Question

Suppose the short run market price a competitive firm faces is birr 9 and the total costof the firm is TC=200+Q+0.02Q2.
a) calculate the short run equilibrium output and profit of the firm
B)drive 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

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

Now, setting Price = Marginal cost,

We get, $9 = 1+0.04 Q$

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

Therefore, Short-run equilibrium output = 200 units

• Profit

Profit = Total revenue - Total Cost

At Q = 200 

 Total Revenue = Price × Quantity

$=9 × 200\\ = Birr 1800$

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

(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

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

$Average \space total \space cost = \frac{ Total \space cost}{Quantity}$

$=\frac{200+ Q + 0.02Q^2}{Q}\\ =\frac{200}{Q}+1+0.02Q$

$Average \space Variable\space cost = \frac{Total \space Variable\space cost}{Quanitty } [ TC = 200 + Q+0.02Q^2 \\where \\FC = 200 \& VC = Q+0.02Q^2)\\ =\frac{Q + 0.02Q^2}{Q}\\ =1 +0.02Q$

• Value at Equilibrium output

Now, At Q = 200

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

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

$Average \space Total\space Cost = 1 + (0.02×200)\\ = 1+4\\ = Birr \space 5$

(C) Producers’ surplus at the equilibrium output.

Producer Surplus = Total revenue - Variable cost 

At Q = 200

Total Revenue = Price × Quantity

$=9 × 200\\ = Birr \space 1800$

Variable cost $= Q+0.02Q^2$

At Q = 200, Variable cost  $= 200 + (0.02×200×200)$

$= Birr \space 1000$

SO,Producers' Surplus $= 1800 − 1000$

$= Birr \space 800$

d)

profit =total revenue -total cost

total revenue=$price \times quantity$

let output be y

profit$=(9\times y)-1200$

$0=9y-1200\\1200=9y\\y=133.33$