American Airlines produces round-trip transportation between Dallas and San Jose using three inputs: capital (planes), labor (pilots, flight attendants, and so on), and fuel. Suppose that American's production function has the following Cobb-Douglas form: T = a Kb Lc Fd = 0.02 K0.25L0.2F0.55, where T is the number of seat-miles produced annually, K is capital, L is labor, and F is fuel.

a. If American currently employs K = 100, L = 500, and F = 20,000, calculate the marginal product associated with K, L, and F.

b. Calculate American's marginal rate of technical substitution between K and F (MRTSKF). What does it show?

c. Does the law of diminishing marginal returns apply for K in the production of seat-miles between Dallas and San Jose? For L or F? Would the law of diminishing marginal returns apply to L if c = 1.2? d. Given that the exponent associated with F is larger than the exponent associated with L, would it be wise for American to spend all its money on either fuel or capital and none on labor? Explain.

e. Does American's production function exhibit constant, increasing, or decreasing returns to scale? Explain. How would your answer change if c = -0.2 instead of 0.2? If c = 1.2?

f. With a functional form of the type above, does the law of diminishing marginal returns imply decreasing returns to scale? Explain. Would decreasing returns to scale imply the law of diminishing marginal returns for each factor?

g. In the real world, do you think that the production of seat miles between Dallas and San Jose is characterized by a multiplicative, Cobb-Douglas technology? If not, explain the nature of the production function that might characterize a typical firm producing seat-miles in this city-pair market.