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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*.