The Dixit-Stiglitz-Krugman Trade Model:
A Geometric Note
Toru Kikuchi∗

Abstract
In this note, we brieﬂy review the now standard Dixit-StiglitzKrugman trade model of monopolistic competition. Furthermore, we
propose a convincing graphical exposition that emphasizes the ﬁrms’
entry-exit process.

∗Toru Kikuchi, Graduate School of Economics, Kobe University, Graduate School of

Economics, Kobe University, 2-1, Rokkodai-cho, Nada-ku, Kobe, Hyogo, 657-8501, Japan.
TEL/FAX: +81-78-803-6838. E-mail: kikuchi@econ.kobe-u.ac.jp1

1

Introduction

Among several competing trade models, the model of monopolistic competition à la Dixit-Stiglitz-Krugman (Krugman 1979, 1980, 1981; Dixit andNorman 1980; Helpman and Krugman 1985) provides an elegant account of
intra-industry trade and plays a major role in the recent literature.1
In his inﬂuential survey, Matsuyama (1995, p. 701) provides the following
deﬁnition of monopolistic competition:
1. The products are diﬀerentiated. Each ﬁrm, as the sole producer of its
own brand, is aware of its monopoly power and sets the price of its
product.
2. The number of ﬁrms (and products) is so large that each ﬁrm ignores
its strategic interactions with other ﬁrms; its action is negligible in theaggregate economy.
3. Entry is unrestricted and takes place until the proﬁts of incumbent
ﬁrms are driven down to zero.
This model is also attractive because increasing returns are internal to theﬁrms, so the problem of multiple equilibria does not arise (as it did in the1

See Helpman (1990), Baldwin et al. (2003, ch. 2), Combes, Mayer and Thisse (2008,

chs. 3–4), and Feenstra (2004, ch. 5) for surveys. In a series of articles, Neary (2001,
2004, 2009) provides an excellent overview of the literature.

2

models of external economies). Furthermore, as Matsuyama has pointed out,by assuming ﬁrms are very small, we don’t have to worry about strategicinteractions between ﬁrms that make any general treatment of oligopolies
impossible. Although this type of model relies heavily on speciﬁc functional
forms (e.g., CES utility), it remains appropriate to model global phenomenausing the monopolistic competition model.
In this note, we present the now standard Dixit-Stiglitz-Krugman trademodel of monopolistic competition. Furthermore, we propose a convincing
graphical exposition that emphasizes the ﬁrms’ entry-exit process. The next
section presents the basic model. The nature of the trading equilibrium isconsidered in Section 3. The eﬀects of factor mobility are brieﬂy reviewed in
Section 4, followed by concluding remarks in Section 5.

2

The Model

Suppose that there are two countries: Home and Foreign. Home (resp. Foreign) is endowed with L (L∗ ) units of labor, which is the only primary factor
of production. The countries have identical tastes and technologies.
Each country produces two consumption goods, Good X and Good Y .
Goods Y is sold in a perfectly competitive market, while Good X is sold ina monopolistically competitive market. Good Y is produced under constant3

returns using only labor; units are chosen such that one unit of labor produces
one unit of output. Wage rates are normalized to unity.
In each country, agents have the following utility function:u = X µ Y 1−µ , 0 < µ < 1,(1)

where Y is the consumption level of Good Y and X is a Good X aggregate,
given by
X=

[ n
∑

]1/ρ
ρ

(ci )

, 0 < ρ < 1,

(2)

i=1

where consumption of each variety is given by ci , n is the number of productvarieties produced in Home, and σ ≡ 1/(1 − ρ) > 1 is the elasticity of
substitution between every pair of Good X varieties, respectively. A lower
value of σ implies that consumers value product diversity more.
The consumer’s utility maximization problem can be solved in two steps.2
First, for a given allocation of spending across goods, maximize X subject
to total spending on the diﬀerentiated products, EX . Second, determine
spending on Good X and Good Y .
For the ﬁrst step, one can check that the demand function for variety i2See, for example, Helpman and Krugman (1985, ch. 6).

