Advanced Micro II - Problem set 2 due date: April, 15th Problem 1

Advanced Micro II - Problem set 2
due date: April, 15th
Problem 1 (2p) Consider the hidden information problem as analyzed during
classes. Redo the first and second best calculations for the case of continuum of
types, i.e. θ ∈ [θ, θ] with density f (θ) > 0 and cdf F (θ). As an axample solve
for f uniform on [θ, θ] and with S(q) = log q.
Problem 2 (2p) Find the optimal contract (first and the second best) for: A =
{0, 1}, q ∈ {0, 10}, π10 (0) = 0.1, π10 (1) = 0.9, U (I, a) = ln(w) − a, where
˜ = 0, and principal is risk neutral. Compute the “loss” from unobservable
U
actions. Illustrate solution on a graph.
Problem 3 (2p) Find the optimal contract (first and the second best) for: A =
{0, 1}, q ∈ {0, 10}, π10 (0) = 0.1, π10 (1) = 0.9, U (I, a) = ln(w) − a, where
˜ = 0, and principal is risk neutral. Compute the “loss” from unobservable
U
actions. Illustrate solution on a graph.
Problem 4 (2p) A risk averse agent (entrapreneur) considers a project with
investment I. Agents has no money and must get them from the principal.
Return on investment is random and equal to V¯ with probability π(e) or V
with probability 1 − π(e), where e ∈ {0, 1} is agent’s effort. Assume V¯ > V
and π(1) > π(0). A contract specifies {¯
z , z}, i.e. an amount to give back as
a function of project’s outcome. Agents utility is u(x) − e, where x is his net
income. Outside option is equal to u
˜.
• write the principal’s problem tha wants to implement e = 1.
• characterize the optimal contract.
• now assume principal wants to implement e = 1 but under limited liability,
i.e. such that only nonnegative payments are allowed (in all cases). What
is the highest I, that he would like to lend? Compare with the FB.
Problem 5 (2p) Similarly to a model presented during classes consider
P a PA
model with two actions but risk averse principal, i.e. with utility
i u(qi −
wi )πi (a), for increasing, differentiable and strictly concave u.
Problem 6 (2p) For a continuum of actions prove that if f (q|a) has MLRP
(q|a)
then ffa(q|a)
is increasing with q.