For many firms, production takes place in two or more different plants whose ope
ID: 1252376 • Letter: F
Question
For many firms, production takes place in two or more different plants whose operating costs can differ. In this question you will be asked to set up a multiple plant monopolist optimization model and economically interpret the marginal condition for optimal production. To this end, let p() be the inverse demand function facing the multiple plant monopolist, where (derivative of )p(y1 + y2 ) < 0, y1 is the output produced at plant 1, and y2 is the output produced at plant 2. Note that there is one good produced by the multiple plant monopolist, its just that now it is produced at two different plants. Let c1() and c2() be the total cost functions for producing the output at plant 1 and plant 2, respectively, where (derivative of) c(y1) > 0 and (derivative of)c2(y2) > 0 are assumed to hold.
a)What would be the multiple plant monopolists profit maximization problem? and how would I derive the FONC? First Order Necessary Condition for that problem?
Any help would be appreciated. Even pointers in the right direction. Thanks :)
Explanation / Answer
We want to maximize profit. Let's call profit V. V = P(y1+ y2)*(y1 + y2) - c1(y1) - c2(y2) Take the derivative with respect to y1 and y2 to find the FONC. y1: (dP/dy1)*(y1 + y2) + P - dc1/dy1 = 0 (1) y2: (dP/dy2)*(y1 + y2) + P - dc2/dy2 = 0 (2) We can divide (1) by (2). Recognize that (dP/dy1) = (dP/dy2) because y1 and y2 are the same product. If there was a differentiated product, this would be different. This simplifies to: dc1/dy1 = dc2/dy2 Produce such that the marginal costs are equal to each other. This is intuitive. If it is cheaper at one plant than the other, then produce more there and less at the more expensive plant until the marginal cost is the same at both plants. Cost is our only consideration because the product produced at one plant affects demand the same way as the product produced at the other. If we had a demand curve and cost curves, we could actually calculate this marginal cost. Then, we could set marginal revenue equal to marginal cost in order to solve for the optimal price and total quantity.
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