Question

1 The Middleman Model

Let’s consider a middleman firm in the business of producing ethanol. Assume that the firm can produce its own corn (the feedstock) or it can purchase corn from third party farmers. The firm processes the corn into ethanol on its own.

To maintain consistency with your notes, let’s call our feedstock Xh (in-house) and Xm (third party). Suppose the inverse demand curve is P = 105 − Q and the production function is Q =f(X) = (Xh + Xm)2. The cost of producing corn in-house is Ch(Xh) = 1.5Xh2. The cost function of a third party producer is Cm(Xm) = Xm2 . Finally, the cost of processing feedstock from either source is Cp(X) = 5(Xh + Xm)2.

Part A

For this middleman, find the:
1. Marginal productivity of the input 2. Marginal revenue of the output
3. Marginal cost of processing
4. Marginal cost of producing in-house 5. Marginal outlay

Part B

First, suppose that the middle man is unable to produce feedstock in-house due to land limitations. He wants to determine the optimal level of Xm.

1. Write down the objective function. 2. Write down the first order condition. 3. Solve for the optimal level of Xm

Part C

Next, assume there are no restrictions. The middle man wants to determine the optimal levels ofXh and Xm.

1. Write down the objective function.

2. Write down the first order condition.

3. Solve for the optimal ratio of in-house to third party feedstock, XhXm

Answer 1