Endogenous Product Di¤erentiation, Market Size
and Prices y
Shon M. Fergusonz
Research Institute of Industrial Economics
April 2012
Abstract
This paper provides a framework to understand how market size a¤ects
…rms’investments in product di¤erentiation in a model of monopolistic competition with horizontal product di¤erentiation. The reduced-form model of endogenous product di¤erentiation proposes that consumers’love of variety makes
them more sensitive to product di¤erentiation e¤orts by …rms, which leads to
more di¤erentiated products in larger markets. The framework predicts an
anti-competitive market size e¤ect without requiring a high concentration of
…rms. Market enlargement can thus lead to higher prices that is not a result of
anti-competitive activities by …rms. In fact, consumers are better o¤ in larger
markets with higher prices because the direct bene…ts of product di¤erentiation
and variety outweigh the adverse indirect e¤ect of product di¤erentiation on
real wages.
JEL Classi…cation Codes: D43, F12, L13, L41.
Keywords: Antitrust, Endogenous Technology, Entry, Market Size E¤ect,
International Trade, Monopolistic Competition, General Equilibrium.
I thank Gregory Corcos, Karolina Ekholm, Rikard Forslid, Henrik Horn, Toshihiro Okubo
Philippe Martin and seminar participants at Stockholm University, the Institute of Industrial Economics and the European Trade Study Group for valuable comments and suggestions. Financial
support from the Social Sciences and Humanities Research Council of Canada (SSHRC) and the
Marianne and Marcus Wallenberg Foundation is gratefully acknowledged.
y
This paper was earlier circulated under the name "A Model of Ideal Di¤erentiation and Trade".
z
[email protected]
1
Economists have shown that larger markets for di¤erentiated goods often provide
consumers with a greater variety of products. Less attention has been paid, however,
to understand the determinants of product di¤erentiation in the …rst place, and how
product di¤erentiation relates to market size. Given that variety itself depends on
the ability of …rms to actively di¤erentiate their products from those of rivals, one
may ask what endogenous product di¤erentiation implies for the number of goods
available, prices and ultimately for consumer welfare.
This paper provides a framework that analyzes how market size a¤ects …rms’investments in product di¤erentiation. The framework predicts that products will be
more di¤erentiated in larger markets by assuming that …rms can choose how much to
di¤erentiate their products. The basic model provides a simple tractable reduced-form
general equilibrium result, with the prediction that product di¤erentiation increases
with market size. Larger markets encourage product di¤erentiation in the model
because the love-of-variety property of the utility function. Love of variety leads individual consumers to consume more varieties and less of each variety in larger markets.
I show that this behavior makes them more sensitive to …rms’spending on product
di¤erentiation, with the prediction that products are more di¤erentiated in larger
markets. This prediction suggests that congestion between varieties need not arise in
larger markets, since …rms can respond by di¤erentiating their products. This prediction also provides a possible explanation for the positive relationship between prices
and market size found by Tabuchi and Yoshida (2000), Roos (2006) and Badinger
(2007).
The theoretical model in this paper is based on monopolistic competition with
endogenous technology choice. I assume that consumer utility follows a generalized
CES utility function originating in Spence (1976). I Following the endogenous sunk
cost literature, I assume that …rms can spend more on …xed costs in order to di¤erentiate their product1 . Firms choose their product di¤erentiation spending from a
1
This contrasts with most of the previous literature on trade and endogenous technology, such as
2
continuum, with a more di¤erentiated product requiring higher …xed cost spending.
The …xed cost spending can be considered to be persuasive advertising or product
development that di¤erentiates one’s own product from that of other …rms. Firms
then set prices via monopolistic competition.
The model outlined in this paper contributes to a new literature on how market
size a¤ects the extent of endogenous product di¤erentiation. While there are several
recent papers that deal with various aspects of product quality in di¤erentiated goods
markets2 , only a few consider the aspect of product di¤erentiation3 . My assumption of
costly product di¤erentiation contrasts with the models of monopolistic competition
by Bertoletti, Fumagalli, and Poletti (2008), Lorz and Wrede (2009) and Zhelobodko,
Kokovin, Parenti, and Thisse (2011) where product di¤erentiation is costless.
A new result stemming from the assumption of costly product di¤erentiation is
that …rm size can either increase or decrease as markets expand, depending on the
responsiveness of consumers to …rms’product di¤erentiation spending. If consumers
are su¢ ciently responsive to product di¤erentiation spending then the model predicts a positive relationship between market size, …rm size and prices, leading to a
positive and concave relationship between market size and entry. Previous empirical
studies typically interpret a concave relationship between market size and entry as a
pro-competitive e¤ect of market size. My framework suggests that market size and
…rm numbers alone do not provide su¢ cient information to make inferences about
competition when product di¤erentiation is endogenous and costly.
