Answer to Question #249717 in Microeconomics for Thom

Question

Given the Cobb- Douglas production function for Preshzo Company limited as follows:

𝑄=K^0.25L^0.75

She estimates that the cost of labour is $3 and the cost of capital $4. You are required to do the following:

a) If she wants to produce 50kg of are output, formulate both the objective and constraint functions for Preshzo Company limited.

b) Find the minimum cost for both labour and Capital required to produce 50 kg units of output.

c) Find the shadow price lambda and interpret the coefficient of lambda.

d) Prove that the maximum unit produced at the lowest cost is 50 kg.

Answer 1

A.). The objective and constraint functions are as follows:

Constraint function: K^0.25L^0.75 = 50

Objective function: C = wL + rK

B.). To derive the minimum cost of labor and capital required to produce 50kg units of output:

MRTS = $\frac{MP_{L} }{MP_{K}} = \frac{w}{r}$

MP_L = $\frac{\partial Q} {\partial L} =$ 0.75^0.25L^-0.25

MP_K = $\frac{\partial Q} {\partial K} =$ 0.25K^-0.75L^0.75

MRTS = $\frac{MP_{L} }{MP_{K}} = \frac{0.75^{0.25} L^{-0.25} }{0.25K^{-0.75} L^{0.75} } = \frac{3K}{L}$

$\frac{3K}{L} = \frac{w}{r} = \frac{3}{4}$

$\frac{3K}{L} = \frac{3}{4}$

L = 4K

Substitute in the Cobb-Douglas function to derive Capital:

50 = K^0.25L^0.75 = K^0.254K^0.75

K = 12.5

L = 4K = 12.5 $\times$ 4 = 50

The minimum cost of labor = 50

The minimum cost of capital = 12.5

C.) Shadow price $λ=\frac{mc_l}{mp_l}=\frac{mc_k}{mp_k}$

λ= marginal cost of increasing fivms productivity with respect to one factor input

$λ = 4(\frac{L}{K})^{0.25}= 4(\frac{50(4^{0.25}}{\frac{50}{4^{0.75}}})^{0.25} \\$

$=4(4^{0.25} \times 4^{0.75})^{0.25} \\ =4(4)^{0.25} \\ λ=4^{1.25} \\$

$mc_K =mc_L$

D.)We have to prove that the max unit is produced at lowest cost is sokg.

$Q=K^{0.25} \times L^{0.75} \\ =(\frac{50}{4^{0.75}})^{0.25} (50(4^{0.25}))^{0.75} \\ = 50(\frac{4^{(0.25)(0.75)}}{4^{(0.25)(0.75)}}) \\$

Q=50units

So, at the lowest cost, the max units produced is 50.