Final answer:
The marginal product of labor can be found by taking the derivative of the production function with respect to labor. The production function exhibits diminishing marginal product because the marginal product of labor decreases as more labor is used. The profit maximization problem involves maximizing profit by setting the derivative of profit with respect to output equal to 0.
Step-by-step explanation:
The marginal product of labor can be found by taking the derivative of the production function with respect to labor. In this case, the production function is y=f(L)=12L^3-1. Taking the derivative, we get the marginal product of labor, MPL = 36L^2-1. This represents the additional output produced from each additional unit of labor.
Diminishing marginal product occurs when the marginal product of a variable input decreases as more of it is used while holding other inputs constant. In this case, the marginal product of labor, MPL, is positive and decreases as more labor is used. Therefore, the production function exhibits diminishing marginal product.
The profit maximization problem can be written as:
- Maximize profit: π = p × y - wL
- Where:
- π is profit
- p is the price of one apple
- y is the level of output
- w is the wage rate
- L is the level of labor
To find the optimal production level, we need to solve for y in the profit maximization problem. Setting the derivative of π with respect to y equal to 0, we can find the optimal production level, y*.The (inverse) supply curve can be plotted by graphing the relationship between the wage rate, w, and the level of labor, L, at the profit-maximizing production level, y*.