CDE

Centre for Development Economics

WORKING PAPER SERIES

COInparing Cournot and Bertrand in a Homogeneous Product Market

KG Dastidar Centre for Development Economics

Delhi School of Economics

Working Paper No 22

Centre for Development Economics Delhi School of Economics

Delhi 110 007 INDIA

COMPAfING COUr(NOT AND BERTRAND IN A HOMOGagtJEOUS PrWDUCT MARKETl

Krishnendu Ghosh Dastidar

Centre for Development Economics Delhi School of Economics

[This paper reexamines some general notions regarding the comparison of Cournot and Bertrand equilibrium outcomes It recasts the Vives (1985) result in a homogeneous product framework and it is shown that the prevailing notion that Bertrand equilibrium involves lower prices and profits than a Cournot equilibrium is not always true especially when costs are asymmetric]

1 I am indebted to Anjan Mukherji for is paper This paper was

revised when I was a visiting fellowmiddot at the Centre for

Development Delhi School of Economics I gratefully acknowledge

the excellent research opportunities being offered there

l

INTRODUCTION

In this paper we will try to reexamine some general notions

regarding the comparison of Cournot and Bertrand equilibrium

outcomes in a homogeneous product market It is a fairly well

known idea that Bertrand (price) competition is more competitive

than Cournot (quantity) competition In fact with a homogeneous

product and constant marginal costs the Bertrand outcomes involves

pricing at marginal costs This is not the case with

differentiated products where margins over marginal costs are

positive even in Bertrand competition

However even in a differentiated product set up Singh and Vives

(1984) provide a thorough comparison of Bertrand and Cournot

equilibria for the special case of constant marginal costs and the

linear system of equations So long as goods are substitutes the

Bertrand equilibrium is more competitive Cheng (1985) provides a

geometric proof of this same result that applies to a more general

class of cost and demand functions Vives (1985) gives fairly

general conditions under which the Cournot equilibria involve

higher prices and profits (and lower welfare) than do Bertrand

equilibria He shows that if the demand structure is symmetric

then (and Bertrand and Cournot equiibria are unique) then prices

and profits are larger and quantities ~inaller in Cournot than in

Bertrand competition (Proposition 1) If Bertrand reaction

functions are upward sloping (and continuous) then even with an

asymmetric demand structure) given any Cournot equilibrium price

vector one can find a Bertrand equilibrium with lower prices In

i the 3ilrlrand equilibrium is unique then it has

lOWEll than any Cournot ~quilibrium (Proposition 2)

Okuguohi (1997) also oompares the equilibrium prices for the

Bertrand and Cournot ies with product differentiation If

all firms have linear demand and cost functions and if in

addition the Jacobian matrix of the demand functions has a

dominant negative diagonal the Cournot equilibrium prices are not

lower than the Bertrand ones Okuguchi also derives the general

condition for the comparison of the Bertrand and Cournot

equilibrium prices when nonlinearities are involved in the cost

andor demand functions The condition ensuring the equilibrium

prices in the Cournot oligopoly to be not lower than those of the

Bertrand oligopoly is shown to be closely related to the global

stability condition for the Bertrand equilibrium prices

We will now reexamine the Vives (1985) results in a homogeneous

product market Consider a n firm oligopoly satisfying the

following assumptions

(a) F(P) is continuous twice continuously differentiable concave

and 3 positive numbers pmax and QTnaX such that F (pmax) 0 and

F(O) Qmax and F (P) lt 0 for 0 s P s pmax

(b) C(Qi) is continuous twice continuosly differentiable and

convex wi th C ( 0) 2 0 and C (Q ) 0 i

(c) In pure price competition typically firm i is viewed as

facing the following demand curve

3

i

o (P P ) 0 if P gt P for some ji i j i j

=1 (P ) if P J P V j and P P m i i j ik i

where k (12 m)

If (P ) if P lt P V j i i j

The above means that if is price is below js price it gets the

whole market whereas if theYcharge the same price they share the

market equally

(d) We assume that in price competition a firm always supplies the

demand it faces

Define the following

7T i

(P) = PF(P) - C(F(P)

~ i

(P) = ~PF(P) -C(F(Praquo m m

where 2 s m s n

P (m) st 7T (p (m)m)iii

-C (0)

P i

(m) st 7T(P(m)m) ~ 1

7T i

(pi

(m) m)

-We know from Dastidar (Forthcoming) that Pi and Pi exist and they

-are unique Also P lt P Also define the followingi i

arg P20 max [PF(P) - C(F(P)]

arg P20 max [~PF (P)2

A A

(e) Assume 7T (pm) 7T (pm) gt -C(O)iii i

Price competition

Suppose that both firms employ price as their strategic variable

4

NovI from De t 1 ci 1 WtJ know th~tt under (ii )

to (e) l e [P I if ] i8 a purte IlltratcOl9Y Nash equil ibrium in pricei J

competition If a firm i guctes a price in that range then it is

best for firm j to quote that price and not undercut it or charge

more To see that consider a firm i quoting a price P E [P bull Pl i i

Quoting the same P yields firm j a payoff equal to 11(1 ) l

quoting more j nets -C(O) and quoting less it gets 11 (P - e) Now i

from lemmas 1 to 8 of Dastidar we know that for any l E [1 1] l i

the following two conditions hold simultaneously

11 (P) c 7r (P) gt 7r (1 - E) V euro gt 0 and 11 (1) - C ( 0 ) i i i i

Since the above two conditions hold for P also our claim follows

Quantity Competition

If both firms employ quantity as its strategic variables then we

have the Cournot equilibrium

p(f) We aSsume that QH (Q) + H (Q) sO where H (Q)

and Q = ~ Q l

i

The above guarantees the existence of Cournot equilibrium (see

Novshek (1985) and Shapiro (1989raquo Let the Cournot price

quantities and profits be denoted by pC Q~ and 7r~ respectively l l

If firms have symmetric costs then the Vives results (Proposition

1 and 2 in Vives (1985raquo follows The reason is as follows

Suppose all firms have same constant marginal costs w Then the

Bertrand equilibrium is unique and 1B w At a Cournot

equilibrium the first order conditions are given by

5

(Q) + II (0) ~ C I (0 ) 0 i 1 2 n i

OR(O) 1- HCQ) ~ w 0 i

If all firms have symmetric costs C () which are strictly convex 1

then the Bertrand equilibrium is necessarily non-unique In fact

any P E [P (n) P (n)] is a Bertrand equilibriumi i

shy

Now consider the Bertrand equilibrium price of P i

(n) We know from

Dastidar (forthcoming) that

shy 7f

) (P (n) n)

1

~ C (0)

shy

and 7f )

(P I n) so - C (0) t V P s P (n) )

A shy

and 11 ]

