Dr. P.V. Viswanath

pviswanath@pace.edu

• Final
• Final Solutions

LUBIN SCHOOL OF BUSINESS
Pace University
MBA 654 Managing & Controlling the Supply Chain (Economics component)
Prof. P.V. Viswanath
Fall 2007

Final

Notes:

1. If your answers are not legible or are otherwise difficult to follow, I reserve the right not to give you any points.
2. If you cheat in any way, I reserve the right to give you no points for the exam, and to give you a failing grade for the course.
3. You may bring in sheets with formulas, but no worked-out examples.
4. You must explain all your answers.
5. Do questions 1 and 2 and any one of questions 3 and 4.
6. You have 1 hour to complete the exam; please make sure to attempt all the questions, so I can give you partial credit, if necessary.

1. (20 points) Suppose the production function for a particular shoe factory is given by q = 10K^2/3L^1/3, 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.

1. What is the optimal amount of labor and capital to use if the manager wants to produce 12.5 million pairs of shoes? What is the total cost of production?
2. If the level of production doubled, would the ratio of labor to capital stay constant or increase? Can you explain why this is so?

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.

1. At what level of spending on movies and plays is Kikuyo's utility maximized? (Assume that she can watch fractions of plays and movies.)
2. What is her marginal rate of substitution of movies for plays at this level?
3. What happens to Kikuyo's marginal rate of substitution of movies for plays if she doubles her budget on entertainment?

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.

1. How many people would cross the bridge if the price were $5? What is the price elasticity of demand at this price?
2. If the price were to be increased to $6, would toll revenue increase or decrease? Base your answer to this question on your answer in part b.

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. What is the equilibrium price and quantity? What is the price elasticity of demand at the equilibrium?
2. What would happen to the price if foreign competition caused demand for domestic televisions to drop by 20%?

Solutions to Final

1.a. The production function is given by q = 10K^2/3L^1/3. 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 = 2.5L^5/3.
If the manager wants to produce 12.5 million pairs of shoes, the amount of labor is given by substituting for q in the modified production function and solving, we get L = 10456.4 hours (approx.). Since we have the relationship, L²=8K, we find that K = 13,667,026 hours.
The total cost of production is (10456.4)(50) + (13,667,026)(200) = $2,733.928 million.
b. If the level of production is doubled, then the use of both labor and capital will increase. However, we already know that L²=8K always if costs are minimized and hence that L/K = 8/L. If L increases, this ratio will fall; hence with an increase in the level of production, the labor/capital ratio has to fall.

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.
b. Her marginal rate of substitution of movies for plays (DP/DM) is 1/3; that is, starting at the level of consumption of 13.33 plays and 40 movies, she will substitute at the rate of 3 movies for one play.
c. The ratio of movies to plays when she is maximizing her utility will always be 3:1. Hence the ratio of her marginal utilities from consuming plays and movies will remain the same. This is because the ratio of marginal utilities from plays and movies is, in this case, simply the inverse of the ratio of the level of consumption of plays and movies.

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.
b. Since the price elasticity of demand is less than one in absolute value, it means that the percentage of fall in demand will be less than the percentage of price rise. Hence revenue would increase if the price were increased.

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.
b. If demand dropped by 20%, the new demand function would be given by 200(0.8) - 0.09(0.8)P, or 160 - -0.072P. Equating the supply and demand functions, we find $465.12. The price will drop, as expected.