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. A consumer with homothetic preferences will have income elasticity equal to 1, for all goods.
2. A firm with increasing returns to scale will have a convex technology set Y.
3. For a price-taking firm with a technology set Y, the netput supply vector will be homogeneous
of degree 0 with respect to commodity prices.
4. In a general equilibrium model with a production sector, Walras’ Law holds if and only if all
markets clear.
5. Consider an excise tax imposed on a normal consumption good of which the supply is
perfectly elastic. The “deadweight loss” measure of efficiency cost will be larger if one uses
Marshallian consumer surplus change than if one uses Hicksian compensating variation, as a
measure of buyer welfare impact.
6. Consider two static 2x2 games, A and B, with complete information and with only equilibria
in mixed strategies, but no equilibria in pure strategies in either game. Assume the only
difference between games A and B is the payoff for the column player in one of the four cells,
while the payoffs for the row player are identical in each cell in both games. Then, both games
have the same mixed-strategy equilibrium.
7. A mixed-strategy Nash equilibrium in a game of complete information can be interpreted as a
pure-strategy Bayesian Nash equilibrium in a closely related game with a little bit of incomplete
information.
8. If there is a unique pure-strategy Nash equilibrium in a static two-player game with complete
information, then the subgame-perfect equilibrium of the equivalent sequential game in which
now one player moves first has to yield the same outcome.
9. Assume two types of workers, A(ble) versus C(hallenged) (type of a specific applicant is
unknown to potential employer): type A is better and has a lower cost a of taking an additional
class than type C with a cost c.
Assume that an employer announces this policy: if an applicant takes more than n classes, she
will be hired as A type and get A money (H), otherwise she will be hired as C type and get C
money (L). Assume 0 < a < c and H > L ≥ 0.
Then there always exists a (potentially non-integer) number of classes n, so that the A types will
take n or more classes, and the C types will take less than n classes.
Part II Section A
Answer ONE of the following two questions.
10. Consider a firm that can produce output y using inputs x1 and x2 using a linear production
function: y = a x1 + b x2.
(a) Describe the isoquants, including marginal rate of technical substitution, and explain the
relationship between the two inputs.
Now suppose the firm’s total expenditure on inputs is NOT linear in usages because the firm is
not a price-taker in the input markets, and the input prices rise as the firm demands more.
Specifically, suppose the two input prices are w1 = s x1 and w2 = t x2, with constants s > 0 and t >
0. Then:
(b) Write an expression for the firm’s total input expenditure and explain how it is non-linear.
(c) Describe the firm’s isocosts. For instance, what will be the isocost for an expenditure level
C, and what can be said about its slope, explained both mathematically and intuitively?
(d) Formulate the firm’s cost minimization problem and state the first-order necessary
conditions.
(e) Is a corner solution possible? Explain.
(f) Use the first-order conditions to give an intuitive “marginal condition” for cost-minimization.
(g) Derive expressions for the firms’ input usages as a function of output level, and for the
firm’s cost function.
11. Suppose an individual has U = w1/2 as a VonNeumann-Morgenstern utility function
describing preferences among wealth levels and wealth gambles.
(a) Is this person risk-averse or risk-loving? Explain.
(b) What is the individual’s preference ranking among the following 5 gambles? Explain, and
give some insight as to why your conclusions are consistent with your answer to part a.
A: 90% chance of w = 900, and 10% chance of w = 400
B: 50% chance of w = 1700 and 50% chance of w = 0
C: 100% chance of w = 850
D: 100% chance of w = 841
E: 100% chance of w = 425.
Now suppose in reality the individual faces gamble A, with the interpretation that current wealth
is w = 900 and there is a 10% chance of a loss of 500 that would reduce wealth to w = 400.
(c) What is the “expected loss”?
(d) What is the maximum amount the person would be willing to pay, for an insurance policy
that would fully protect against the loss? Is that amount larger or smaller than your answer to c,
and why? Explain the difference.
(e) If the insurance policy has a “deductible” amount of 200, so for a loss of 500 it would only
reimburse 300, what will be the individual’s willingness to pay for that policy?
(f) Why might an insurance company be willing to offer to the individual one or both of the
policies described above?
Part II Section B
Answer ONE of the following two questions.
12. Two bidders i = 1, 2 have private values vi that are independently drawn from the uniform
distribution on [0, 1] for a single object.
(i) Assume player i’s utility function is given by ui = vi – ti, if the player receives the object for
payment ti and ui = 0 otherwise.
(a) What are the equilibrium bids in an English auction (also called ascending auction)?
What are the expected revenues?
(b) Solve for the symmetric equilibrium bidding functions bi(vi) in a first-price sealed-bid
auction by obtaining the first-order conditions that optimal bidding must satisfy and
solving the resulting equation (hint: remember what bi(0) is equal to). What are the
expected revenues?
(c) Which auction yields higher expected revenues?
(ii) Now assume
if player i receives the objects (ui = 0 if not). Repeat parts a, b,
and c from above, and provide intuition. Does the Revenue Equivalence Theorem hold?
13. Consider a model with two profit-maximizing firms producing output q1 and q2 of a
homogenous good (total production Q=q1 + q2). Assume demand is linear (p(Q) = 1-Q) and
costs are given by Ci(qi) = ciqi for i = 1,2.
(a) Assume the two firms choose their quantities simultaneously. What are the profitmaximizing quantities? What is the market price in this case? Why is the resulting allocation of
production inefficient?
(b) What happens to total output and efficiency if one of the two firms over time becomes more
efficient in its production (in the sense that its constant marginal cost decreases), while the other
firm stays the same? Note that there are several different cases, depending on which firm
becomes more efficient.
(c) In this part, assume firms choose prices (instead of quantities) and have unlimited capacities
to do so. Since both firms’ products are homogeneous, all customers buy from the cheapest firm.
What is the equilibrium price and quantity in this case (distinguish between non-identical firms
[c1 < c2] and identical firms [c1 = c2])? Is this situation efficient? Why is the situation with
identical firms often called a paradox?
(d) One short-coming of the model in part (c) is that both firms have unlimited capacity. In this
part (d), now assume that each firm faces a (exogenous) capacity constraint ki, which means that
Ci(qi) = ciqi for all qi≤ki and = ∞ for all qi>ki. To keep things simple, assume here both capacities
are the same, k1=k2=k, and both firms have the same marginal cost c = c1 = c2. What is the new
equilibrium (in pure strategies) if firms choose profit-maximizing prices? Careful, the
equilibrium depends on assumptions about the size of k (why?). Also, assume efficient
rationing—if two firms charge two different prices, then the consumers with the highest
valuation purchase the good from the firm that offers the lower price, and the remaining
consumers (if there are any) constitute the residual demand for the firm charging the higher
price.
(e) Finally, assume capacities are endogenous (and do not necessarily have to be the same). For
this part, assume c = c1 = c2 = 0. Also assume that each firm’s capacity constraint is ki, and can
be acquired at a marginal cost of ck ∈ [3/4, 1]. For this situation, prove the statement “the
Cournot equilibrium (from part (a)) can be thought of as the result of a game where firms first
choose capacity and then compete in prices.”