ECON301 QUESTION

ECON 301: Microeconomics Winter 2016

1 CONTINUE

Problem Set 2 Due at the start of class on January 28. You may work with your teammates, but you must turn in your own version. All solutions should be neatly and clearly written. Be sure to label any graphs and clearly indicate your reasoning. Each part of each question is worth five (5) points; the total possible points is 150. 1. Suppose that the local moving company DrexelHaul has the following production function:

q = 6L0.25K0.5 where q is the number of boxes moved per day, L is labor (measured in worker hours), and K is the number of moving trucks.

a. Does DrexelHaul’s production function exhibit decreasing, constant, or increasing returns

to scale? Explain your answer using math, and describe in words what it means to have the type of returns to scale you find.

b. Suppose that the number of trucks is fixed in the short run at 16. What is the short-run production function?

c. Using the short-run production function from (b), calculate the marginal product of labor. d. Does the short-run production function from (b) exhibit diminishing marginal returns to

labor? Show why it does or does not using math, and explain in words what diminishing marginal returns to an input means.

e. Using the short-run production function from (b), graph the relationship between output q and labor L (putting output on the y-axis and quantity of labor on the x-axis). In a separate figure, graph the relationship between the marginal product of labor and labor (putting the marginal product of labor on the y-axis and quantity of labor on the x-axis). Note: these graphs do not need to be to scale.

f. Now suppose that we are in the long run, such that both capital and labor are adjustable (i.e., K is no longer fixed at 16). What is the marginal rate of technical substitution (MRTS) for DrexelHaul’s production function?

g. Suppose that DrexelHaul develops a new truck routing system that changes its production function to

q = 8L0.25K0.5 Does this new technology constitute neutral technical progress, non-neutral technical progress, or neither? Explain your answer.

h. Calculate the MRTS given DrexelHaul’s new production function. How does it compare to that which you calculated for the old production function in part (f)?

i. Now suppose that DrexelHaul adopts new “driverless” technology for its trucks that changes its production function from that in part (g) to

q = 8L0.25K Does this new technology constitute neutral technical progress, non-neutral technical progress, or neither? Explain your answer.

j. Calculate the MRTS given DrexelHaul’s new production function. How does it compare to that which you calculated for the old production functions in parts (f) and (h)?

ECON 301: Microeconomics Winter 2016

2 CONTINUE

2. Suppose that Drexenture, a new consulting company, has the following production function:

ececq  2

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where q is the number of reports produced, c denotes the number of fresh college graduates, and e denotes the number of experienced consultants. a. What is the marginal product of college graduates? What is the marginal product of

experienced consultants? b. Calculate the marginal rate of technical substitution (MRTS) between college graduates

and experienced consultants for Drexenture. c. Suppose that college graduates cost $60K per year and experienced consultants cost

$120K per year. Also assume that Drexenture has fixed costs amounting to $50K (including rent on office space, utilities, etc.). Given these costs and the production function above, determine the optimal production ratio; i.e., the optimal number of college graduates per experienced consultant Drexenture has on staff.

d. Given the optimal production ratio, determine the cost-minimizing numbers of college graduates and experienced consultants Drexenture should use to produce any given amount of output q; i.e., determine c and e as a function of q. Hint: you will need the quadratic formula to solve this; if you need a refresher on it, see http://en.wikipedia.org/wiki/Quadratic_equation. Only the positive square root will be relevant in this case.

e. Using your solutions for c and e, derive Drexenture’s optimal total cost function. f. Suppose that Drexenture wants to produce 65 reports in the next year. Given your

answers above, what are the optimal numbers of c and e to employ to produce 65 reports? What is the total cost of producing this number of reports given that you are producing it as efficiently as possible?

g. Suppose that Drexenture wants to produce 240 reports in the next year. Given your answers above, what are the optimal numbers of c and e to employ to produce 240 reports? What is the total cost of producing this number of reports given that you are producing it as efficiently as possible?

h. Using the total cost function derived in part (e), derive the marginal cost function. How much does the 65th report cost to produce? How much does the 240th report cost to produce?

i. Explain if and why there is (or is not) a difference in the cost of producing the 65th report as compared to the cost of producing the 240th report.

ECON 301: Microeconomics Winter 2016

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3. You are in charge of a large firm, Drexanto, which has developed a new variety of corn for farmers that is highly resistant to disease and bugs. Suppose that the production function for this corn is

25.025.0)93(10 weuq  where q represents tons of corn, u represents hours of unskilled labor, e represents hours of skilled (engineer) labor, and w represents tons of water.

a. What is the marginal product of unskilled labor hours? What is the marginal product of

skilled labor hours? What is the marginal product of water? b. Suppose Drexanto’s costs are $15 per hour for unskilled labor and $27 per hour for

skilled labor. Show mathematically and explain why, given these costs and the marginal products for u and e you calculated in part (a), that you will not use any unskilled labor.

c. Suppose water costs $3 per ton. Given this and the cost of skilled labor given above (and that Drexanto will use zero unskilled labor hours), determine the optimal ratio of skilled labor hours to tons of water to use.

d. Given the optimal production ratio from part (c), determine the cost-minimizing quantities of skilled labor hours and tons of water to use to produce any given amount of output q; i.e., determine e and w as a function of q.

e. Using the input prices from parts (b) and (c) together with your solutions for e and w from part (d), write out the optimal total cost function for Drexanto. In doing so, assume Drexanto has fixed costs of $300.

f. Using the total cost function derived in part (e), derive the marginal cost function. g. Suppose that Drexanto plans to produce 30 tons of corn. What is the total cost of

producing this amount of corn given that they are producing it as efficiently as possible, and how many units of each input (e and w) should they use to the minimize costs?

h. Using the total cost function derived in part (e), write down the equation determining Drexanto’s economic profits (which will be a function of the price per ton of corn p and the number of tons sold q).

i. Based on extensive market research, your staff thinks that the company can sell the corn at a price p = $12 per ton. How many tons of corn would Drexanto want to supply at that price?

j. What are Drexanto’s economic profits if the corn is sold at a price of $12? k. Suppose that Drexanto incurs its fixed cost, but then discovers that the price p at which

they can sell the corn is only $6 per ton. Determine the optimal quantity of corn to supply at that price and the resulting economic profits. Should Drexanto stay in business or shutdown in the short run? Explain your answer in words.