War and Peace:The Two Faces of Competition

Rodolphe Dos Santos FerreiraBETA, University of Strasbourg

13th Summer School on Economic History, Philosophy,and History of Economic Thought, Acqui Terme, 1-8 September 2010

Principal references

• Dos Santos Ferreira, R., ‘‘Guerre et paix: Variations sur la notion de concurrence au siècle de Cournot’’, Revue économique, 2004.

• d’Aspremont, C. and Dos Santos Ferreira, R., ‘‘Price-quantity competition with varying toughness’’, Games and Economic Behavior, 2009.

Cooperation and conflictin the process of exchange

• Cooperation stems from mutual advantage opportunities.• But mutual advantage generally coexists with conflicting

interests.

• This coexistence of mutual advantage and conflicting interests, in the context of exchange in particular, can be seen whether we adopt a cooperative or a non-cooperative analytical approach.

• The opposition between the two approaches should not be associated with the opposition between converging and diverging interests.

Cooperation and conflictin the process of exchange

• Cooperation stems from mutual advantage opportunities.• But mutual advantage generally coexists with conflicting

interests.

• This coexistence of mutual advantage and conflicting interests, in the context of exchange in particular, can be seen whether we adopt a cooperative or a non-cooperative analytical approach.

• The opposition between the two approaches should not be associated with the opposition between converging and diverging interests.

Cooperation and conflictin the process of exchange

• Cooperation stems from mutual advantage opportunities.• But mutual advantage generally coexists with conflicting

interests.

• This coexistence of mutual advantage and conflicting interests, in the context of exchange in particular, is easy to identify whether we adopt a cooperative or a non-cooperative analytical approach.

• The opposition between the two approaches should not be associated with the opposition between converging and diverging interests.

Cooperation and conflictin the process of exchange

• Cooperation stems from mutual advantage opportunities.• But mutual advantage generally coexists with conflicting

interests.

• This coexistence of mutual advantage and conflicting interests, in the context of exchange in particular, is easy to identify whether we adopt a cooperative or a non-cooperative analytical approach.

• The opposition between the two approaches should not be associated with the opposition between converging and diverging interests.

Initial allocation

Contract curve ->

I1

I2

• It is mutually advantageous to move from the initial allocation to any allocationin the interior of the lens-shaped region between indifference curves I1 and I2.• Mutual advantage opportunities are exhausted only on the contract curve.• Along this curve, the two interests are conflicting.

EDGEWORTH (1881)

0

I1

I2

x0 (initial allocation)

x x1

x2

In this (once-) replicated economy, the allocation x on the contract curve (symmetricwith respect to each type of consumers) can be blocked by a coalition consisting ofa consumer of type 1, receiving x1 and leaving his competitor of the same type withthe initial resources x0, plus two consumers of type 2, each receiving ( – x2), with2x2 = x1 + x0 .

As the economy is replicated again and again, and as the consumers becomecorrespondingly more and more insignificant with respect to the size of theeconomy, the core (the set of allocations that cannot be blocked) shrinks andis eventually reduced to the (perfectly) competitive allocation(s).

‘Is it peace or war?’ asks the lover of ‘Maud’,*of economic competition, and answers hastily:It is both, pax or pact between contractors during contract, war, when some of the contractors without the consent of others recontract.

F. Y. Edgeworth, Mathematical Psychics, 1881.

* Alfred Tennyson, Maud (1855/56)

Cooperation and conflictin competition between producers

COURNOT (1838)• Non-cooperative approach (contrary to Edgeworth, 1881): each

producer ‘‘goes his own way’’ when taking decisions.

• Conflict: each producer’s market share can only increase at the expense of his rivals’ market shares.

• Cooperation: producers know that ‘‘the price is necessarily the same’’, so that price undercutting will always be matched. Hence, price manipulation by each producer concerns in fact competition against other sectors, aimed at changing market size or, given his rivals’ sales targets, his residual demand and, ultimately, his market share.

