Answer to Question #249381 in Economics of Enterprise for Thom

Question

Given The Cobb- Douglas Production Function For Preshzo Company Limited As Follows: 𝑄=K0.25L0.75 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

Answer 1

Solution:

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