Question

Given the production function Q = 30K 0.7 L 0.5 and input prices r = 20 and w = 30.
( a) Determine an equation for the expansion path
( b) What is the efficient input combination for an output rate of Q = 200? For 500?

Answer 1

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Answer on Question #73879 - Economics - Microeconomics

Given the production function $Q = 30K^{0.7}L^{0.5}$ and input prices $r = 20$ and $w = 30$.

(a) Determine an equation for the expansion path

(b) What is the efficient input combination for an output rate of $Q = 200$? For 500?

Answer.

a)

Expansion path is determined by the condition

$$\frac{MP_L}{MP_K} = \frac{w}{r}$$

And,

$$MP_L = (30K^{0.7}L^{0.5})' = 30 \times 0.5 \times K^{0.7}L^{-0.5} = \frac{15K^{0.7}}{L^{0.5}}$$ $$MP_K = (30K^{0.7}L^{0.5})' = 30 \times 0.7 \times K^{-0.3}L^{0.5} = \frac{21L^{0.5}}{K^{0.3}}$$

So,

$$\frac{\frac{15K^{0.7}}{L^{0.5}}}{\frac{21L^{0.5}}{K^{0.3}}} = \frac{30}{20}$$ $$\frac{15K^{0.7} \times K^{0.3}}{L^{0.5} \times 21L^{0.5}} = \frac{30}{20}$$ $$\frac{5K}{7L} = \frac{3}{2}$$ $$10K = 21L$$

The equation of expansion path is

$$10K - 21L = 0$$

b)

We have to solve the system of equations

$$30K^{0.7}L^{0.5} = 200$$ $$10K = 21L$$

And, if $K=2.1L$ then

$$30 \times (2.1L)^{0.7}L^{0.5} = 200$$ $$30 \times (2.1L)^{0.7}L^{0.5} = 200$$ $$50.43L^{1.2} = 200$$ $$L = 3.15$$ $$K = 2.1 \times 3.15 = 6.62$$

The efficient input combination for an output rate of $Q = 200$ is

L=3.15

K=6.62

The efficient input combination for an output rate of $Q = 500$ is calculated from

$$50.43L^{1.2} = 500$$ $$L = 6.76$$ $$K = 2.1 \times 3.15 = 14.2$$

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