Characterize the optimal compensation plan

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Reference no: EM131206214

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:

1447_Fig.jpg

Assume mo > 0 and σo > 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 Cy - 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 0x 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.

Reference no: EM131206214

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