suppose that the cost function of the honey farm is C_H(H, A)  H ²/100  3A and the cost function of the apple orchard is C_A(H, A)  A²/100, where H and A are the number of units of honey and apples produced respectively.The price of honey is $7 and the price of apples is $5 per unit.Let A₁ be the output of apples if the firms operate independently, and let A₂ be the output of apples if the firms are operated by a profit-maximizing single owner.

A) A₁  125 and A₂  250.
B) A₁  A₂  250.
C) A₁  200 and A₂  250.
D) A₁  250 and A₂  400.
E) A₁  350 and A₂  250.

Question

Charlene Louder · Accepted Answer

Final Answer : D, Explanation :  To find the output of apples for each scenario, we need to maximize the profit function of the honey farm and the apple orchard.For the honey farm:πH = revenue - costπH = 7H - C H (H, A) Taking the derivative of πH with respect to H and setting it equal to 0 gives:7 - ∂C H (H, A) /∂H = 0∂C H (H, A) /∂H = 7/3Similarly, for the apple orchard:πA = 5A - C A (H, A) Taking the derivative of πA with respect to A and setting it equal to 0 gives:5 - ∂C A (H, A) /∂A = 0∂C A (H, A) /∂A = 5/2If the firms operate independently, then we have:∂C H (H, A 1 ) /∂H = 7/3∂C A (A 1 , H) /∂A = 5/2Solving the above two equations simultaneously will give us A 1  = 250.If the firms are operated by a profit-maximizing single owner, then we have:Total profit = πH + πATotal profit = 7H + 5A - (C H (H, A)  + C A (H, A) )We need to maximize this function with respect to both H and A. By taking partial derivative with respect to H and A, we get:7 - ∂C H (H, A) /∂H = 05 - ∂C A (H, A) /∂A = 0Solving the above two equations simultaneously will give us A 2  = 400.Therefore, the answer is D) A 1  = 250 and A 2  = 400.

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