Instructions
Answer ALL questions in Part I.
Choose and answer only one question in Part II.A. and only one question in Part II.B.
All answers should be graduate-level. Clearly explain and show all your work whether or not the
question formally asks for explanation and/or asks you to indicate the steps in your analysis.
As microeconomics faculty may not be available during the exam, please make sure to state
clearly any assumptions that you find absolutely necessary to answer a question. In general, you
should aim to minimize making additional assumptions.
PART I.
Respond to each of the nine statements in this part by stating whether it is
TRUE, FALSE, or UNCERTAIN, and provide a justification for your
answer.
1. If a consumer’s preferences display Diminishing Marginal Rate of Substitution, then any
utility function which represents those preferences must be concave.
2. For goods x1 and x2, if x2 is a net complement of x1 and a gross substitute for x1, then x2
must be a Giffen good.
3. For a long-run equilibrium in a competitive industry to exist, all firms in the industry
must have constant returns to scale technology.
4. For a competitive firm using a decreasing-returns-to-scale technology, the conditional
factor demand curve for input i, xi(w, y), is always downward-sloping with respect to its
own price. (Note that w is a vector of input prices and y is a given level of output.)
5. Consider an economy with 2 households (A and B) and 2 commodities (x1 and x2).
Assume that household A has a utility function of the form uA(x1A,x2A) = min(0.75x1A,x2A)
and household B has a utility function of the form uB(x1B,x2B) = min(0.25x1B,x2B).
Assume that individual endowments (e(x1,x2)) are eA = (1,α) and eB = (1,1-α).
Unless α=0.5, there is no equilibrium with strictly positive prices.
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6. Consider a Walrasian exchange economy with 2 agents (A and B) and 4 commodities
(x1, x2,x3,x4) . Suppose the two agents have the following utility functions:
UA(x1A, x2A, x3A, x4A) = log(x1A) + log(x2A) + log(x3A) + log(x4A)
UB(x1B, x2B, x3B, x4B) = – 1/x1B – 1/x2B – 1/x3B – 1/x4B
The First Theorem of Welfare Economics can be used to show that whenever
endowments are such that
eA + eB = (
), all equilibrium allocations must satisfy xlh= xl’h for h = (A,B) and
all l, l’ = 1, 2, 3, 4.
7. Suppose student/workers are of two types (i ∈ {L, H}), and that there are at least two
potential employers. The incremental profits earned from hiring a worker of type i is πi,
with πH > πL. Suppose that events unfold as follows:
(1) Students observe their types and employers do not observe those types;
(2) Students choose an education level, e, which is costly and observed by employers, but
is not productive;
(3) Employers offer prospective workers (currently students) a wage, w, where the offer
is contingent on observed education (Bertrand competition);
(4) Students accept an employer’s offer and become workers.
Assume that the utility for a student/worker of type i receiving a wage w is given by
U(w,e) = w – ci(e). Assume: cH(e) = e2, and cL(e) = e.
The payoff for an employer of a type i student/worker paying a wage w is (πi – w).
There are separating equilibria if πH - πL<1, and no separating equilibria if πH - πL>1.
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8. Consider an infinitely repeated version of the 3×3 game illustrated below.
Assume that both players have the same discount factor. The lowest discount factor, δ*,
for which it is possible to sustain the symmetric cooperative outcome, (U, L), using Nash
reversion is 1/7.
9. In the game below, (M,A, μ=1) and (L,B, μ>1/4) are pure strategy Perfect Bayesian
Equilibria (PBE), and there are no mixed strategy PBE.
R
1
1,1
L
M
[μ]
A
3,1
2
B
[1- μ]
B
A
3,4
5,2
0,1
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Part II Section A
Answer ONE of the following two questions
10. Suppose Roberto’s preferences concerning consumption and labor are represented by the
utility function U(X,L) = 2ln(X) + ln(120 - L) where X is consumption in each period, L
is labor supplied in each period, and 0 ≤ L ≤ 120.
I. a. Assuming Roberto’s wage is w and his non-wage income is N, represent
mathematically Roberto’s optimization problem. Also provide a graph summarizing
the problem.
b. Derive Roberto’s consumption demand function X*(w, N) and labor supply function
L*(w, N). Be sure to examine the possible cases of an interior solution (L* > 0) and a
corner solution (L* = 0).
c. Discuss the effects of w and N on the optimizing amounts of X and L.
d. Assuming that w = 1, identify the values of X* and L*, when N = 30, N = 45 and N =
60.
