Suppose you are a corn farmer in Wisconsin. You can grow two different types of corn: (1) regular field corn, which you can sell to the local grain elevator for the going market price, and (2) artisanal popcorn, which you can sell directly to local consumers as a specialty food product. You are the only farmer in the area who grows local popcorn, so you can essentially set the price for popcorn. However, there is limited demand for local popcorn: the more you choose to produce, the lower the price you will be able to command. You have a total of 360 acres on your farm, and you are trying to decide how many acres to plant to popcorn and how many acres to plant to field corn to maximize your profits. In order to determine the market demand for popcorn, you have gathered a data set of local popcorn markets in previous years across a number of communities. For each observation, you observe the total amount of popcorn produced and sold (measured in bushels) as well as the market price for popcorn (measured in dollars per bushel) that cleared the market. However, there is one more wrinkle here: popcorn consumers prefer longer corn cobs to shorter ones. Therefore, your data set also includes information about the average corn cob length (measured in inches) for each price-quantity pair. Use the information below and your own analysis of the available data to determine how to allocate acreage on your farm. The questions in this problem set should walk you through the necessary steps. INFORMATION It costs $850 per acre to grow popcorn, but only $400 per acre to grow field corn. You can grow 250 bushels of field corn per acre, but only 120 bushels of popcorn per acre. The market price for field corn is $4.00/bushel. Based on the specific variety of popcorn you grow on your farm, you expect to harvest popcorn cobs that are 8 inches long, on average. QUESTIONS 1. Begin by analyzing the market demand for popcorn. To do this, run a regression of popcorn price on cob length and quantity. a. Estimated intercept, rounded to the nearest whole number ($/bu): ________ (1 pt) b. Coefficient on cob length, rounded to the nearest tenth ($/bu): ________ (1 pt) c. Coefficient on quantity produced, rounded to four decimal places ($/bu): _______ (1 pt) d. Use your answers to parts a-c, along with any other necessary information, to generate a formula for the inverse demand curve for local popcorn. The formula should take the form: P=a+bQ, with numbers instead of a and b: _______________________ (2 pts)

2. If you produce 40,000 bushels of popcorn, at what price per bushel would you be able to sell it all? (Round your answer to the nearest cent, if necessary): ______ (2 pts) 3. If you grow 100 acres of popcorn, at what price per bushel would you be able to sell all the popcorn you produce? (Round your answer to the nearest cent, if necessary): ______ (2 pts) 4. If you grow 100 acres of popcorn and 200 acres of field corn, how much profit will you earn? (Round your answer to the nearest dollar): ______ (2 pts) 5. Solve the nonlinear programming problem to determine how many acres of field corn and popcorn to grow to maximize profits. a. In the optimal solution, how many acres will you plant to popcorn? (Round your answer to the nearest tenth of an acre): ______ (1 pt) b. How many acres will you plant to field corn? (Round your answer to the nearest tenth of an acre): ______ (1 pt) c. How much will you earn in profits? (Round your answer to the nearest dollar): ______ (1 pt) 6. Now, lets introduce some uncertainty to this problem. Specifically, suppose you believe there is a 75% chance that you will be able to plant all 360 acres of your farm and a 25% chance that part of your farm will flood in the spring and you will only be able to plant 250 acres. In addition, and independently, you think there is a 50% chance the price of field corn will be $4.00/bushel, and a 50% chance the price of field corn will be $3.50/bushel. (For the purposes of this problem, assume that you will know the price of field corn with certainty by the time you have to plant.) a. Suppose the farm floods (so you can plant a maximum of 250 acres) and the price of field corn is $3.50/bushel. Re-solve the nonlinear programming problem under these assumptions. i. In the optimal solution, how many acres will you plant to popcorn? (Round your answer to the nearest tenth of an acre): ______ (1 pt) ii. How many acres will you plant to field corn? (Round your answer to the nearest tenth of an acre): ______ (1 pt) iii. How much will you earn in profits? (Round your answer to the nearest dollar): ______ (1 pt) b. Suppose the farm floods, but the price of field corn is $4.00/bushel. How much will you earn in profits? (Round your answer to the nearest dollar): ______ (2 pts) c. Suppose the farm does not flood, but the price of field corn is $3.50/bushel. How much will you earn in profits? (Round your answer to the nearest dollar): ______ (2 pts) d. Use your answers to 5.c., 6.a.iii., 6.b., and 6.c. to calculate your expected profits given the uncertainty. (Round your answer to the nearest dollar): ______ (2 pts) e. What is the standard deviation of your profits given the uncertainty? (Round your answer to the nearest dollar): ______ (2 pts) *** If I cannot find your work for an answer above in an attached file (.R file, .xlsx file, etc.), you will not receive credit for it even if the final number is correct.***