# where is the labour input and K the capital input per unit of time

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```							                                    TRENT UNIVERSITY

DEPARTMENT OF ECONOMICS

ECON 3250H

FINAL EXAMINATION

April 2011

Dr. M. Arvin                                                                2010-2011

Instructions

There are FIVE questions in this examination. You are to attempt THREE questions. If you
do more than three questions, only the first three will be considered. All questions are of
equal value (but not necessarily their parts).

In answering each question you must explain all your steps and derivations carefully.

You are not allowed to consult any notes or books during the examination period.
Department-approved scientific calculators are allowed. Sharing of calculators during the
exam is strictly forbidden. No other electronic devices are allowed. Cell phones should be
left at the front of the exam room. Paper and electronic dictionaries cannot be used as per

number on the front of all your examination booklets.

BOOKLET IDENTIFYING ITS NUMBER AT THE FRONT COVER OF THE
BOOKLET.

There is a time limit of 3 HOURS. Allocate your time between the questions wisely. Good
luck!
1.   Suppose a profit maximizing automobile manufacturer produces its output in two
plants, one in the U.S. and the other in Canada. The total costs of producing in the
two plants are identical, except that the output produced in the U.S. is subject to a per
unit tax, t. Suppose the two total cost functions are TC US  QUS / 2  QUS  1  tQUS
2

and   TC CAN  QCAN / 2  QCAN  1 . The firm’s demand function is P=26-QT ,
2

where QT is total output in U.S. and Canada.

i)     Find the first-order conditions for this problem.

ii)     Find the reduced form solutions for optimum values of QUS and QCAN .

iii)    Show that the second-order condition for this problem is satisfied.

iv)    By partially differentiating your reduced form solutions, describe (both
mathematically and in words) the effect of a tax change (i.e., a change in t) on
optimum output in the U.S. and Canada.

v)     Now find the same comparative static results by totally differentiating your
first-order conditions, rearranging them, writing them in a matrix form, etc.

2.   Consider a firm with a production function

Q(, K )  120K

where  is the labour input and K the capital input per unit of time. The firm has set
aside a budget of \$100 for production per unit of time and faces a wage rate of \$5 for
labour and a rental rate of \$2 for capital.

i)      By setting up a constrained maximization problem and using a Lagrangian,
determine the firm’s optimal output.

ii)     Show that the second-order condition for the problem in i) is satisfied.

iii)    Without performing another optimization problem predict how much the
firm’s optimal output changes by (and in which direction) if the firm had

iv)     Suppose this firm tries to make a different decision. Continue to assume that
it has the same production function as above and that it faces the same factor
input prices as above. The firm now decides that it wants to produce an
output of 30,000 units per unit of time. What is the least cost of producing
this output per unit of time? (Note: You must set up and solve another
constrained optimization problem.)

v)      Show that the second-order condition for the problem in iv) is satisfied.

vi)     Without performing another optimization problem predict the firm’s new
minimum cost if the firm had decided to produce an output of 30,001 units

3.   Consider a consumer who has Cobb-Douglas preferences

U = x1½ x2½           ,
x1 >0 ,    x2 >0

and faces the usual budget constraint

M = P1 x1 + P2 x2 .

Obviously income and prices are exogenously determined, and the consumer wishes
to maximize her utility.

i)      Using a Lagrangian, write the first-order conditions for the consumer’s
problem.

ii)     Prove that the second-order condition is satisfied in this problem.

iii)    Prove that the consumer’s indifference curves in this problem are strictly
convex.

iv)     Prove that the consumer’s utility function in this problem is quasiconcave.

4.   A student wishes to allocate her available study time of 60 hours per week between two
subjects in such a way as to maximize her grade average. Let gi be the expected grade
in subject i as a function of study time in that subject (i = 1,2). Assume

g1 = 20 + 20t1 ½

g2 = − 80 + 3t2 .

Obviously ti (i = 1,2) is the study time for subject i .

i)     By setting up a constrained optimization problem and using a Lagrangian, find
how much study time this student allocates for each subject per week. Also, by
working out her subject grades figure out which subject she is having a tougher
time with. (Note: the maximization problem must reflect the information
exactly as described above; i.e., the student is maximizing her grade average,
etc.)

ii)    Show that the second-order condition for this problem is satisfied.

iii)   Without performing another optimization problem predict this student’s grade
average if she had spent one more hour per week studying. Explain your
5.   Answer all parts of this question. Each part is an independent question.

i)     At the end of 2008, world oil reserves were estimated to be 1950 billion
barrels (see www.theoildrum.com). During 2008, about 29.3 billion barrels of
oil were consumed (see CIA World Factbook, 2008). Over the past decade,
oil consumption has been increasing on average at about 1% per year (see
www.eia.doe.gov/emeu/international/oilconsumption.html). Assuming yearly
oil consumption increases at this rate in the future, how long will it take
before we run out of oil reserves (assuming the 2008 estimates do not
change)? [Hint: Write the 2009, 2010, etc. oil consumption estimates up to a
future year “n”; then determine “n” given that you know how much oil
reserves we have left. Relevance to EC 3250H material? You have to use
logarithms at some point; obviously I cannot tell you where exactly.]

ii)    A company is considering investing \$30 million now in a project that is
expected to yield a profit of \$12 million per year for 4 years. This profit is
earned in a continuous fashion over the course of each year. If the company
discounts future earnings at a rate of 15% and does so in a continuous manner,
determine if the company should undertake this project.

iii)   The daily demand and supply functions for rye flour at a local health food
market are

p  60  q  .004 q 2 ,     0  q  75 ,

and

p  10  .3q  .002 q 2 ,    0  q  75,

respectively. Here q represents hundreds of pounds of flour and p
represents dollars per hundred pounds. Find the consumer’s surplus at market
equilibrium.

iv)    Show that the Euler’s Theorem holds for the production function

Q  51 / 4 K 4 / 5 .

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