E604 - Advanced Microeconomic Theory
Final Exam, 150 points.
Davis, Spring 2004
Answer all questions. Please take no more than 6 hours to complete this exam, and use
only your text as a reference. The test should be delivered to my (by mail, by hand or by
email) by 8:00 a.m. Wednesday May 12, 2004. (Note: I increased the timeframe only to
make you more comfortable. You should be able to finish considerably more quickly)
1. (30 points total) Consider two consumers with the following linear demand curves
for Hamburgers
Q1
Q2
=
=
20
30
-
2.5PH +
1.5PH +
.1I1
.05I2
-
2.25PB
1.75PB
PH
Ii
PB
=
=
=
Price of hamburgers
Individual i’s income (in thousands of dollars)
Price of Mugs of Beer
Where
a. Suppose I1 = 30, I2 = 60 and PB = 1. Construct the Market Demand Curve for
hamburgers (Assume that consumers 1 and 2 constitute the entire market for
hamburgers). (6 points)
b. Graph the demand curve. Label this demand curve D1 (6 points)
c. Suppose the price of beer PB increases to $3. Identify the new market demand curve.
Plot this new demand curve on your original graph. Label it D2. Explain the position of
D2 relative to D1. Intuitively, does this change make sense?
d. Suppose that the price of hamburgers is $5 (use PB = $1).
i) Identify the own price elasticity of demand. In what direction would the firm
change price to increase revenues? (6 points)
ii), Identify the cross price elasticity of demand price. What does your answer
suggest about the relationship between beers and hamburgers? Is this reasonable?
(6 points)
e. Suppose that an income change affects both consumers by the same (proportional)
amount, , as would happen, for example with a flat income tax rate increase. With such
an assumption, could you calculate an income tax elasticity? Explain why or why not. (6
points. Note: No elasticity calculations are necessary here)
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2. (24 points total) Consider a person with a current wealth of $50,000 who faces a
20% chance of losing $15,000 (say via automobile theft). Suppose that the utility
index is U(W) = ln(W).
a. Calculate the expected utility for this person over this outcome. (6 points)
b. Suppose this person is offered an insurance policy that would cover the
$15,000 in the event of a loss. The price of the policy is $1,500. Would the
person improve his or her well-being b purchasing the policy? Explain why or
why not. (6 points)
c. What is the maximum amount this individual would pay for such a policy? (6
points)
d. Suppose now a second consumer, identical to the first, but with an initial
income of $100,000. Would this person pay less or more insurance? Why? (6
points)
3. (24 points total) Manufacturing granite countertops requires three inputs, Labor,
(L) Stone Cutting Machines (K) and space (F). Suppose that the production
function for these inputs is the following
Q = 8K¼ L¼ F½
Suppose that F is fixed at 25 (square yards of space)
a. Identify the marginal products of Labor and Capital. Do labor and capital
exhibit a diminishing marginal productivity? (6 points)
b. Develop an expression for the rate of technical substitution. Show that this
function exhibits a diminishing marginal RTS (6 points. For specificity, w.r.t.
labor).
c. Holding F fixed, does this function exhibit constant, increasing or diminishing
returns to scale? Why? (6 points)
d. Develop an expression for , the elasticity of substitution. In intuitive terms,
what does  imply about the shape of the production isoquant? (6 points)
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4. (36 points total) Consider the production function
Q = 3K1/3 L1/3
Suppose that capital can be rented at a price v per hour and Labor at a price w per
hour.
a. Develop a total cost function for the firm (12 points).
b. Graph this relationship when w = 9 and v = 9. (6 points -Hint, let Q run from 0 to about
16 in steps of 1)
c. Suppose now that K is fixed at K*. Develop the short run cost (STC) curve for the
firm. (6 points)
d. On your above graph, with w=9 and v=9, plot the STC for K* =3, 6 and 9. (6 points)
e. Apply the envelope theorem to your short run cost function to show the relationship
between STC and TC (6 points)
5. (36 points total) Profit Maximization. Consider a pizza shop with a demand curve
q
=
60
-
2P
Suppose that the TC function for pizza is TC = 100 + Q2/4
a. Identify the profit maximizing number of pizzas, as well as the profit-maximizing price
and maximum profits. (6 points)
b. Identify analytically the MC curve and the SATC and SAVC curves. (6 points)
c. Identify the shutdown point for the firm, and the breakeven point for the firm. (6
points)
d. Use your P, MR, MC, SATC and SAVC functions to illustrate the profit maximizing
output. (6 points)
e. At the profit maximizing price, what is the price elasticity of demand? (Hint: Use MR 6 points))
f. Analytically, and in the above figure identify the producer surplus at the profit
maximizing price. How do producer surplus and profits differ here? (6 points)
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