A Decomposition of Ricardian Trade Gains
under External Economies of Scale
Toru Kikuchi∗
Graduate School of Economics, Kobe University, Kobe 657-8501, Japanand
Ngo Van Long
Department of Economics, McGill University, Montreal H3A 2T7, Canada

March 29, 2010

Abstract
Although the one-factor Ricardian trade model with external economies
of scale plays a significant role for the understanding of important tradeissues under increasing returns, it lacks a compelling graphical representation. We propose a convincing graphical exposition that uses both the
PPF and a labor market graph.
JEL classification: F10
Keywords: Trade Gains; Ricardian Trade Model; External Economies of
Scale; Graph of the Labor Market

1

Introduction

Increasing returns to scale (IRS) generated by external economies have becomea main feature of many economic models that deal with trade gains, endogenous growth, multiplicity of equilibria, and indeterminacy in dynamic models
of growth1 . The simplest route to understand a basic mechanism in these models is the one-sector Ricardian trade model with IRS. Yet, this model lacks acompelling graphical representation.2 The purpose of this note is to offer sucha representation. Our experience indicates that this graphical approach helps
the reader to gain, almost effortlessly, a clear understanding of the effect of IRS
on trade gains.
∗ E-mail addresses: kikuchi@econ.kobe-u.ac.jp (Kikuchi), ngo.long@mcgill.ca (Long).
1 Jones (1968), Herberg and Kemp (1969), Kemp (1969, ch. 8), Melvin (1969), Panagariya(1981) and Ethier (1982) are key contributions to the IRS literature in the static trade theory.
Helpman and Krugman (1985, ch. 3) provides a comprehensive survey. For indeterminacy inmodels of trade and growth with IRS, see Nishimura and Shimomura (2002).
2 Francois and Nelson (2002) is a notable exception. They develop a general graphicalframework for division of labor models.

1

In most textbooks on international economics, the analysis of trade gains
under external economies of scale relies either on the utilization of Production Possibility Frontier (PPF) diagrams or partial equilibrium diagrams.3 Oneof the great traditions in the analysis of international trade is the expositionof canonical models (e.g., Ricardo, Heckscher-Ohlin-Samuelson) using simple
graphical representations. It is useful both providing an intuitive grasp of thetheory and for pedagogical purposes. We believe that the focus on the extensive
use of trade models with external economies of scale should be accompanied bya canonical geographical representation.
In this note, we propose the use of a combination of the PPF graph anda labor market graph to facilitate the understanding of Ricardian trade gains
under external economies of scale.4 By using these graphs, one can easily decompose Ricardian trade gains under external economies of scale. In addition,for educational purposes, the inclusion of labor market graphs in a lecture on
the Ricardian Model lays a useful groundwork for later lectures which deal with
the Specific-Factors Model and International Factor Movements.

2

Analysis

Consider a Ricardian economy which produces two goods (good 1 and good 2)
with one factor of production, labor.5 Let Yi and Li denote respectively theoutput and employment levels of industry i.
Good 1 is produced in a competitive industry under external economies ofscale. The output of the representative firm (employing `1 units of labor) is
given byδ
y1 = (L1 ) `1 , δ ≥ 0,(1)
δwhere the firm takes (L1 ) as exogenous. The firm perceives that its marginalδ
product of labor is (L1 ) which it takes as a constant. Aggregating over all thefirms in industry 1, we getδ

Y1 = (L1 ) L1 = L1+δ≡ Lε1 ,1ε≡1+δTherefore at the industry level, the marginal product of labor is εLδ1 > Lδ1 if
δ > 0. We call εLδ1 the marginal social product of labor in sector 1, and Lδ1 themarginal private product of labor, as perceived by the firms. Here ² is the degreeof external economies of scale: ² = 1 corresponds to the case without external
economies of scale. Let aLi denote the amount of labor needed to produce one
unit of good i. In sector 1, aL1 is defined as follows:aL1 (L1 ) ≡

³
´−δ
L1
1/ε
−δ
= (L1 ) = Y1
Y1

(2)

3 See Bowen, Hollander, and Viaene (1998), Caves, Frankel, and Jones (2002, ch. 3) for theformer examples, while Krugman and Obstfeld (2008, ch. 6), for the latter example.
4 Kikuchi and Long (2010) apply these methods to the standard Ricardian model without
external economies of scale.
5 The model is taken from Ethier (1982). See also Panagariya (1981).

