Question

A firm uses labour and capital to produce output according to the production function ??(??, ??) = 4??0.5??0.5, where L is the number of units of labour and K is the units of capital. The cost of labour is $40 a unit and the cost of capital is $10 a unit.

a) On a graph, draw an isocost line for this firm, showing combinations of capital and labour that cost $400 and another isocost line showing combinations that cost $200. What is the slope of these isocost lines?

b) Suppose that the firm wants to produce its output in the cheapest possible way. Find the number of capital it would use per worker.

c) On your graph, sketch the production isoquant corresponding to an output of 40 units. Calculate the amount of labour and capital used to produce 40 units in the cheapest possible way given the factor prices. Calculate the cost of producing 40 units at these factor prices.

d) How many units of labour and capital would the firm use to produce y units in the cheapest possible way? How much would this cost?

Answer 1

Q = f(L, K) = 4L^0.5K^0.5

(a)

Isocost equation: C = wL + rK = 40L + 10K

When C = 400,

400 = 40L + 10K

40 = 4L + K.........(1)

When L = 0, K = 40 (vertical intercept) and when K = 0, L = 40/4 = 10 (horizontal intercept).

When C = 200,

200 = 40L + 10K

20 = 4L + K.........(2)

When L = 0, K = 20 (vertical intercept) and when K = 0, L = 20/4 = 5 (horizontal intercept).

In following graph, AB & CD are isocosts showing (1) and (2) respectively.

40 2020

(b)

Cost is minimized when MPL/MPK = w/r = 40/10 = 4

MPL = $\dpi{80} \partial$ Q/$\dpi{80} \partial$L = 4 x 0.5 x (K/L)^0.5

MPK = $\dpi{80} \partial$ Q/$\dpi{80} \partial$K = 4 x 0.5 x (L/K)^0.5

MPL/MPK = K/L = 4

(c)

Since K/L = 4,

K = 4L

When Q = 40,

4L^0.5K^0.5 = 40

L^0.5K^0.5 = 10

Squaring,

LK = 100

L x 4L = 100

L² = 25

L = 5

K = 4 x 5 = 20

C = 40 x 5 + 10 x 20 = 200 + 200 = 400

In above graph, isoquant is Q0.

(d)

When output is y,

4L^0.5K^0.5 = y

L^0.5K^0.5 = y/4

Squaring,

LK = y² / 16

L x 4L = y² / 16

L² = y² / 64

L = y / 8

K = 4 x (y/8) = y / 2

C = 40 x l + 10 x K = 40 x (y/8) + 10 x (y/2) = 5y + 5y = 10y