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Implicit function theorem to maximize profit

Assume that firm A produces good G using only labor. Therefore, the firm's output is a function of the quantity of labor hired (i.e. output = q(L)).

Assume further that this firm receives a price (p) for good G and pays laborers a wage (w) that are both constant, and that the firm pays a constant health care cost (h) for each worker. If total revenue is calculated as pq(L), and the total cost of paying all laborers is measured by the equation TC = (w + h)L, we have a profit function that looks like this:  = pq(L) - (w + h)L. To maximize profits, firm A must adjust L until profits are maximized at L*.

a. From the profit equation, derive an equation for the marginal product of labor (i.e. dq/dL) and then show whether dq/dL is positive or negative. Use the implicit function theorem to determine how changes in h affect L (i.e. dL/dh).
Assume that d2q/dL2 < 0.

Assume that output and price have the following values: q=3*(3rd root of L) and p = $9, and that the firm's total costs take the following form: (w + h)L = (6+h)(3rd root of L^4)

b. Given this new information, maximize the firm's profits and find L*.

Solution Preview

a) to maximize Profit = p&#61482;q(L) - (w + h)L in terms of L
first order condition: d(Profit)/dL = 0, i.e. p*dq/dL - (w+h) = 0 (1)
then we have dq/dL = (w+h)/p (1)
since w, h, p are all ...

Solution Summary

Detailed steps are provided for everything. The response is very well written and easy to understand as well. Overall, an excellent response. It is an excellent response for students who want to understand the concepts and then use the same concepts to solve similar problems in the future.

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