Monopolistic Competition when Income Matters
Paolo Bertoletti and Federico Etro
University of Pavia and Ca’Foscari University, Venice
Hitotsubashi University, March 6, 2014
Purpose
I
We propose an alternative microfoundation to models of
imperfect competition and product di¤erentiation with and
without endogenous entry
Purpose
I
We propose an alternative microfoundation to models of
imperfect competition and product di¤erentiation with and
without endogenous entry
I
Alternative to the Dixit-Stiglitz (1977) microfoundation.
Purpose
I
We propose an alternative microfoundation to models of
imperfect competition and product di¤erentiation with and
without endogenous entry
I
Alternative to the Dixit-Stiglitz (1977) microfoundation.
I
Separable utility à la Dixit-Stiglitz is widely applied in trade
(Krugman, 1980; Melitz, 2003) and macroeconomics
(New-Keynesian models, Endogenous entry models) under
CES preferences.
Purpose
I
We propose an alternative microfoundation to models of
imperfect competition and product di¤erentiation with and
without endogenous entry
I
Alternative to the Dixit-Stiglitz (1977) microfoundation.
I
Separable utility à la Dixit-Stiglitz is widely applied in trade
(Krugman, 1980; Melitz, 2003) and macroeconomics
(New-Keynesian models, Endogenous entry models) under
CES preferences.
I
It is also useful to study Cournot competition with product
di¤erentiation.
Background
I
D-S assume additively separable preferences (“direct
additivity”):
U = ∑nj=1 u (xj )
Background
I
D-S assume additively separable preferences (“direct
additivity”):
U = ∑nj=1 u (xj )
I
Separability is restrictive, because there are many other
symmetric but non-separable preferences U = U (x)
Background
I
D-S assume additively separable preferences (“direct
additivity”):
U = ∑nj=1 u (xj )
I
Separability is restrictive, because there are many other
symmetric but non-separable preferences U = U (x)
I
Direct additivity implies that the Marginal Rate of
Substitution u (xi ) /u (xj ) between any two varieties does not
depend on the consumption of other varieties xk .
Background
I
D-S assume additively separable preferences (“direct
additivity”):
U = ∑nj=1 u (xj )
I
Separability is restrictive, because there are many other
symmetric but non-separable preferences U = U (x)
I
Direct additivity implies that the Marginal Rate of
Substitution u (xi ) /u (xj ) between any two varieties does not
depend on the consumption of other varieties xk .
I
Most applications focus on CES preferences with θ 2 (1, ∞):
U = ∑nj=1 xj
θ 1
θ
Monopolistic competition à la Dixit-Stiglitz
I
Equilibrium in the CES case:
EL
( θ 1) F
,
q=
θ 1
θF
c
where p = price, n = number of …rms and q = xL = …rm
production, E = income, L = population, c = marginal cost
and F = …xed cost. A double marke size (number of
consumers) generates double number of goods, with same
price (and quantity per …rm), but only with CES. Income does
not a¤ect price and quantity, not just with CES
p=
θc
,
n=
Monopolistic competition à la Dixit-Stiglitz
I
Equilibrium in the CES case:
EL
( θ 1) F
,
q=
θ 1
θF
c
where p = price, n = number of …rms and q = xL = …rm
production, E = income, L = population, c = marginal cost
and F = …xed cost. A double marke size (number of
consumers) generates double number of goods, with same
price (and quantity per …rm), but only with CES. Income does
not a¤ect price and quantity, not just with CES
General case (Zhelobodko et al., 2012, E.; Bertoletti-Epifani,
2012; Bertoletti-Etro, 2014, Econ. Bulletin):
p=
I
p=
θc
θ (x )c
,
θ (x ) 1
where θ (x ) =
,
n=
n=
EL
,
θ (x )F
u 0 (x )/xu 00 (x ).
q=
( θ (x )
1) F
c
Monopolistic competition à la Dixit-Stiglitz
I
Equilibrium in the CES case:
EL
( θ 1) F
,
q=
θ 1
θF
c
where p = price, n = number of …rms and q = xL = …rm
production, E = income, L = population, c = marginal cost
and F = …xed cost. A double marke size (number of
consumers) generates double number of goods, with same
price (and quantity per …rm), but only with CES. Income does
not a¤ect price and quantity, not just with CES
General case (Zhelobodko et al., 2012, E.; Bertoletti-Epifani,
2012; Bertoletti-Etro, 2014, Econ. Bulletin):
p=
I
p=
I
θc
θ (x )c
,
θ (x ) 1
,
n=
n=
EL
,
θ (x )F
q=
( θ (x )
1) F
c
where θ (x ) = u 0 (x )/xu 00 (x ).
