Question

1.    Given the following production function Q = 20L²K and the unit prices of labor and capital to be Birr 48 and 12 respectively, then

a. What combination of labor and capital maximizes output with a cost of Birr 720?

b. What is the maximum output?

c. Show the output maximizing condition graphically.

Answer 1

a.

For solving this question, we will use Lagrange method

$Q = 20L^2K$

The equation for isocost line is $48L + 12K = 720$

So, using Lagrange method,

$L = 20L^2K - \lambda (48L + 12K - 720)$

Differentiating w.r.t L and K 

$20\times2LK - \lambda(48) = 0 ..................................... (1)$

$20L^2 - \lambda(12) = 0 ............................................. (2)$

Dividing equation (1) by (2),

$\frac{2K}{L }= \frac{48}{12}$

$K = 2L$

Putting this value of K in isocost line

$48L + 12(2L) = 720$

$72L = 720$ $L = 10$

and$K = 2\times 10 = 20$

So, combination of labor that maximizes output is L = 10 and K = 20 units.

b.

Maximum output can be calculated by putting value of K and L in production function Q

So,

$Q = 20(10)^2(20) = 40000$

So, maximum output = 40000

c.

The graph below is showing the maximizing condition.

AB is the isocost line. Intercept on X axis is calculated by$\frac{720}{48} = 15$ and on Y axis $\frac{720}{12} = 60$ .

Q is the production function curve.

Point E is showing the point where we will have maximum production of 40000, using L = 10 and K = 20 units. 

The maximizing condition is that the slope of isocost line and production curve should be equal.