1. Suppose that UPS has the following production function:

q = 10L^0.75K^0.75

where q is the number of packages delivered per day, L is labor (measured in thousands of worker hours), and K is the number of trucks (measured in thousands).

a. Does UPS' 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 thousand trucks. 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 for UPS' production function?

g. Suppose that improvements to package tracking software changes UPS' production function to

q = 20L^0.75K^0.75

Does this new technology constitute neutral technical progress, non-neutral technical progress, or neither? Explain your answer.

h. Calculate the marginal rate of technical substitution given UPS' new production function.

How does it compare to that which you calculated for the old production function in part (f)?

i. Now suppose that UPS designs a new truck routing system that changes its production function from that in part (g) to

q = 20L^0.75K

Does this new technology constitute neutral technical progress, non-neutral technical progress, or neither? Explain your answer.

j. Calculate the marginal rate of technical substitution given UPS' new production function.

How does it compare to that which you calculated for the old production function in parts (f) and (h)?

2. Suppose that Drexelabra, a new app company, has the following production function:

q = 2c + 5e + 1/2c.e

where q is the number of apps produced, c denotes the number of fresh college graduates, and e denotes the number of experienced programmers.

a. What is the marginal product of college graduates? What is the marginal product of experienced programmers?

b. Calculate the marginal rate of technical substitution between college graduates and experienced programmers for Drexelabra.

c. Suppose that college graduates cost $60K per year and experienced programmers cost $150K per year. Assuming that Drexelabra has fixed costs amounting to $50K (including rent on office space, utilities, etc.), write down Drexelabra's total costs.

d. Given the production function and costs given above, determine the optimal production ratio; i.e., the optimal number of college graduates per experienced programmer Drexelabra has on staff.

e. Given the optimal production ratio, determine the cost-minimizing numbers of college graduates and experienced programmers Drexelabra should use to produce any given amount of output q; i.e., determine c and e as a function of q.

f. Substitute your solutions for c and e from part (e) back into your total costs equation from (c) to arrive at Drexelabra's optimal total cost function.

g. Suppose that Drexelabra wants to produce 25 apps in the next year. Given your answers above, what are the optimal numbers of c and e to employ to produce 25 apps? What is the total cost of producing this number of apps given that you are producing it as efficiently as possible?

h. Suppose that Drexelabra wants to produce 60 apps in the next year. Given your answers above, what are the optimal numbers of c and e to employ to produce 25 apps? What is the total cost of producing this number of apps given that you are producing it as efficiently as possible?

i. Using the total cost function derived in part (g), derive the marginal cost function. How much does the 25^th app cost to produce? How much does the 60^th app cost to produce?

j. Explain if and why there is (or is not) a difference in the cost of producing the 25^th app as compared to the cost of producing the 60th
app.

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

q = 120(2u + 5e)^0.1w^0.4

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 productivity of unskilled labor hours? What is the marginal productivity of skilled labor hours? What is the marginal productivity of water?

b. Suppose Drexanto's costs are $10 per hour for unskilled labor and $40 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 skilled labor.

c. Suppose water costs $10 per ton. Given this and the cost of unskilled labor given above (and that Drexanto will use zero skilled labor hours), determine the optimal ratio of unskilled labor hours to tons of water to use.

d. Given the optimal production ratio from part (c), determine the cost-minimizing quantities of unskilled labor hours and tons of water to use to produce any given amount of output q; i.e., determine u and w as a function of q. Note: at this step, and only at this step, please round numbers to the nearest integer.

e. Using the input prices from parts (b) and (c) together with your solutions for u and w from part (d), write out the optimal cost function for Drexanto. In doing so, assume Drexanto has fixed costs of $10,000.

f. Using the total cost function derived in part (e), derive the marginal cost function.

g. Suppose that Drexanto plans to produce 5600 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 should they use to minimize costs?

h. Using the total cost function derived in part (g), 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. The company believes that they can sell the corn at a price p = $50 per ton. How many tons of corn would Drexanto be willing to supply at that price?

j. What are Drexanto's economic profits if the corn is sold at a price of $50?

k. Suppose that Drexanto incurs its fixed cost, but then discovers that the price p at which they can sell the corn is only $5 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.