Transcript Bertrand Equilibrium with Increasing Marginal Costs

Anticompetitive consequence of
the nationalization of a public
enterprise in a mixed duopoly
OT2012
1
Nationalization of a private firm
yields collusive outcome in a
Bertrand duopoly
(1) Price Leadership Revisited (JoE 2011, joint work with
Daisuke Hirata).
(2) On the Uniqueness of Bertrand Equilibrium
(Operation Research Letters 2010, with Daisuke Hirata)
(3) Welfare Implication of Asymmetric Regulation in
Mixed Bertrand Duopoly (Economics Letters 2012)
(4) Price vs. Quantity in a Mixed Duopoly (Economics
Letters 2012, with Akira Ogawa).
OT2012
2
Bertrand Competition
OT2012
3
rationing rule
If P1<P2, only firm 1 supplies D(P1).
If P1>P2, only firm 2 supplies D(P2).
If P1=P2 , each firm supplies D(P1)/2.
D(P) is decreasing in P.
OT2012
4
rationing rule
P1<P2→D1=D(P1), D2=max{D(P2)-Y1, 0}
P1>P2→D2=D(Ｐ2 ), D1=max{D(P1)-Y2, 0}
P1=P2→D1=D(P1)/2+max{D(P2 )/2-Y2, 0}
Suppose that firm 1 names a lower price. It can
choose its output Y1 , which is not larger than
D1=D(P1), and then firm 2 can choose its output
Y2, which is not larger than the remaining
demand D2= D2=max{D(P2)-Y1, 0}.
OT2012
5
Bertrand Paradox
Symmetric Duopoly, Homogeneous Product Market,
Constant Marginal Costs, Price Competition,
Simultaneous-Move Game
→Perfect Competition (MC=P) ~ Bertrand Paradox
OT2012
6
Bertrand Equilibrium with
Increasing Marginal Costs
P
MC of firm 1
D
supply curve derived
from the marginal cost
curves of the two firms
0
OT2012
Y
7
Non-Existence of Pure Strategy
Equilibrium under Increasing
Marginal Costs
Symmetric Duopoly, Homogeneous Product Market,
Increasing Marginal Costs, Price Competition,
Simultaneous-Move Game
→No Pure Strategy Equilibrium ~ Edgeworth Cycle
OT2012
8
Pure Strategy Symmetric
Bertrand Equilibrium
Ｐ
Ｄ supply curve derived
from the marginal cost
curves of the two firms
0
OT2012
Ｙ
9
Bertrand Equilibrium with
Increasing Marginal Costs
Suppose that P1=P2=MC1=MC2 at a pure strategy
equilibrium. →We derive a contradiction
Suppose that firm 1 deviate from the strategy above
and raises its price
→Firm 2 has no incentive to increase its output since its
output before the deviation is the best given P2.
→Given Y2, firm 1 obtains the residual demand.
→Since P1=MC1>MR1 before the deviation, a slight
increase of P1 must increase the profit of firm 1, a
contradiction.
OT2012
10
Pure Strategy Symmetric
Bertrand Equilibrium
Ｐ
Ｄ supply curve derived
from the marginal cost
curves of the two
firms
0
OT2012
Ｙ
11
Pure Strategy Symmetric
Bertrand Equilibrium
Ｐ
Ｄ
0
OT2012
supply curve
derived from the
marginal cost
curves of two firms
Ｙ
12
pure strategy symmetric Bertrand
Equilibrium
Suppose that P1=P2>MC1=MC2 at a pure strategy
equilibrium. →We derive a contradiction
Suppose that firm 1 deviates from the strategy above
and reduces its price slightly.
→Firm 1 can increase its demand (demand elasticity is
infinite. Since P1>MC1 , the deviation increases the
profit of firm 1, a contradiction.
⇒No symmetric Bertrand equilibrium exists.
OT2012
13
pure strategy asymmetric
Bertrand Equilibrium
P
MC of firm 2
P1
P2
0
OT2012
D
supply curve
derived from
the marginal
cost curves of
two firms
Y
14
The deviation increases the
profit of firm 2, a contradiction
MC of firm 2
P
P1
P2*
P2
0
OT2012
D
Y2 Y2*
supply curve
derived from
the marginal
cost curves of
two firms
Y
15
pure strategy asymmetric
Bertrand Equilibrium
P
P1
P2
MC of firm 2
D
0
OT2012
supply curve
derived from
the marginal
cost curves of
two firms
Y
16
The deviation of firm 1 increases
the profit of firm 1, a contradiction
The profit of firm 1 is zero, and it has incentive to
name the price slightly lower than the rival's
⇒Neither symmetric nor asymmetric pure strategy
Bertrand equilibrium exists.
OT2012
17
Edgeworth Cycle
Consider the symmetric Bertrand duopoly. Consider the
following capacity constraint.
Marginal cost of firm i is c if Yi ≦K and ∞ otherwise.
