University of Hong Kong
ECON6021 Microeconomic Analysis
Homework assignment 1
Due date: October 11, 2003
Answer all questions in this problem set. While discussion with your fellow classmates is
encouraged (it is a very effective way of learning!), you have to write up your own script.
Plagiarism is a serious offence. For the part on consumer choice problems, in case the other good is
not explicitly specified, you can call it as “all other goods” (AOG), and treat it as if a single good
with a unit price equal to one. For the part on production and cost, if you find it necessary, you can
check out definitions of short run, long run, average cost, marginal cost, etc. from Chapter 10 in
Frank.
I hope you all enjoy this homework assignment!
Math
1. “The upper contour sets of a concave utility function are always convex.” Please explain.
2. Which of the following are positive monotonic transformations? (a) v=2u – 13; (b) v = 1/u2; (c) v = 1/u2; (d) v= ln u; (5) v = - e-u; (6) v = u2
Consumer Choice Problems:
1. "I like beer, but I cannot tell the difference between Brand X and Brand Y. A case of 24
bottles is all I want in a week. I would not drink any more even if it were free." Show
graphically and explain all you can about my tastes in the form of:
a. Indifference curves on axes depicting Brand X and Brand Y beers.
b. Indifference curves on axes depicting beer and all other goods.
c. Indifference curves on axes depicting beer and some good that is often consumed
along with beer (e.g., pretzels).
d. Discuss each diagram in terms of the unit amounts on the axes and the MRS.
2. In a two-good world a consumer has a utility function U = X Y. Suppose his income is $800
and PX = $4 and PY = $2. Derive the consumer's demand curve for each good and find his
purchase of X and Y.
3. Suppose the local telephone company charges 10 cents for each of the first 40 calls per
month from a home phone and 5 cents for each additional call. Alternatively, it could he
compelled to charge 7 cents for each call (whether less or more than 40). Suppose N is the
number of calls at which consumers spend the same amount on both pricing systems.
Calculate N and then use indifference curves and budget lines to analyze the following
questions: Which price system will make "a customer" better off? Under what condition or
conditions will the consumer be indifferent between the two systems?
4. Consider the following scheme devised by the parent of an overweight child to reduce the
child's consumption of candy bars. The parent offers to sell candy bars to the child at their
market price P plus a parental tax t. To induce the child to cooperate, the parent offers to
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increase the child's allowance by S dollars. If the child accepts the deal, he/she agrees not
to buy candy from anyone else.
e. Draw a diagram in which you identify the minimum value of S which will induce
the child to cooperate. Explain.
f. Assuming the parent does increase the child's allowance by that amount and the
child accepts the offer, will the child's consumption of candy bars increase or
decrease? Explain.
g. Will the amount of the tax the parent collects from the child be larger or smaller
than the increase in the child's allowance? Explain.
5. Alice is given two choice problems. In problem 1, she needs to choose between two lotteries:
A and B. Lottery A gives a 0.33 chance of $27,500, a 0.66 chance of $24,000, and a 0.01
chance of nothing. Lottery B gives $24,000 for sure. In problem 2, she needs to choose
between two lotteries: C and D. Lottery C gives a 0.33 chance of $27,500 and 0.67 chance
of nothing. Lottery D gives 0.34 chance of $24,000 and a 0.66 chance of nothing. It turns
out that Alice chooses B over A in the first problem and C over D in the second problem.
Explain why Alice’s choices are inconsistent with the von-Neumann-Morgenstern
expected utility theory.
Production Problems
1. (Aggregate Production Function) Consider an industry in which all firms have the same
constant returns to scale technology: Q = KcLd where K and L are capital and labour
employed in that industry. Suppose all firms are maximizing profits and no firm has market
power. Let K = 200 and L = 400. There are 20 firms in the industry. Eighteen of them
altogether hire 160 units of K and 320 units of L. The nineteenth firm hires 25 units of K.
a. Calculate the amount of L employed by the nineteenth firm. Calculate the amount
of K and L employed by the twentieth firm.
b. Suppose the rental rate of K is v = $20/unit while the wage rate of L is w = $10/unit.
Calculate c and d.
c. What are the total rentals and total wages paid out by the industry? How are these
values related to the coefficients c and d?
d. What is the total output in the industry?
2. Suppose that a firm's fixed proportion production function is given by q = min(5K, 10 L),
and that the rental rates for capital and labour are given by v = 1, w = 3.
a. Calculate the firm's long-run total, average, and marginal cost curves.
b. Suppose that K is fixed at 10 in the short run. Calculate the firm's short-run total,
average, and marginal cost curves. What is the marginal cost of the 10th unit? The
50th unit? The 100th unit?
3. Professor Smith and Professor Jones are going to produce a new introductory textbook. As
true scientists, they have laid out the production function for the book as Q = S 0.5 J 0.5,
where Q = the number of pages in the finished book, S = the number of working hours
spent by Smith, and J the number of hours spent working by Jones. Smith values his labour
as $3 per working hour. He has spent 900 hours preparing the first draft. Jones, whose
labour is valued at $12 per working hour, will revise Smith's draft to complete the book.
a. How many hours will Jones have to spend to produce a finished book of 150 pages?
Of 300 pages? Of 450 pages?
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b. What is the marginal cost of the 150th page of the finished book? Of the 300th
page? Of the 450th page?
4. Suppose that a firm's production function is given by the Cobb-Douglas function Q=L c L d,
(where c and d > 0), and that the firm can purchase all the K and L it wants in competitive
markets at rental rates of r and w, respectively.
a. Show that cost minimization requires rK/c = wL/d.
b. Assume cost minimization, show total costs can be expressed as a function of q, r,
and w of the form?
TC = BQ 1 /( c + d ) w d /( c + d ) r c /( c + d ) ,
where B is a constant depending on c and d.
c. Show that if c + d = 1, TC is proportional to q.
d. Calculate the firm’s marginal cost curve.
5. A college student who is cramming for final exams has only six hours of study-time
remaining. Her goal is to get as high an average grade as possible in three subjects:
economics, mathematics, and statistics. She must decide how to allocate her time
among the subjects. According to the best estimates, her grade in each subject will
depend upon the time allocated to it according to the following schedule. How
should the student allocate her time? How did you get the answer? (Be sure to use
an economic approach)
Economics
mathematics
Hours of
study
grade
0
1
20
45
Hours of
study
0
1
Statistics
grade
40
52
Hours of
study
0
1
grade
80
2
65
2
62
2
90
95
3
75
3
71
3
97
4
83
4
78
4
98
5
6
90
92
5
6
83
86
5
6
99
99
End of Homework Assignment
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