II. Now suppose that N is uncertain in each period, with a 50% chance of N = 30, and
a 50% chance of N = 60. The expected value of N is 45. Assume the utility function
U(X,L) serves as a Von Neumann-Morgenstern utility function for the purposes of
expected utility calculation. Suppose Roberto must commit to the quantity L before
knowing the value of N. Once N is known, the amount of X is determined by the budget
equation.
a. Assuming w = 1, reformulate Roberto’s optimization problem for this case.
(Suggestion: instead of forming a Lagrangean function, use the constraint to
substitute for X in the objective function, making it an unconstrained problem in one
variable, L.)
b. State the first-order necessary condition for L* in the uncertainty case. Would it be
optimal for Roberto simply to optimize as if N has its expected value, N = 45?
Explain.
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11. Consider a 2-commodity production world with hay (H) and manure (M) as the
commodities. Consider two possible cases:
(i)
Hay can be used to produce manure according to the production function
M = H/2, and
(ii)
manure can be used to produce hay according to the production function
H = ln (M+1),
a. For each case, derive a function for the marginal product of the input.
b. How would you characterize each production process regarding returns to scale?
c. Present this production situation in a “netput” or “net output” model, by constructing a
carefully detailed graph in R2 showing the comprehensive production possibilities set
(i.e., technology set) where input quantities are modeled as negative output quantities.
Label your graph carefully and identify some representative points that help locate the set
and its boundaries.
d.
Explain whether and why the technology set satisfies the following properties:
(i) Convexity, (ii) Divisibility, (iii) Additivity, and (iv) Irreversability
e. Assuming profit-maximization, derive the producer’s “netput supply” curve for hay, as a
function of the price ratio PH/PM. Carefully identify critical threshold values and other
helpful points on the curve. Fully explain your derivation.
f. In the same way, derive the producer’s “netput supply” curve for manure, as a function of
the price ratio PM/PH..
g. Discuss the shapes of the supply curves. Do they have the expected characteristics,
relative to generalized theoretical principles and your answers in part b.? Explain.
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Part II Section B
Answer ONE of the following two questions
12. Consider an asymmetric all-pay auction between two bidders (i ∈ {1,2}), whose values
are common knowledge. The good is worth v1 to Bidder 1 and v2 to Bidder 2 where
v2 > v1. All bidders simultaneously submit sealed-bids bi which must be non-negative.
The highest bidder wins the good and all bidders pay their bids. In the event of a tie, the
good is given to Bidder 2.
a. What bidding strategies are strictly dominated? Can you iteratively eliminate any
more strictly dominated strategies?
b. Are there any pure-strategy Nash equilibria (PSNE)? If so, characterize the set of
PSNE. If not, explain why not.
c. We now consider mixed-strategy Nash equilibrium (MSNE) where both bidders
randomize. At a MSNE, would Bidder 1 ever bid strictly higher than v with positive
probability? Would Bidder 2 ever bid strictly higher than v with positive probability?
d. Consider a MSNE where Bidder i randomly selects a bid based on the cumulative
distribution function Fi. You can assume that the support of Fi is [0, v1] and that Fi is
continuous. Write an expression for 2’s payoff from bidding x in [0, v1], fixing F1,
and an expression for 1’s payoff from bidding x in [0, v1], fixing F2.
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13. Three roommates (1, 2, and 3) are considering redecorating their shared house. They
must decide which color to paint the walls. The paint store is having a sale on four
colors: azure, hazel, indigo and mauve. The sale price is $15.30 for enough paint to
complete the job. The three roommates have a gross willingness to pay for the four
colors as follows:
1
2
3
azure hazel indigo mauve
$10
$9
-$1
$8
-$4
$9
$7
$5
$4
-$3
$5
$3
do nothing
0.05
0.05
0.05
a. If the roommates used a demand-revealing process, which alternative would they
select?
b. Suppose the roommates use approval voting, all vote sincerely, and all agree that the
painting cost should be divided equally among them. Which outcome will be
selected?
c. Suppose that the roommates have decided to use pair-wise majority rule voting, and
have elected person 2 to construct an agenda, subject to the requirement that the last
vote should be the winner from the preceding votes versus the status quo (do
nothing). Continue to assume that the painting cost (if incurred) will be divided
equally among them. Which outcome will win, assuming that person 2 constructs the
agenda that is best for him?
d. Suppose that the group has used some process to select hazel. Before the meeting of
the roommates adjourns, however, one player makes side offers to the others which
cause them to agree unanimously to change their decision to another outcome. What
is that outcome, what were the offers, and who made them?
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