Question #47996

Each firm in a perfectly competitive market has a total cost function given by
C(q) = 10q – 2q² + q3, market’s aggregate demand function is given by q(p) = 376 – 8p.
a) Calculate each firm’s supply function in the short-run.
b) What quantity produces each firm in the short-run if p = 42?
c) How many firms are active in this market in the short-run if p = 42?
d) Calculate the quantity produced by every firm in the long-run and the price at which it offers this quantity.
e) How many firms are active in this market in the long-run?

Expert's answer

C(q) = 10q – 2q² + q^3, q(p) = 376 – 8p.

a) Calculate each firm’s supply function in the short-run.

In the short-run supply equals marginal cost, so:

S = MC = C' = 10 - 4q + 3q^2

b) What quantity produces each firm in the short-run if p = 42?

If p = 42, according to profit-maximization rule MC = MR = p = 42, so:

10 - 4q + 3q^2 = 42

3q^2 - 4q - 32 = 0

q = (4 + 20)/6 = 4 units

c) How many firms are active in this market in the short-run if p = 42?

If p = 42, quantity demanded Qd = 376 - 8*42 = 40

If every firm produces 4 units and total demand is 40 unitsm then there is 40/4 = 10 firms.

d) Calculate the quantity produced by every firm in the long-run and the price at which it offers this quantity.

In the long-run profit is maximized, when P = ATC = MC = MR.

ATC = TC/q = 10 - 2q + q^2

ATC = MC, so:

10 - 2q + q^2 = 10 - 4q + 3q^2

2q^2 - 2q = 0

q(q - 1) = 0

q1 = 0, q2 = 1

p1 = $10, p2 = 10 - 2*1 + 1 = $9

e) How many firms are active in this market in the long-run?

If q = 0, then there will be no firms in the market.

If q = 1, there will be 40/1 = 40 firms.

a) Calculate each firm’s supply function in the short-run.

In the short-run supply equals marginal cost, so:

S = MC = C' = 10 - 4q + 3q^2

b) What quantity produces each firm in the short-run if p = 42?

If p = 42, according to profit-maximization rule MC = MR = p = 42, so:

10 - 4q + 3q^2 = 42

3q^2 - 4q - 32 = 0

q = (4 + 20)/6 = 4 units

c) How many firms are active in this market in the short-run if p = 42?

If p = 42, quantity demanded Qd = 376 - 8*42 = 40

If every firm produces 4 units and total demand is 40 unitsm then there is 40/4 = 10 firms.

d) Calculate the quantity produced by every firm in the long-run and the price at which it offers this quantity.

In the long-run profit is maximized, when P = ATC = MC = MR.

ATC = TC/q = 10 - 2q + q^2

ATC = MC, so:

10 - 2q + q^2 = 10 - 4q + 3q^2

2q^2 - 2q = 0

q(q - 1) = 0

q1 = 0, q2 = 1

p1 = $10, p2 = 10 - 2*1 + 1 = $9

e) How many firms are active in this market in the long-run?

If q = 0, then there will be no firms in the market.

If q = 1, there will be 40/1 = 40 firms.

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