With Cournot or Bertrand competition and CES, an additional
competition e¤ect for trade (Etro, 2013, Scand.J.E.) and RBC
(Etro-Colciago, 2010, Econ.Journ.)
The Model
I
We consider a di¤erent microfoundation, based on di¤erent
preferences
The Model
I
We consider a di¤erent microfoundation, based on di¤erent
preferences
I
We look at the indirect utility (dual approach)
The Model
I
We consider a di¤erent microfoundation, based on di¤erent
preferences
I
We look at the indirect utility (dual approach)
I
and assume additively separable indirect utility (“indirect
additivity”):
The Model
I
We consider a di¤erent microfoundation, based on di¤erent
preferences
I
We look at the indirect utility (dual approach)
I
and assume additively separable indirect utility (“indirect
additivity”):
I
n
V =
∑v
j =1
pj
E
with v > 0, v 0 < 0 and v 00 > 0 and some regularity
conditions. E is income of the consumer.
The Model
I
We consider a di¤erent microfoundation, based on di¤erent
preferences
I
We look at the indirect utility (dual approach)
I
and assume additively separable indirect utility (“indirect
additivity”):
I
n
V =
∑v
j =1
pj
E
with v > 0, v 0 < 0 and v 00 > 0 and some regularity
conditions. E is income of the consumer.
I
By Hicks (1969) and Samuelson (1969) we know that direct
additivity and indirect additivity represent two distinct classes
of well-behaved preferences with only one case in common:
CES.
Direct demand function
I
Indirect additivity (∑nj=1 =
Rn
V =
0
if you like):
∑v
pj
E
Direct demand function
I
Indirect additivity (∑nj=1 =
Rn
V =
I
0
if you like):
∑v
pj
E
The Roy identity generates the direct demand function of
each consumer:
0
v
∂V /∂pi
xi =
=
∂V /∂E
∑ v0
pi
E
pj
E
pj
E
Direct demand function
I
Indirect additivity (∑nj=1 =
Rn
V =
I
0
if you like):
∑v
pj
E
The Roy identity generates the direct demand function of
each consumer:
0
v
∂V /∂pi
xi =
=
∂V /∂E
∑ v0
I
pi
E
pj
E
pj
E
Indirect additivity implies that the relative demand of two
varieties xi /xj does not depend on the price of other varieties
pk .
Direct demand function
I
Indirect additivity (∑nj=1 =
Rn
V =
I
0
if you like):
∑v
pj
E
The Roy identity generates the direct demand function of
each consumer:
0
v
∂V /∂pi
xi =
=
∂V /∂E
∑ v0
pi
E
pj
E
pj
E
I
Indirect additivity implies that the relative demand of two
varieties xi /xj does not depend on the price of other varieties
pk .
I
The denominator µ = ∑ v 0
monopolistic competition.
pj
E
pj
E
< 0 is taken as given in
Direct demand function
I
Indirect additivity (∑nj=1 =
Rn
V =
I
0
if you like):
∑v
pj
E
The Roy identity generates the direct demand function of
each consumer:
0
v
∂V /∂pi
xi =
=
∂V /∂E
∑ v0
pi
E
pj
E
pj
E
I
Indirect additivity implies that the relative demand of two
varieties xi /xj does not depend on the price of other varieties
pk .
I
The denominator µ = ∑ v 0
monopolistic competition.
