Problem 4. Farmer Hoglund has discovered that on his farm, he can get 30 bushels of corn per acre if he applies no fertilizer. When he applies N pounds of fertilizer to an acre of land, the marginal product of fertilizer is 1 - N/200 bushels of corn per pound of fertilizer. 1. Write down a function that states Farmer Hoglund’s yield per acre as a function of the amount of fertilizer he uses. (Note: you will need a bit of calculus integral here.) 2. If the price of corn is $3 a bushel and the price of fertilizer is $p per pound (where p < 3), how many pounds of fertilizer should he use per acre in order to maximize profits? (Hint: you can directly solve for the optimal N by setting up the profit function and find the optimal N using a first order condition, or use the optimal condition to maximize profit: MR=MC). 3. Hoglund’s neighbor, Skoglund, has better land than Hoglund. In fact, for any amount of fertilizer that he applies, he gets exactly twice as much corn per acre as Hoglund would get with the same amount of fertilizer. How much fertilizer will Skoglund use per acre when the price of corn is $3 a bushel and the price of fertilizer is $p a pound? (Hint: Start by writing down Skoglund’s marginal product of fertilizer as a function of N.
MP of fertilizer is 1-N/200. he can get 30 bushels of corn per acre
Hoglund’s yield per acre = 30+ N(1-N/200) = 30 + N-N2/400
Farmer Hoglund will choose fertilizer in an amount at which the marginal product of fertilizer multiplied by the price of corn is equal to its cost
This condition is the ﬁrst order condition for proﬁt maximization. Solving for the optimal amount we get
Let the production function of Hoglund be
y = fH (N)
Then the production function of Skoglund, fS(N), is
Therefore, the marginal product of fertilizer for Skoglund is
where we know that
The optimal choice of fertilizer for Skoglund is found by setting the marginal product of fertilizer times the corn price equal to its cost
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