(P t p) gt - C(O) V P i

(n) s P lt pmax

Again the first order conditions for Cournot equilibrium are as

follows

QH (Q) + H(Q) C (Q) = 0 i = 1 2 n ) i

Therefore QC

i

Q~ gt 0 =gt H(QC) gt C (Qc) since H () lt 0 -----(1)] i

We claim that QC gt 0 =gt 1Ic gt -C(O) i i

Suppose on the contrary that 11 C

s -C(O)i

Now C(O) - C(Qc) gt QC C (QC) since C( is strictly convex iii

c c C cTherefore -Q C (Q) lt -Q H(Q)

iii

9 C (QC) gt H(Qc) which contradicts (1) i

6

Henae our claim follows

Note that any P e (Pi (n) P(n)l is a Bertrand equilibrium Since

pC P (n) there is always a Bertrand equilibrium price which is i

lower than the Cournot price Therefore the Vives (1985

Proposition 1) is valid in a homogeneous product market where

firms have symmetric costs

However proposition 2 is not always valid in an asymmetric market

We produce below a simple but rather extreme example to

illustrate this point Consider a homogeneous product duopoly

where the demand is given by the following

p 10 - (Q + Q ) 1 2

and costs are given by C (0) = ~Q2 and C (Q) = 5Q2 11 21 22 2

Routine calculations lead to the following A

pC 447 pm = 203 pm = 809 P 2 Ii 307 P 507 and 1 2 1 1 2

P 150172

It should be noted that P~ is the monopoly price of the ith firm

From the analysis in Dastidar (forthcoming) it is evident that the

Bertrand equilibrium is unique and is given by pe = pm = 203 1

Firm 1 undercuts firm 2 and since its monopoly price is less than A

p (the least that 2 will charge) it remains the only operating2

firm in the market We may note that pC lt pB That is we have

shown that even when Bertrand and Cournot equilibrium are uniqu~

it is not necessary that Bertrand equilibrium is more competitive

7

than Coumot tquilbriutn especially if costs are very asymmetric

n other words Vives results are not always valid in a homogeneous

product market where firms have asymmetric costs

8

REFERENCES

Cheng L (1985) Comparing Bertrand and Cournot Equilibria A Geometric Approach Rand Journal of Economics 16 146-152

Daatidar KG (forthcoming) On the Existence of Pure Strategy Bertrand Equilibrium Eccmomic Theory

Novshek W (1985) On the Existence of Cournot Equilibria Rev-ielv of Economic Studies 52 85-98

Okuguchi K (1987) Equilibrium Prices in the Bertrand and Cournot Oligopolies Journal of Economic Theory 42 128-139

Shapiro C (1989) Theories of Oligopoly Behaviour in Schmalensee R and Willig RD (ed) Handbook of Industrial Organisation (vol 1) North Holland

Singh N and Vives X (1984) Price and Quantity Competition in a Differentiated Duopoly Rand Journal of Economics 15 546-554

Vives X (1985) On the Efficiency of Bertrand and Cournot Equilibria with Product Differentiation Journal of Economic Theory 36 166-175

COMPAfING COUr(NOT AND BERTRAND IN A HOMOGagtJEOUS PrWDUCT MARKETl

Krishnendu Ghosh Dastidar

Centre for Development Economics Delhi School of Economics

[This paper reexamines some general notions regarding the comparison of Cournot and Bertrand equilibrium outcomes It recasts the Vives (1985) result in a homogeneous product framework and it is shown that the prevailing notion that Bertrand equilibrium involves lower prices and profits than a Cournot equilibrium is not always true especially when costs are asymmetric]

1 I am indebted to Anjan Mukherji for is paper This paper was

revised when I was a visiting fellowmiddot at the Centre for

Development Delhi School of Economics I gratefully acknowledge

the excellent research opportunities being offered there

l

INTRODUCTION

In this paper we will try to reexamine some general notions

regarding the comparison of Cournot and Bertrand equilibrium

outcomes in a homogeneous product market It is a fairly well

known idea that Bertrand (price) competition is more competitive

than Cournot (quantity) competition In fact with a homogeneous

product and constant marginal costs the Bertrand outcomes involves

pricing at marginal costs This is not the case with

differentiated products where margins over marginal costs are

positive even in Bertrand competition

However even in a differentiated product set up Singh and Vives

(1984) provide a thorough comparison of Bertrand and Cournot

equilibria for the special case of constant marginal costs and the

linear system of equations So long as goods are substitutes the

Bertrand equilibrium is more competitive Cheng (1985) provides a

geometric proof of this same result that applies to a more general

class of cost and demand functions Vives (1985) gives fairly

general conditions under which the Cournot equilibria involve

higher prices and profits (and lower welfare) than do Bertrand

equilibria He shows that if the demand structure is symmetric

then (and Bertrand and Cournot equiibria are unique) then prices

and profits are larger and quantities ~inaller in Cournot than in

Bertrand competition (Proposition 1) If Bertrand reaction

functions are upward sloping (and continuous) then even with an

asymmetric demand structure) given any Cournot equilibrium price

vector one can find a Bertrand equilibrium with lower prices In

i the 3ilrlrand equilibrium is unique then it has

lOWEll than any Cournot ~quilibrium (Proposition 2)

Okuguohi (1997) also oompares the equilibrium prices for the

Bertrand and Cournot ies with product differentiation If

all firms have linear demand and cost functions and if in

addition the Jacobian matrix of the demand functions has a

dominant negative diagonal the Cournot equilibrium prices are not

lower than the Bertrand ones Okuguchi also derives the general

condition for the comparison of the Bertrand and Cournot

equilibrium prices when nonlinearities are involved in the cost

andor demand functions The condition ensuring the equilibrium

prices in the Cournot oligopoly to be not lower than those of the

Bertrand oligopoly is shown to be closely related to the global

stability condition for the Bertrand equilibrium prices

We will now reexamine the Vives (1985) results in a homogeneous

product market Consider a n firm oligopoly satisfying the

following assumptions

(a) F(P) is continuous twice continuously differentiable concave

and 3 positive numbers pmax and QTnaX such that F (pmax) 0 and

F(O) Qmax and F (P) lt 0 for 0 s P s pmax

(b) C(Qi) is continuous twice continuosly differentiable and

convex wi th C ( 0) 2 0 and C (Q ) 0 i

(c) In pure price competition typically firm i is viewed as

facing the following demand curve

3

i

o (P P ) 0 if P gt P for some ji i j i j

=1 (P ) if P J P V j and P P m i i j ik i

where k (12 m)

If (P ) if P lt P V j i i j

The above means that if is price is below js price it gets the

whole market whereas if theYcharge the same price they share the

market equally

(d) We assume that in price competition a firm always supplies the

demand it faces

Define the following

7T i

(P) = PF(P) - C(F(P)

~ i

(P) = ~PF(P) -C(F(Praquo m m

where 2 s m s n

P (m) st 7T (p (m)m)iii

-C (0)

P i

(m) st 7T(P(m)m) ~ 1

7T i

(pi

(m) m)

