a) Suppose a firm's average cost curve is described by the equation AC = 2q2 - 16q + 90. At what output level does the marginal cost curve cross the average cost curve?

b) Explain the relationship between the shape of the marginal cost curve and the marginal product of labor curve.

c) Suppose the cost of producing milkshakes is C = 0.333Q^3 - 3Q^2 + 15Q + 50. What is the equation for marginal cost? At what point (output) is marginal cost minimized? What is the minimum marginal cost? In your work, make sure you show that both the first order and second order conditions are satisfied at the point where the marginal cost is minimized.

d) A production function is q = 100 * min (L, 5K), where q is output per hour, L is the number of workers, and K is the number of machines. a. Draw the isoquants for this production function. b. Draw the total product of labor, average product of labor, and marginal product of labor curves for this production function for some fixed level of capital.

e) Haskel and Sadun (2012) estimated the production function for U.K. supermarkets is Q = L^0.23 K^0.10 M^0.66 , where L is labor, K is capital, and M is materials. What kind of returns to scale do these production functions exhibit? Provide explanations.