Question

Consider a producer making choices over two inputs, labour (l) and capital (k) with prices w = 3 and r = 1. The production technology is f(l, k) = l + 3k.

What is the marginal rate of technical substitution (MRTS)? Is there diminishing MRTS?

Find the input demands in long-run (as a function of output level)?

Find the long-run total cost, marginal cost, and average cost functions?

Do the properties of a typical cost function hold for the long run total cost function (show any two properties)?

Answer 1

Consider a producer making choices over two inputs, labour (l) and capital (k) with prices w = 3 and r = 1. The production technology is f(l, k) = l + 3k.

What is the marginal rate of technical substitution (MRTS)? Is there diminishing MRTS?

MRTS is the ratio of MPL and MPK. Here the two inputs have a fixed rate of substitution which is MRTS = 1/3 that is, 1 unit of capital is equally productive to three units of labor. MRTS is fixed and not diminishing

Find the input demands in long-run (as a function of output level)?

Equimarginal principle indicates that if MPL/w > MPK/r then the firm should used only labor if they are perfect substitutes. Here we see that MPL/w = 1/3 and MPK/r = 3. Since MPL/w < MPK/r, firm should use only capital in the long run given the prices

L* = 0, K* = Q/3 are the long run demand functions given the prices

For general prices w and r, we have the demand functions

L = Q, K = 0 , when MPL/w > MPK/r, that is, r/w > 3

L = 0, K = Q/3, if r/w < 3

Any combination that satisfies Q = L + 3K if r/w = 3

Find the long-run total cost, marginal cost, and average cost functions?

Cost function is C = wL + rK

C = wQ when   r/w > 3 ..........(in this case with r/w = 1/3, cost function is C = 3Q)

C = rQ/3,  when r/w < 3 .........(in this case with r/w = 1/3, cost function is C = Q/3)

C = wL + rK = wQ + rQ/3 ........(in this case with r/w = 1/3, cost function is C = 3Q + Q/3)

Do the properties of a typical cost function hold for the long run total cost function (show any two properties)?

Long run cost function has no fixed cost. Here there is no fixed cost so this property is satisfied. Long run ATC envelops the short run ATC. It is U-shaped as higher Q increases cost function