2Thus a greater employment level L1 results in a lower value of aL1 .
Good 2 is the numeraire good. It is produced by a competitive industry
under constant-returns-to-scale technology, so that aL2 is a constant.
Then the PPF is represented by
1/²

aL1 Y1 + aL2 Y2 = (Y1 )+ aL2 Y2 = L(3)where L is the total amount of labor. With ² > 1, the marginal social cost ofgood 1 in terms of good 2 falls as Y1 increases; this gives rise to a convex PPF,
V P , depicted at Figure 1 (a).6 Also, the marginal social cost of good 1 in termsof good 2 is lower than its private counterpart.7
Let µ be the expenditure share on good 1. In autarky, the relative price ofgood 1 is obtained as follows. First, from the definition of µ, and using the facts
that in the Ricardian model national income equals the wage bill, wL, and thatthe price equals labour cost per unit of output, so that w = p2 /aL2 , we canwrite
p1 Y1 = µ(p1 Y1 + p2 Y2 ) = µwL = µL

p2
aL2

Thus in at the autarkic equilibrium
Y1 =

p2
(µL)
p1 a2LThe equalization of the price ratio to the labour cost ratio gives
µ

p1
p2

or

Finally, we obtain

¶A
=

1−ε
1
aL1 (LA1 ³ 1/ε ´−δ
1)
=
=
Y1
Y1 ε
aL2
a2L
aL2

µ
¶ 1−ε
ε
1−ε
p1 aL2p2(µL) ε=p2
p1 aL2µ¶11−εp1 aL2 ε= (µL) ε
p2
µ

p1
p2

¶A

1−²

=

aL1 (LA
(µL)
1)
=
aL2
aL2

,

(4)

where the superscript A represents the autarkic equilibrium. The autarkic priceratio is obtained as line V A.
6 When there are more factors, the shape of the PPF will in general be determined by thetension between (1) differences in factor intensities which tend to make the PPF concave, and(2) external economies of scale which tends to make the PPF convex. See, Herberg and Kemp
(1969), Kemp (1969, ch. 8), Melvin (1969), and Markusen and Melvin (1984) for discussion.
−δ/e
7 The marginal social cost of good 1 in terms of good 2 is (1/εaand the marginalL2 )Y1−δ/eprivate cost of good 1 in terms of good 2 is (1/aL2 )Y1
produced along the PPF.

3

. Both fall as more of good 1 is

Now let us add one more panel for a better understanding: Figure 1 (b)depicts labor allocation between sectors. The horizontal axis represents thetotal labor force, L. The quantity of worker employed in sector 1 (resp. sector
2) is measured from the left (resp. right). The left (resp. right) vertical axis
shows the wage rate in the sector 1 (resp. sector 2). Initially, in the autarkic
equilibrium, wage rates are equalized between sectors and O1 L1 workers are
hired in sector 1, while L1 O2 workers are hired in sector 2. The total income(in terms of the numeraire, good 2) is shown by the shaded rectangle.
For later use, we add a curve that represents the relationship between thewage rate (value of average product of labour) in sector 1 and the hypothetical
employment level in that sector, L1 , for a given price p1 . This hypotheticalcurve is denoted by w̃1 (L1 ; p1 ):w̃1 (L1 ; p1 ) ≡p1= p1 L²−1
1 .
aL1 (L1 )(5)

