Question

Assume the Cobb-Douglas production function
is Q= L ^0.75 K^0.5 and if price of labor per day is 5 birr and price of capital per day is 10 birr; and if total outlay (cost budget) per day is 400 birr,
A.Find L and K that maximize out put
B.What is the maximum out put the equilibrium L* and K*

Answer 1

a)

Given 

$Q =L^{0.75} K^{0.5} ......... (1)$

Price of labor = 5 birr per day 

Price of capital = 10 birr per day 

Total cost budget = 400 birr per day

Budget line or isocost line 

$5L+10K=400 ......(2)$

Maximize $Q =L^{0.75} K^{0.5}$

Subject to constraints $5L+10K=400$

Using lagrangian method

$X =L^{0.75} K^{0.5} +λ(400-5L-10K)$

FOC

$\frac{dX}{dL}=0, \frac{dX}{dK}=0, \frac{dX}{dλ}=0$

Differentiating Lagrangian function with respect to L

$\frac{dX}{dL}=0\\⇒0.75L^{−0.25}K^{0.5}=5λ ..........(3)\\ Now\\ \frac{dX}{dK}=0\\⇒0.5L^{0.75}K^{−0.5}=10λ ..........(4)\\ and \\ \frac{dX}{dλ}=0\\⇒5L+10K=400 ..........(2)$

Dividing eq 3 and 4

$⇒\frac{0.75L^{−0.25}K^{0.5}}{0.5L^{0.75}K^{−0.5}}=\frac{5λ}{10λ}\\⇒\frac{3 K}{2L}=\frac{1}{2}\\⇒K=\frac{L}{3}$

Putting value of K in eq 2

$5L+10(\frac{L}{3}) = 400\\⇒\frac{15+10}{3}L=400\\⇒L=400×\frac{3}{25}\\⇒L=48\\ And\space K = \frac{48}{3} = 16$

The optimal level of L is 48 and K is 16 that maximizes the output. 

(b)

Maximum output will be

$Q^*=48^{0.75}×16^{0.5}\\⇒Q^*= 72.944$