Suppose that two duopoly firms have constant marginal costs. Firm 1 faces the de

Question

Suppose that two duopoly firms have constant marginal costs. Firm 1 faces the demand function:

q1 = 60 – 2p1 + p2

And has a constant marginal cost of 30 per unit.

The demand Firm 2 faces is:

q2 = 75 – 2p2 + p1

Firm 2’s marginal cost is 15 per unit.

- Solve for the Bertrand Equilibrium. Compute equilibrium prices and profits.

- Describe the Bertrand paradox in one or two sentences and name two solutions to this paradox. Relate the outcome in (b) to the discussion of solutions of the Bertrand paradox.

Explanation / Answer

Profit1 = (60 – 2p1 + p2)(P1-30) = 60P1 - 2p12 +p1p2 - 1800 +60p1 -30p2 = 120p1 - 2p12 + p1p2 - 1800 +30p2

now dprofit1/dp1 = 0

dprofit1/dp1 = 120 - 4p1 + p2 = 0

p2 = 4p1 - 120 ............ eq1

now profit2 = (75 – 2p2 + p1) (p2-15) = 75p2 - 1125 - 2p22 +p1p2 + 30p2 - 15p1 =105p2 - 1125 - 2p22 +p1p2 - 15p1

now dprofit2/dp2 = 0

pdprofit/dp2 = 105 - 4p2 + p1 = 0

p1 = 4p2 - 105 .........eq 2

from eq 1 and eq 2

p1 = 4(4p1 - 120) - 105 = 16p1 - 480 - 105

p1 = 39

thus

p2 = 4(39) - 120 = 36

hence

q1 = 60 - 2(39) + 36 = 18 units

q2 = 75 - 2(36) + 39 = 42 units.

profit1 = 120*(39) - 2*(39)2 + (39*36) - 1800 +30*(36) = 2322

profit2 = 105*(36) - 1125 - 2*(36)2 +(39*36)- 15*(39) = 882

The bertrand paradox is the situation where firms reach equilibrium wherein they charge equal prices equal to marginal cost.

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