4

can be written as3p−σiEX(PX )1−σ() ()
pi −σ EX
=
,
PX
PX

ci =

(3)

where PX is the price index of Good X, which is dual to X:4
PX =

[ n
∑

](ρ−1)/ρ
ρ/(ρ−1)

(pi )

=

i=1

[ n
∑

]1/(1−σ)
1−σ

(pi )

.

(4)

i=1

Now we turn to the problem of ﬁnding the optimal spending on Good X,
EX . EX can be obtained by solving the following problem:
max u = X µ Y 1−µ ,s.t. PX X + Y= E,

where E represents national income. Then, one can obtain
EX = µE.

(5)

Substituting this back into (3), one can obtain the demand function:
ci =
3

p−σ
i
µE.
(PX )1−σ(6)Note that this function is log-linear in own price, pi , and total spending on Good X,

EX , both deﬂated by a price index of Good X.
4Note that PX is deﬁned in terms of negative exponents (σ > 1). See, Neary (2001, p.
537) on this point.

5It is important to note that the demand function perceived by the typicalﬁrm is not (6) but rather:5c = φp−σ , φ = µE(PX )σ−1 ,(7)with the intercept φ assumed to be taken as given by the ﬁrm.6 Figure 1(a)shows the constant-elasticity demand curve described by equation (7).
Note also that we can express maximized utility as a function of income
and the price index for Good X, giving the indirect utility function:
V = µµ (1 − µ)1−µ

E
1−µ E
µ
.
µ = µ (1 − µ)
(PX )
P

(8)

The termP ≡ (PX )µis the cost-of-living index in Home.7
Now turn to the production of each variety. Each product is supplied bya monopolistically competitive ﬁrm. Before starting production, α units oflabor are required as a ﬁxed cost of production. Then, β units of labor are
required as a marginal cost of production. Thus, the toal cost function of

5

Hereafter, the subscript i is dropped for simplicity.

6

Neary (2001, p. 538) and Helpman (2006, p. 593).
Fujita, Krugman, and Venables (1999, p. 48). Baldwin et al. (2003, p. 15) call it a7

“perfect” price index in that real income deﬁned with P is a measure of utility.

6the typical ﬁrm becomes8
T C = α + βx,(9)where x is the output level. This implies a horizontal marginal cost (M C)
curve at the level β, and an average cost (AC) curve which is a rectangular
hyperbola with respect to the vertical axis and the marginal cost curve. These
curves are also illustrated in Figure 1(a).
Given a Dixit-Stiglitz speciﬁcation with constant elasticity σ, each ﬁrm
sets its price as
p=

σ
β.
σ−1

(10)

With free entry and exit, the level of output that generates zero proﬁts isgiven byx̄ =α(σ − 1).
β(11)

It is important to note that the (long-run) equilibrium output of each ﬁrm
is constant.
Now let us add one more panel for a better understanding. Figure 1(b)depicts the relationship between the total number of varieties, n, and thedemand level for each variety, c. In the present setting the total expenditure

8

Note that the wage rate is normalized to unity.

7

for Good X is constant:9
npc = µE = µL.

(12)

Substituting the pricing rule (10) into this and rearranging, one can obtainthe following relationship:c=

1 (σ − 1) µL
.
n σ
β

(13)

This demand condition (i.e., budget constraint) is depicted as hyperbola CCin panel (b).
On the other hand, the zero-proﬁt condition implies that each ﬁrm mustsell at least x̄ in the long run. This is depicted as the horizontal line ZZ. In
equilibirum, then, the following condition must hold for each variety:c = x̄.

(14)

By combining these conditions, the equilibrium number of varieties isobtained:
nA =

µL
,
ασ

(15)

where the superscript A represents the autarky (i.e., no international trade)
equilibrium value. Thus the autarky equilibrium value of the cost-of-living
9

Since free entry ensures that proﬁts will be zero in the long run, the national income

consists only of wage income.