I
Market demand is given by qi = xi L where L is number of
consumers (marke size)
pj
E
pj
E
< 0 is taken as given in
Some examples of direct demands
I
CES: v (p ) = p 1
θ
delivers:
qi =
pi θ EL
∑ pj1
θ
Some examples of direct demands
I
CES: v (p ) = p 1
θ
delivers:
qi =
I
EXPONENTIAL: v (p ) = e
qi =
pi θ EL
∑ pj1
τp
θ
delivers log-linear demand:
e
∑e
τp i
E
EL
τp j
E
pj
(notice the di¤erence from the Logit, which has no income
e¤ects)
Some examples of direct demands
I
ADDILOG: v (p ) = (a
demand when γ = 1:
p )1 +γ delivers the linear perceived
qi =
a
∑ a
pi
E
pj
E
EL
pj
Some examples of direct demands
I
ADDILOG: v (p ) = (a
demand when γ = 1:
p )1 +γ delivers the linear perceived
qi =
I
a
∑ a
pi
E
pj
E
DISPLACED CES: v (p ) = (p + b )1
qi =
∑
pi
E +b
pi
E +b
EL
pj
ϑ
ϑ
delivers:
EL
ϑ
pj
Monopolistic Competition (Dual)
I
Monopolistic competition implies that there are so many …rms
that the impact of each price on the marginal utility of income
is negligible: µ is taken as given
Monopolistic Competition (Dual)
I
Monopolistic competition implies that there are so many …rms
that the impact of each price on the marginal utility of income
is negligible: µ is taken as given
I
Pro…t can be written as:
πi =
( pi
c )v
µ
0
pi
E
L
F
where c > 0 and F > 0 are respectively marginal and …xed
costs.
Monopolistic Competition (Dual)
I
Monopolistic competition implies that there are so many …rms
that the impact of each price on the marginal utility of income
is negligible: µ is taken as given
I
Pro…t can be written as:
πi =
( pi
c )v
µ
0
pi
E
L
F
where c > 0 and F > 0 are respectively marginal and …xed
costs.
I
The demand elasticity is θ (pi /E )
pi /E , not on µ and L.
v 00 p i
v 0E
> 0: it depends on
Pricing
I
The FOC is:
pe c
1
=
e
e
p
θ pE
Pricing
I
The FOC is:
I
The optimal price is always independent from the number of
varieties.
pe c
1
=
e
e
p
θ pE
Pricing
I
The FOC is:
I
The optimal price is always independent from the number of
varieties.
I
However, if θ 0 > (<) 0, the optimal price grows (decreases)
with income because …rms face a more (less) rigid demand.
pe c
1
=
e
e
p
θ pE
Pricing
I
The FOC is:
I
The optimal price is always independent from the number of
varieties.
I
However, if θ 0 > (<) 0, the optimal price grows (decreases)
with income because …rms face a more (less) rigid demand.
I
Rationale for procyclical markups in macro, for pricing to
market in trade
pe c
1
=
e
e
p
θ pE
Endogenous Entry Equilibrium
I
Dual:
pe c
1
=
e ,
pe
θ pE
EL
n =
e ,
F θ pE
e
e
q =F
θ
pe
E
c
1
Endogenous Entry Equilibrium
I
Dual:
pe c
1
=
e ,
pe
θ pE
I
EL
n =
e ,
F θ pE
e
e
q =F
θ
pe
E
1
c
Notice that
epL = eqL = 0
and
enL = 1
which generalizes the classical result by Krugman (1980)
concerning market size with CES preferences: pure gains from
variety without any competitive e¤ect on prices and …rm size.
Endogenous Entry Equilibrium
I
Dual:
pe c
1
=
e ,
pe
θ pE
I
EL
n =
e ,
F θ pE
e
θ
q =F
pe
E
1
c
Notice that
epL = eqL = 0
I
e
and
enL = 1
which generalizes the classical result by Krugman (1980)
concerning market size with CES preferences: pure gains from
variety without any competitive e¤ect on prices and …rm size.