Cooperation and conflictin competition between producers

COURNOT (1838)• Non-cooperative approach (contrary to Edgeworth, 1881): each

producer ‘‘goes his own way’’ when taking decisions.

• Conflict: each producer’s market share can only increase at the expense of his rivals’ market shares.

• Cooperation: producers know that ‘‘the price is necessarily the same’’, so that price undercutting will always be matched. Hence, price manipulation by each producer concerns in fact competition against other sectors, aimed at changing market size or, given his rivals’ sales targets, his residual demand and, ultimately, his market share.

Cooperation and conflictin competition between producers

COURNOT (1838)• Non-cooperative approach (contrary to Edgeworth, 1881): each

producer ‘‘goes his own way’’ when taking decisions.

• Conflict: each producer’s market share can only increase at the expense of his rivals’ market shares.

• Cooperation: producers know that ‘‘the price is necessarily the same’’, so that price undercutting will always be matched. Hence, price manipulation by each producer concerns in fact competition against other sectors, aimed at changing market size or, given his rivals’ sales targets, his residual demand and, ultimately, his market share.

• Program of producer k:max p [F(p) – ∑i≠k qi ] – Ck(F(p) – ∑i≠k qi )

or, equivalently (in Cournot’s formulation),max F 1 (qk + ∑i≠k qi ) qk – Ck(qk)

• FOC:p [1 – sk / F(p)] = Ck‘ (qk)

where sk = qk / (qk + ∑j≠k qj ) is k’s market share

and F(p) is the demand elasticity at p.• As the market share tends to zero (as k’s output becomes

insignificant with respect to market size), the price is less and less manipulable and tends to marginal cost at equilibrium: competition becomes indefinite.

• Program of producer k:max p [F(p) – ∑i≠k qi ] – Ck(F(p) – ∑i≠k qi )

or, equivalently (in Cournot’s formulation),max F 1 (qk + ∑i≠k qi ) qk – Ck(qk)

• FOC:p [1 – sk / F(p)] = Ck‘ (qk)

where sk = qk / (qk + ∑i≠k qi ) is k’s market share

and F(p) is the demand elasticity at p (in absolute value).• As the market share tends to zero (as k’s output becomes

insignificant with respect to market size), the price is less and less manipulable and tends to marginal cost at equilibrium: competition becomes indefinite.

• Program of producer k:max p [F(p) – ∑i≠k qi ] – Ck(F(p) – ∑i≠k qi )

or, equivalently (in Cournot’s formulation),max F 1 (qk + ∑i≠k qi ) qk – Ck(qk)

• FOC:p [1 – sk / F(p)] = Ck‘ (qk)

where sk = qk / (qk + ∑i≠k qj ) is k’s market share

and F(p) is the demand elasticity at p (in absolute value).• As the market share tends to zero (as k’s output becomes

insignificant with respect to market size), the price is less and less manipulable and tends to marginal cost at equilibrium: competition eventually becomes indefinite.

Cooperation and conflictin competition between producers

BERTRAND (1883)• Peace: the common interest of the producers would be collusion, a

solution put aside by Cournot.• War: Bertrand erroneously describes Cournot’s analysis as depicting

price undercutting by each producer in order to divert customers from his rivals. He thus attributes to Cournot the description of a price war, a process without solution:

‘‘no solution is possible, the [price] fall would have no limit.’’

• In the standard (erroneous) interpretation of Bertrand, the equilibrium outcome in a sector where each producer has the same linear cost function is the competitive outcome, with zero profits, as soon as there are at least 2 competitors: the Bertrand paradox.

Cooperation and conflictin competition between producers

BERTRAND (1883)• Peace: the common interest of the producers would be collusion, a

solution put aside by Cournot.• War: Bertrand erroneously describes Cournot’s analysis as depicting

price undercutting by each producer in order to divert customers from his rivals. He thus attributes to Cournot the description of a price war, a process without solution:

‘‘no solution is possible, the [price] fall would have no limit.’’

