Final Answer:
1. The marginal product of labor (MP_L) is 4. Yes, the production function exhibits diminishing marginal product because as more units of labor (L) are employed, the additional output (marginal product) generated by each additional unit of labor decreases.
2. The profit maximization problem is to maximize profit, given the production function y=f(L)=4L, wage w=1, and demand function x∗(pa)=pa100. The optimal production level y∗(pa) is determined by equating marginal cost (which is the wage divided by the marginal product of labor) to the inverse demand function.
Explanation:
In the profit maximization problem, the marginal product of labor (MP_L) is the derivative of the production function with respect to labor (L), which is 4. This indicates that each additional unit of labor contributes 4 units of output, and as more labor is added, the additional output diminishes.
The profit maximization equation is formed by equating the marginal cost (wage divided by MP_L) to the inverse demand function: 1/4L = pa/100. Solving for L gives the optimal level of labor, and substituting it back into the production function provides the optimal production level (y∗).
The equilibrium price and quantity are determined by the intersection of the inverse demand and supply curves. The inverse demand is given by x∗(pa)=pa100, and the supply is the production function y=f(L)=4L. Solving for pa at the intersection provides the equilibrium price, and substituting it into either the demand or supply function gives the equilibrium quantity.