Other results (pricing to market and undershifting):
epE ? 0 and enE ? 1 i¤ θ 0 (p e /E ) ? 0
epc 7 1 and enc 7 0 i¤ θ 0 (p e /E ) ? 0
Two new examples with closed form solutions
I
The (negative) exponential demand qi = e
pe = c +
E
,
τ
ne =
E 2L
,
F (cτ + E )
τp i
E
L/µ delivers:
qe =
Fτ
E
Two new examples with closed form solutions
I
The (negative) exponential demand qi = e
pe = c +
I
E
,
τ
ne =
E 2L
,
F (cτ + E )
The linear demand case qi = a
pe =
c + aE
,
2
ne =
pi
E
τp i
E
L/µ delivers:
qe =
Fτ
E
L/µ delivers:
(aE c ) EL
,
F (aE + c )
qe =
2F
aE c
Two new examples with closed form solutions
I
The (negative) exponential demand qi = e
pe = c +
I
ne =
E 2L
,
F (cτ + E )
The linear demand case qi = a
pe =
I
E
,
τ
c + aE
,
2
ne =
ϑ (c + bE )
,
ϑ 1
ne =
pi
E
+b
L/µ delivers:
Fτ
E
qe =
L/µ delivers:
(aE c ) EL
,
F (aE + c )
The displaced CES case qi =
pe =
pi
E
τp i
E
ϑ
qe =
2F
aE c
L/µ delivers:
(c + ϑbE ) EL
,
ϑF (c + bE )
qe =
F ( ϑ 1)
c + bE
Direct Utility Functions
I
How did the direct utility look like? Not separable, but how?
Direct Utility Functions
I
I
How did the direct utility look like? Not separable, but how?
We can recover it by duality..
Direct Utility Functions
I
I
I
How did the direct utility look like? Not separable, but how?
We can recover it by duality..
The exponential demand derives from the direct utility:
U=
∑ xi
exp
τ + ∑nj=1 xj ln xj
∑nj=1 xj
Direct Utility Functions
I
I
I
How did the direct utility look like? Not separable, but how?
We can recover it by duality..
The exponential demand derives from the direct utility:
U=
I
∑ xi
exp
τ + ∑nj=1 xj ln xj
∑nj=1 xj
The linear demand case derives from the direct utility:
U=
(a ∑ xj 1)2
∑ xj2
Direct Utility Functions
I
I
I
How did the direct utility look like? Not separable, but how?
We can recover it by duality..
The exponential demand derives from the direct utility:
U=
I
∑ xi
exp
The linear demand case derives from the direct utility:
U=
I
τ + ∑nj=1 xj ln xj
∑nj=1 xj
(a ∑ xj 1)2
∑ xj2
The displaced CES case derives from the direct utility:
∑ xj
U=
ϑ 1
ϑ
ϑ
ϑ 1
1 + b ∑ xj
Social Optimum and ine¢ cient entry
I
The best allocation solves the following problem:
max n v
n,p
under the resource constraint EL
p
E
n (cq + F ).
Social Optimum and ine¢ cient entry
I
The best allocation solves the following problem:
max n v
n,p
under the resource constraint EL
I
p
E
n (cq + F ).
FOCs deliver:
p
c
p
where η (p/E )
1
=
1+η
v 0p
vE
p
E
,
n =
h
EL
F 1+η
p
E
> 0 is the elasticity of v ( ).
i
Social Optimum and ine¢ cient entry
I
The best allocation solves the following problem:
max n v
n,p
under the resource constraint EL
I
n (cq + F ).
FOCs deliver:
p
c
p
where η (p/E )
I
p
E
1
=
1+η
v 0p
vE
p
E
,
n =
h
EL
F 1+η
p
E
i
> 0 is the elasticity of v ( ).
Excess entry arises if and only if η 0 > 0, as in the exponential
and linear examples (CES delivers the optimal equilibrium)
Bertrand competition and endogenous entry
I
Suppose that the number of …rms is limited and strategic
interactions play a role
Bertrand competition and endogenous entry
I
I
Suppose that the number of …rms is limited and strategic
interactions play a role
In a Bertrand setting, considering the actual demand, each
…rm i chooses its price pi to maximize:
πi =
( pi
c )v
n
∑
v0
0
pj
E
pi
E
L
F
pj
E
j =1
where the denominator is not taken as given.
Bertrand competition and endogenous entry
I
I
Suppose that the number of …rms is limited and strategic
interactions play a role
In a Bertrand setting, considering the actual demand, each
…rm i chooses its price pi to maximize:
πi =
( pi
c )v
n
∑
v0
0
pi
E
pj
E
L
F
pj
E
j =1
I
where the denominator is not taken as given.
