At the beginning of month t, farmer Smith has xt bushels of wheat in his warehouse. He has the opportunity to sell wheat at a price st dollars per bushel and can buy wheat at pt dollars per bushel....

At the beginning of month t, farmer Smith has xt bushels of wheat in his warehouse. He has the opportunity to sell wheat at a price st dollars per bushel and can buy wheat at pt dollars per bushel. Farmer Smith’s warehouse can hold at most C units at the end of each month.

a Formulate a recursion that can be used to maximize the total profit earned during the next T months.

b Let ft(x_t) be the maximum profit that can be earned during months t, t  +1, . . . , T, given that x_t bushels of wheat are in the warehouse at the beginning of month t. By working backward, show that for appropriately chosen constants

a_t and b_t,

f_t(x_t) = a_t + b_tx_t

c During any given month, show that the profit maximizing policy has the following properties: (1) The amount sold during month t will equal either xt or zero. (2) The amount purchased during a given month will be either zero or sufficient to bring the month’s ending stock to C bushels.