The general equilibrium approach allows for a welfare analysis that considers the
e¤ects of product di¤erentiation on the real wage and love for variety. In the welfare
analysis I show that product di¤erentiation has two countervailing e¤ects. While
Markusen and Venables (1997), Ekholm and Midelfart (2005) and Bustos (2010) assume a trade-o¤
between lower marginal costs and higher …xed costs.
2
Prominent examples include Lin and Saggi (2002), Baldwin and Harrigan (2011), Simonovska
(2010) and Khandelwal (2010).
3
The model in this paper describes endogenous horizontal product di¤erentiation, in contrast to
the models of vertical product di¤erentiation in the Industrial Organization literature. Prominent
examples include Shaked and Sutton (1987), Mazzeo (2002) and Berry and Waldfogel (2010).
3
individuals like to consume varieties that are more di¤erentiated, there is also a negative e¤ect due to higher price-cost markups. I am able to show, however, that the
bene…cial e¤ect of product di¤erentiation and variety outweighs the adverse e¤ect
of higher markups in larger markets. The model thus predicts that market size has
an unambiguously positive e¤ect on consumer welfare. I also compare the market
equilibrium with the social optimum and show that the market equilibrium leads to
an underinvestment in product di¤erentiation compared to the social planner equilibrium. The social planner does not charge a markup, which allows it to di¤erentiate
products more than the market equilibrium.
I extend the basic model to include …rm heterogeneity. In this case …rms are
assumed to vary with respect to their e¢ ciency in di¤erentiating their product, which
leads to a distribution of prices in the economy and a lower bound on prices. In larger
markets the least e¢ cient …rms with the lowest prices exit, which increases the lower
bound price and hence the average price. I also extend the basic model to include
two countries and trade costs. A new prediction regarding trade liberalization in this
model is that price-cost markups and …xed cost spending is highest at an intermediate
level of per-unit trade costs. The intuition behind this result is that …rms’ total
trade costs are greatest when per-unit trade costs are at an intermediate level. Since
a lower elasticity of substitution reduces the negative impact of total trade costs
on export demand, …rms have the strongest incentive to di¤erentiate their product
and reduce their substitution elasticity when total trade costs make up the largest
proportion of …rms’ output. The model thus predicts that prices can either rise
or fall with trade liberalization. This result can help to make sense of the mixed
evidence concerning whether prices fall when trade liberalizes and markets enlarge,
as is found by Revenga (1997), Tre‡er (2004) and Feenstra (2006). The prediction
that the elasticity of substitution is decreasing with market size and trade costs can
also reconcile the Broda and Weinstein (2006) …nding that elasticities of substitution
4
have decreased over time. Moreover, the result that prices can increase when markets
expand may provide a valuable insight for economists that study anti-competitive
behaviour, as this paper shows that higher prices can occur in the absence of anticompetitive activities by …rms. In the mechanism described here, …rms’higher prices
are simply a response to consumers’preferences.
The rest of the paper is organized as follows: The basic model in a closed economy and the e¤ect of market size on product di¤erentiation, entry and welfare are
presented in Section 1. A comparison of the market solution to the social optimum
is given in Section 2. Firm heterogeneity is added to the model in Section 3. The
model is expanded to include two countries and trade costs in Section 4. Conclusions
follow in Section 5.
1
Basic Model
I begin by describing the model in a closed economy. The economy is composed of a
continuum of monopolistically competitive …rms N indexed by i 2 [0; N ] and there is
no strategic interaction between …rms. The representative consumer is endowed with
one unit of labor.
1.1
Consumer Preferences:
The representative consumer’s utility maximization problem for di¤erentiated goods
is de…ned as:
max U =
ci
Z
1
m
N
s.t.
ci di
i
Z
N
pi ci di = w
(1)
0
0
where ci is the quantity of good i consumed by the representative consumer. The
utility for di¤erentiated goods is based on the generalized CES utility function by
Spence (1976), where
i
2 (0; 1) is a …rm-speci…c parameter that determines the
5
price elasticity of demand for good i,4
m
2 (0; 1) is a preference parameter that is
not …rm-speci…c. pi is the price of good i and w is the representative consumer’s
wage. The demand for a good by a representative consumer is thus:
w
ci (pi ; i ;
m ; N;
)=
RN
0
where M =
RN
0
ci i di and
M
1
m
1
m
M
i
i
pi
m
pi
1
1
1
m
(2)
1
1
1
i
pi
i
di
is the representative consumer’s marginal utility of income.
