Final Exam: Solution
Date: June 11, 2009
Subject: Advanced Microeconomics I (ECO600E) Professor: Yosuke YASUDA
1. True or False (15 points)
Answer whether each of the following statements is true (T) or false (F). You do NOT need to explain the reason.
(a) A binary relation % is said to be a preference relation if it is “complete” and
“monotone.”
(b) Hicksian demand of the good is always weakly decreasing in its own price. (c) The vNM (von Neumann-Morgenstern) utility function is de…ned over “lotter-
ies,” not over “prizes.” Answer:
(a) F (b) T (c) F
2. Consumer Theory (25 points)
A consumer gets utility from 2 sources: drinking (measured in liters x) and time spent on the phone (measured in hours y). Each liter of drink costs $4 and each hour on the phone costs $4. She has a total of $120 available for spending. Her utility function is given by:
u(x; y) = xy (a) Solve the utility maximization problem.
Answer:
Since the utility function has Cobb-Douglas form (with equal weight to each good), the expenditure of one good is identical to that of the other. Therefore, this consumer spends $60 each, which concludes
x= y = 60 15 = 4.
(b) The health authorities are putting up a program to cut down alcohol consump- tion. They propose a quota that allows to consume a maximum of 8 liters. What are the optimal choices under this new scenario?
Answer:
Clearly, it is optimal for her to consume 8 liters of alcohol. Therefore, x= 8; y = 120 4 8
4 = 22.
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(c) Suppose the price of alcohol is reduced from $4 to $3. Then, what are her optimal choices if the quota is not imposed?
Answer:
Since the optimal expenditure is $60 each, x= 60
3 = 20; y = 60
4 = 15.
(d) What are the optimal choices when the price of alcohol is $3 but the quota (of 8 liters) is imposed?
Answer:
As in (b), she will consume 8 liters of alcohol, so x= 8; y = 120 3 8
4 = 24.
(e) Find the amount of quota q that would make the consumer indi¤erent between the scenario (a) (no quota, price is $4) and (d) (quota of q, price is $3). Answer:
She will consume alcohol at the same amount of quota as long as q 15. 152 = q120 3q
4 ) q = x = 10; y = 120 43 10 = 22:5. 3. Producer Theory (20 points)
A …rm’s production function is given by:
f(x1; x2) = pmin(x1; x2)
Let w1; w2 > 0 be the input prices for good x1 and x2 respectively. Then, answer the following questions.
(a) Set up the cost minimization problem. Answer:
x1min;x2 0w1x1+ w2x2 s.t. pmin(x1; x2) y
(b) Solve this cost minimization problem you describe in (a), and derive the cost function, c(w1; w2; y).
Answer:
From the form of cost function, it is clear that conditional input demand becomes
x1 = x2 = y 2.
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Then, the cost function is written as
w1y2+ w2y2 = (w1+ w2)y2.
(c) Let p be the output price. Set up the pro…t maximization problem using your answer in (b).
Answer:
maxy 0 py (w1 + w2)y 2
(d) Solve the pro…t maximization problem in (c), and derive the pro…t function, (p; w1; w2).
Answer:
The …rst order condition becomes d
dy = p 2(w1+ w2)y = 0 ) y = 2(w p
1+ w2)
; = p
2
4(w1+ w2). 4. Uncertainty (10 points)
Suppose that an individual can either exert e¤ort or not. The cost of e¤ort is c. Her initial wealth is 100. Her probability of facing a loss 75 (that is, her wealth becomes 25) is 13 if she exerts e¤orts and 23 if she does not. Her wealth will not change with the rest of probability in each scenario. Let u(x) be her vNM utility function.
(a) Express her expected utilities in each scenario, i.e., exerting e¤ort or not, by using u(x). You can assume that her expected utility is additively separable between e¤ort cost and (probabilistic) monetary outcome, i.e., E[u(x)] c. Answer:
Let UE, UN be the expected utility when exerting e¤ort and not, respectively. UE = 1
3u(25) + 2
3u(100) c UN = 2
3u(25) + 1
3u(100)
(b) Assume u(x) = px. For what values of c will she exert e¤ort? Answer:
UE UN = 1
3fu(100) u(25)g c
= 1 3f
p100 p25g c= 5
3 c
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Therefore, the individual will exert e¤ort (that is, UE UN) if c 53.
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