Chapter 5: Short Run Price Competition Price competition (Bertrand - - PDF document

Chapter 5: Short Run Price Competition

• Price competition (Bertrand competition)
• A1. Firms meet only once in the market.
• A2. Homogenous goods.
• A3. No capacity constraints.
• Bertrand paradox: same outcome as competitive

market...

• Solution: relax the assumptions....

– Repeated interaction (Chapter 6) – Product Differentiation (Chapter 7) – There exist capacity constraints;

∗ Cournot Equilibrium

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1 The Bertrand Paradox

• Duopoly, n = 2
• Because identical goods, consumers buy from the

supplier that charges the lowest price.

• Market demand: q = D(p)
• Marginal cost: c
• Firm i’s demand is

Di(pi, pj) =        D(pi) if pi < pj

1 2D(pi) if pi = pj

if pi > pj

• Firm i’s profit is

Πi(pi, pj) = (pi − c)Di(pi, pj)

Definition A Nash equilibrium in price (Bertrand equilibrium) is a pair of prices (p∗

1, p∗ 2) such that each

firm i’s price maximizes the profit of i, given the other firm’s price.

Πi(p∗

i, p∗ j) ≥ Πi(pi, p∗ j) for i = 1, 2 and for any pi.

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• The Bertrand Paradox (1883):

The unique equilibrium is p∗

1 = p∗ 2 = c.

• Firms price at MC and make no profit.
• If asymmetric marginal costs c1 < c2, it is no longer

an equilibrium. – p = c2 (firm 1 sets price lower than c2 to get the whole market) – firm 1 makes Π1 = (c2 − c1)D(c2) and firm 2 has no profit. 3

2 Solutions to Bertrand Paradox

2.1 Repeated Interaction (relax A1)

• Chapter 6
• If firms meet more than once, (p∗

1, p∗ 2) = (c, c) is not

the only equilibrium.

• Collusive behavior can be sustain by the threat of

future losses in a price war.

2.2 Product Differentiation (relax A2)

• Chapter 7
• With homogenous product: at equal price, con-

sumers are just indifferent between goods.

• If goods are differentiated: (p∗

1, p∗ 2) = (c, c) is no

longer an equilibrium.

• Example: spacial differentiation

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2.3 Capacity Constraints (relax A3)

• Edgeworth solution (1887).
• If firms cannot sell more than they are capable
• f producing: (p∗

1, p∗ 2) = (c, c) is no longer an

equilibrium.

• Why? If firm 1 has a production capacity smaller

than D(c), firm 2 can increase his price and get a positive profit.

• Example: 2 hotels in a small town, number of beds

are fixed in SR.

• The existence of a rigid capacity constraint is

a special case of a decreasing returns-to-scale technology.

• Why? Firm 1 has a marginal cost of c up to the

capacity constraint and then MC = ∞. 5