Question

7. Suppose that cost function is of a firm is given by C=Q ³ -4Q ² +14Q+60. Then , determine

a. Fixed Cost function and AFC at Q=2

b. TVC function and AVC at Q=2

c. MC function and MC at Q=2

d. Minimum average cost

8. Suppose Q gives the production function Q=150KL and the price of labor and capital is 2.5 and 6 birr respectively. If the total outlays of the firm is 3000 Birr. Determine the level of employment of both inputs that maximizes output.

Answer 1

Solution:

7.). a.). Fixed Cost Function and AFC at Q = 2

C = Q³ – 4Q² + 14Q + 60

Fixed Cost Function = 60

Average Fixed Cost (AFC) = $\frac{TFC}{Q}$ = $\frac{60}{2}$ = 30

 

b.). TVC function and AVC at Q = 2:

TVC function = Q³ – 4Q² + 14Q

AVC = $\frac{TVC}{Q}$ = Q³ – 4Q² + 14Q$\div$ Q = Q² – 4Q + 14 = 2² – 4(2) + 14 = 4 – 8 + 14 = 10

AVC = 10

 

c.). MC function and MC at Q=2:

MC function = $\frac{\partial TC} {\partial Q}$ = 3Q² – 8Q + 14

MC = 3Q² – 8Q + 14 = 3(2²) – 8(2) + 14 = 12 – 16 + 14 = 10

MC = 10

 

d.). Minimum average cost:

Derive average cost:

Average cost = Q³ – 4Q² + 14Q + $\frac{60}{Q}$ = Q² – 4Q + 14 + $\frac{60}{Q}$

AC = Q² – 4Q + 14 + $\frac{60}{Q}$

$\frac{\partial AC} {\partial Q}$ = 2Q – 4 – $\frac{60}{Q^{2} }$

2Q – 4 – $\frac{60}{Q^{2} }$ = 0

Q = 3.94

Minimum Average Cost = 3.94

 

8.). Derive MP_L and MP_K

MP_L = $\frac{\partial Q} {\partial L}$ = 150K

MP_K = $\frac{\partial Q} {\partial K}$ = 150L

Maximization is where: $\frac{MP_{L} } {MP_{K} }$ = $\frac{w}{r}$

w = 2.5

r = 6

$\frac{150K}{150L} = \frac{2.5}{6}$

K = 2.39L

 

Q = 150KL

3,000 = 150 (2.39L) (L)

3000 = 358.5L²

L = 2.89

K = 2.39L = 2.39 (2.89) = 6.9

K = 6.9

The level of employment of both inputs that maximize output (L, K) = 2.89, 6.9