• In the standard (erroneous) interpretation of Bertrand, the equilibrium outcome in a sector where each producer has the same linear cost function is the competitive outcome, with zero profits, as soon as there are at least 2 competitors: the Bertrand paradox.

Cooperation and conflictin competition between producers

BERTRAND (1883)• Peace: the common interest of the producers would be collusion, a

solution put aside by Cournot.• War: Bertrand erroneously describes Cournot’s analysis as depicting

price undercutting by each producer in order to divert customers from his rivals. He thus attributes to Cournot the description of a price war, a process without solution:

‘‘no solution is possible, the [price] fall would have no limit.’’

• In the standard (erroneous) interpretation of Bertrand, the equilibrium outcome in a sector where each producer has the same linear cost function is the competitive outcome, with zero profits, as soon as there are at least 2 competitors: the Bertrand paradox.

Cooperation and conflictin competition between producers

• For Cournot (as for Edgeworth), competition is both peace and war.

• Bertrand: competition is either peace or relentless war.

• Jevons (1871) and Walras (1874): (perfect or absolute) free competition is only peace.The competitive outcome is achieved, even when there is a single producer (and possibly a single consumer), hence when there is in fact no competition, through a behavioural rule: giving up price manipulation, i.e. price taking. This is also competition in the sense of Arrow and Debreu (1954).

Cooperation and conflictin competition between producers

• For Cournot (as for Edgeworth), competition is both peace and war.

• For Bertrand, competition is either peace or relentless war.

• Jevons (1871) and Walras (1874): (perfect or absolute) free competition is only peace.The competitive outcome is achieved, even when there is a single producer (and possibly a single consumer), hence when there is in fact no competition, through a behavioural rule: giving up price manipulation, i.e. price taking. This is also competition in the sense of Arrow and Debreu (1954).

Cooperation and conflictin competition between producers

• For Cournot (as for Edgeworth), competition is both peace and war.

• For Bertrand, competition is either peace or relentless war.

• For Jevons (1871) and for Walras (1874), (perfect, or absolute) free competition is only peace.The competitive outcome is achieved, even when there is a single producer (and possibly a single consumer), hence when there is in fact no competition, through a behavioural rule that amounts for producers to give up price manipulation for strategic purposes.Notice that this is also competition (price taking behaviour) in the sense of Arrow and Debreu (1954).

Cooperation and conflictin competition between producers

Walras (1874)• The rule for a monopolistic sector is “that the selling price be higher

than the cost and that producers make the highest possible profit.”

• The rule for a freely competitive sector is “that each product have a sole price in the market, which should be equal to the cost price, so that producers make neither profit nor loss.”This is achieved through the following behavioural rule: “an increase in the quantity of products whose selling price is higher than their cost, and a decrease in the quantity of those whose cost is higher than their price.” No profit maximization is assumed.

A Cournot-Bertrand competition game

• Each producer i=1,…,n chooses a pair (pi,qi) where pi is a list price at which producer i commits to serve the whole demand [Bertrand, as viewed by Vives (1999)] and qi is the quantity to produce in advance, which he is ready to sell at any positive discount price [Cournot].

A Cournot-Bertrand competition game

• Each producer i=1,…,n chooses a pair (pi,qi) where pi is a list price at which producer i commits to serve the whole demand [Bertrand, as viewed by Vives (1999)] and qi is the quantity to produce in advance, which he is ready to sell at any positive discount price [Cournot].

qi

pi

A Cournot-Bertrand competition game

• If there is excess demand at the lowest list price, this excess demand is allocated to firms having set this price according to an ex ante sharing rule: (s1(p,q),…,sn(p,q)), s.t.