-We know from Dastidar (Forthcoming) that Pi and Pi exist and they

-are unique Also P lt P Also define the followingi i

arg P20 max [PF(P) - C(F(P)]

arg P20 max [~PF (P)2

A A

(e) Assume 7T (pm) 7T (pm) gt -C(O)iii i

Price competition

Suppose that both firms employ price as their strategic variable

4

NovI from De t 1 ci 1 WtJ know th~tt under (ii )

to (e) l e [P I if ] i8 a purte IlltratcOl9Y Nash equil ibrium in pricei J

competition If a firm i guctes a price in that range then it is

best for firm j to quote that price and not undercut it or charge

more To see that consider a firm i quoting a price P E [P bull Pl i i

Quoting the same P yields firm j a payoff equal to 11(1 ) l

quoting more j nets -C(O) and quoting less it gets 11 (P - e) Now i

from lemmas 1 to 8 of Dastidar we know that for any l E [1 1] l i

the following two conditions hold simultaneously

11 (P) c 7r (P) gt 7r (1 - E) V euro gt 0 and 11 (1) - C ( 0 ) i i i i

Since the above two conditions hold for P also our claim follows

Quantity Competition

If both firms employ quantity as its strategic variables then we

have the Cournot equilibrium

p(f) We aSsume that QH (Q) + H (Q) sO where H (Q)

and Q = ~ Q l

i

The above guarantees the existence of Cournot equilibrium (see

Novshek (1985) and Shapiro (1989raquo Let the Cournot price

quantities and profits be denoted by pC Q~ and 7r~ respectively l l

If firms have symmetric costs then the Vives results (Proposition

1 and 2 in Vives (1985raquo follows The reason is as follows

Suppose all firms have same constant marginal costs w Then the

Bertrand equilibrium is unique and 1B w At a Cournot

equilibrium the first order conditions are given by

5

(Q) + II (0) ~ C I (0 ) 0 i 1 2 n i

OR(O) 1- HCQ) ~ w 0 i

If all firms have symmetric costs C () which are strictly convex 1

then the Bertrand equilibrium is necessarily non-unique In fact

any P E [P (n) P (n)] is a Bertrand equilibriumi i

shy

Now consider the Bertrand equilibrium price of P i

(n) We know from

Dastidar (forthcoming) that

shy 7f

) (P (n) n)

1

~ C (0)

shy

and 7f )

(P I n) so - C (0) t V P s P (n) )

A shy

and 11 ]

(P t p) gt - C(O) V P i

(n) s P lt pmax

Again the first order conditions for Cournot equilibrium are as

follows

QH (Q) + H(Q) C (Q) = 0 i = 1 2 n ) i

Therefore QC

i

Q~ gt 0 =gt H(QC) gt C (Qc) since H () lt 0 -----(1)] i

We claim that QC gt 0 =gt 1Ic gt -C(O) i i

Suppose on the contrary that 11 C

s -C(O)i

Now C(O) - C(Qc) gt QC C (QC) since C( is strictly convex iii

c c C cTherefore -Q C (Q) lt -Q H(Q)

iii

9 C (QC) gt H(Qc) which contradicts (1) i

6

Henae our claim follows

Note that any P e (Pi (n) P(n)l is a Bertrand equilibrium Since

pC P (n) there is always a Bertrand equilibrium price which is i

lower than the Cournot price Therefore the Vives (1985

Proposition 1) is valid in a homogeneous product market where

firms have symmetric costs

However proposition 2 is not always valid in an asymmetric market

We produce below a simple but rather extreme example to

illustrate this point Consider a homogeneous product duopoly

where the demand is given by the following

p 10 - (Q + Q ) 1 2

and costs are given by C (0) = ~Q2 and C (Q) = 5Q2 11 21 22 2

Routine calculations lead to the following A

pC 447 pm = 203 pm = 809 P 2 Ii 307 P 507 and 1 2 1 1 2

P 150172

It should be noted that P~ is the monopoly price of the ith firm

From the analysis in Dastidar (forthcoming) it is evident that the

Bertrand equilibrium is unique and is given by pe = pm = 203 1

Firm 1 undercuts firm 2 and since its monopoly price is less than A

p (the least that 2 will charge) it remains the only operating2

firm in the market We may note that pC lt pB That is we have

shown that even when Bertrand and Cournot equilibrium are uniqu~

it is not necessary that Bertrand equilibrium is more competitive

7

than Coumot tquilbriutn especially if costs are very asymmetric

n other words Vives results are not always valid in a homogeneous

product market where firms have asymmetric costs

8

REFERENCES

Cheng L (1985) Comparing Bertrand and Cournot Equilibria A Geometric Approach Rand Journal of Economics 16 146-152

Daatidar KG (forthcoming) On the Existence of Pure Strategy Bertrand Equilibrium Eccmomic Theory

Novshek W (1985) On the Existence of Cournot Equilibria Rev-ielv of Economic Studies 52 85-98

Okuguchi K (1987) Equilibrium Prices in the Bertrand and Cournot Oligopolies Journal of Economic Theory 42 128-139

Shapiro C (1989) Theories of Oligopoly Behaviour in Schmalensee R and Willig RD (ed) Handbook of Industrial Organisation (vol 1) North Holland

Singh N and Vives X (1984) Price and Quantity Competition in a Differentiated Duopoly Rand Journal of Economics 15 546-554

Vives X (1985) On the Efficiency of Bertrand and Cournot Equilibria with Product Differentiation Journal of Economic Theory 36 166-175

INTRODUCTION

In this paper we will try to reexamine some general notions

regarding the comparison of Cournot and Bertrand equilibrium

outcomes in a homogeneous product market It is a fairly well

known idea that Bertrand (price) competition is more competitive

than Cournot (quantity) competition In fact with a homogeneous

product and constant marginal costs the Bertrand outcomes involves

pricing at marginal costs This is not the case with

differentiated products where margins over marginal costs are

positive even in Bertrand competition

However even in a differentiated product set up Singh and Vives

(1984) provide a thorough comparison of Bertrand and Cournot

equilibria for the special case of constant marginal costs and the

linear system of equations So long as goods are substitutes the

Bertrand equilibrium is more competitive Cheng (1985) provides a

geometric proof of this same result that applies to a more general

class of cost and demand functions Vives (1985) gives fairly

general conditions under which the Cournot equilibria involve

higher prices and profits (and lower welfare) than do Bertrand

equilibria He shows that if the demand structure is symmetric

then (and Bertrand and Cournot equiibria are unique) then prices

and profits are larger and quantities ~inaller in Cournot than in

Bertrand competition (Proposition 1) If Bertrand reaction

functions are upward sloping (and continuous) then even with an

asymmetric demand structure) given any Cournot equilibrium price

vector one can find a Bertrand equilibrium with lower prices In

i the 3ilrlrand equilibrium is unique then it has

lOWEll than any Cournot ~quilibrium (Proposition 2)

Okuguohi (1997) also oompares the equilibrium prices for the

Bertrand and Cournot ies with product differentiation If

all firms have linear demand and cost functions and if in

addition the Jacobian matrix of the demand functions has a

dominant negative diagonal the Cournot equilibrium prices are not

lower than the Bertrand ones Okuguchi also derives the general

condition for the comparison of the Bertrand and Cournot

equilibrium prices when nonlinearities are involved in the cost

andor demand functions The condition ensuring the equilibrium

prices in the Cournot oligopoly to be not lower than those of the

Bertrand oligopoly is shown to be closely related to the global

stability condition for the Bertrand equilibrium prices

We will now reexamine the Vives (1985) results in a homogeneous

product market Consider a n firm oligopoly satisfying the

following assumptions

(a) F(P) is continuous twice continuously differentiable concave

and 3 positive numbers pmax and QTnaX such that F (pmax) 0 and

F(O) Qmax and F (P) lt 0 for 0 s P s pmax

(b) C(Qi) is continuous twice continuosly differentiable and

convex wi th C ( 0) 2 0 and C (Q ) 0 i

(c) In pure price competition typically firm i is viewed as

facing the following demand curve

3

i

o (P P ) 0 if P gt P for some ji i j i j

=1 (P ) if P J P V j and P P m i i j ik i

where k (12 m)