If K is sufficiently large, the equilibrium outcome is same
as the Bertrand model with constant marginal cost.
If K is sufficiently small, then the equilibrium price is
derived from 2K=D(P). Firms just produce the upper
limit output K.
Otherwise →No pure strategy equilibrium
(a similar problem under increasing marginal cost case
appears) ~a special case of increasing marginal cost.
OT2012
18
rationing rule under supply
obligation
If P1<P2, only firm 1 supplies D(P1).
If P1>P2, only firm 2 supplies D(P2).
If P1=P2 , each firm supplies D(P1)/2.
OT2012
19
Bertrand Equilibrium with
Increasing Marginal Costs
P
PE
MC of firm 1
D
supply curve derived
from the marginal cost
curves of two firms
0
OT2012
Y
20
Bertrand Equilibrium with
Increasing Marginal Costs
In the equilibrium both firms name P＝ PE and obtain
the demand D(PE)/2.
Suppose that firm 1 raises its price.→The profit is zero,
so it has no incentive for raising its price.
Suppose that firm 1 reduces its price. →It obtains the
demand D(P1). Since PE =C1'(D(PE)/2), the profit is
maximized given the price. Since C' is increasing,
PE D(PE)/2 - C1(D(PE)/2) > P1D(P1) - C1(D(P1)) .
OT2012
21
Continuum Equilibrium
Both higher and lower prices than the perfectly
competitive price can be equilibrium prices.
Define PH by PHD(PH)/2 - C1(D(PH)/2) = PHD(PH) C1(D(PH)).
If P1> PH, then P1D(P1)/2 - C1(D(P1)/2) < P1D(P1) C1(D(P1)).
Define PL by PLD(PL)/2 - C1(D(PL)/2) = 0.
If P1> PL, then P1D(P1)/2 - C1(D(P1)/2) < 0.
Any price P ∈(PL, PH) can be an equilibrium price.
OT2012
22
Bertrand Equilibrium with
Increasing Marginal Costs
P
D
PH
PL
supply curve derived
from the marginal cost
curves of two firms
0
Continuum Equilibrium
OT2012
Y
23
Indeterminacy of Bertrand Equilibria
Hirata and Matsumura (2010)
Does this result (indeterminacy of equilibria) depend on
the assumption of homogeneous product?
p1=a-q1-bq2
p2=a-q2-bq1 b∈(-1,1]
b>0 supplementary products
b=1 homogeneous product
b represents the degree of product differentiation.
If b =1, a continuum of equilibria exists.
If b∈(0,1), the equilibrium is unique and it converges to
Walrasian as b →1.
It is also true under more general demand function.
OT2012
24
Homogeneous Product Market
P2
P1
0
OT2012
D2
Y2
25
Differentiated Product Market
P2
P1
0
OT2012
D2
Y2
26
Bertrand Equilibrium with
Increasing Marginal Costs
S
P
D
0
OT2012
supply curve derived
from the marginal cost
curves of the two firms
Y
27
Asymmetric Supply Obligation
We observe supply obligations in many markets,
such as postal service (overnight delivery),
electric power distribution, natural gas
distribution, telecom, water supply, and so on.
However, in most cases, this obligation is imposed
to only one firm (usually a dominant firm).
In mixed oligopoly, only the public firm has this
obligation.
OT2012
28
Asymmetric Supply Obligation
Consider a duopoly private market.
Suppose that only one firm (firm1) has this supply
obligation.
→No Pure Strategy Equilibrium exists.
OT2012
29
Pure Strategy Symmetric
Bertrand Equilibrium?
Question: Does firm 2 have an incentive for changing its
price?
Ｐ
Ｄ
PW
0
OT2012
supply curve derived
from the marginal cost
curves of the two
firms
Ｙ
30
Pure Strategy Symmetric
Bertrand Equilibrium?
Question: Does firm 1 have an incentive for changing its
price?
Ｐ
Ｄ
PW
0
OT2012
supply curve derived
from the marginal cost
curves of the two
firms
Ｙ
31
Pure Strategy Symmetric
Bertrand Equilibrium?
Question: Does firm 2 have an incentive for changing
its price?
Ｐ
Ｄ supply curve derived
from the marginal cost
curves of the two firms
0
OT2012
Ｙ
32
Mixed Duopoly
Question:Suppose that firm 1 (2) is a welfare(profit)maximizing public (private) firm. Does firm 1 have an
incentive for changing its price?
Ｐ
PW
0
OT2012
Ｄ supply curve derived
from the marginal cost
curves of the two
firms
Ｙ
33
Mixed Duopoly
Question:Suppose that firm 1 (2) is a welfare(profit)maximizing public (private) firm. Does firm 2 have an
incentive for changing its price?
Ｐ
PW
0
OT2012
Ｄ supply curve derived
from the marginal cost
curves of the two
firms
Ｙ
34
Without Supply Obligation in
Mixed Oligopoly
The Asymmetric Obligation + Mixed Oligopoly yield the
first best outcome.
