Suppose that a firm has the production function F(K, L) = K^1/4 L^1/4, where K a
ID: 3011782 • Letter: S
Question
Suppose that a firm has the production function F(K, L) = K^1/4 L^1/4, where K and L respectively denote capital and labor. Let r = 2 be the price of K and w = 2 be the price of L. Assume that both K and L are variable. Find the greatest output that the firm can produce at a total cost of C. What are the total, marginal, and average costs as a function of output Q? What is the elasticity of the total cost with respect to output? Explain whether the cost structure displays economies or diseconomies of scale.Explanation / Answer
(4) Production Function, F(K,L) = K1/4L1/4
=> Q = K1/4L1/4
Total cost can be calculates as C = w*L + r*K = 2L + 2K
(a) We will get maximum output when (MPL / MPK) = w / r = 2/2 = 1
Since MPL = dQ / dL = 0.25 x K1/4 / L3/4 ........(1)
MPK = dQ / dK = 0.25 x L1/4 / K3/4.........(2)
Divide 1 by 2, we get,
MPL / MPK = (0.25 x K1/4 / L3/4) / (0.25 x L1/4 / K3/4) = K / L = 1
=> K and L are equal
Q = L1/4 x L1/4 = L1/4 = K1/4
=> L = K = Q2 ........(3)
From total cost, C = 2*L + 2*K = 2*L + 2*L = 4L
From 3, C = 4 x Q2 ........(4)
Q = C1/2 / 2 ............(5)
This will be the greatest output when C is the total cost.
(b) Marginal cost, MC = dC / dQ
From 4, MC = 8Q ..............(6)
Average cost = Total Cost / Production = C/Q = 4Q .............(7)
(c) Elasticity, E = MC / Average Cost
From 6 and 7,
E = 8Q / 4Q = 2
(d) Average Total Cost = 4Q
Hence we can say that as output goes on, average total cost will also increase, displays diseconomies of scale.
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