Question

Assume that the manufacturing of cellular phones is a perfectly competitive indus-try. The market demand for cellular phones is described by a linear demand function:QD=6000−50P9. There are fifty manufacturers of cellular phones. Each manufacturerhas the same production costs. These are described by long-run total and marginal costfunctions ofTC(q)=100+q2+10q,andMC(q)=2q+10.a. Show that a firm in this industry maximizes profit by producingq=P−102.b. Derive the industry supply curve and show that it isQS=25P−250.c. Find the market price and aggregate quantity traded in equilibrium.d. How much output does each firm produce? Show that each firm

Answer 1

(a) A perfectly competitive firm produces at P =MC

P = 2q +10

=> P -10 = 2q
=> q = (P-10)/2

Hence, each firm produces q =(P-10)/2

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(b) P = MC is a supply function of a perfectly competitive firm.

P = MC

=> P = 2q +10

=> P -10 = 2q
=> q = (P-10)/2

Individual firm supply function: q = (P-10)/2

There are 50 identitical firms;

Industry supply curve: Qs = 50q

=> Qs = 50[(P-10)/2]

=> Qs = 50[(0.5P - 5)]

=> Qs = 25P - 250

Industry supply curve: Qs = 25P - 250

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(3) Market demand curve: Qd = (6000 - 50P)/9

Market supply curve: Qs = 25P - 250

At market equilibrium point; Qd = Qs

=> (6000 - 50P)/9 = 25P - 250

=> 6000 - 50P = 9(25P - 250)

=> 6000 - 50P = 225P - 2250

=> 6000 + 2250 = 225P + 50P

=> 8250 = 275P

=>P = (8250 / 275)

=> P = 30

Equilibrium market price is 30

and Qd = Qs = Q at equilibrium

=> Q = 25P - 250

=> Q = 25(30) - 250

=> Q = 750 - 250

=> Q = 500

Equilibrium qggregate quantity is 500 units.

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(4) Individual firm supply function: q = (P-10)/2

Put P =30

=> q = (30 -10)/2

=> q = 10

Each firm will produce 10 units.

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Profit of each firm:

Profit = TR - TC

=> Profit = Pq - (100 + q2 + 10q)

=> Profit = (30 * 10) - (100 + 10² + 10 *10)

=> Profit = 300 - (100 + 100 + 100)

=> Profit = 300 -300

=>Profit = 0

Each firm earns zero profit in equilibrium.

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