8

index becomes:()
A µ/(1−σ)()(µL µ/(1−σ) σβ
p =
P = n
ασ
σ−1
)
(
)(
L
dP A
µ
−
=
.
A
P
dL
σ−1
A

)µ

,

(16)

It is important to note that the cost-of-living index is a decreasing functionof the labor endowment: the larger country can support a greater numberof varieties of diﬀerentiated products than the smaller country.10 Note alsothat as the share of Good X, µ, becomes larger and/or product diﬀerentiation
matters more (i.e., σ is smaller), the impact of a change in labor endowment
on the price index becomes larger.
In panel (b), the autarky equilibrium is obtain as the intersection of
curve CC and curve ZZ, point A. This graphical exposition provides aeasier understanding for comparative statics analysis. Let us consider, forexample, an increase in the labor endowment, L. In this case, the hyperbola
CC moves upward to C 0 C 0 . Then, in the short run, each ﬁrm can sell morethan the zero-proﬁt output x̄: each ﬁrm earns positive proﬁts. This situation
is depicted as point A0 .
However, responding to positive proﬁts, new ﬁrms enter into the Good X
sector. Since consumers spread their income among every variety, demandfor each variety becomes lower. This change is shown by the arrow in panel
10

Fujita, Krugman and Venables (1999, pp. 56–57) call this the price index eﬀect.

9

(b). In the long run, each ﬁrm sells x̄ again: changes in the level of thelabor endowment L lead to adjustments in industry output via changes in
the number of ﬁrms only.11

3

Trading Equilibrium

Suppose that the two countries open their goods markets: the eﬀect will be
the same as if each country had experienced an increase in its labor force.12
The product market equilibrium requires that the demand for each product is equal to the zero-proﬁt output level:c + c∗ = x̄,(17)where c∗ represents the demand for a Home product in Foreign. Adding (13)and its Foregin counterpart, the LHS of (17) can be obtained as follows:c + c∗ =µ(L + L∗ ).
(n + n∗ )p(18)

Substituting this and (10) into (17), one can obtain the total number of varieties in the trading equilibrium, which is the sum of the number of varieties
in the autarky equilibrium,
N T ≡ nT + n∗T =11Neary (2001, p. 539).

12

See, Krugman (1979) on this point.

µ(L + L∗ )= nA + n∗A ,
ασ

10

(19)

where superscript T indicates a trading equilibrium value. Opening trade
can be interpreted as an expansion of market size.
Now we can show the impact of trade liberalization in Figure 2. Let us
take the case of L = L∗ . Panel (b) shows the relationship between N andc, while panel (a) is its Foreign counterpart. As in the case of autarky, thedemand condition (i.e., budget constraint) is depicted by hyperbolas CC andC ∗ C ∗ . The production equilibrium in each country is depicted by point A
and point A∗ , respectively.
Suppose that the opening of trade does not aﬀect the production structure. On the other hand, since consumers now face twice as many productvarieties (from nA to N T = 2nA ), demand for each product becomes halved(the increase from x̄ to x̄/2). Because each country specializes in a diﬀerent range of diﬀerentiated products, intra-industry trade in Good X occurs.
Home consumers’ consumption point moves from point A to point B. Thus,
the total import volume of Foreign varieties is shown by the shaded rectangle. Although the (wage) income level in terms of the numeraire remains
unchanged, an increase in the number of product varieties implies that the
cost-of-living index becomes lower:µ

P T = (PXT ) = (N T )µ/(1−σ) µµ/(1−σ) µp < (nA )µp = (PXA ) = P A .

(20)

Note that an increasing availability of diﬀerentiated products leads to a lower
11cost of obtaining each unit of utility, u, although the price of each productremains constant.

4

Factor Mobility

Now suppose that there are impediments to trade in goods, but economic
integration makes it possible for some workers to migrate across countries.13
Workers migrate toward the country where the equilibrium real wage ishigher. Using (16), one can deﬁne the real wage rate in one country,
(

1
µL
=
P
ασ

)µ/(σ−1) (

σ−1
σβ

)µ

.