In a symmetric Bertrand equilibrium:
[θ (p B /E )
1+
pB c
EL
=
pB
pB
θ E
1 ]F
,
nB =
EL
Fθ
F
pB
E
+1
Bertrand competition and endogenous entry
I
I
Suppose that the number of …rms is limited and strategic
interactions play a role
In a Bertrand setting, considering the actual demand, each
…rm i chooses its price pi to maximize:
πi =
( pi
c )v
n
∑
v0
0
pi
E
pj
E
L
F
pj
E
j =1
I
where the denominator is not taken as given.
In a symmetric Bertrand equilibrium:
[θ (p B /E )
1+
pB c
EL
=
pB
pB
θ E
I
1 ]F
,
nB =
EL
Fθ
F
pB
+1
E
nB > ne and thus excess entry is more likely in Bertrand than
in monopolistic competition. The competitive e¤ect of L is
restored.
Aggressive Leaders and implications for Competition Policy
I
The model belongs to the class of "aggregative" games with
endogenous entry (Etro, 2006, Rand; 2008, EJ): neutrality of
the price/commitments of Stackelberg leaders on µ and the
strategy of followers.
Aggressive Leaders and implications for Competition Policy
I
The model belongs to the class of "aggregative" games with
endogenous entry (Etro, 2006, Rand; 2008, EJ): neutrality of
the price/commitments of Stackelberg leaders on µ and the
strategy of followers.
I
Leaders always choose p e < p B , thereby reducing the
equilibrium number of …rms with respect to nB .
Aggressive Leaders and implications for Competition Policy
I
The model belongs to the class of "aggregative" games with
endogenous entry (Etro, 2006, Rand; 2008, EJ): neutrality of
the price/commitments of Stackelberg leaders on µ and the
strategy of followers.
I
Leaders always choose p e < p B , thereby reducing the
equilibrium number of …rms with respect to nB .
I
This is neutral on consumer welfare with CES preferences
(Etro, 2008; Anderson et al., 2012), but raises (decreases)
consumers welfare if η 0 > (<) 0.
Aggressive Leaders and implications for Competition Policy
I
The model belongs to the class of "aggregative" games with
endogenous entry (Etro, 2006, Rand; 2008, EJ): neutrality of
the price/commitments of Stackelberg leaders on µ and the
strategy of followers.
I
Leaders always choose p e < p B , thereby reducing the
equilibrium number of …rms with respect to nB .
I
This is neutral on consumer welfare with CES preferences
(Etro, 2008; Anderson et al., 2012), but raises (decreases)
consumers welfare if η 0 > (<) 0.
I
Applications to competition policy: vertical contracts with low
wholesale price below the marginal cost, bundling to
strengthen price competition in the secondary market, other
incentive contracts increase CS with η 0 > 0, mergers to
increase prices reduce CS with η 0 > 0
Extensions and applications
1. Outside good à la D-S:
V =
E
p0
γ
∑v
pj
E
1 γ
Extensions and applications
1. Outside good à la D-S:
V =
I
All goes through.
E
p0
γ
∑v
pj
E
1 γ
Extensions and applications
1. Outside good à la D-S:
V =
E
p0
γ
∑v
pj
E
1 γ
I
All goes through.
I
The …rst best requires marginal cost pricing.
Extensions and applications
1. Heterogenous consumers:
Vh =
∑ vh
pj
Eh
Extensions and applications
1. Heterogenous consumers:
Vh =
I
∑ vh
pj
Eh
All goes through with
1
pe c
=
with e
θ (p, C )
e
e
p
θ (p e , C )
Z
h
θh
p
Eh
ω h dC (h )
Extensions and applications
1. Heterogenous consumers:
Vh =
I
I
∑ vh
pj
Eh
All goes through with
1
pe c
=
with e
θ (p, C )
e
e
p
θ (p e , C )
Z
h
θh
p
Eh
ω h dC (h )
a) market size is neutral
b) if θ 0 > 0, a change of the distribution according to the
likelihood-ratio dominance raises prices and number of …rms
more than with respect to the increase of the average income
c) a mean preserving spread of the income distribution
decreases (raises) prices and the mass of active …rms if the
demand elasticity is convex (concave) in the price.
