Final answer:
The marginal productivity of labor (MPL) is derived by taking the partial derivative of the production function with respect to labor, resulting in β/L. In a competitive market, the firm's profit maximizing level of employment is when the MPL equals the going market wage, such as $12 in the given scenario.
Step-by-step explanation:
To derive the marginal productivity of labor from the given production function f(L, K) = αln(K) + βln(L) with α and β as constants, we need to calculate the partial derivative of the function with respect to labor (L). The marginal productivity of labor (MPL) is the change in output (Q) resulting from a one-unit change in labor, holding capital (K) constant. The derivative of ln(L) with respect to L is 1/L, hence the MPL is β/L.
Regarding the application to a competitive market, a firm will continue to hire additional workers until the MPL multiplied by the price of output is equal to the going market wage. If the wage is $12 and we assume that the price of output is 1 for simplicity, the firm will hire workers up to the point where MPL = $12. This is the firm's profit maximizing level of employment.