If (P ) if P lt P V j i i j

The above means that if is price is below js price it gets the

whole market whereas if theYcharge the same price they share the

market equally

(d) We assume that in price competition a firm always supplies the

demand it faces

Define the following

7T i

(P) = PF(P) - C(F(P)

~ i

(P) = ~PF(P) -C(F(Praquo m m

where 2 s m s n

P (m) st 7T (p (m)m)iii

-C (0)

P i

(m) st 7T(P(m)m) ~ 1

7T i

(pi

(m) m)

-We know from Dastidar (Forthcoming) that Pi and Pi exist and they

-are unique Also P lt P Also define the followingi i

arg P20 max [PF(P) - C(F(P)]

arg P20 max [~PF (P)2

A A

(e) Assume 7T (pm) 7T (pm) gt -C(O)iii i

Price competition

Suppose that both firms employ price as their strategic variable

4

NovI from De t 1 ci 1 WtJ know th~tt under (ii )

to (e) l e [P I if ] i8 a purte IlltratcOl9Y Nash equil ibrium in pricei J

competition If a firm i guctes a price in that range then it is

best for firm j to quote that price and not undercut it or charge

more To see that consider a firm i quoting a price P E [P bull Pl i i

Quoting the same P yields firm j a payoff equal to 11(1 ) l

quoting more j nets -C(O) and quoting less it gets 11 (P - e) Now i

from lemmas 1 to 8 of Dastidar we know that for any l E [1 1] l i

the following two conditions hold simultaneously

11 (P) c 7r (P) gt 7r (1 - E) V euro gt 0 and 11 (1) - C ( 0 ) i i i i

Since the above two conditions hold for P also our claim follows

Quantity Competition

If both firms employ quantity as its strategic variables then we

have the Cournot equilibrium

p(f) We aSsume that QH (Q) + H (Q) sO where H (Q)

and Q = ~ Q l

i

The above guarantees the existence of Cournot equilibrium (see

Novshek (1985) and Shapiro (1989raquo Let the Cournot price

quantities and profits be denoted by pC Q~ and 7r~ respectively l l

If firms have symmetric costs then the Vives results (Proposition

1 and 2 in Vives (1985raquo follows The reason is as follows

Suppose all firms have same constant marginal costs w Then the

Bertrand equilibrium is unique and 1B w At a Cournot

equilibrium the first order conditions are given by

5

(Q) + II (0) ~ C I (0 ) 0 i 1 2 n i

OR(O) 1- HCQ) ~ w 0 i

If all firms have symmetric costs C () which are strictly convex 1

then the Bertrand equilibrium is necessarily non-unique In fact

any P E [P (n) P (n)] is a Bertrand equilibriumi i

shy

Now consider the Bertrand equilibrium price of P i

(n) We know from

Dastidar (forthcoming) that

shy 7f

) (P (n) n)

1

~ C (0)

shy

and 7f )

(P I n) so - C (0) t V P s P (n) )

A shy

and 11 ]

(P t p) gt - C(O) V P i

(n) s P lt pmax

Again the first order conditions for Cournot equilibrium are as

follows

QH (Q) + H(Q) C (Q) = 0 i = 1 2 n ) i

Therefore QC

i

Q~ gt 0 =gt H(QC) gt C (Qc) since H () lt 0 -----(1)] i

We claim that QC gt 0 =gt 1Ic gt -C(O) i i

Suppose on the contrary that 11 C

s -C(O)i

Now C(O) - C(Qc) gt QC C (QC) since C( is strictly convex iii

c c C cTherefore -Q C (Q) lt -Q H(Q)

iii

9 C (QC) gt H(Qc) which contradicts (1) i

6

Henae our claim follows

Note that any P e (Pi (n) P(n)l is a Bertrand equilibrium Since

pC P (n) there is always a Bertrand equilibrium price which is i

lower than the Cournot price Therefore the Vives (1985

Proposition 1) is valid in a homogeneous product market where

firms have symmetric costs

However proposition 2 is not always valid in an asymmetric market

We produce below a simple but rather extreme example to

illustrate this point Consider a homogeneous product duopoly

where the demand is given by the following

p 10 - (Q + Q ) 1 2

and costs are given by C (0) = ~Q2 and C (Q) = 5Q2 11 21 22 2

Routine calculations lead to the following A

pC 447 pm = 203 pm = 809 P 2 Ii 307 P 507 and 1 2 1 1 2

P 150172

It should be noted that P~ is the monopoly price of the ith firm

From the analysis in Dastidar (forthcoming) it is evident that the

Bertrand equilibrium is unique and is given by pe = pm = 203 1

Firm 1 undercuts firm 2 and since its monopoly price is less than A

p (the least that 2 will charge) it remains the only operating2

firm in the market We may note that pC lt pB That is we have

shown that even when Bertrand and Cournot equilibrium are uniqu~

it is not necessary that Bertrand equilibrium is more competitive

7

than Coumot tquilbriutn especially if costs are very asymmetric

n other words Vives results are not always valid in a homogeneous

product market where firms have asymmetric costs

8

REFERENCES

Cheng L (1985) Comparing Bertrand and Cournot Equilibria A Geometric Approach Rand Journal of Economics 16 146-152

Daatidar KG (forthcoming) On the Existence of Pure Strategy Bertrand Equilibrium Eccmomic Theory

Novshek W (1985) On the Existence of Cournot Equilibria Rev-ielv of Economic Studies 52 85-98

Okuguchi K (1987) Equilibrium Prices in the Bertrand and Cournot Oligopolies Journal of Economic Theory 42 128-139

Shapiro C (1989) Theories of Oligopoly Behaviour in Schmalensee R and Willig RD (ed) Handbook of Industrial Organisation (vol 1) North Holland

Singh N and Vives X (1984) Price and Quantity Competition in a Differentiated Duopoly Rand Journal of Economics 15 546-554

Vives X (1985) On the Efficiency of Bertrand and Cournot Equilibria with Product Differentiation Journal of Economic Theory 36 166-175

i the 3ilrlrand equilibrium is unique then it has

lOWEll than any Cournot ~quilibrium (Proposition 2)

Okuguohi (1997) also oompares the equilibrium prices for the

Bertrand and Cournot ies with product differentiation If

all firms have linear demand and cost functions and if in

addition the Jacobian matrix of the demand functions has a

dominant negative diagonal the Cournot equilibrium prices are not

lower than the Bertrand ones Okuguchi also derives the general

condition for the comparison of the Bertrand and Cournot

equilibrium prices when nonlinearities are involved in the cost

andor demand functions The condition ensuring the equilibrium

prices in the Cournot oligopoly to be not lower than those of the

Bertrand oligopoly is shown to be closely related to the global

stability condition for the Bertrand equilibrium prices

We will now reexamine the Vives (1985) results in a homogeneous

product market Consider a n firm oligopoly satisfying the

following assumptions

(a) F(P) is continuous twice continuously differentiable concave

and 3 positive numbers pmax and QTnaX such that F (pmax) 0 and

F(O) Qmax and F (P) lt 0 for 0 s P s pmax

(b) C(Qi) is continuous twice continuosly differentiable and

convex wi th C ( 0) 2 0 and C (Q ) 0 i

(c) In pure price competition typically firm i is viewed as

facing the following demand curve

3

i

o (P P ) 0 if P gt P for some ji i j i j

=1 (P ) if P J P V j and P P m i i j ik i

where k (12 m)