(21)

That is, in the presence of internal scale economies, a larger country oﬀers agreater number of diﬀerentiated products and thus the real wage rate becomes
higher than in the smaller country.
In this setting, workers migrate from the smaller country to the larger
country. Thus, the size of the larger country will expand, while the size ofthe smaller country will shrink. The point is that there will be a cumulative
process in which the wide range of diﬀerentiated products attracts workers,
and immigration will enhance further expansion of the range of diﬀerentiatedproducts.
13

Krugman (1979, pp. 477–478), Helpman and Krugman (1985, ch. 11), Matsuyama(1995, pp. 712–713).

12Figure 3 illustrates the allocation of labor between countries. The horizontal axis represents the total labor force in the world economy, L + L∗ .
The quantity of labor employed in Home (resp. Foreign) is measured fromthe left (resp. right). The left (resp. right) vertical axis shows the real
wage rate (21) in Home (resp. Foreign). Initially, in the autarkic equilibriumwith identical labor endowments (L = L∗ ), wage rates are equalized between
countries. The relationship in Home between the total labor force and thereal wage rate is depicted with the curve ω:(µL
1ω(L) ≡=P
ασ)µ/(σ−1) (

σ−1
σβ

)µ

.

(22)

Likewise, the relationship in Foreign is depicted with the curve ω ∗ .
Now let us describe the process of labor movement. If some workers movefrom Foreign to Home, it raises the real wage rate in Home, while lowering thereal wage rate in Foreign. This wage gap further stimulates labor movementfrom Foreign to Home. Note that this movement hurts those left behindin Foreign (i.e., the smaller country). While the Home wage rate increases
along the ω curve, the Foreign counterpart decreases along the ω ∗ curve.
This provides a striking contrast with the case of trade in goods, in which
all workers gain and those in the small country gain the most.14
Until now, we have concentrated on the case with identical technologies
14

Matsuyama (1995, p. 712).

13

between countries. Now let us brieﬂy review what happens if both ﬁxed andvariable costs are higher in one country.15 In this case, it is clearly desirable
that all worker should move to the other country. But if the inferior country
starts with a large enough share of the labor endowment, migration may
move in the wrong direction.16

As in the case of external economies, theworld economy may be trapped into a Pareto inferior situation.17

5

Concluding Remarks

In this paper, we have brieﬂy reviewed the now standard Dixit-StiglitzKrugman model of monopolistic competition. In particular, we have proposed a convincing graphical exposition that emphasizes the ﬁrms’ entry-exit
process, which facilitates the understanding of several topics such as determinants of equilibrium and existence of intra-indutry trade. Although thistractable model of monopolistic competition relies heavily on speciﬁc functional forms, it will remain as one of the key ingredients of trade models for
15

Krugman (1979, p. 478). Matsuyama and Takahashi (1998) present a model of two

regional economies with similar features.
16
Related to this, in the case of trade in goods, Lancaster (1980, pp. 167–168) notes thata size diﬀerence between countries may become a source of “false comparative advantage.”That is, autarky relative prices do not serve as reliable predictors of trade patterns.
17
Note also that this model is similar to the models of standard setting in the Industrial
Organization literature. See, for example, Chou and Shy (1990).

14

internal scale economies.18 Note that, since this model is quite special, one
should view it as a complement rather than a substitute for the other modelsof trade (e.g., trade models for external economies).

References[1] Baldwin, Richard, Rikard Forslid, Philippe Martin, Gianmarco Ottaviano, and Frederic Robert-Nicoud (2003) Economic Geography and Public Policy, Princeton, NJ: Princeton University Press.
[2] Chou, Chien-fu and Oz Shy (1990) “Network Eﬀects without Network
Externalities,” International Journal of Industrial Organization, Vol. 8,
pp. 259–270.
[3] Combes, Pierre-Philippe, Thierry Mayer, and Jacques-Francois Thisse(2008) Economic Geography: The Integration of Regions and Nations,
Princeton, NJ: Princeton University Press.

