Discuss the nature of the optimal compensation contract and

Question

Question 1:

Consider an agency in which the outcome x is not publicly reported and the only public (verified) information that is reported is either y₁ or y = (y₁, y₂). x, y₁ and y₂ are jointly normally distributed, and the two signals have been normalized so that they have unit variance:

Assume m_o > 0 and σ_o > 0, Le., the agent's action influences the expected payoff and the payoff is uncertain. There are five other parameters: m₁, m₂, ∈[0, ∞] and ρ₀₂, ρ₀₂ ∈(+1, -1).

Action a is a choice variable and the set of actions A is a compact and convex set in [0, ∞).

Assume that the agent is strictly risk and effort averse, with

u₁(c,a) = -exp[-r(c - a²)]           c ∈ [0, ∞

Consider the following two settings:
(I) The principal is risk neutral and "owns" x.

(II) The principal is risk neutral and the agent "owns" x.

Required:

Answer the following questions for settings I and II,

(a) For what range of values for m₁ and p₀₁, is reporting y, strictly preferred to no report,

(b) For what range of values for m₁, m₂, p₀₁, p₀₂, and p₁₂ is reporting y = (y₁, y₂) strictly preferred to reporting only y₁.

Provide intuitive explanations of your results.

Question 2:

Consider the following agency problem:

Principal preferences:   risk neutral

Agent preferences:       u1(c,a) = 1/γ C^y - a²       y ∈(0, 1), c ∈ [0, ∞), a ∈ [0, ∞)

Beliefs:                         Φ(x|a) = x/a² exp [-x/a]   (i.e., Gamma of order 2)

Assume that x = y + ε, where y and ε are independently distributed with exponential distributions with mean a. Hence,

Φ(x|a) = 1/a² ₀∫^x exp[-x-y/a]exp[-y/a]dy = x/a²exp[-x/a]

Required:

(a) Derive Φ(x, y|a).

(b) Assume that both x and y are contractible information.

(i) Characterize the optimal compensation plan c(x, y).

(ii) Show either that y is or is not incentive informative given x.

(iii) Give an intuitive explanation of your result.

(c) Assume x is always contractible, but y is only contractible if the principal pays an information cost K. The principal can commit to an "investigation" strategy in which he specifies the values of x for which he will obtain y. Discuss the nature of the optimal compensation contract and investigation strategy for inducing a given action a > 0.