If (P ) if P lt P V j i i j

The above means that if is price is below js price it gets the

whole market whereas if theYcharge the same price they share the

market equally

(d) We assume that in price competition a firm always supplies the

demand it faces

Define the following

7T i

(P) = PF(P) - C(F(P)

~ i

(P) = ~PF(P) -C(F(Praquo m m

where 2 s m s n

P (m) st 7T (p (m)m)iii

-C (0)

P i

(m) st 7T(P(m)m) ~ 1

7T i

(pi

(m) m)

-We know from Dastidar (Forthcoming) that Pi and Pi exist and they

-are unique Also P lt P Also define the followingi i

arg P20 max [PF(P) - C(F(P)]

arg P20 max [~PF (P)2

A A

(e) Assume 7T (pm) 7T (pm) gt -C(O)iii i

Price competition

Suppose that both firms employ price as their strategic variable

4

NovI from De t 1 ci 1 WtJ know th~tt under (ii )

to (e) l e [P I if ] i8 a purte IlltratcOl9Y Nash equil ibrium in pricei J

competition If a firm i guctes a price in that range then it is

best for firm j to quote that price and not undercut it or charge

more To see that consider a firm i quoting a price P E [P bull Pl i i

Quoting the same P yields firm j a payoff equal to 11(1 ) l

quoting more j nets -C(O) and quoting less it gets 11 (P - e) Now i

from lemmas 1 to 8 of Dastidar we know that for any l E [1 1] l i

the following two conditions hold simultaneously

11 (P) c 7r (P) gt 7r (1 - E) V euro gt 0 and 11 (1) - C ( 0 ) i i i i

Since the above two conditions hold for P also our claim follows

Quantity Competition

If both firms employ quantity as its strategic variables then we

have the Cournot equilibrium

p(f) We aSsume that QH (Q) + H (Q) sO where H (Q)

and Q = ~ Q l

i

The above guarantees the existence of Cournot equilibrium (see

Novshek (1985) and Shapiro (1989raquo Let the Cournot price

quantities and profits be denoted by pC Q~ and 7r~ respectively l l

If firms have symmetric costs then the Vives results (Proposition

1 and 2 in Vives (1985raquo follows The reason is as follows

Suppose all firms have same constant marginal costs w Then the

Bertrand equilibrium is unique and 1B w At a Cournot

equilibrium the first order conditions are given by

5

(Q) + II (0) ~ C I (0 ) 0 i 1 2 n i

OR(O) 1- HCQ) ~ w 0 i

If all firms have symmetric costs C () which are strictly convex 1

then the Bertrand equilibrium is necessarily non-unique In fact

any P E [P (n) P (n)] is a Bertrand equilibriumi i

shy

Now consider the Bertrand equilibrium price of P i

(n) We know from

Dastidar (forthcoming) that

shy 7f

) (P (n) n)

1

~ C (0)

shy

and 7f )

(P I n) so - C (0) t V P s P (n) )

A shy

and 11 ]

(P t p) gt - C(O) V P i

(n) s P lt pmax

Again the first order conditions for Cournot equilibrium are as

follows

QH (Q) + H(Q) C (Q) = 0 i = 1 2 n ) i

Therefore QC

i

Q~ gt 0 =gt H(QC) gt C (Qc) since H () lt 0 -----(1)] i

We claim that QC gt 0 =gt 1Ic gt -C(O) i i

Suppose on the contrary that 11 C

s -C(O)i

Now C(O) - C(Qc) gt QC C (QC) since C( is strictly convex iii

c c C cTherefore -Q C (Q) lt -Q H(Q)

iii

9 C (QC) gt H(Qc) which contradicts (1) i

6

Henae our claim follows

Note that any P e (Pi (n) P(n)l is a Bertrand equilibrium Since

pC P (n) there is always a Bertrand equilibrium price which is i

lower than the Cournot price Therefore the Vives (1985

Proposition 1) is valid in a homogeneous product market where

firms have symmetric costs

However proposition 2 is not always valid in an asymmetric market

We produce below a simple but rather extreme example to

illustrate this point Consider a homogeneous product duopoly

where the demand is given by the following

p 10 - (Q + Q ) 1 2

and costs are given by C (0) = ~Q2 and C (Q) = 5Q2 11 21 22 2

Routine calculations lead to the following A

pC 447 pm = 203 pm = 809 P 2 Ii 307 P 507 and 1 2 1 1 2

P 150172

It should be noted that P~ is the monopoly price of the ith firm

From the analysis in Dastidar (forthcoming) it is evident that the

Bertrand equilibrium is unique and is given by pe = pm = 203 1

Firm 1 undercuts firm 2 and since its monopoly price is less than A

p (the least that 2 will charge) it remains the only operating2

firm in the market We may note that pC lt pB That is we have

shown that even when Bertrand and Cournot equilibrium are uniqu~

it is not necessary that Bertrand equilibrium is more competitive

7

than Coumot tquilbriutn especially if costs are very asymmetric

n other words Vives results are not always valid in a homogeneous

product market where firms have asymmetric costs

8

REFERENCES

Cheng L (1985) Comparing Bertrand and Cournot Equilibria A Geometric Approach Rand Journal of Economics 16 146-152

Daatidar KG (forthcoming) On the Existence of Pure Strategy Bertrand Equilibrium Eccmomic Theory

Novshek W (1985) On the Existence of Cournot Equilibria Rev-ielv of Economic Studies 52 85-98

Okuguchi K (1987) Equilibrium Prices in the Bertrand and Cournot Oligopolies Journal of Economic Theory 42 128-139

Shapiro C (1989) Theories of Oligopoly Behaviour in Schmalensee R and Willig RD (ed) Handbook of Industrial Organisation (vol 1) North Holland

Singh N and Vives X (1984) Price and Quantity Competition in a Differentiated Duopoly Rand Journal of Economics 15 546-554

Vives X (1985) On the Efficiency of Bertrand and Cournot Equilibria with Product Differentiation Journal of Economic Theory 36 166-175

i

o (P P ) 0 if P gt P for some ji i j i j

=1 (P ) if P J P V j and P P m i i j ik i

where k (12 m)

If (P ) if P lt P V j i i j

The above means that if is price is below js price it gets the

whole market whereas if theYcharge the same price they share the

market equally

(d) We assume that in price competition a firm always supplies the

demand it faces

Define the following

7T i

(P) = PF(P) - C(F(P)

~ i

(P) = ~PF(P) -C(F(Praquo m m

where 2 s m s n

P (m) st 7T (p (m)m)iii

-C (0)

P i

(m) st 7T(P(m)m) ~ 1

7T i

(pi

(m) m)

