ans: If the agent has all the bargaining power, the agent receives all of the social surplus under full information. Since P max’s social surplus under full info, the agent will choose the same outcome under full info as does P; the only difference will be in the distribution of the surplus – all will now go to A rather than P. AIf A has all the bargaining power because of competition between principals, then there will still be the issue of providing the correct incentives for A once the contract is signed, so the ICC constraint will still be relevant. The other constraint will be the zero profit constraint for P.
2. (17 ) Consider a moral hazard insurance model. Let the consumer's utility of wealth be , let her initial wealth be , and suppose there are two loss levels, and . There are two levels of effort, and . The consumer's disutility of effort is given by the function d(e), where d(0)=0 and d(1)=1/3. Finally, suppose that the loss probabilities are given by the following table:









  1. (2) Verify that the probabilities in the table satisfy the MLRP.


  1. (2) Find the consumer's reservation utility assuming that there is only one insurance company and that the consumer's only other option is to self-insure.

Ans: reservation utility will depend on effort; consumer chooses the effort which gives the higher EU without insurance:

  • if low effort, EU =

  • if high effort, EU =

Since EU is higher with high effort, the consumer will choose to expend high effort, and reservation utility will by 26/3.

  1. (3) What effort level will the consumer exert if no insurance is available?

Ans: high effort; see (2)

  1. (3) Show that if effort is observable, then it is optimal for the insurance company to offer a policy that induces high effort.

Ans: If effort is observable, then insurance company will offer full insurance, based on effort. If the insurance company is a monopoly, the consumer will get only their reservation utility, even wilh full insurance – all of the surplus from the risk sharing will go to the firm.

  • if low effort, firm charges p such that

  • if high effort, firm charges p such that

Firm’s expected profit?

  • if low effort, Eprofit = 36-2x51/3=2

  • if high effort, Eprofit = 19-51/3=2

Since offering a policy which induces low effort does not lead to higher profits, it is optimal for the company to offer a policy which induces high effort.
Note: this approach assumed that if the individual did not buy full insurance for low effort, they would choose low effort with no insurance . If instead they chose high effort, the premium which utility from full insurance with low effort to the reservation utility would yield negative expected profits for the monopoly insurer.

Hence inducing high effort would be strictly preferred by the firm.

  1. (2) Show that the policy in (d) will not induce high effort if effort is not observable.

Ans: If the firm offers full insurance, at whatever price, when effort is not observable the consumer max’s EU by minimizing the cost of effort – so supplying zero effort.

  1. (3) Find the optimal policy under asymmetric information.

Ans: optimal policy is an amount of insurance, I, and a per unit price p, which max’s expected profit. If the policy induces high effort, the firm’s expected profit is = ; high effort will be induced if the policy satisfies

i) ICC:

ii) IR:

If both these constraints are satisfied as equalities, the simplify to



Together these imply that (I,p) = : less than full insurance, at a premium greater than the probability of loss for high effort.

  1. (2) Compare the company's profits and the consumer's utility when effort is observable, and when it is not.

Ans: in both cases the consumer gets their reservation utility, so any cost of the asymmetric information is borne by the firm. When effort is observable, expected profits =2; when effort is not observable, expected profits

= pI-(1-p)I =

3. (15) Workers in many occupations face compensation packages that have two components: a fixed salary, B, and a piece rate b, for each unit of output produced or sold. Total income, y, is then given by y=B+bq, where q is output. Suppose that output is produced from effort according to the production function q=e . The employer receives a payoff of S(q)=2q from output. The worker has a cost of effort function , and a reservation utility of zero. Both employer and employee are risk neutral, with payoffs and , and the employer has all the bargaining power.
a) (5) Compute the first best level of output, the compensation package, and the payoffs of the worker and the employer. Explain why your calculations are correct, and illustrate in a diagram.
Ans: In first best, if P has all bargaining power, A’s utility = reservation utility = 0. Thus y=, for whatever e P chooses. P chooses e to maximize:

, First order conditions for P’s problem yield e=q=2; thus y=2, and P has utility of 4-2=2.

The compensation package here? Any number of packages would work; some examples:


ii) ; in this case the employee chooses e to maximize

, so e=2.
Suppose now that the production function has a random component, so q=e+r, where r is uniformly distributed on the interval [-0.5, 0.5] .

Diagram: in e-y space. Worker has indifference curves given by , for some constant k, so slope is is slope of IC at any given (e,y) pair. Thus, worker’s IC has positive slope, slope increases as effort increases, and higher utility corresponds to IC’s above and to the left of any given point. Reservation utility is zero, so minimum feasible combinations are on the IC for k=0.

Principal has indifference curves given by , for some constant m, so slope of P’s IC is . Higher utility for P corresponds to points below and to the right of any given point in (e,y) space.
Optimal contract will have A on reservation utility curve. One such contract is the point of tangency between this IC for A and an IC for P; this has (e,y) = (2, 2).

b) (5) Suppose now that effort is unobservable. Derive the level of output, and the compensation package chosen by the employer, and the payoffs of the worker and the employer. Explain why your calculations are correct, and illustrate in a diagram.


First, consider worker’s problem: choose e to max (here, E(q(e) denotes the expected value of q, given e). if the worker accepts the contract, the optimal solution for the worker is to choose : this is the incentive compatibility constraint the principal must face. The worker will accept the contract so long as . As the principal has all the bargaining power, this constraint will bind in equilibrium, and the worker will obtain EU=0; this means that .

Now consider the principal’s problem: choose {B,b} to maximize

Substituting the constraints into the objective function, and using the optimal choices for e and B above, leaves the principal maximizing solving this for b yields b=2, thus B = -2. This is one of the solutions above.

Diagram? As explained above.
Notice that this contract has the properties of a franchise contract: an upfront lump sum paid by A to the employer, and A’s income depends on A’s effort. In this context, another interpretation of the lump sum payment is as a minimum amount of output/sales which much be achieved before the worker starts to earn a commission on sales.

c) (5) Suppose now that effort is unobservable, as in (b), but the worker is

risk averse. Using no more than one half page (single spaced), explain how this change in risk aversion would affect the compensation package and the optimal output. (NB: you do not need to calculate the optimal output.)

Ans: In the contract above A bears all the risk of the venture. If A is risk averse, the variability in y generated by this risk bearing will lower A’s EU, and hence lower A’s choice of e for this contract. In order to induce higher effort on the part of A, P needs to transfer some of the surplus to A, in the form of reduced variability – so the proportion of A’s income accounted for by the lump-sum component rises. This reduces the surplus available to P, and reduces the optimal effort induced in the optimal contract.

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