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 


 D(pi) if pi < pj
Di(pi, pj ) = 21 D(pi) if pi = pj


 0
if p > p
i
j
• 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,
Ci0(qi) > 0 and Ci00(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 q 1 = S1(p1)
3.1.1 The efficient-rationing rule (parallel)
• Suppose q 1 < D(p1)
• Residual demand(for firm 2 is
D(p2) − q 1 if D(p2) > q 1
e
D2(p2) =
0
otherwise
• 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) − q 1
D(p1)
• Residual demand for 2 is
e 2(p2) = D(p2) D(p1) − q 1
D
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 ≤ q i
• Investment: c0 ∈ [ 34 , 1] unit cost of acquiring the
capacity q i.
• Marginal cost of producing is 0 up to q i and then it is
∞.
• Rule is efficient rationing rule.
• Price that maximizes (gross) monopoly profit
Maxp(1 − p)
p
is pm = 1/2 and thus Πm = 1/4.
• Thus the (net) profit of firm i is at most 1/4 − c0q i and
is negative for q i ≥ 1/3.
• Assume that qi ∈ [0, 1/3]
• The unique Nash equilibrium is p∗ = 1 − (q 1 + 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 − q j ) where q is the quantity
8
sold by firm i at price p. Furthermore
∂Πi
∂q = 1 − 2q − q j
2
i
and ∂∂qΠ2 < 0
– For q = q i, 1 − 2q i − q j > 0 as q i < 1/3 and
q j < 1/3.
– Hence, increasing p above p∗ is not optimal (i.e.
lowering q below q i).
• It is as if the 2 firms produce at the capacities, and an
auctioneer equals supply and demand.
• For q i ∈ [0, 1/3], i = 1, 2, the firms’ reduced form
profit functions are the exact Cournot forms (Levitan
and Shubik (1972))
• Gross
Πig (q i, q j ) = (1 − (q i + q j ))q i
• Net
Πig (q i, q j ) = (1 − (q i + q j ) − c0)q i
• 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,
only in mixed strategies.
3.3 Ex ante Investment and ex post price
competition
• A two stage-game in which
1. firms simultaneously choose q i
2. Firms observe the capacities and simultaneously
choose pi.
• is equivalent to a one-stage game in which firms choose
quantities q i and an auctioneer determines the market
price that clears the market p = P (q 1 + q 2); 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 differentiable.
• FOCi:
∂Ci(qi)
∂P (.)
qi
+ P (qi + qj ) −
=0
∂qi
∂qi
⇒ qi = Ri(qj ) Best response function of i to qj
• Lerner index is
qi
αi
Q
$i = P 1 =
ε
− Q ∂P (.)
∂qi
• 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 ) =
for i, j = 1, 2 and i 6= j.
1 − ci − qj
2
• What is the Cournot equilibrium?
1 − 2ci + cj
3
• What are the profits of the firms?
qi∗ =
(1 − 2ci + cj )2
Π =
9
i
• A firm’s output decreases with its MC , ∂q
∂ci < 0.
• A firm’s output increases with its competitor MC ,
∂qi
∂cj > 0.
i
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
Pn
• Q = i=1 qi
• P (Q) = 1 − Q
• Ci(qi) = cqi
• FOCi:
∂Ci (qi )
qi ∂P∂q(.)
+
P
(Q)
−
∂qi = 0
i
⇒ qi + 1 − Q − c = 0
• and Lerner index
αi
$i =
ε
• Symmetric equilibrium Q = nq , and thus
1−c
q=
n+1
13
• Market price is
p = 1 − nq = c +
1−c
n+1
• and the profit of each firm is
(1 − c)2
Π=
(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
Pn
qi
αi = Q where i = 1, ..., n and i=1 αi = 1
Concentration indices:
• m-firms concentration ratio (m < n)
Pm
Rm = 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
output of the industry.
• Herfindahl index
Pn 2
RH = 10, 000 i=1 αi
– 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