-We know from Dastidar (Forthcoming) that Pi and Pi exist and they

-are unique Also P lt P Also define the followingi i

arg P20 max [PF(P) - C(F(P)]

arg P20 max [~PF (P)2

A A

(e) Assume 7T (pm) 7T (pm) gt -C(O)iii i

Price competition

Suppose that both firms employ price as their strategic variable

4

NovI from De t 1 ci 1 WtJ know th~tt under (ii )

to (e) l e [P I if ] i8 a purte IlltratcOl9Y Nash equil ibrium in pricei J

competition If a firm i guctes a price in that range then it is

best for firm j to quote that price and not undercut it or charge

more To see that consider a firm i quoting a price P E [P bull Pl i i

Quoting the same P yields firm j a payoff equal to 11(1 ) l

quoting more j nets -C(O) and quoting less it gets 11 (P - e) Now i

from lemmas 1 to 8 of Dastidar we know that for any l E [1 1] l i

the following two conditions hold simultaneously

11 (P) c 7r (P) gt 7r (1 - E) V euro gt 0 and 11 (1) - C ( 0 ) i i i i

Since the above two conditions hold for P also our claim follows

Quantity Competition

If both firms employ quantity as its strategic variables then we

have the Cournot equilibrium

p(f) We aSsume that QH (Q) + H (Q) sO where H (Q)

and Q = ~ Q l

i

The above guarantees the existence of Cournot equilibrium (see

Novshek (1985) and Shapiro (1989raquo Let the Cournot price

quantities and profits be denoted by pC Q~ and 7r~ respectively l l

If firms have symmetric costs then the Vives results (Proposition

1 and 2 in Vives (1985raquo follows The reason is as follows

Suppose all firms have same constant marginal costs w Then the

Bertrand equilibrium is unique and 1B w At a Cournot

equilibrium the first order conditions are given by

5

(Q) + II (0) ~ C I (0 ) 0 i 1 2 n i

OR(O) 1- HCQ) ~ w 0 i

If all firms have symmetric costs C () which are strictly convex 1

then the Bertrand equilibrium is necessarily non-unique In fact

any P E [P (n) P (n)] is a Bertrand equilibriumi i

shy

Now consider the Bertrand equilibrium price of P i

(n) We know from

Dastidar (forthcoming) that

shy 7f

) (P (n) n)

1

~ C (0)

shy

and 7f )

(P I n) so - C (0) t V P s P (n) )

A shy

and 11 ]

(P t p) gt - C(O) V P i

(n) s P lt pmax

Again the first order conditions for Cournot equilibrium are as

follows

QH (Q) + H(Q) C (Q) = 0 i = 1 2 n ) i

Therefore QC

i

Q~ gt 0 =gt H(QC) gt C (Qc) since H () lt 0 -----(1)] i

We claim that QC gt 0 =gt 1Ic gt -C(O) i i

Suppose on the contrary that 11 C

s -C(O)i

Now C(O) - C(Qc) gt QC C (QC) since C( is strictly convex iii

c c C cTherefore -Q C (Q) lt -Q H(Q)

iii

9 C (QC) gt H(Qc) which contradicts (1) i

6

Henae our claim follows

Note that any P e (Pi (n) P(n)l is a Bertrand equilibrium Since

pC P (n) there is always a Bertrand equilibrium price which is i

lower than the Cournot price Therefore the Vives (1985

Proposition 1) is valid in a homogeneous product market where

firms have symmetric costs

However proposition 2 is not always valid in an asymmetric market

We produce below a simple but rather extreme example to

illustrate this point Consider a homogeneous product duopoly

where the demand is given by the following

p 10 - (Q + Q ) 1 2

and costs are given by C (0) = ~Q2 and C (Q) = 5Q2 11 21 22 2

Routine calculations lead to the following A

pC 447 pm = 203 pm = 809 P 2 Ii 307 P 507 and 1 2 1 1 2

P 150172

It should be noted that P~ is the monopoly price of the ith firm

From the analysis in Dastidar (forthcoming) it is evident that the

Bertrand equilibrium is unique and is given by pe = pm = 203 1

Firm 1 undercuts firm 2 and since its monopoly price is less than A

p (the least that 2 will charge) it remains the only operating2

firm in the market We may note that pC lt pB That is we have

shown that even when Bertrand and Cournot equilibrium are uniqu~

it is not necessary that Bertrand equilibrium is more competitive

7

than Coumot tquilbriutn especially if costs are very asymmetric

n other words Vives results are not always valid in a homogeneous

product market where firms have asymmetric costs

8

REFERENCES

Cheng L (1985) Comparing Bertrand and Cournot Equilibria A Geometric Approach Rand Journal of Economics 16 146-152

Daatidar KG (forthcoming) On the Existence of Pure Strategy Bertrand Equilibrium Eccmomic Theory

Novshek W (1985) On the Existence of Cournot Equilibria Rev-ielv of Economic Studies 52 85-98

Okuguchi K (1987) Equilibrium Prices in the Bertrand and Cournot Oligopolies Journal of Economic Theory 42 128-139

Shapiro C (1989) Theories of Oligopoly Behaviour in Schmalensee R and Willig RD (ed) Handbook of Industrial Organisation (vol 1) North Holland

Singh N and Vives X (1984) Price and Quantity Competition in a Differentiated Duopoly Rand Journal of Economics 15 546-554

Vives X (1985) On the Efficiency of Bertrand and Cournot Equilibria with Product Differentiation Journal of Economic Theory 36 166-175

NovI from De t 1 ci 1 WtJ know th~tt under (ii )

to (e) l e [P I if ] i8 a purte IlltratcOl9Y Nash equil ibrium in pricei J

competition If a firm i guctes a price in that range then it is

best for firm j to quote that price and not undercut it or charge

more To see that consider a firm i quoting a price P E [P bull Pl i i

Quoting the same P yields firm j a payoff equal to 11(1 ) l

quoting more j nets -C(O) and quoting less it gets 11 (P - e) Now i

from lemmas 1 to 8 of Dastidar we know that for any l E [1 1] l i

the following two conditions hold simultaneously

11 (P) c 7r (P) gt 7r (1 - E) V euro gt 0 and 11 (1) - C ( 0 ) i i i i

Since the above two conditions hold for P also our claim follows

Quantity Competition

If both firms employ quantity as its strategic variables then we

have the Cournot equilibrium

p(f) We aSsume that QH (Q) + H (Q) sO where H (Q)

and Q = ~ Q l

i

The above guarantees the existence of Cournot equilibrium (see

Novshek (1985) and Shapiro (1989raquo Let the Cournot price

quantities and profits be denoted by pC Q~ and 7r~ respectively l l

If firms have symmetric costs then the Vives results (Proposition

1 and 2 in Vives (1985raquo follows The reason is as follows

Suppose all firms have same constant marginal costs w Then the

Bertrand equilibrium is unique and 1B w At a Cournot

equilibrium the first order conditions are given by

5

(Q) + II (0) ~ C I (0 ) 0 i 1 2 n i

OR(O) 1- HCQ) ~ w 0 i

If all firms have symmetric costs C () which are strictly convex 1

then the Bertrand equilibrium is necessarily non-unique In fact

any P E [P (n) P (n)] is a Bertrand equilibriumi i

shy

Now consider the Bertrand equilibrium price of P i

(n) We know from

Dastidar (forthcoming) that

shy 7f

) (P (n) n)

1

~ C (0)

shy

and 7f )

(P I n) so - C (0) t V P s P (n) )

A shy

and 11 ]

