# Using the 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&#61655;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: &#61552; = p&#61655;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*.

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#### Solution Preview

...positive, dq/dL must be positive.

Then we let (1) to be f(h, L) = p*dq/dL - (w+h) = 0
From this we find df / dh = -1 And df / dL = p*d2q / dL2
The implicit function theorem: dL/dh = - (df ...