Extensions and applications
1. Heterogenous costs à la Melitz: market size is neutral, but
changes in income induce selection e¤ects
Extensions and applications
1. Heterogenous costs à la Melitz: market size is neutral, but
changes in income induce selection e¤ects
I
c is distributed according to G (c ). Entry cost Fe
Extensions and applications
1. Heterogenous costs à la Melitz: market size is neutral, but
changes in income induce selection e¤ects
I
c is distributed according to G (c ). Entry cost Fe
I
Condition for marginal active …rm:
[p (ĉ )
ĉ ]
v 0 (p (ĉ )/E )L
=F
µ
Extensions and applications
1. Heterogenous costs à la Melitz: market size is neutral, but
changes in income induce selection e¤ects
I
c is distributed according to G (c ). Entry cost Fe
I
Condition for marginal active …rm:
[p (ĉ )
I
ĉ ]
v 0 (p (ĉ )/E )L
=F
µ
Condition for endogenous entry:
Z ĉ
c
[ π v (c )
F ] dG (c ) = Fe
Extensions and applications
1. Heterogenous costs à la Melitz: market size is neutral, but
changes in income induce selection e¤ects
I
c is distributed according to G (c ). Entry cost Fe
I
Condition for marginal active …rm:
[p (ĉ )
I
v 0 (p (ĉ )/E )L
=F
µ
Condition for endogenous entry:
Z ĉ
c
I
ĉ ]
[ π v (c )
F ] dG (c ) = Fe
market size is neutral but higher income increases all prices
and makes less productive …rms able to survive (an
anti-selection e¤ect) if θ 0 > 0
Extensions and applications
1. Two-country model à la Krugman (assume θ 0 > 0)
First case: no transport costs, di¤erent countries
Extensions and applications
1. Two-country model à la Krugman (assume θ 0 > 0)
First case: no transport costs, di¤erent countries
I
…rms adopt a higher price in the richer country, and
international trade does reduces the mass and increases the
size of …rms in the richer country
Second case: identical countries with transport costs
Extensions and applications
1. Two-country model à la Krugman (assume θ 0 > 0)
First case: no transport costs, di¤erent countries
I
…rms adopt a higher price in the richer country, and
international trade does reduces the mass and increases the
size of …rms in the richer country
Second case: identical countries with transport costs
I
trade opening reduces the markup on the exported goods and
the mass of …rms in each country relative to autarky
Extensions and applications
1. Two-country model à la Krugman (assume θ 0 > 0)
First case: no transport costs, di¤erent countries
I
…rms adopt a higher price in the richer country, and
international trade does reduces the mass and increases the
size of …rms in the richer country
Second case: identical countries with transport costs
I
trade opening reduces the markup on the exported goods and
the mass of …rms in each country relative to autarky
I
a reduction in transport costs reduces the price of exports but
increases their markups, and therefore induces the creation of
new traded goods
Extensions and applications
1. Two-country model à la Krugman (assume θ 0 > 0)
First case: no transport costs, di¤erent countries
I
…rms adopt a higher price in the richer country, and
international trade does reduces the mass and increases the
size of …rms in the richer country
Second case: identical countries with transport costs
I
trade opening reduces the markup on the exported goods and
the mass of …rms in each country relative to autarky
I
a reduction in transport costs reduces the price of exports but
increases their markups, and therefore induces the creation of
new traded goods
I
richer countries trade between themselves more than poorer
countries
Extensions and applications
1. New-Keynesian macroeconomics à la Blanchard-Kyotaki:
nominal rigidities are ampli…ed with Bertrand competition
Extensions and applications
1. New-Keynesian macroeconomics à la Blanchard-Kyotaki:
nominal rigidities are ampli…ed with Bertrand competition
2. Endogenous quality à la Sutton: price and quality still
independent of market size
Extensions and applications
1. New-Keynesian macroeconomics à la Blanchard-Kyotaki:
nominal rigidities are ampli…ed with Bertrand competition
2. Endogenous quality à la Sutton: price and quality still
independent of market size
3. Generalized EMS under any symmetric non-separable
preferences
Conclusions
I
The dual assumption of Indirect Additivity introduces a new
setting into monopolistic competition.
Its results generalize properties of the CES case concerning
the impact of the market size.
The neutrality of income disappears and a rationale for pricing
to market emerges.
New simple models of price competition with and without free
entry can be derived from indirectly additive preferences
(exponential and linear demand).