(1 point) A certain commodity, which we call corn, is grown by many farmers, but the amount of corn harvested by every farmer depends on the weather. sunny weather yields more com than cloudy weather during the growing season. All corn is harvested simultaneously, and the price per bushel is determined by a market demand function. Suppose the price demand function P P.(D) is such that Pa(0) = 10 dollars and Pa(500000) = 5 dollars, and P.(D) is linear in the demand in bushels. Then (1) price in terms of demand Pa(D) = 10-0.00001D dollars Through supply and demand equality, the demand equals the total crop size. The amounts produced on different farms are all perfectly correlated and there are a total of 100 farms, and thus D = 100C. Then (2) price in terms of production P (C) = 10-0.001C dollars Then from selling the crop, each farmer has (3) revenue from production R (C) = (10-0.001C)C dollars This shows that the revenue is a nonlinear function of the underlying variable C. Because of the weather. C is random and each farmer faces nonlinear risk. Can a farmer hedge this risk in advance by participating in the futures market for corn? Since the farmer is ultimately going to sell his corn harvest at the (risky) spot price, it might be prudent to sell some corn now at a known price in the futures market. Indeed, if the farmer knew exactly how much corn he would produce, and only the price were uncertain, he could implement an equal and opposite policy by shorting this amount in the corn futures market. Let T be the expected corn production. Let P = P() be the expected price at future time T. Suppose the farmer enters into h com future contracts, where h 0 bushels of corn at time I for per bushel, and h> 0 represents purchase of h bushels of corn at time T for per bushel. (4) profit or loss for h futures 0.1C-0.00001C^2 dollars ( write h as h. Pas P. P as Pbar) Replacing P with P (C) and with Pp() gives (5) profit or loss for h futures 0.1C-0.000010^2 dollars ( write h as h. Cas C, C as Cbar) The total revenue from corn production and from corn futures is (6) revenue Rtot (Ch) = 0.1C-0.00001C-2 dollars ( write h as h. Cas C, as Cbar) For fixed h, the (partial) derivative with respect to C is (7) Rot(C,h) = = 0.1C-0.00001C^2 dollars/bushel (write h as h. Casc, as Cbar) To arrange for the total revenue to be insensitive to changes in corn production levels near the expected production level, the farmer requires Rot(C,h) = 0 at C = 7 This leads to the number of corn futures (8) h = 0.1C-0.000010^2 (write C as Cbar) which is an unexpected result because for any in the range 0 0 which would have the farmer buying corn in the future