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=10(3u+9e)^.25 w^.25 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