Question

4. Suppose a firm’s technology is described by a production function where the quantity produced is equal to the square root of capital times labor (that is, Q = √(L●K)

a. Using graph paper, draw the isoquant for output = 8 units.

b. Suppose the firm has $128 to spend on its variable inputs and the price of L and K is $8. On the same graph on which you drew the isoquant, draw the isocost line. What is the optimal combination of labor and capital?

c. Suppose the price of labor increases to $16 and the price of capital decreases to $4. draw the new isocost line and determine the new optimal combination of labor and capital.

d. Repeat c) for price of labor at $4 and capital at $16.

Answer 1

Q = (LK)

(a) When Q = 8,

(LK) = 8

LK = 64

K = 64 / L

Data table used for graph:

L	K
1	64
2	32
4	16
8	8
16	4
32	2
64	1

Isoquant graph:

(b) Total cost (TC) ($) = 8L + 8K

128 = 8L + 8K

16 = L + K

When L = 0, K = 16 (Vertical intercept) & when K = 0, L = 16 (Horizontal intercept). In above graph, AB is the isocost line.

TC is minimized when MPL / MPK = PL / PK = 8/8 = 1

MPL = Q / L = (1/2) x (K / L)^1/2

MPK = Q / K = (1/2) x (L / K)^1/2

MPL / MPK = K / L = 1

K = L

Substituting in TC function,

16 = L + L = 2L

L = 8

K = L = 8

(c) New TC ($): 128 = 16L + 4K, or 32 = 4L + K

When L = 0, K = 32 (Vertical intercept) & when K = 0, L = 32/4 = 8 (Horizontal intercept). In above graph, CD is the new isocost line.

MPL / MPK = K / L = 16 / 4 = 4

K = 4L

Substituting in TC function,

32 = 4L + 4L = 8L

L = 4

K = 4L = 4 x 4 = 16

(d) New TC ($): 128 = 4L + 16K, or 32 = L + 4K

When L = 0, K = 32/4 = 8 (Vertical intercept) & when K = 0, L = 32 (Horizontal intercept). In above graph, EF is the new isocost line.

MPL / MPK = K / L = 4/16 = 1/4

L = 4K

Substituting in TC function,

32 = 4K + 4K = 8K

K = 4

L = 4K = 4 x 4 = 16