3. Assume the production function is Q = 4K0.5L0.5, and K = 4 in the short run. Find the marginal product of labor. Using derivatives, show that the production faces diminishing marginal return to labor. If the market wage is w, express the optimal number of labor (L) in terms of w. Are they positively or negatively related, and why?

Suppose a production function is given by the equation Q = LVK. 1. Graph the isoquants corresponding to Q = 10, and Q = 20. 2. MP = √K and MP = 0.5(L/VK). Find the Marginal Rate of Technical Substitution LK (MRTSLK). Do these isoquants exhibit diminishing marginal rates of technical substitution? (Does the slope get flatter as L increases?) 3. The cost of labor is w = $5 per labor hour and the cost of capital is r = $15 per machine hour. What is the equation of the $1500 iso-cost curve?

B. Suppose that your production process is characterized by the production function x = f(e) = 100 In ( + 1). For purposes of this problem, assume w > 1 and p > 0.01. a. Set up your profit-maximization problem. b. Derive the labor demand function. C. The labor demand curve is the inverse of the labor demand function with p held fixed. Can you demonstrate what happens to this labor demand curve when p changes? d. Derive the output supply function. e. The supply curve is the inverse of the supply function with w held fixed. What happens to this supply curve as w changes? (Hint: Recall that In x = y implies e' = x, where e ≈ 2.7183 is the base of the natural log.) f. Suppose p = 2 and w profit will you make? = 10. What is your profit-maximizing production plan, and how much