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

1

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.

2

• 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 no

longer an 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, consumers

are just indifferent between goods.

• If goods are differentiated: (p∗

1, p∗ 2) = (c, c) is no longer

an equilibrium.

• Example: spacial differentiation

4

2.3 Capacity Constraints (relax A3)

• Edgeworth solution (1887).
• If firms cannot sell more than they are capable of

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

3 Decreasing Returns-to-scale and Capacity Constraints

3.1 Rationing rules

• Cost of production Ci(qi) is increasing and convex,

C0

i(qi) > 0 and C00 i (qi) ≥ 0, for any qi.

• If firm 1 has a capacity constraint, there exists a residual

demand for 2. It depends on the rationing rule.

• p1 < p2
• Supply of firm 1 is q1 = S1(p1)

3.1.1 The efficient-rationing rule (parallel)

• Suppose q1 < D(p1)
• Residual demand for firm 2 is

e D2(p2) = ( D(p2) − q1 if D(p2) > q1

• therwise
• It is as if the most eager consumers buy from 1, others

from 2.

• Efficient because maximizes consumers’ surplus.

6

3.1.2 The proportional-rationing rule (randomized)

• All consumers have the same probability of being

rationed.

• Demand that cannot be satisfied at p1: D(p1) − q1
• Thus fraction of consumers that cannot buy at p1

(probability of not being able to buy from 1) is

D(p1) − q1 D(p1)

• Residual demand for 2 is

e D2(p2) = D(p2)D(p1) − q1 D(p1)

• Not efficient for consumers.
• However, firm 2 prefers this rule because his residual

demand is higher for each price.

3.2 Price Competition

• Decreasing returns-to-scale softens price competition.

Both firms’ prices exceed the competitive price.

• Demand function is D(p) = 1 − p
• Inverse demand function is P = P(q1+q2) = 1−q1−q2

7

• The two firms have capacity constraints qi ≤ qi
• Investment: c0 ∈ [3

4, 1] unit cost of acquiring the

capacity qi.

• Marginal cost of producing is 0 up to qi and then it is

∞.

• Rule is efficient rationing rule.
• Price that maximizes (gross) monopoly profit

Max

p

p(1 − p)

is pm = 1/2 and thus Πm = 1/4.

• Thus the (net) profit of firm i is at most 1/4 − c0qi and

is negative for qi ≥ 1/3.

• Assume that qi ∈ [0, 1/3]
• The unique Nash equilibrium is p∗ = 1 − (q1 + q2)
• Proof:

– Is it worth charging a lower price? NO because of the capacity constraints. – Is it worth charging a higher price? Profit of i if price

p ≥ p∗ is Πi = q(1 − q − qj) where q is the quantity

8

sold by firm i at price p. Furthermore

∂Πi ∂q = 1 − 2q − qj

and ∂2Πi

∂q2 < 0

– For q = qi, 1 − 2qi − qj > 0 as qi < 1/3 and

qj < 1/3.

– Hence, increasing p above p∗ is not optimal (i.e. lowering q below qi).

• It is as if the 2 firms produce at the capacities, and an

auctioneer equals supply and demand.

• For qi ∈ [0, 1/3], i = 1, 2, the firms’ reduced form

profit functions are the exact Cournot forms (Levitan and Shubik (1972))

• Gross

Πig(qi, qj) = (1 − (qi + qj))qi

• Net

Πig(qi, qj) = (1 − (qi + qj) − c0)qi

• Same result with the proportional rationing rule

(Beckman (1967)). 9

• The assumption of large investment cost insures small

capacities, and thus we find a solution.

• For larger capacities: no equilibrium in pure strategies,
• nly in mixed strategies.

3.3 Ex ante Investment and ex post price competition

• A two stage-game in which
• 1. firms simultaneously choose qi
• 2. Firms observe the capacities and simultaneously

choose pi.

• is equivalent to a one-stage game in which firms choose

quantities qi and an auctioneer determines the market price that clears the market p = P(q1 + q2); Cournot (1838).

• Examples of ex ante choice of scale:

– Hotel (cannot adjust its capacity), – Vendor of perishable food. 10

4 Cournot Competition

• Firms choose the quantities simultaneously.
• Each firm maximizes his profit

Πi(qi, qj) = qiP(qi + qj) − Ci(qi)

• where Πi(.) is strictly concave in qi and twice differen-

tiable.

• FOCi:

qi ∂P(.) ∂qi + P(qi + qj) − ∂Ci(qi) ∂qi = 0 ⇒ qi = Ri(qj) Best response function of i to qj

• Lerner index is

$i =

qi Q

−P

Q 1

∂P(.) ∂qi

= αi ε

• where αi is firm i’s market share .
• Thus the Lerner index is

– proportional to the firm’s market share, – inversely proportional to the elasticity of demand. 11

4.1 Case of linear demand

Duopoly

• P(Q) = 1 − Q
• Ci(qi) = ciqi
• What are the reaction functions?

Ri(qj) = 1 − ci − qj 2

for i, j = 1, 2 and i 6= j.

• What is the Cournot equilibrium?

q∗

i = 1 − 2ci + cj

3

• What are the profits of the firms?

Πi = (1 − 2ci + cj)2 9

• A firm’s output decreases with its MC, ∂qi

∂ci < 0.

• A firm’s output increases with its competitor MC,

∂qi ∂cj > 0.

12

• For more general demand and cost functions, these 2

conditions are true:

• a. If reaction curves are downward sloping (quantities

are strategic substitutes)

• b. if reaction curves cross only once, and slope of

|R2| <slope of |R1| .

Generalization to n firms

• Q = Pn

i=1 qi

• P(Q) = 1 − Q
• Ci(qi) = cqi
• FOCi:

qi

∂P(.) ∂qi + P(Q) − ∂Ci(qi) ∂qi

= 0 ⇒ qi + 1 − Q − c = 0

• and Lerner index

$i = αi ε

• Symmetric equilibrium Q = nq, and thus

q = 1 − c n + 1

13

• Market price is

p = 1 − nq = c + 1 − c n + 1

• and the profit of each firm is

Π = (1 − c)2 (n + 1)2

• If n → ∞, thus p → c.
• Exercises 5-3, 5-4, 5-5

14

5 Concentration Indices

• Structure - conduct - performance paradigm
• Structure: concentration of the market
• Market share of i is

αi = qi

Q where i = 1, ..., n and Pn i=1 αi = 1

Concentration indices:

• m-firms concentration ratio (m < n)

Rm = Pm

i=1 αi

– where α1 ≥ ... ≥ αm ≥ ... ≥ αn – If Rm → 0, low concentration (many firms) – If Rm → 1, high concentration (few firms): m (usually m = 4) or less firms produce all of the

• utput of the industry.
• Herfindahl index

RH = 10, 000 Pn

i=1 α2 i

– US department of Justice uses this index. The limit for antitrust case is 1,800. – RH → 0, low concentration – RH → 10, 000 high concentration 15

6 Conclusion

16