(P t p) gt - C(O) V P i

(n) s P lt pmax

Again the first order conditions for Cournot equilibrium are as

follows

QH (Q) + H(Q) C (Q) = 0 i = 1 2 n ) i

Therefore QC

i

Q~ gt 0 =gt H(QC) gt C (Qc) since H () lt 0 -----(1)] i

We claim that QC gt 0 =gt 1Ic gt -C(O) i i

Suppose on the contrary that 11 C

s -C(O)i

Now C(O) - C(Qc) gt QC C (QC) since C( is strictly convex iii

c c C cTherefore -Q C (Q) lt -Q H(Q)

iii

9 C (QC) gt H(Qc) which contradicts (1) i

6

Henae our claim follows

Note that any P e (Pi (n) P(n)l is a Bertrand equilibrium Since

pC P (n) there is always a Bertrand equilibrium price which is i

lower than the Cournot price Therefore the Vives (1985

Proposition 1) is valid in a homogeneous product market where

firms have symmetric costs

However proposition 2 is not always valid in an asymmetric market

We produce below a simple but rather extreme example to

illustrate this point Consider a homogeneous product duopoly

where the demand is given by the following

p 10 - (Q + Q ) 1 2

and costs are given by C (0) = ~Q2 and C (Q) = 5Q2 11 21 22 2

Routine calculations lead to the following A

pC 447 pm = 203 pm = 809 P 2 Ii 307 P 507 and 1 2 1 1 2

P 150172

It should be noted that P~ is the monopoly price of the ith firm

From the analysis in Dastidar (forthcoming) it is evident that the

Bertrand equilibrium is unique and is given by pe = pm = 203 1

Firm 1 undercuts firm 2 and since its monopoly price is less than A

p (the least that 2 will charge) it remains the only operating2

firm in the market We may note that pC lt pB That is we have

shown that even when Bertrand and Cournot equilibrium are uniqu~

it is not necessary that Bertrand equilibrium is more competitive

7

than Coumot tquilbriutn especially if costs are very asymmetric

n other words Vives results are not always valid in a homogeneous

product market where firms have asymmetric costs

8

REFERENCES

Cheng L (1985) Comparing Bertrand and Cournot Equilibria A Geometric Approach Rand Journal of Economics 16 146-152

Daatidar KG (forthcoming) On the Existence of Pure Strategy Bertrand Equilibrium Eccmomic Theory

Novshek W (1985) On the Existence of Cournot Equilibria Rev-ielv of Economic Studies 52 85-98

Okuguchi K (1987) Equilibrium Prices in the Bertrand and Cournot Oligopolies Journal of Economic Theory 42 128-139

Shapiro C (1989) Theories of Oligopoly Behaviour in Schmalensee R and Willig RD (ed) Handbook of Industrial Organisation (vol 1) North Holland

Singh N and Vives X (1984) Price and Quantity Competition in a Differentiated Duopoly Rand Journal of Economics 15 546-554

Vives X (1985) On the Efficiency of Bertrand and Cournot Equilibria with Product Differentiation Journal of Economic Theory 36 166-175

(Q) + II (0) ~ C I (0 ) 0 i 1 2 n i

OR(O) 1- HCQ) ~ w 0 i

If all firms have symmetric costs C () which are strictly convex 1

then the Bertrand equilibrium is necessarily non-unique In fact

any P E [P (n) P (n)] is a Bertrand equilibriumi i

shy

Now consider the Bertrand equilibrium price of P i

(n) We know from

Dastidar (forthcoming) that

shy 7f

) (P (n) n)

1

~ C (0)

shy

and 7f )

(P I n) so - C (0) t V P s P (n) )

A shy

and 11 ]

(P t p) gt - C(O) V P i

(n) s P lt pmax

Again the first order conditions for Cournot equilibrium are as

follows

QH (Q) + H(Q) C (Q) = 0 i = 1 2 n ) i

Therefore QC

i

Q~ gt 0 =gt H(QC) gt C (Qc) since H () lt 0 -----(1)] i

We claim that QC gt 0 =gt 1Ic gt -C(O) i i

Suppose on the contrary that 11 C

s -C(O)i

Now C(O) - C(Qc) gt QC C (QC) since C( is strictly convex iii

c c C cTherefore -Q C (Q) lt -Q H(Q)

iii

9 C (QC) gt H(Qc) which contradicts (1) i

6

Henae our claim follows

Note that any P e (Pi (n) P(n)l is a Bertrand equilibrium Since

pC P (n) there is always a Bertrand equilibrium price which is i

lower than the Cournot price Therefore the Vives (1985

Proposition 1) is valid in a homogeneous product market where

firms have symmetric costs

However proposition 2 is not always valid in an asymmetric market

We produce below a simple but rather extreme example to

illustrate this point Consider a homogeneous product duopoly

where the demand is given by the following

p 10 - (Q + Q ) 1 2

and costs are given by C (0) = ~Q2 and C (Q) = 5Q2 11 21 22 2

Routine calculations lead to the following A

pC 447 pm = 203 pm = 809 P 2 Ii 307 P 507 and 1 2 1 1 2

P 150172

It should be noted that P~ is the monopoly price of the ith firm

From the analysis in Dastidar (forthcoming) it is evident that the

Bertrand equilibrium is unique and is given by pe = pm = 203 1

Firm 1 undercuts firm 2 and since its monopoly price is less than A

p (the least that 2 will charge) it remains the only operating2

firm in the market We may note that pC lt pB That is we have

shown that even when Bertrand and Cournot equilibrium are uniqu~

it is not necessary that Bertrand equilibrium is more competitive

7

than Coumot tquilbriutn especially if costs are very asymmetric

n other words Vives results are not always valid in a homogeneous

product market where firms have asymmetric costs

8

REFERENCES

Cheng L (1985) Comparing Bertrand and Cournot Equilibria A Geometric Approach Rand Journal of Economics 16 146-152

Daatidar KG (forthcoming) On the Existence of Pure Strategy Bertrand Equilibrium Eccmomic Theory

Novshek W (1985) On the Existence of Cournot Equilibria Rev-ielv of Economic Studies 52 85-98

Okuguchi K (1987) Equilibrium Prices in the Bertrand and Cournot Oligopolies Journal of Economic Theory 42 128-139

Shapiro C (1989) Theories of Oligopoly Behaviour in Schmalensee R and Willig RD (ed) Handbook of Industrial Organisation (vol 1) North Holland

Singh N and Vives X (1984) Price and Quantity Competition in a Differentiated Duopoly Rand Journal of Economics 15 546-554

Vives X (1985) On the Efficiency of Bertrand and Cournot Equilibria with Product Differentiation Journal of Economic Theory 36 166-175

Henae our claim follows

Note that any P e (Pi (n) P(n)l is a Bertrand equilibrium Since

pC P (n) there is always a Bertrand equilibrium price which is i

lower than the Cournot price Therefore the Vives (1985

Proposition 1) is valid in a homogeneous product market where

firms have symmetric costs

However proposition 2 is not always valid in an asymmetric market

We produce below a simple but rather extreme example to

illustrate this point Consider a homogeneous product duopoly

where the demand is given by the following

p 10 - (Q + Q ) 1 2

and costs are given by C (0) = ~Q2 and C (Q) = 5Q2 11 21 22 2

Routine calculations lead to the following A

pC 447 pm = 203 pm = 809 P 2 Ii 307 P 507 and 1 2 1 1 2

P 150172

It should be noted that P~ is the monopoly price of the ith firm

From the analysis in Dastidar (forthcoming) it is evident that the

Bertrand equilibrium is unique and is given by pe = pm = 203 1

Firm 1 undercuts firm 2 and since its monopoly price is less than A

p (the least that 2 will charge) it remains the only operating2

firm in the market We may note that pC lt pB That is we have

shown that even when Bertrand and Cournot equilibrium are uniqu~

it is not necessary that Bertrand equilibrium is more competitive

7

than Coumot tquilbriutn especially if costs are very asymmetric

n other words Vives results are not always valid in a homogeneous

product market where firms have asymmetric costs

8

REFERENCES

Cheng L (1985) Comparing Bertrand and Cournot Equilibria A Geometric Approach Rand Journal of Economics 16 146-152

