Question

2. A firm’s production function is given by q= L^1/2+ K. The price of labour is fixed at w = 1, and the price of capital is fixed at r = 8.

a. Find the firm’s marginal rate of technical substitution.

b. Suppose both labour and capital can be varied by the firm, and that the firm wishes to produce q units of output. Use the answer to (a) to find the cost-minimising amounts of labour and capital (as functions of q). You may assume, here and for the other parts of this question, that 4≥ q  .

c. Use the answer to (b) to find the firm’s cost function, and from this, its marginal and average cost functions.

d. For which values of q does this firm face economies of scale? Diseconomies of scale?

e. Explain in words the way this firm expands production as output rises.

Answer 1

Solution:

q = L^1/2 + K

a. Marginal rate of technical substitution, MRTS = MPL/MPK, where MPL is the marginal product of labor and MPK is marginal product of capital.

So, MPL = = (1/2)L^{1/2 - 1} = 1/(2L^1/2)

And MPK = = 1

Thus, MRTS = (1/(2L^1/2))/1 = 1/(2L^1/2)

b. Cost minimizing output combination occurs where ratio of marginal productivities equal the ratio of input prices. So, MRTS = wage rate/rental rate

1/(2L^1/2) = 1/8

2L^1/2 = 8

So, L = (8/2)² = 16 units

And using the production function, then q = 16^1/2 + K

q = 4 + K

So, K = q - 4

(Notice the quasi linearity of the production function. This implies that optimal level of L will always be 16 units, unless of course any of the input prices change)

c. Total cost, C = w*L + r*K

So, C(q) = 1*16 + 8*(q - 4)

C(q) = 8q - 16

Then, marginal cost function becomes: MC(q) = = 8

Average cost function is: AC(q) = C(q)/q

AC(q) = (8q - 16)/q = 8 - 16/q

d. Economies of scale is achieved when by increasing the production level, q, the average cost decreases. Opposite holds for discontinued of scale: discontinued of scale is achieved when by increasing the production level, average cost also increases. So to solve this part, we need to find that output level which minimizes average cost.

Using derivatives, = 0 - (-1)*16/q² = 16/q²

So q = 0

For all levels of q, the firm experiences diseconomies of scale.