Answer to Question #283101 in Microeconomics for Meselech

Question

Suppose that the SAC curve function of a firm is given by TC = 4Q³+2Q²+3Q²+Q+20

A. Find the expression of TFC and TVC

B. Derive the expression of AFC,AVC,AC and MC

C. Find the level of out put that minimize MC and AVC

D. Find the minimum value of MC and AVC

Answer 1

Solution:

A.). Total Fixed Cost (TFC) = 20

Total Variable Cost (TVC) = 4Q³+2Q²+3Q²+Q

 

B.). Average Fixed Cost (AFC) = $\frac{Total \; Fixed \; Cost}{Q} = \frac{20}{Q}$

Average Variable Cost (AVC) = $\frac{Total \; Variable \; Cost}{Q} = \frac{4Q^{3}+2Q^{2} +3Q^{2}+Q }{Q} = 4Q^{2}+2Q + 3Q$

Average Cost (AC) = $\frac{Total \; Cost}{Q} = \frac{4Q^{3}+2Q^{2} +3Q^{2}+Q +20 }{Q} = 4Q^{2}+2Q + 3Q +\frac{20}{Q}$

Marginal Cost (MC) = $\frac{\partial TC} {\partial Q} = 12Q^{2}+4Q + 6Q$

 

C.). The level of output that minimizes MC:

Marginal Cost (MC) = $\frac{\partial TC} {\partial Q} = 12Q^{2}+4Q + 6Q$

Derive the Derivative of MC and set MC to zero and solve for Q:

$\frac{\partial MC} {\partial Q} = 24Q+4 + 6$ = 0

24Q = -10

Q = -0.42

 

The level of output that minimizes AC:

Average Variable Cost (AVC) = 4Q²+2Q+3Q

Derive the Derivative AVC and set AVC to zero and solve for Q:

$\frac{\partial AVC} {\partial Q}$ = 8Q + 2 + 3 = 0

8Q = -5

Q = -0.63

 

D.). Find the minimum value of MC and AVC:

The minimum value of MC:

MC = 12Q² + 4Q + 6Q

MC = 12(-0.42²) + 4(-0.42) + 6(-0.42) = 2.1168 – 1.68 – 2.52 = -2.08

The minimum value of MC = -2.08

 

The minimum value of AVC:

AVC = 4Q²+2Q+3Q

AVC = 4(-0.63²) + 2(-0.63) + 3(-0.63) = 1.5876 – 1.26 – 1.89 = -1.56

The minimum value of AVC = -1.56