One can also derive an expression for the marginal utility of income:
=
M
1
m
m
As in Krugman (1979), …rms will take
1.2
1.2.1
1
i ci
i
1
pi
.
(3)
as given.
Technology:
Production Technology
Labor is the only input in this economy, and each …rm’s total labor requirement li
includes an endogenously determined …xed labor cost Fi , and an exogenous variable
amount of labor cost
in the production process:
li = Fi + xi
(4)
where xi is the total quantity demanded of good i.
4
Since i is a parameter in the consumer’s utility function it may be argued that it is not
observable. I assume that …rms are able to invest in product di¤erentiation, which is assumed to
have a mapping into i .
6
1.2.2
Endogenous Product Di¤erentiation
The model in this paper assumes that the …xed cost, Fi , is a function of the preference
parameter
i
and imposes the following assumption:
ASSUMPTION 1:
For all
i
2 (0; 1),
(5)
Fi = Fi ( i )
is twice continuously di¤erentiable in
i,
with the following properties:
Fi0 ( i ) < 0; Fi00 ( i ) > 0; lim+ Fi ( i ) = 1; lim Fi ( i ) = 0:
i !0
i !1
The assumption is simply that costs are increasing in product di¤erentiation and
convex.
The concept that …xed costs a¤ect a consumer demand follows the work of Sutton
(1991) and Dixit (1979). Di¤erentiating one’s own product from others (i.e. lowering
the preference parameter,
i)
requires higher …xed costs. These …xed costs could
be persuasive advertising or product development that di¤erentiates a …rm’s own
product from that of other …rms. I do not assume a functional form at the moment,
only that it is upward sloping and convex as
i
decreases, so that …xed cost spending
exhibits decreasing returns. I later assume a particular function form for Fi ( i )
which satis…es all of these properties and allows for an analytical solution, but many
properties can be shown with this more general assumption. I refer to (5) as the
"advertising function" throughout the rest of the paper.
1.2.3
Markup Pricing
Firms enter, then they choose their optimal level of product di¤erentiation, then
they set prices via monopolistic competition. The equilibrium is found by backward
7
induction. Each …rm sets price in order to maximize pro…t, and takes endogenous
…xed costs, Fi , as given:
i
= p i xi
wFi ( i ) .
w xi
(6)
Firms maximize (6) with respect to pi , yielding following …rst order condition:
w
pi =
.
(7)
i
One thus obtains markup pricing, with a constant markup for a given
is endogenous, however, since
i
The markup
is an endogenous variable chosen by the …rm. Firms
will take (7) into consideration when choosing
1.2.4
i.
i.
Optimal Product Di¤erentiation
Each …rm chooses their preference parameter,
i,
to maximize operating pro…ts less
the …xed cost to di¤erentiate. Firm i’s demand is the sum of consumer demands,
xi = Lci , where L is the number of consumers in the economy. Operating pro…ts are
concave in
i,
so a maximum exists as long as Fi ( i ) is chosen such that pro…ts are
non-negative. Using (2), the demand for …rm i’s product is:
Lw
xi (pi ; i ; N; L; ) =
RN
0
Firms take
pi
M
1
m
1
M
m
i
i
pi
m
1
m
1
1
.
1
1
1
i
pi
i
(8)
di
, the marginal utility of income, and the denominator in (8) as
given when setting their preference parameter. The …rst order condition for product
di¤erentiation is derived by substituting (7) and (8) into (6), then maximizing …rm
8
pro…ts with respect to
(pi ( i )
i:
2
6
w ) xi (pi ; i ) 4
1
2
i)
(1
+ (1
ln
2
2
i)
1
m
M
1
m
ln
i
+
1
w
1
(1
i) i
3
7
0
5 = wF ( i )
(9)
We can obtain a simpler expression describing the optimal level of product di¤erentiation by substituting (3) into (9) and rearranging:
xi (pi ; i )
xi
1
+
= F 0 ( i) .
L
i
ln
i
(10)
Note that (10) equates the marginal revenue and marginal cost of reducing product
di¤erentiation, which explains why both sides of (10) are negative (as long as xi =L
is small enough). Given Assumption 1, the marginal cost of product di¤erentiation
increases and approaches in…nity as
1.2.5
i
decreases towards zero.
Equilibrium with Identical Firms
The rest of this section assumes that …rms are identical, meaning that they will all
choose the same level of product di¤erentiation. Firms enter until pro…ts equal zero
for each …rm. Combining the markup pricing condition, (7), and the zero pro…t
condition, p = w + wF=x, one obtains an expression for output per …rm:
x( ) =
F( )
1
.
