C3016, Economics of Information. Coursework 2
Deadline: To be handed by Thursday 26th October
Problem 1 (25% of the mark of this coursework)
Show using the standard Principal Agent model studied in the lectures that if both the
principal and the agent are risk averse, and if effort is verifiable, then the optimal
rp
dw

remuneration scheme for a given level of effort should verify that
where
dx rp  ra
rp is the principal’s Pratt risk aversion measure and ra is the agent’s Pratt risk
aversion measure (Hint: obtain the first order condition for xi and differentiate it with
respect to xi as if xi was continuous).
Interpret this condition in term of the risk that both principal and agent bear.
Compute the optimal remuneration scheme for a given level of effort if both
principal’s and agent’s utility function exhibit constant absolute risk aversion (that
this, B( x  w)  q *exp(rp ( x  w)), U ( w)   g *exp(ra w) ). What would be a
typical name for this remuneration scheme?
Problem 2 (25% of the mark of this coursework)
The following describes a principal-agent problem between a shop owner (principal)
and shop assistant (agent). For the agent, U = w^0.5 - e and reservation utility is 4.
Effort can be low or high, and sales can be {low, medium and high}. The three
possible levels of sales are equally probable. The agent’s cost of high effort is 1 while
the cost of low effort is 0. The principal is risk neutral and wants to maximize his
expected revenue borne of the cost of contracting the agent. The following table
presents the revenues for the principal under each contingency:
Effort / State of
nature
Low
High
Low sales
Medium sales
High sales
50
50
100
500
500
500
Moreover, there is a minimum wage rule, and in consequence all wages must be at
least 9.
Assume that the Principal wishes the Agent to exert high effort. Write the problem
that the Principal solves. Find the optimal remuneration schedule if there is no
asymmetric information (do not forget to prove mathematically that the
participation constraint is binding). What is the Agent’s utility?
Assume that the Principal wishes the Agent to exert low effort. Find the optimal
remuneration schedule if there is no asymmetric information (do not forget to
prove mathematically that the participation constraint is binding). What is the
Agent’s utility? Should the Principal induce low or high effort? Write the optimal
contract.
Problem 3 (25% of the mark of this coursework)
An employee can choose to exert either high or low effort. If the effort exerted is
high, the probability that the production is defective is 0.25. This probability will be
10
0.75 if effort exerted is low. The employee utility function is U ( w, e)  100   v ,
w
where w is his remuneration and v=2 is effort exerted is high and v=0 if effort exerted
is low. The production is worth £20 if it is perfect, but it is worth £0 if it is defective.
The employer (principal) is risk neutral. The employee’s reservation utility is 0.
Assume that effort is verifiable. Write the problem that the principal solves for each
level of effort. Prove that the participation constraint is binding. Give the intuition
behind this result.
Compute the optimal remuneration scheme for each level of effort. What is the
optimal contract?
Problem 4 (25% of the mark of this coursework)
Consider a special case of the model analysed in the lectures in which there are only
three possible results: high (H), medium (M), or low (L). Denote by pH(e) the
probability of obtaining H when effort exerted is e. Denote by pM(e) the probability
of obtaining M when effort exerted is e. Consequently, when effort exerted is e, the
probability of obtaining L is 1- pH(e)- pM(e). The principal’s utility function is B(x-w)
and the agent’s utility function is U(w)-v(e). The agent’s reservation utility is 0.
Assume that effort is verifiable.
Write the problem that the principal solves for each level of effort. Prove that the
participation constraint is binding. Give the intuition behind this result.
Write which conditions will satisfy the optimal remuneration scheme for a given level
of effort. Which of these conditions imply that the solution is Pareto Efficient?
Describe the risk sharing properties of the optimal remuneration scheme for a given
level of effort for the following cases: (1) Principal is risk neutral and agent is risk
averse, (2) Agent is risk neutral and principal is risk averse.