Are both indispensable?
Symmetric obligation yields the first best in both mixed
and private duopolies (known results). ~but the
equilibrium is not unique.
Neither mixed and private duopoly yields the first best
under asymmetric obligation on the private firm.
Mixed Oligopoly without Supply Obligation
It is obvious that the first best is not achieved.
OT2012
35
Without Supply Obligation in
Mixed Oligopoly
Mixed Oligopoly without Supply Obligation
It is obvious that the first best is not achieved. What is
the equilibrium outcome?
First I think that (as well as in private duopoly) no pure
strategy equilibrium exists.
→My conjecture turns out to be wrong.
⇒Monopoly outcome
OT2012
36
Model
Homogeneous Product Market, Mixed Duopoly,
Firm 0~welfare maximizer, Firm 1~profit maximizer,
common cost function~ increasing marginal cost,
Firms independently choose their prices.
The firm naming lower price chooses its output, and
then the other firm chooses it output.
When firms name the same price, the private chooses
its output under the constraint y1 ≦D(P)/2 and the
public chooses its output y0 ≦D(P)-y1.
OT2012
37
Behavior of Firm 0
Firm 0 prefers y1= y0 for production efficiency.
However, as long as y1 is positive, Y=D(p1).
Firm 0 can choose the stand alone best outcome where
its price is equal to its marginal cost.
(Let y0* denote the output of the public monopolist).
Firm 0 chooses the latter if and only if y1= y0 =D(p1)/2
yields the larger welfare the stand alone best.
OT2012
38
Limit Pricing by Firm 1
Firm 1 chooses the price either the price which
maximizes p1 D(p1)/2 -c1( D(p1)/2 ) ~collusive
pricing
or
chooses the price which yields W(D(p1)/2, D(p1)/2)=
W(y0*, 0) ~Limit Pricing.
Either Collusive Pricing or Limit Pricing appears in
equilibrium.
OT2012
39
Model
Homogeneous Product Market, Mixed Duopoly,
Firm 0~welfare maximizer, Firm 1~profit maximizer,
constant marginal cost, cost difference between
public and private firms
Firms independently choose their prices.
The firm naming lower price chooses its output, and
then the other firm chooses it output.
When firms name the same price, the private chooses
its output under the constraint y1 ≦D(P)/2 and the
public chooses its output y0 ≦D(P)-y1.
OT2012
40
Behavior of Firm 1
As long as p0>p1>c1, firm 1 chooses y1=D(p1).
OT2012
41
Behavior of Firm 0
Firm 0 prefers firm 1’s production rather than its own
production.
Firm 0 can choose y0=D(c0) (public monopoly).
It can choose y0=0 and then y1=D(p1).
OT2012
42
Limit Pricing by Firm 1
Firm 1 chooses the price either the price which
maximizes p1 D(p1) -c1D(p1) ~monopoly pricing
or
chooses the price which yields W(0, D(p1))= W(D(c0)*,
0) ~Limit Pricing.
Either Monopoly Pricing or Limit Pricing appears in
equilibrium.
OT2012
43
Implication
Supply obligation to the public firm is reasonable. If this
obligation is abolished without privatization, it can
produces huge welfare loss.
OT2012
44
important property
Under Bertrand competition, the public firm (welfare
maximizer) becomes less aggressive because its
aggressive behavior reduces the resulting production
level of the private rival.
Cf Under the Cournot competition, the output level of
the private firm is given exogenously when the public
firm chooses its output. Thus, its aggressive behavior
does not reduces the rival’s output.
OT2012
45
mixed Bertrand and mixed Cournot
Competition is less severe under mixed Bertrand
competition than under the mixed Cournot
competition, contrasting to the standard results in
private oligopoly.~ Under Bertrand competition, the
public firm (Ghosh and Mitra, 2010 Letters).
OT2012
46
Endogenous Choice of PriceQuantity Contract
Firms choose whether to adopt price contract or
quantity contract, and then choose the prices or
quantities.
Singh and Vives (1984) showed that choosing the
quantity (price) contract is a dominant strategy for
each firm if the goods are substitutes (complements).
Intuition (substitutable goods case)： Choosing a
price contract increases the demand elasticity of the
rival, resulting in a more aggressive action of the rival.
OT2012
47
Endogenous Choice of PriceQuantity Contract in Mixed Duopoly
For the private firm, choosing a price contract increases
the demand elasticity of the rival, resulting in a less
aggressive action of the rival (substitutable goods case).
Thus, the private firm has an incentive to choose the
price contract, as opposed to the private duopoly.
For the public firm, choosing a price contract increases
the demand elasticity of the rival, resulting in a more
aggressive action of the rival . Thus, the public firm has
an incentive to choose the price contract.
→Bertrand competition appears in Mixed Duopoly
(Matsumura and Ogawa, 2012)
OT2012
48