Boğaziçi University
EC 306 Problem Set 3
1. Let the utility of a worker whose base productivity is θ0 ∈ {1, 17
3 } over wages w and education e be given
by
u(w, e|θ0 ) = w −
e2
θ0
Suppose that the opportunity cost of employment is r(θ) = 0 for both types and let P {θ0 = 17
3 } = 3/4.
Moreover, assume that education determines final productivity of a worker as in the following function.
θ = (1 +
√
e)θ0
Consider the screening game in which Nature first selects θ0 and the worker who observes her own type.
Then two firms simultaneously make contract menu (wage-education) offers to the worker who accepts
one of the offers.
(a) Find the separating PBE?
2. Consider a principle-agent problem. Principle is a risk neutral profit maximizer, agent has a utility
function in the following form:
u(w, a) =
√
w−a
where w is wage received at the end of the period and a is the effort level. Once the agent is hired, she can
choose high or low effort, aL = 0, aH = 5. Agent has reservation utility as her opportunity cost, u = 10.
Revenue of the firm, π ∈ {200, 2000}, is a random variable which has a distribution depends on effort
level, f (π|a), such that f (π = 2000|aL ) = 31 and f (π = 2000|aH ) = α.
(a) Find the optimal contract to implement low effort.
(b) Find the minimum value of α to design an optimal contract implementing high effort.
(c) Find the optimal contract implementing high effort if α = 53 .
3. Consider an environment with two agents and 3 alternatives, i.e. I = 2 and X = {x, y, z}. Both agents
have two types, Θ1 = {θ1 , θ10 }, Θ2 = {θ2 , θ20 }. Let their preferences be:
1 (θ10 )
a-b
c
Consider the social choice function
1 (θ1 )
a
c
b
(
b
f (θ) =
a
2 (θ20 )
a-b
c
2 (θ2 )
a
b
c
if θ = (θ10 , θ20 )
otherwise.
(a) Is f (θ) ex-post efficient?
(b) Is f (θ) strategy-proof?
(c) Consider a direct mechanism that truthfully implements f (θ). Show that truth telling is not the
unique weakly dominant strategy for agents.
(d) For the direct mechanism considered in part (c), show that if an agent selects untruthful weakly
dominant strategy, then f (θ) is not implemented.
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4. Consider an auction setup with three potential buyers with individual valuations (θ1 , θ2 , θ3 ). Assume that
for each i, θi is independently and uniformly distributed over [0, 1] and observable by bidder i only.
(a) Write down necessary and sufficient conditions for ex-post efficiency.
Consider the following social choice function f (θ) = (y1 (θ), y2 (θ), y3 (θ), t1 (θ), t1 (θ), t1 (θ)), in which
(
1 if θi > max{θj }
yi (θ) =
0 otherwise.
2
ti (θ) = − θi yi (θ)
3
Item is sold to the potential buyer with the highest valuation and only that buyer pays
2
3
of his valuation.
(b) Show that Revenue Equivalence Theorem holds for the direct mechanism implementing f (θ) and the
second price auction (show that necessary conditions for Revenue Equivalence Theorem are satisfied).
(c) Verify that Revenue equivalence theorem holds for the direct mechanism implementing the social
choice function above and the second price auction (show that expected revenues are equal).
(d) Is f (θ) ex-post efficient? If your answer is yes, then prove. Otherwise what kind of trick you would
use to generate an ex-post efficient social choice function.
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