Dr. P.V. Viswanath
|Home/ MBA 662/ Exams/|
SCHOOL OF BUSINESS
1. (20 points) Suppose the production function for a particular shoe factory is given by q = 10K2/3L1/3, where K is hours of machine time, and L is hours of labor input, and q is the output in pairs of shoes. Machine time rents for $10,000 per week, while labor costs are $50 per hour. Machines can be used about 50 hours per week.
2. (30 points) Kikuyo has decided to spend $800 on entertainment. She likes movies, but she likes plays as well. Movies cost $10 a pop, while plays cost thrice as much. Suppose Kikuyo's utility function is given by u(M,P) = MP.
3. (20 points) Suppose you're in charge of a tollbridge that costs essentially nothing to operate. The demand for bridge crossings, Q, is given by P = 15-Q/2.
4. (20 points) Suppose the demand for domestic televisions is given by D = 200 - 0.09P, while the supply is given by S = 80 + 0.1P, where price is in dollars, and demand/supply are in millions of sets.
1.a. The production function is given by q = 10K2/3L1/3.
The marginal product of capital (obtained by differentiating the production
function with respect to K) is (20/3)K-1/3. The marginal product
of labor is (10/3)L-2/3. Costs are minimized when the ratio
of marginal products equals the ratio of prices. Hence, costs are minimized
at the levels of K,L, where [(20/3)K-1/3]/[(10/3)L-2/3]
= 200/50. (Note that machines can be used for 50 hours per week and cost
$10,000 per week.) Solving this equation, we find L2=8K. That
is to say, we should always use labor and capital in such a way that L2=8K.
Substituting this relationship in the production function, we find the
modified production function q = 2.5L5/3.
2.a. The utility function is given by u(M,P) = MP. Hence the marginal
utilities of movies and plays is given by P and M respectively. Hence
utility is maxmized where the ratio of marginal utilities is equal to
the ratio of prices. That is, where P/M = Price(movies)/Price(plays) =
10/30 = 1/3, or where 3P = M. Given this relationship and the prices of
plays and movies, Kikuyo's total spending is given by 10M + 30P = 30P
+ 30P = 60P. Since Kikuyo wants to spend $800 in all, she will watch 800/60
or 13.33 plays and 40 movies. That is, she will spend $400 on movies and
$400 on plays. Intuitively, this is obvious -- since M and P contribute
to utility in exactly the same fashion, she will spend the same amount
on each of them.
3. a. The demand function is given by P = 15-Q/2. Hence if the price
is 5, the demand would be Q=20. The price elasticity is given by (DQ/Q)(DP/P)
= (DQ/DP)(P/Q) =
-2(5/20) = -0.5.
4. a. We can compute the equilibrium price and quantity by solving the
demand and supply equations. In this case, we find that P=$631.58, while
D = 143.16m. sets. The price elasticity of demand is computed by (DQ/Q)(DP/P)
= (DQ/DP)(P/Q) =
-0.09(631.58/143.16) = -0.397.