A Cournot-Bertrand competition game

• We thus obtain, given the cost functions Ci, a Cournot-Bertrand competition game in prices and quantities, with payoff functions (for i=1,…,n):

Oligopolistic equilibrium

• An oligopolistic equilibrium is a Nash equilibrium (p*,q*) of the Cournot-Bertrand competition game which satisfies in addition the credibility condition: si(p*,q*) = 0 for any i (no firm should be obliged to sell at equilibrium more than it would spontaneously wish to do):

Canonical characterisationof oligopolistic equilibrium

• The pair (p*,q*) is an oligopolistic equilibrium iff (pi*,qi*) solves, for any i,

and satisfies in addition the credibility condition.

• The first of the two constraints concerns the market share.

• The second constraint concerns the market size.

Parameterization of the set of equilibria:An index of competitive toughness

• There is a continuum of oligopolistic equilibria, in particular those which lead to the Cournot, Bertrand (for linear cost functions), and perfect competition outcomes (but not to the collusive solution).

• i : Lagrange multiplier associated with the constraint on market share

i : Lagrange multiplier associated with the constraint on market size

i = i /(i + i ) : index of competitive toughness

• Lerner index of the degree of monopoly:

= 0 : Cournot outcome; = 1 : competitive outcome.

Parameterization of the set of equilibria:An index of competitive toughness

• There is a continuum of oligopolistic equilibria, in particular those which lead to the Cournot, Bertrand (for linear cost functions), and perfect competition outcomes (but not to the collusive solution).

• i : Lagrange multiplier associated with the constraint on market share

i : Lagrange multiplier associated with the constraint on market size

i = i /(i + i ) : index of competitive toughness

• Lerner index of the degree of monopoly:

= 0 : Cournot outcome; = 1 : competitive outcome.

Parameterization of the set of equilibria:An index of competitive toughness

• There is a continuum of oligopolistic equilibria, in particular those which lead to the Cournot, Bertrand (for linear cost functions), and perfect competition outcomes (but not to the collusive solution).

• i : Lagrange multiplier associated with the constraint on market share

i : Lagrange multiplier associated with the constraint on market size

i = i /(i + i ) : index of competitive toughness

• Lerner index of the degree of monopoly:

= 0 : Cournot outcome; = 1 : competitive outcome.

Asymmetric duopoly: market share of the technological leader (i=1)

1=1/3

1=1/6

2=1/32=1/6

1= 2

Linear demand and costs

2=0

1=0

• The index of competitive toughness is endogenous: it signs an equilibrium of which we do not know how it is selected.

• It can however be exogenized in different ways corresponding to distinct interpretations. We can in particular distinguish models of competition that emphasize either the peaceful side or the warlike side of the competitive interaction, very much as we can climb a mountain on either its south or its north faces.

Reaching the competitive summitvia the south face

• The pacific face of competition: i as the weight put by the altruistic producer i on collective

interest.

The payoff function of producer i is taken as the arithmetic mean of his own profit and, with relative weight i, of the sum of his rivals’ profits plus the consumers’ surplus:

=> same FOC as in the Cournot-Bertrand competition game.

Reaching the competitive summitvia the north face

• The warlike face of competition: i as the probability that producer i choose an aggressive conduct.

Two-stage duopoly game:

1) Each producer i chooses a pair (pi,qi), where pi is a list price and qi a quantity to produce in advance. However, contrary to the preceding game, producer i commits to serve the demand only up to quantity qi (Bertrand-Edgeworth).

2) With probability 1 i producer i adopts a compromising conduct, sticking to his list price pi and selling min{qi,D(pi) qj}.

With probability i he adopts an aggressive conduct, supplying at the discount price min{pi,pj} the whole quantity he can actually sell at that price (possibly beyond his residual demand, if his discount price is the lowest).

Reaching the competitive summitvia the north face

• It is easy to show that, under the assumption of linear costs, we obtain at equilibrium Lerner’s index:

This index tends to its value in the canonical model when tends to zero.