Suppose that ranchers can buy up to a collective total of 200 cows and run them

Question

Suppose that ranchers can buy up to a collective total of 200 cows and run them on the open range. The value of each cow after it has grazed is given by V= $2000-10C, where C is the total number of cows on the range. The cost of buying each cow is $1000.

a. What is the producer surplus in a perfectly competitive equilibrium?

b. What number of cows would maximize the societal surplus from the ranching industry?

c. What tax would need to be imposed on each cow if a benevolent government sought to maximize the social surplus from the ranching industry?

Explanation / Answer

A. We can find the optimal number of cows for the firm by setting the value equal to marginal cost. 2000-10C = 1000 1000 = 10C C = 100 We can find the producer surplus for the optimal situation by taking the integral of the marginal value minus the marginal cost from 0 to 100 cows. This will add up each producer surplus from each cow. PS = Int[0,100] {2000-10C - 1000}dC PS = Int[0,100] {1000-10C}dC PS = 1000C - 5C^2 | [0,100] PS = 1000*100 - 5*(100^2) PS = $50,000 B. From before, C = 100 Perfectly competitive industries maximize societal surplus so long as no externalities are present. The problem doesn't give us any information about externalities. C. A tax of T=0 should be imposed. That is, no tax needs to be imposed in order to maximize social surplus because a perfectly competitive market already maximizes social surplus in the absence of externalities. There is not a problem with market power because the capacity constraint is nonbinding. That is, the capacity constraint is 200, but a perfectly competitive industry will only produce 100. So, the capacity constraint cannot limit the number of firms and create market power, which would allow firms to price above marginal cost and create dead weight loss.

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