(11)
The functional form of F ( ) can be chosen such that x ( ) is an increasing, decreasing,
or constant function of .
The full employment of labor condition determines the number of …rms in the
manufacturing industry:
N=
L
.
F ( )+ x
9
(12)
Overall, equations (5), (7), (10), (11) and (12) make up the basic model. This
includes the same equations as Krugman (1980) for pro…t maximization in price, zero
pro…ts, and full employment of labor, plus (5) and (10). The unknowns are p, x, N ,
and F .
The …rst order condition for optimal product di¤erentation in the symmetric equilibrium is found by substituting (11) into (10):
"
F( )
ln
1
!
F( )
1
L
+
1
#
= F0 ( ).
(13)
Equation (13) provides an implicit solution for the equilibrium level of product differentiation, , which is a function of market size L, marginal cost
and the form
of the advertising function. The implicit nature of the …rst order condition makes it
di¢ cult to show that the solution has a unique maximum. For the sake of simplicity
we proceed by choosing a functional form of the advertising function that yields an
analytical solution with a unique maximum.
1.3
An Analytical Solution
One can fully solve for a symmetric solution to the basic model by using a speci…c
functional form for F ( i ). I thus impose:
ASSUMPTION 2:
F( )=
where
1
;
2 (0; 1)
(14)
> 0 is a parameter. This functional form has all of the properties in
Assumption 1. We can now …nd an analytical solution to the system of equations
(3), (7), (11), (12), (10) and (14).
Lemma 1 Suppose that Assumption 2 holds. Then the unique pro…t maximizing
symmetric equilibrium price, quantity, number of …rms, level of product di¤erentation
10
and …xed costs are given by the following expressions:
p=
w
ln
2
L
x=
2
N =L
=
F =
L
ln
2
ln
ln
L
L
2
2
Proof. See appendix.
This particular formulation of the advertising function is tractable because it
eliminates the problem of having a logged
term in (13). The analytically solvable
model is also characterized by product di¤erentiation and prices increasing in , the
marginal cost parameter. Thus more expensive materials lead to higher prices and
more product di¤erentiation. A graphical illustration of (13) is given in …gure 1 using
this particular advertising function.
1.4
Market Size E¤ect
The model predicts that …rms’spending on product di¤erentiation is a¤ected by market size, which has important implications for prices and entry. The next proposition
describes how market size a¤ects equilibrium product di¤erentiation.
Proposition 1 Suppose that Assumption 1 holds. Then product di¤erentiation is
increasing in market size, i.e.:
d
< 0.
dL
Proof. See appendix.
11
The intuition for this result falls directly from the love-of-variety property of the
generalized CES utility function. The elasticity of substitution a¤ects the concavity
of utility for each variety and a greater concavity of utility increases consumers marginal utility of consumption at low levels of consumption. Larger markets encourage
product di¤erentiation because consumers consume more varieties and less per variety
per capita. Consumers thus move down their utility curves for each variety and become more sensitive to …rms’spending on product di¤erentiation. Firms respond by
spending more on product di¤erentiation, which makes their demand curve more inelastic. This leads …rms to spend more on product di¤erentiation via the relationship
speci…ed in equation (5). Prices rise via the markup pricing rule (7).
The intuition for the market size e¤ect is is illustrated in …gure 2. Spending on
product di¤erentiation leads to a more concave utility curve, changing from U (c;
to U (c;
1 ).
1)
The e¤ect of more concave utility on the marginal utility of income
(the slope of the utility curve) depends on consumers’ level of consumption, which
ultimately depends on market size since consumers purchase more varieties and less
per variety in larger markets. In a small market (L = L2 ) product di¤erentiation
leads to a decrease in the marginal utility of income from (L2 ;
1)
to (L2 ;
2)
while
in a large market (L = L1 ) it leads to an increase in the marginal utility of income
from (L1 ;
1)
to (L1 ;
2 ).
It is important to note that the connection between market size and product
di¤erentiation is not caused by …rms selling more units in larger markets. In fact, the
analytical solution given in Lemma 1 illustrates that x =
= , a constant. Larger
markets are instead characterized by a greater amount of …rms, due to the free entry
condition in the model. The marginal bene…t of product di¤erentiation is a¤ected not
directly by the number of units a …rms can sell but rather via the indirect mechanism
whereby consumers put a greater value on variety in larger markets.
