Question

Suppose the short run market price a competitive firm faces is Birr 9 and the total cost of the firm is: TC = 200 + Q + 0.02Q 2 . Answer the questions that follow.

(A) Calculate the short run equilibrium output and profit of the firm.

(B) Derive the MC, ATC, and AVC and calculate the values at the short run equilibrium output.

(C) Calculate the producers’ surplus at the equilibrium output.

(D) Find the output level that will make the profit of the firm zero.

Answer 1

(A) Short-run equilibrium output and profit of the firm:

• Short-run equilibrium output

Short-run equilibrium condition : Price = Marginal cost = Marginal revenue

We have, Price = Birr 9

Marginal cost = change in total cost due to quantity

         $=\frac{d(Total cost \space t)} {d(quantity)}\\ = \frac{d( 200 + Q + 0.02Q^2)} {d( Q)}\\ = 1 + 0.04 Q$   

Now, setting Price = Marginal cost,

 We get, $9 = 1+0.04 Q$

  $0.04 Q = 8\\ Q = \frac{8} {0.04} 80.04 = 200\space units.$

 Therefore, Short-run equilibrium output = 200 units

• Profit

Profit = Total revenue - Total Cost

At Q = 200 

 Total Revenue = Price × Quantity

            $=9 × 200 \\ = Birr 1800$

$Total \space Cost = 200 + Q + 0.02Q^2\\ = 200 + 200 + 0.02× ×(200)^2\\ = Birr 1,200 Thus, \\Profit = 1800 - 1200 \\ = Birr 600$

(B) Value of MC, ATC, and AVC at the short-run equilibrium output.

• MC, ATC, AVC

Marginal cost = change in total cost due to change in quantity

            $=\frac{d(Total cost \space t)} {d(quantity)}\\ = \frac{d( 200 + Q + 0.02Q^2)} {d( Q)}\\ = 1 + 0.04 Q$

Average total cost

$=\frac{Total \space cost}{Quantity}\\ =\frac{200+ Q + 0.02Q^2}Q\\ =\frac{200}{Q}+1+0.02Q$

Average Variable cost $=\frac{Total Variable cost t} {Quantity}$

$TC = 200 + Q+0.02Q^2 \\where\\ FC = 200 \& VC = Q+0.02Q^2)\\ =\frac{Q + 0.02Q ^2} {Q}\\ =1 +0.02Q$

• Value at Equilibrium output

Now, At Q = 200

Marginal cost 

$1 + (0.04 × × 200)\\ = 1 + 8 \\ =Birr 9$

Average Total Cost

$\frac{200}{200}+1+ (0.02×200)\\ = 1+1+4\\ = Birr 6$

Average Total Cost

$1 + (0.02×200)\\ = 1+4\\ = Birr 5$

(C) Producers’ surplus at the equilibrium output.

Producer Surplus = Total revenue - Variable cost 

At Q = 200

Total Revenue = Price × Quantity

$9 × 200\\ = Birr 1800$

Variable cost $=Q+0.02Q^2$

At Q = 200, Variable cost

 $= 200 + (0.02×200×200)\\ = Birr 1000$

SO,Producers' Surplus

$= 1800 − 1000\\ = Birr 800$

d) output will also be zero