18

In his inﬂuential contribution, Melitz (2003) has proposed an extension of the Dixit-

Stiglitz-Krugman model that makes it possible to work with heterogeneous ﬁrms in termsof their marginal labor input requirement. See Helpman (2006) for a survey of the relevant
literature.

15

[4] Dixit, Avinash K. and Victor Norman (1980) The Theory of International
Trade: A Dual, General Equilibrium Approach. Cambridge: Cambridge
University Press.
[5] Dixit, Avinash K. and Joseph E. Stiglitz (1977) “Monopolistic Competition and Optimum Product Diversity,” American Economic Review, Vol.
67, pp. 297–308.
[6] Feenstra, Robert C. (2004) Advanced International Trade: Theory andEvidence, Princeton, NJ: Princeton University Press.
[7] Fujita, Masahisa, Paul Krugman and Anthony J. Venables (1999) The
Spatial Economy: Cities, Regions, and International Trade. Cambridge,
MA: MIT Press.
[8] Helpman, Elhanan (1990) “Monopolistic Competition in Trade Theory,”
Special Papers in International Finance, No. 16.
[9] Helpman, Elhanan (2006) “Trade, FDI and the Organization of Firms,”
Journal of Economic Literature, Vol. 44, pp. 589–630.
[10] Helpman, Elhanan and Paul R. Krugman (1985) Market Structure andForeign Trade: Increasing Returns, Imperfect Competition and the International Economy. Cambridge, MA: MIT Press.

16[11] Krugman, Paul (1979) “Increasing Returns, Monopolistic Competition,and International Trade,” Journal of International Economics, Vol. 9,
pp. 469–479.
[12] Krugman, Paul (1980) “Scale Economies, Product Diﬀerentiation, and
the Pattern of Trade,” American Economic Review, Vol. 70, pp. 950–959.
[13] Krugman, Paul (1981) “Intraindustry Specialization and the Gains from
Trade,” Journal of Political Economy, Vol. 89, pp. 959–974.
[14] Lancaster, Kelvin (1980) “Intraindustry Trade under Perfect Monopolistic Competition,” Journal of International Economics, Vol. 10, pp.
151–175.
[15] Matsuyama, Kiminori (1995) “Complementarities and Cumulative Processes in Models of Monopolistic Competition,” Journal of Economic Literature, Vol. 33, pp. 701–729.
[16] Matsuyama, Kiminori and Takaaki Takahashi (1998) “Self-Defeating
Regional Concentration,” Review of Economic Studies, Vol. 65, pp. 211–234.
[17] Melitz, Marc J. (2003) “The Impact of Trade on Intra-Industry Reallocation and Aggregate Industry Productivity,” Econometrica, Vol. 71, pp.
1695–1725.
17

[18] Neary, Peter J. (2001) “Of Hype and Hyperbolas: Introducing the New
Economic Geography,” Journal of Economic Literature, Vol. 39, pp. 536–
561.
[19] Neary, Peter J. (2004) “Monopolistic Competition and International
Trade Theory,” in Brackman, Steven and Ben J. Heijdra (eds.) The Monopolistic Competition Revolution in Retrospect, Cambridge, Cambridge
University Press.
[20] Neary, Peter J. (2009) “Putting the ‘New’ into New Trade Theory: Paul
Krugman’s Nobel Memorial Prize in Economics,” Scandinavian Journalof Economics, Vol. 111, pp. 217–250.

18

Figure 1
(a)

(b)

p

c
C’
C
D
AC

A’

x
Z

A

MC
x

n^A

C’
Z
C

n

Figure 2
c*

(b)
C*
A*

Z
C*
n*

B*

(a)

c
C
x
x/2

A

Z
B
C

n

Figure 3
ω*

ω

ω*

L

ω

L*