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Dirk Bergemann
Department of Economics
Yale University
Economics 121b: Intermediate Microeconomics
Final Exam – Suggested Solutions
1. Both moral hazard and adverse selection are products of asymmetric in-
formation, where one or more of the parties in the transaction involved
do not possess some relevant information that the other parties do. Moral
hazard refers to hidden actions, while adverse selection refers to hidden
information.
The example discussed in class demonstrating moral hazard was that of
the chef who could produce a high or low quality meal at high or low cost
respectively, but some patrons could not observe the quality until after
purchasing it, giving him incentive to produce the low quality meal. The
example for adverse selection is that of the car dealer who knows the value
of the car to be sold and the buyer who does not; due to uncertainty about
the value of the car, the buyer does not always make a trade, even when
it would benefit him to do so.
Constrainedoptimizationisaproblemwhereapartydesirestominimizeor
maximizesomeobjectivefunction, but must satisfy some constraints while
doing so. A classic example of constrained optimization is maximizing
utility subject to a budget constraint:
max u(x ,x )
x ,x 1 2
1 2
s.t.p x +p x ≤ I.
1 1 2 2
First order conditions mark the points at which the derivative of the objec-
tive function with respect to each control variable is zero. In the example:
∂u(x ,x )
1 2 =0
∂x
1
∂u(x ,x )
1 2 =0.
∂x
2
In an unconstrained optimization problem, these points mark potential
solutions. In a constrained optimization problem, however, the constraint
must be incorporated into the objective function before taking the deriva-
tives. One method for doing so is the method of substitution, which as-
sumes the constraint is binding and substitutes it directly into the objec-
tive function, transforming the problem into unconstrained optimization.
1
The first order conditions in the example become:
I −p x
x (x ) = 1 1
2 1 p
2
∂u(x ,x (x )) ∂u(x ,x (x )) ∂x (x )
1 2 1 + 1 2 1 2 1 =0.
∂x ∂x (x ) ∂x
1 2 1 1
A second method is the Lagrangian method, which adds a penalty term
that incorporates the constraint into the objective function, transform-
ing the problem into unconstrained optimization. The augmented utility
function and the first order conditions in the example become:
U(x ,x ,λ) = u(x ,x )+λ(I −p x −p x )
1 2 1 2 1 1 2 2
∂U(x ,x ,λ)
1 2 =0
∂x
1
∂U(x ,x ,λ)
1 2 =0
∂x
2
∂U(x ,x ,λ)
1 2 =0.
∂λ
2. (a) The profit function of firm 1 is
π (x ,x ) = (a−b(x +x ))x .
1 1 2 1 2 1
The profit function of firm 1 is
π (x ,x ) = (a−b(x +x ))x .
2 1 2 1 2 2
Given the quantity x1 produced by firm 1, firm 2 chooses x2 to max-
imize its profit π (x ,x ):
2 1 2
max (a−b(x +x ))x
x 1 2 2
2
s.t. x≥0.
The best response function of firm 2 to x is
1
a−bx
x (x ) = 1.
2 1 2b
∗ ∗
(b) The strategy profile (x ,x ) is a Nash equilibrium in pure strategies
1 2
∗
if given x , firm 1 has no incentive to deviate to producing at level
2
∗ ∗
other than x . In the same way, given x , firm 2 has no incentive to
1 1
deviate. The following conditions characterizes the equilibrium:
∗ ∗ ∗
π (x ,x ) ≥ π (x ,x ) ∀x ≥0
1 1 2 1 1 2 1
π (x∗,x∗) ≥ π (x∗,x ) ∀x ≥0.
2 1 2 1 1 2 2
2
∗
(c) From the conditions of Nash equilibrium, we learn that x is firm 1’s
1
∗ ∗ ∗
best response to x . And x is also the best response by firm 2 to x .
2 2 1
Using the result from part 2.1, we have
∗
a−bx
∗ 1
x =
2 2b
∗
a−bx
∗ 2
x = .
1 2b
The solution of the equations is
∗ a
x =
2 3b
∗ a
x = .
1 3b
Correspondingly, the profits of firm 1 and 2 are
2
π = a
1 9b
2
π = a .
2 9b
(d) We need to choose x to maximize the sum of firm 1 and firm 2’s
profits,
max (a−b(x+x))x+(a−b(x+x))x
x
s.t. x≥0.
The solution is a
∗
x = 4b.
This level is lower than the equilibrium production level in 2.3. This
is because when each firm increases its production, it pushes down
the price of the good for both firms. In Nash equilibrium, firms do
not take this effect into account and tends to produce more than the
collusion level x∗.
(e) i. Given the production level of firm 1 x , firm 2 best responds by
1
producing
x (x ) = a−bx1
2 1 2b
Since both firms know that the spy exists, firm 1 knows that his
production choice x determines firm 2’s production level and
1
the price as well. In this case, the profit of firm 1 at production
level x1 is
π (x ) = (a−b(x +x (x )))x
1 1 1 2 1 1
a−bx
= (a−b(x + 1))x .
1 2b 1
3
ii. The first order condition with respect to x1 is
a − bx − bx =0,
2 2 1 2 1
which gives a
spy
x = .
1 2b
Substituting this quantity into firm 2’s best response function,
we have a
spy
x = .
2 4b
iii. Firm 1 produces more than in the Nash equilibrium and firm 2
produces less. The profits of firm 1 and firm 2 are
2
π = a
1 8b
2
π = a .
2 16b
In this situation, firm 1 benefits from firm 2’s spy and firm 2
loses. This is because spy enables firm 1 to credibly commit
to a relatively high production level. This is as if firm 1 could
move first. By confirming quantity early, firm 1 can force firm 2
to produce less by producing a relatively large number of units
first.
3. (a) See figure 3a.
(b) See figure 3b.
AsδB increases, the indifference curves shift downwards, as shown in
the diagram.
As p increases, the budget line rotates around the endowment (1,1),
2
becoming steeper.
(c) Setting Bill’s marginal rate of substitution equal to the ratio of prices
yields:
1
xb 1
1 =
δ
B p
xB 2
2
Solving yields
B B
δ x =p x
B 1 2 2
Plugging this into the budget constraint yields
(1+δ )xB =1+p
B 1 2
4
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