While such an anti-competitive e¤ect of market size does not hold in all contexts,
12
the prediction that prices are higher in larger markets has been con…rmed by various
studies. Tabuchi and Yoshida (2000) …nd that doubling city size in Japan reduces
real wages by approximately 7-12%, while Roos (2006) …nds that consumer prices
are higher in regions of Germany with larger populations. Badinger (2007) …nds that
the European Union’s Single Market Program led to lower markups in manufacturing
and construction but higher markups in service industries.
This new evidence on the pattern of prices does not square with any workhorse
models of monopolistic competition in the International Trade and New Economic
Geography literature5 . However, it is related to recent theoretical contributions in
the Industrial Organization literature, such as Amir and Lambson (2000) and Chen
and Riordan (2007, 2008).
The prediction that prices increase in market size is the opposite conclusion of
the "ideal variety" approach to modeling monopolistic competition, whether product
di¤erentiation is exogenous as in Lancaster (1979, 1980) or endogenously determined
as in Weitzman (1994). The reason for this discrepancy is that product di¤erentiation
imposes a disutility on consumers in the Ideal Variety approach, while it enhances
consumers’ utility in the "love of variety" approach. Adding endogenous product
di¤erentation in the manner described above can thus provide a very di¤erent result
compared to previous models.
The predictions here are relevant for goods with a propensity for endogenous
product di¤erentiation, which explains why it con…cts with the evidence from the
market for ready-mixed concrete by Syverson (2004). Ready-mixed concrete is a
homogeneous product, which makes switching between suppliers easier when markets
5
The intra-industry trade models of Krugman (1980) and Helpman and Krugman (1987) employ
Dixit and Stiglitz (1977) preferences and exhibit no market size or distance e¤ect on prices. The
heterogeneous …rm model by Melitz (2003) exhibits no market size e¤ects and average export prices
that are decreasing in distance. Models based on Melitz and Ottaviano (2008) assume quadratic
utility and exhibit "pro-competitive e¤ects", whereby markups and hence prices are lower in larger
markets due to greater competition. Models based on Baldwin and Harrigan (2011) exhibit the
desired characteristic that higher-priced goods are sold to more distant countries, but lack any
market size e¤ect.
13
are more dense. In contrast, my framework deals with markets for di¤erentiated
products, and going without a particular good is more di¢ cult in larger markets
because goods are more di¤erentiated.
The relationship between the number of …rms, market size and product di¤erentiation can be seen by substituting (11) into (12):
N=
L (1
)
.
F( )
(15)
Equation (15) illustrates that the e¤ect of market size on variety depends not only
on the direct e¤ect of L but also on F and . In contrast to Krugman (1980), the
number of …rms need not increase linearly with market size. It is unclear whether
product di¤erentiation has a positive or negative e¤ect on entry without making
a further assumption about the functional form of F ( ). However, one can make
a more precise statement about how variety changes with market size by totally
di¤erentiating (15) with respect to L:
0
B
1
dN
B
=
B1
dL
F( )@
|
1
F0 ( ) d L
d L
+
dL
F ( ) dL
{z
}
E¤ect of endogenous product di¤erentiation
1
C
C
C
A
(16)
One can see in (16) that variety is a¤ected by market size via three distinct channels.
The …rst channel is the linear relationship between market size and variety that occurs
when product di¤erentiation is exogenous. The second channel is the e¤ect of market
size on variety via product di¤erentiation, which is positive since Proposition 1 showed
that d =dL is negative. The third channel is the e¤ect of market size on variety due
to …xed cost spending, which depends on the elasticity of the advertising function and
the market size e¤ect on product di¤erentiation. The third channel is negative since
F 0 ( ) is negative by Assumption 1. While the …rst channel is the linear relationship
between market size and entry found in Krugman (1980), other two channels capture
14
the two countervailing e¤ects of product di¤erentiation on variety. On the one hand,
product di¤erentiation on its own has positive e¤ects on variety, but the …xed costs
associated with increasing product di¤erentiation reduces variety by leading to fewer
larger …rms. These e¤ects can be summarized in the following proposition:
Proposition 2 If the advertising function is su¢ ciently inelastic then …rm size
increases with market size and entry increases with market size.
F0 ( )
dl ( )
?0,
7
dL
F( )
dN
?0,1
dL
d L
dL
1
(17)
1
F0 ( )
F( )
?0
(18)
Proof. See appendix.
