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 of a competitive firm is derived at the point where MR = MC.

The market equilibrium price for a competitive firm = MR

MR = 9

MC = Derivative of the Total Cost relative to quantity:

TC = 200 + Q + 0.02Q²

MC = $\frac{\partial TC} {\partial Q} = 1 + 0.04Q$

Set MR = MC:

MR = MC

9 = 1 + 0.04Q

9 – 1 = 0.04Q

8 = 0.04Q

Q = $\frac{8} {0.04} = 200$

Q = 200

The short-run equilibrium output of the firm = 200

Profit = TR – TC

TR = P x Q = 9 x 200 = 1,800

TC = 200 + 200 + 0.02(200²)

TC = 200 + 200 + 800 = 1,200

Profit = 1,800 – 1,200 = 600

The profit of the firm = 600

B.). The values of MC, ATC, and AVC are derived as follows:

Short-run equilibrium output (Q) = 200

MC = $\frac{\partial TC} {\partial Q} = 1 + 0.04Q$ = 1 + 0.04(200) = 1 + 8 = 9

MC = 9

ATC = $\frac{TC}{Q} = \frac{200 + Q + 0.02Q^{2} }{Q} = \frac{200 + 200 + 0.02(200)^{2} }{200} = 1 + 1 +4 = 6$

ATC = 6

AVC = $\frac{VC}{Q} = \frac{ Q + 0.02Q^{2} }{Q} = \frac{200 + 0.02(200)^{2} }{200} = 1 + 4 = 5$

AVC = 5

C.). Producer’s surplus at the equilibrium output:

Produce’s surplus = $\frac{1}{2} (200\times 9) = 0.5\times 1800 = 900$

Produce’s surplus = 900

D.). Derive the output level that will make the profit of the firm zero:

Set MC = ATC

MC = 9

ATC = $\frac{200}{Q} +0.02Q$

9 = $\frac{200}{Q}$ + 0.02Q

Multiply both sides by 100:

900 = $\frac{20000}{Q}$ + 2Q

Divide both sides by Q:

900Q = 20000 + 2Q²

900Q – 2Q² – 20000 = 0

Q = 23

The output level that will make the profit of the firm zero = 23