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

							                                    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.
Unsupported answers will receive no credit.


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
your course outline.


Please make sure your answers are neat and legible and that you have your name and student
number on the front of all your examination booklets.


PLEASE WRITE THE ANSWER TO EACH QUESTION IN A SEPARATE EXAM
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
             decided to set aside $101 instead for production. Explain your answer.

     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
             per unit of time instead. Explain your answer.


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
            answer.
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 .