The results in Proposition 2 infer that endogenous product di¤erentiation can
lead to a positive and concave relationship between market size and entry if …rm
size increases with market size. The relationship between …rm size and market size
depends on the elasticity of the advertising function. If the advertising function is
su¢ ciently elastic then …rms are induced to spend more on product di¤erentiation
as the market grows to such an extent that they increase in size, which leads to a
concave relationship between entry and market size. This result contrasts with the
oligopoly framework by Bresnahan and Reiss (1991) that infers increased competition and lower markups when entry is increasing and concave in market size. For
example, Campbell and Hopenhayn (2005) found that retailing …rms are larger in
larger cities, and concluded that …rms were larger due to falling markups in larger
markets, although the authors had no direct evidence on markups. My framework
suggests, however, that market size and …rm numbers or …rm size alone do not provide su¢ cient information to make inferences about competition. The endogenous
sunk cost literature argues a similar point. In contrast to Sutton (1991), however, my
framework predicts an anti-competitive market size e¤ect without requiring a high
15
concentration of …rms. This result di¤ers from the model by Zhelobodko, Kokovin,
Parenti, and Thisse (2011), which predicts that …rm size increases with market size
only in the pro-competitive case. Including a …xed cost to di¤erentiate means that
…rm size is increasing with market size even in the anti-competitive case.
1.5
Welfare E¤ect of Market Size
I use the direct utility function for manufactures, M , to examine the welfare e¤ects
of market size. In the symmetric equilibrium with
m
= , and substituting (11) and
(12) into (1) using c = x=L provides an expression of utility in terms of market size
and product di¤erentiation:
U (L; ) = L
1
1
1
F( )
1
(19)
x( )
The direct e¤ect of L on utilty from manufactures is positive. The e¤ect of product
di¤erentation on utility, however, is unclear because reductions in have two countervailing e¤ects: utility becomes more concave in consumption, but prices increase as
markups widen. This contrasts with Krugman (1980) that assumes exogenous technology, where prices are constant and welfare e¤ects occur exclusively via increased
variety. One can show, however, that utility is increasing as products become more
di¤erentiated and this leads to utility increasing in market size. These results are
summarized in the following proposition:
Proposition 3 Utility per-capita is monotonically increasing in market size given
the advertising function in Assumption 2 as long as
large and
> 0 and L > 0 are su¢ ciently
> 0 is su¢ ciently small:
@U
70,
@
1 7 ln
Proof. See appendix.
16
2L
ln L
!
:
Utility per-capita is increasing as products become more di¤erentiated as long as
the market size, L, is large enough. Endogenous product di¤erentiation modeled in
this way thus has a net positive e¤ect on welfare when the market expands, despite
the adverse e¤ect of higher markups.
2
Market Equilibrium vs. Social Optimum
An important task is to compare the level of product di¤erentiation and variety
provided in the market equilibrium with the social optimum. The social optimum is
constrained only by the full employment condition and the technologies for producing
the di¤erentiated good and for di¤erentiating these goods as de…ned by Assumption
1. The social planner maximizes the representative consumer’s utility by choosing
the optimal variety, output per …rm and level of product di¤erentiation in the di¤erentiated goods industry, subject to the full employment of labor constraint and the
advertising function:
maxU [N (N; ) x] s.t. N (F ( ) + x) = L
x;N;
where
(N; ) represents the "taste for variety", which captures consumers’love of
variety:
(N; ) =
MN (1; :::; 1)
MN (x; :::; x)
=
M1 (N x)
N
The concept of separating the love of variety e¤ect from the markup condition originates in Benassy (1996). I extend Benassy’s formulation so that the love of variety
depends not only on the number of varieties but also on how di¤erentiated they are
from each other. Di¤erentiating with respect to x, N , and
U 0 [ (N; )] =
17
yields, respectively:
(20)
U 0 x (N; ) + xN
U0
@
@N
=
(F ( ) + x)
@
x = F0 ( )
@
(21)
(22)
where U 0 is the partial derivative of U . The solutions to equations (20), (21) and (22)
give xopt , N opt , and
opt
. Dividing (20) by (21) we obtain:
1
x
=
@
F ( )+ x
1 + @N
N
(N; )
(23)
This result, originating in Benassy (1996), illustrates how the socially optimal production decision depends crucially on consumer’s love of variety. Dividing (20) and
(21) by (22) respectively and combining these with (23) provides a new expression
that describes the social planner’s optimal choice of product di¤erentiation:
@
@
@
@N
F0 ( )
=N
F( )
(24)
The social planner sets the optimum level of product di¤erentiation where the marginal rate of substitution between product di¤erentiation and variety equals the percentage change in the cost to di¤erentiate all available goods.
