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 = aK^bL^cF^d = 0.02 K^0.25 L^0.2 F^0.55,

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 products associated with K, L, and F. 

b. What is American’s marginal rate of technical substitution (MRTS) between K and L? How about the MRTS between K and F? Should American try to ensure that all its MRTSs are equal? Explain.

c. Does the law of diminishing returns apply to K in the production of seat- miles between Dallas and San Jose by American? to L or F? Explain. Would the law of diminishing returns apply to L if c = - 0.2 instead of 0.2? 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. Does the law of diminishing returns imply decreasing returns to scale? Explain. Would decreasing returns to scale imply the law of diminishing returns? 

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.