Initially suppose that both countries have the same production function, namely q = K.3L.7. For this production function, MPL = .7K.3L-.3 and MPK = .3K.-7L.7.
Derive (with all steps shown) the long-run total cost function for each country. (Given that both countries have the same production function, one derivation will do, and then simply write out the two cost functions, one for each country. Of course, the variables in the cost function for Nepal will have “n” subscripts, and for the U.S. “us” subscripts, though sometimes as for output level in the next sentence, the variables may have the same value.)
Make explicit use of the resulting functions to prove that for any given output level, the total cost of producing that output level in Nepal is less than the total cost of producing it in the U.S.
(b) Now suppose that there is a “technology coefficient” A in the production function for each country, so that the production function is AnK.3L.7 for Nepal and analogously for the U.S., and where An < Aus . (This sort of technology coefficient is referred to as “factor-neutral”.) Assume that for any production function that has a technology coefficient, A > 1.)
For a production function of the form q = AK.3L.7, the marginal products are MPL = .7AK.3L-.3 and MPK = .3AK-.7L.7.
With explicit reference to the steps in the derivation of the cost functions in part (a), show rigorously how the technology coefficients enter into the respective cost functions of the two countries.
Hint: You may find that the technology coefficient “cancels out” in some step in the derivation. That does not mean that it cancels out in every step of the derivation.
(c) Making rigorous use of the cost functions derived in part (b), (that is, the cost functions for the two countries after you have rigorously introduced the technology coefficients), discuss again how the cost of production of any given quantity of gravel in Nepal compares to that in the U.S.
To derive the long-run total cost function for each country, we need to go through several steps. Let's start by deriving the cost function for the production function q = K^0.3L^0.7.
Step 1: Derive the marginal product of labor (MPL):
MPL = 0.7K^0.3L^(-0.3)
Step 2: Derive the marginal product of capital (MPK):
MPK = 0.3K^(-0.7)L^0.7
Step 3: Set up the cost-minimization problem:
Minimize Total Cost (TC) = wL + rK subject to q = K^0.3L^0.7
Step 4: Solve the cost-minimization problem:
Differentiate the production function with respect to L:
0.7K^0.3L^(-0.3) = λw
Differentiate the production function with respect to K:
0.3K^(-0.7)L^0.7 = λr
Solve the two equations to find the optimal values of L and K. Let's call the optimal values L* and K*.
Step 5: Substitute the optimal values back into the production function to get the minimized cost:
q = K*^0.3L*^0.7
Step 6: Express the minimized cost in terms of the optimal values and the input prices:
TC = wL* + rK*
Now, let's write out the cost functions for each country, with "n" subscripts representing Nepal and "us" subscripts representing the U.S.
Cost Function for Nepal: TCn = wLn + rKn
Cost Function for the U.S.: TCus = wLus + rKus
For any given output level (q), the total cost of producing that output level in Nepal (TCn) is less than the total cost of producing it in the U.S. (TCus). This is due to the fact that Nepal's cost function (TCn) is based on the same production function but with different input prices. As long as the input prices (w and r) in Nepal are lower than those in the U.S., the total cost of production in Nepal will be less.