Comparing the social planner and market equilibrium is eased by assuming a
particular utility function. In the case of CES utility the love of variety expression
is
(N; ) = N
1
1
. Plugging this into (24) pins down the socially optimal level of
product di¤erentiation:
F( ) 1
F( )
ln
= F0 ( ).
1
L (1
)
(25)
Comparing equations (13) and (25), it is di¢ cult to say without making further
assumptions whether the market equilibrium di¤erentiates products more or less than
the social optimal. The di¤erence between these equations illustrates, however, that
18
the social optimal and equilibrium equilibrium need not be equivalent. One can show,
however, that the market equilibrum results in less than the socially optimal product
di¤erentiation when the advertising function is given by Assumption 2 if
L >
.
This result, although dependent upon a particular advertising function, illustrates
a key property of the model. The social planner does not charge a markup, which
allows it to di¤erentiate products more than is possible in the market equilibrium.
3
Firm Heterogeneity
The basic model can easily be extended to capture product di¤erentiation heterogeneity across …rms. The pro…t of …rm i is:
i
where
i
A lower
= (pi ( i )
w ) xi (pi ; i )
wFi ( i ;
i)
is a parameter that determines the …rm’s e¢ ciency in product di¤erentiation.
i
makes the …rm more e¢ cient in di¤erentiating its product. Following
Melitz (2003), …rms draw their e¢ ciency parameter from a random distribution.
Since operating pro…ts are concave in
this means that low-e¢ ciency …rms will
i
make negative pro…ts and leave the market. The "cuto¤" …rm D with an e¢ ciency
draw
D
such that is indi¤erent between leaving the market and remaining earns zero
pro…ts:
(pD (
D)
w ) xD (pD ;
D)
= wFD (
D;
D)
The …rst order condition for the cuto¤ …rm is given by:
FD (
1
D)
D
"
ln
F(
D)
D
1
D
L
!
+
1
D
#
= FD0 (
D)
As given by Proposition 1 and abstracting from general equilibrium e¤ects, a larger
market will be characterised by a cuto¤ …rm with a greater degree of di¤erentiation.
19
This model thus predicts that higher-priced goods are sold in larger markets.
4
4.1
Two Countries and Trade Costs
Setting, Preferences and Technology
Extending the basic model to a two country model with trade costs yields new results
regarding …rms’technology choice and the pattern of trade. There are two industries:
a di¤erentiated goods industry characterized by increasing returns to scale and a
constant returns industry. Preferences and the …rms’problem are otherwise identical
to the basic model. I assume that iceberg trade costs between the two countries,
whereby
units must be shipped in order for one unit to arrive at its destination.
I assume that the markets are of equal size and that …rms are identical. Moreover,
I normalize the price of the agricultural good to unity, which equalizes the wage in
both country. These simpli…cations allow us to more easily see the e¤ect of trade
costs on equilibrium product di¤erentiation.
4.2
The Trade Friction E¤ect
The e¤ect of trade costs on the equilibrium level of product di¤erentiation is a unique
property of the model. The …rst order condition for product di¤erentiation under the
special case where country sizes are identical (i.e. L = L ) is:
2 0
F( )
F( )4 @
1
ln
1
L 1+
|
{z
1
1
"market size e¤ect"
1
A+
3
15
}
" 1
#
1
ln
F( )
= F0 ( ).
1
2
(1
) 1+ 1
|
{z
}
(26)
"trade friction e¤ect"
This equation e¤ectively divides the …rst order condition into two parts, a "market
size e¤ect" and a "trade friction e¤ect". The "market size e¤ect" is almost identical
to the left hand side of the …rst order condition in the basic model, (13), except for the
20
additional term 1 +
1
1
multiplying L in the denominator. This term equals 1 under
in…nite trade costs and 2 under free trade, since free trade between two countries of
equal size e¤ectively doubles the market.
The "trade friction e¤ect" is an additional term that is not present in the …rst order condition under autarky. If trade costs per unit are zero or in…nite then no trade
occurs and the trade friction e¤ect will disappear. This term is positive for intermediate trade costs, reaching a single maximum. The marginal revenue of decreasing
is thus maximized at some intermediate level of trade costs. Trade frictions thus
a¤ect more than just market potential in the model; the friction itself enhances the
marginal revenue of product di¤erentiation. The intuition is that lowering
abates
the loss of demand due to "melting", and the marginal bene…t from this activity is
greatest when "melting" is greatest (i.e. intermediate trade costs).
