Consider a sales associate who is working for a clothing retailer. For each unit of effort
e, the sales associate sells q = 10e + ? shirts, where ? is a normally distributed random
variable with mean 0 and variance, 2 = 3. The associate’s cost of effort is c(e) = e2
2 ,
her outside option is C = 5.
The clothing firm that has employed the associate is risk-neutral. Suppose that the firm
offers the associate piece rate of the form w = r + a · q, where r is the base salary and a
is a commission rate (0 ? a ? 1).
a) Consider the case of symmetric information, where the firm can observe the associate’s effort.
i) Set up the function E( ¯w) describing the associate’s expected payoff. Here, w¯
denotes the wage net of effort cost.
ii) The sales associate is risk-averse. Her utility can be described by ua( ¯w) =
ew¯ = exp(w¯) with coefficient of absolute risk aversion = 2. This
means that the risk premium R that compensates her for the riskiness in her
payments is given by R =
2 a22. Set up the function C( ¯w) describing the
certainty equivalent of the associate’s payof
Problem Set 3.pdf Methods in Institutional Economics: Problem Set 3 Winter 2019 Substitute Exam for Erasmus Students As announced in the lecture, we grant Erasmus students the option of taking a sub- stitute exam to account for travel timing, lower credit point requirements etc. This problem set is part of this substitute exam (the other part being the term paper). The hand-in deadline is 10:30 on 13th of December. You can either hand in by email (schul-
[email protected]) or personally before the beginning of the tutorial session. There will be no time extensions. Exercise 1 Consider a sales associate who is working for a clothing retailer. For each unit of effort e, the sales associate sells q = 10e+ ✓ shirts, where ✓ is a normally distributed random variable with mean 0 and variance, �2 = 3. The associate’s cost of effort is c(e) = e 2 2 , her outside option is C = 5. The clothing firm that has employed the associate is risk-neutral. Suppose that the firm offers the associate piece rate of the form w = r+ a · q, where r is the base salary and a is a commission rate (0 a 1). a) Consider the case of symmetric information, where the firm can observe the asso- ciate’s effort. i) Set up the function E(w̄) describing the associate’s expected payoff. Here, w̄ denotes the wage net of effort cost. ii) The sales associate is risk-averse. Her utility can be described by ua(w̄) = e��w̄ = exp(��w̄) with coefficient of absolute risk aversion � = 2. This means that the risk premium R that compensates her for the riskiness in her payments is given by R = �2a 2�2. Set up the function C(w̄) describing the certainty equivalent of the associate’s payoff. 1 iii) Set up the function describing the firm’s expected net profit E(⇡̄). iv) Set up the firm’s optimization problem. Which constraint(s) does it contain? Will it/they be binding at optimality? v) Solve the maximization problem to find the optimal effort level e and com- mission rate a. vi) Finally, calculate the optimal base salary r and the principal’s and agent’s expected utility. b) Now consider the case of asymmetric information where the firm cannot observe the associate’s effort. i) Set up the sales associate’s decision problem and solve it. This way you receive the incentive constraint. ii) Set up the firm’s optimization problem. Which constraint(s) does it contain? Will it/they be binding at optimality? iii) Solve the maximization problem to find the optimal commission rate a. Which effort level follows from it? iv) Finally, calculate the optimal base salary r and the principal’s and agent’s expected utility. c) Compare your results in task a and task b. i) When is the firm better off? ii) When is the sales associate better off? iii) What can be said about total welfare? Why? Exercise 2 Suppose that the owner -the principal- of a firm has hired a manager -the agent- to perform a project. The gross profits of the firm depend on the effort that the manager exerts and are given by: ⇡ = e, where e denotes the manager’s effort. The manager’s cost of effort is c = 12e 2 and his reservation utility is ūa = 0. The principal offers the manager a remuneration, w, of the form: w = r + a⇡, where r is a fixed payment (fee) and a is a profit share (0 a 1). The manager’s utility function is: ua = w � 12e 2. The principal’s net profits are given by: ⇡ = ⇡ � w. a) Consider first the case of certain results. The principal cannot directly observe effort that the manager is supplying, but since the manager’s effort produces defi- nite and quantifiable results on the level of profits, the actual level of effort exerted by the manager can be inferred from realized profits ⇡. i) Calculate the equilibrium effort level of the manager. 2 ii) Determine the optimal profit share and fee that the principal offers. iii) Calculate the gross profits, the net profits of the principal and the wage. iv) Briefly interpret what kind of contract arose at optimality. Consider now the case of uncertain results. Profits, ⇡ are determined not only by the manager’s effort, e, but also by a random variable, ✓, that is, ⇡ = e+ ✓, where ✓ is normally distributed with mean 0 and variance, �2 = 1. The principal is risk-neutral. That means he maximizes expected profits. The manager is risk-averse with coefficient of absolute risk aversion, � = 3. This means that maximizing his utility is equivalent to maximizing certainty equivalent C(ua(w̄) = r + ae� 1 2 e2 � � 2 a2�2 = r + ae� 1 2 e2 � 3 2 a2 . b) Consider the case of symmetric information. The principal can directly observe the effort level and base contracts on this. Determine the optimal profit share, the manager’s effort level and the fee. Calculate the equilibrium wage and the expected net profits of the principal. Briefly explain your results. c) Consider the case of asymmetric information. The principal can no longer observe the manager’s effort level directly. Determine the optimal profit share, the man- ager’s effort level and the fee. Calculate the equilibrium expected net profits of the principal. Compare your results with those obtained in tasks a and b, briefly explain. 3