Daatidar KG (forthcoming) On the Existence of Pure Strategy Bertrand Equilibrium Eccmomic Theory

Novshek W (1985) On the Existence of Cournot Equilibria Rev-ielv of Economic Studies 52 85-98

Okuguchi K (1987) Equilibrium Prices in the Bertrand and Cournot Oligopolies Journal of Economic Theory 42 128-139

Shapiro C (1989) Theories of Oligopoly Behaviour in Schmalensee R and Willig RD (ed) Handbook of Industrial Organisation (vol 1) North Holland

Singh N and Vives X (1984) Price and Quantity Competition in a Differentiated Duopoly Rand Journal of Economics 15 546-554

Vives X (1985) On the Efficiency of Bertrand and Cournot Equilibria with Product Differentiation Journal of Economic Theory 36 166-175

than Coumot tquilbriutn especially if costs are very asymmetric

n other words Vives results are not always valid in a homogeneous

product market where firms have asymmetric costs

8

REFERENCES

Cheng L (1985) Comparing Bertrand and Cournot Equilibria A Geometric Approach Rand Journal of Economics 16 146-152

Daatidar KG (forthcoming) On the Existence of Pure Strategy Bertrand Equilibrium Eccmomic Theory

Novshek W (1985) On the Existence of Cournot Equilibria Rev-ielv of Economic Studies 52 85-98

Okuguchi K (1987) Equilibrium Prices in the Bertrand and Cournot Oligopolies Journal of Economic Theory 42 128-139

Shapiro C (1989) Theories of Oligopoly Behaviour in Schmalensee R and Willig RD (ed) Handbook of Industrial Organisation (vol 1) North Holland

Singh N and Vives X (1984) Price and Quantity Competition in a Differentiated Duopoly Rand Journal of Economics 15 546-554

Vives X (1985) On the Efficiency of Bertrand and Cournot Equilibria with Product Differentiation Journal of Economic Theory 36 166-175

REFERENCES

Cheng L (1985) Comparing Bertrand and Cournot Equilibria A Geometric Approach Rand Journal of Economics 16 146-152

Daatidar KG (forthcoming) On the Existence of Pure Strategy Bertrand Equilibrium Eccmomic Theory

Novshek W (1985) On the Existence of Cournot Equilibria Rev-ielv of Economic Studies 52 85-98

Okuguchi K (1987) Equilibrium Prices in the Bertrand and Cournot Oligopolies Journal of Economic Theory 42 128-139

Shapiro C (1989) Theories of Oligopoly Behaviour in Schmalensee R and Willig RD (ed) Handbook of Industrial Organisation (vol 1) North Holland

Singh N and Vives X (1984) Price and Quantity Competition in a Differentiated Duopoly Rand Journal of Economics 15 546-554

Vives X (1985) On the Efficiency of Bertrand and Cournot Equilibria with Product Differentiation Journal of Economic Theory 36 166-175

CENTRE FOR DEVELOPMENT ECONOMICS WORKING PAPER SERIES

Number

1

2

3

4

5

6

7

8

9

10

11

Author(s)

Kaushik Basu Arghya Ghosh Tridip Ray

M N Murty Ranjan Ray

V Bhaskarmiddot Mushtaq Khan

V Bhaskar

Bishnupriya Gupta

Kaushik Basu

Partha Sen

Partha Sen

Partha Sen Arghya Ghosh Abheek Barman

V Bhaskar

V Bhaskar

Title

The Babu and The BoxwalJah Managerial Incentives and Government Intervention (Jan 1994)

Optimal Taxation and Resource Transfers (Feb 1994)

in a Federal Nation

PrivatizatStudy of Bangladesh

ion The

and Employment Jute Industry

(Mar 1994)

A in

Distributive Justice and The Control of Global Warming (Mar 1994)

The Great Depression and Brazils Capital Goods Sector A Re-examination (Apr 1994)

Where There Is No Economist Some Institutional and Legal Prerequisites of Economic Reform in India (May 1994)

An Example of Welfare Reducing Tariff Under Monopolistic Competition (May 1994)

Environmental Policies and NorthshySouth Trade A Selected Survey of The Issues (May 1994)

The Possibility of Welfare Gains with Capital Inflows in A Small Tariff shyRidden Economy (June 1994)

Sustaining Inter-Generational Altruism when Social Memory is Bounded (June 1994)

Repeated Games with Almost Perfect Monitoring by Privately Observed Signals (June 1994)

NUmber

12

13

14

15

16

17

18

19

20

21

22

Authors

S Nandeibam

Kaushik Basu

Kaushik Basu

S Nandeibam

MrinalDatta Chaudhuri

S Nandeibam

D Jayaraj S Subramanian

K Ghosh Dastidar

Kaushik Basu

Partha Sen

K Ghosh Dastidar

Coalitional Power Structure in Stochastic Social Choice Functions with An qnrestricted Preference Domain (June 1994)

The Axiomatic S~ructure of Knowledge And Perception (July 1994)

Bargaining with Set Valued Disagreement (July 1994)

A Note on Randomized Social Choice and Random Dictatorships (July 1994)

Labour Markets As Social Institutions in India (July 1994)

Moral Hazard in a Principal-Agent (s) Team (July 1994)

Caste Discrimination in the Distribution of Consumption Expenditure in India Theory and Evidence (August 1994)

Debt Financing with Limited Liability and Quantity Competition (August 1994)

Industrial Organization and Developing Economies August 1994

Theory

Immiserizing Growth in a Model of Trade with Monopolisitic Competition (August 1994)

Comparing Cournot and Bertrand-in a Homogeneous Product Marketmiddot (Sept~ 1994)

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NUmber

12

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Authors

S Nandeibam

Kaushik Basu

Kaushik Basu

S Nandeibam

MrinalDatta Chaudhuri

S Nandeibam

D Jayaraj S Subramanian

K Ghosh Dastidar

Kaushik Basu

Partha Sen

K Ghosh Dastidar

Coalitional Power Structure in Stochastic Social Choice Functions with An qnrestricted Preference Domain (June 1994)

The Axiomatic S~ructure of Knowledge And Perception (July 1994)

Bargaining with Set Valued Disagreement (July 1994)

A Note on Randomized Social Choice and Random Dictatorships (July 1994)

Labour Markets As Social Institutions in India (July 1994)

Moral Hazard in a Principal-Agent (s) Team (July 1994)

Caste Discrimination in the Distribution of Consumption Expenditure in India Theory and Evidence (August 1994)

Debt Financing with Limited Liability and Quantity Competition (August 1994)

Industrial Organization and Developing Economies August 1994

Theory

Immiserizing Growth in a Model of Trade with Monopolisitic Competition (August 1994)

Comparing Cournot and Bertrand-in a Homogeneous Product Marketmiddot (Sept~ 1994)

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