It can be helpful to analyze this result within a trade liberalization context. If
trade costs are gradually lowered from autarky to free trade,
…rst decreases, then
increases as trade costs approach zero. Similarly, F and p …rst increase to a maximum at some intermediate level of trade costs, then decrease as trade costs approach
zero. This contrasts with the monotonic market size e¤ects that one observes in
the basic closed economy model. The result that product di¤erentiation e¤ects are
strongest at intermediate trade costs is akin to new economic geography literature,
where agglomeration forces are strongest at intermediate trade costs.
When trade costs are low the model predicts that prices increase with distance,
which agrees with the export price literature, such as Baldwin and Harrigan (2011)
and Ottaviano and Mayer (2008). The result that the elasticity of substitution is
decreasing can also reconcile the Broda and Weinstein (2006) result that the elasticity
of substitution has been trending downwards at the same time as the process of
"globalization" has expanded markets and reduced trade barriers.
21
5
Conclusion
I take a new look at the determinants of product di¤erentiation using a monopolistic competition framework. The model allows for …rms to endogenously choose
from a continuous set of technologies by creating a trade-o¤ between …xed costs and
product di¤erentiation. This assumption is consistent with …xed costs that represent
persuasive advertising or product development that di¤erentiate one’s own product
from others in the eyes of consumers. Fixed costs, markups, and output per …rm are
increasing functions of market size, a characteristic that agrees with the literature.
The model thus generates "endogenous markups" that are a direct result of …rms’
optimizing behavior.
This reduced-form approach to modelling endogenous product di¤erentiation may
provide a useful policy insight to economists who analyze the anti-competitive e¤ects
of trade liberalization and economic integration. In particular, this paper illustrates
that larger markets may lead to higher prices that are not caused by …rms’ anticompetitive behavour. The mechanism in this paper instead suggests that consumers
preferences for di¤erentiated goods
The mechanism of endogenous product di¤erentiation described in this paper may
be part of the reason why we do not always see pro-competitive e¤ects in di¤erentiated
goods markets. This model can be applied to many issues, including growth, trade,
and economic geography. The prediction that markups increase with market size
may be considered somewhat controversial, since the "conventional wisdom" is that
markups will decrease as market size increases, …rms enter, and the competition
becomes tougher. However, the recent empirical evidence on prices suggests that procompetitive e¤ects of market size need not hold, especially in di¤erentiated products.
As Jean Tirole (1988, p.289) puts it, "Though it will be argued that advertising may
foster competition by increasing the elasticity of demand (reducing "di¤erentiation"),
it is easy to …nd cases in which the reverse is true." It is hoped that this paper has
22
given some theoretical foundation to this argument.
2
MC
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
theta
-2
-4
-6
-8
MR
-10
Figure 1: Graphical illustration of existance of maximum solution, F ( ) =
L = 100,
=1
Figure 2: E¤ect of product di¤erentiation on marginal utility of income,
large- vs. small-market
23
1
,
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A
A.1
Mathematical Appendix
Proof of Lemma 1:
Substituting into (13) yields the following …rst order condition:
ln
L
+2
1
=0
with a unique solution for . The analytical solutions for the remaining endogenous
variables can then be found by substituting the solution for
into (5), (7), (11) and
(12). The second order condition reveals that the solution yields a unique a maximum:
2
2
27
< 0:
Proof of Proposition 1:
Totally di¤erentiating and rearranging (10) we obtain an expression for the e¤ect of
market size on product di¤erentiation:
d
=
dL
1 F( )
L 1
@
@
F( )
1
F( )
ln
1
+
L
1
F 00 ( )
The denominator is negative by Assumption 1 when L is su¢ cienly large. Thus
d =dL < 0.
Proof of Proposition 2:
The result for …rm size is found by substituting (11) into (4) and rearranging. The
result for …rm entry is obtained rearranging (16). Inspection of (17) and (18) reveals
that if dl ( ) =dL > 0 then dN=dL > 0.
Proof of Proposition 3:
The total di¤erentiatial is:
dU
@U
@U d
=
+
dL
@L
@ dL
It is clear from (19) that @M=@L > 0. @ =@L < 0 due to Proposition 1. The partial
derivative of indirect utility with respect to
@U
=
@
1
L (1
)
F( )
1
ln
is:
L (1
)
F0 ( )
+
(1
F( )
F( )
In order for indirect utility to be decreasing in
@U
F0 ( )
1
70,
7
@
F( )
1
28
) .
the following condition must hold:
ln
L (1
)
.
F( )
Substituting (14) into this condition, simplifying, then substituting the analytical
solution for
yields a new condition that can be easily evaluated:
@U
70,
@
@U=@L > 0 as long as
1 7 ln
2L
ln L
!
> 0 and L > 0 are su¢ ciently large and
small.
29
> 0 is su¢ ciently