The dotted increasing curve in Figure 1 (b) depicts w̃1 (L1 , pA
1 ). Since economies
of scale are external to firms (and workers), this curve is not perceived by them.8
Note that as ² approaches unity this curve approaches a horizontal line.
Now let us move to the trading situation with a fixed free-trade price ratioT(p1 /p2 ) (Figure 2). If the terms of trade are given by the slope of the line P D(i.e., the free-trade relative price of good 1 is higher than autarkic price ratio),the economy specializes in good 1 at point P . Assume that consumption occurs
at point C located on the consumption possibility frontier P D, so that CBP
is the trade triangle (the country exports BP units of good 1 and imports BC
units of good 2).
Now we can decompose the movement toward the trading equilibrium into
two steps, which is in line with the traditional separation of trade gains into consumption and production gains.9 Firstly, suppose that in the short run, labor
allocation is fixed, and thus the economy is staying at the autarky production
Tpoint, A, while it is able to trade at the terms of trade pT ≡ (p1 /p2 ) . Superscript T denotes the trading equilibrium. Then, the consumption possibility
frontier expands from V AP to V 0 AP as in Figure 2 (a).10 In Figure 2 (b), thischange can be illustrated as an increase in the wage rate for the workers emATA
ployed in sector 1 [i.e., from pA
1 /aL1 (L1 ) to p1 /aL1 (L1 )]. Their total increasein wage income is represented by the dotted rectangle in Panel (b), which isshown as V V 0 in Panel (a). It is also important to note that, as the result ofmoving from autarkic price to the free-trade price, the curve of hypotheticalTwage in sector 1, w̃1 (L1 , p1 ), shifts up from w̃1 (L1 , pA
1 ) to w̃1 (L1 , p1 ).
Next, let us consider the (long-run) labor movement between sectors. Since
sector 1 offers a higher wage rate, workers will gradually move from sector 2
to sector 1 (as shown by the arrows in Figure 2). Due to external economies
8 Alternatively, we may assume that all agents know this relationship but they do not act

on it because individually they are atomistic.
9 See, for example, Dixit and Norman (1980, pp. 71–72).
10 At the production point A, under free trade consumers will consume more of good 2 than
before, because good 2 becomes relatively cheaper than under autarky, pT > pA .

4

of scale, each unit of labor moved from sector 2 to sector 1 generates a larger
output increase than the previous unit, which implies a higher wage rate, i.e. a
movement along the w̃1 (L1 , pT1 ) curve. In Panel (b), the eventual distribution oflabor force will be one of complete sepecialization (with O1 O2 workers in sector1), which corresponds to the production point P in Panel (a). The effect of thislabor movement among sectors is shown as the expansion of the consumption
possibility frontier from V 0 AP to DP in Panel (a), reflected by the sum of
the horizontally shaded area and the vertically shaded area in Panel (b). It is
important to note that, by utilizing a labor market graph, the gains from labor
movement can be decomposed into (1) direct gains from labor movement (thehorizontally shaded area) and (2) indirect gains via external economies of scale(the vertically shaded area). Note that the direct gains from labor movement
also exists under constant-returns-to-scale technology. Note also that, the higher
is the numerical value of ², the bigger are the indirect gains.
In terms of the numeraire, trade gains are measured by V D in Panel (a),and by the sum of the dotted rectangle, the horizontally shaded rectangle, andthe vertically shaded rectangle in Panel (b).
We must now find out what happens when pT1 is lower than pA
1 (Figure 3).
In this case the curve of hypothetical wage in sector 1, w̃1 (L1 , p1 ), shifts downTfrom w̃1 (L1 , pA
1 ) to w̃1 (L1 , p1 ). The wage rate in sector 1 becomes lower andworkers will gradually moves from sector 1 to sector 2 (shown by the arrows
in Figure 3). The eventual distribution of labor force will be one with O1 O2
workers in sector 2, which corresponds to the production point V in Panel (a).
The total income (in terms of the numeraire, good 2) is shown by the sum of theshaded area and the horizontally shaded rectangle, which is same as the total
income under the autarky equilibrium. Since the price of good 1 becomes lower
in in terms of the numeraire, this implies an increase in total real income.
Now we can restate Ethier’s (1982) interesting result on trade gains under
external economies of scale.
Proposition 1 (Ethier):A small country entering into international trade will
be driven to specialize in that commodity with the lower autarkic relative price.
Regardless of which commodity that is, the small country will gain from free
trade relative to autarky.
Figure 3 also illustrates the possibility of multiple equilibria that can be
Pareto-ranked. Although when pT1 is lower than pA
1 the economy gains relative
to its autarky welfare level (at A) by specializing in good 2, it can obtain alarger gain by economy specializing in good 1 if the following condition holds:w̃1 (L; pT1 ) >

pT2
.
aL2

(6)

(recall that by normalization, pT2 = pA
2 = 1). By shifting labor force toward
sector 1 instead of sector 2, the economy can obtain the indirect gain via external
economies of scale. When the above inequality holds, the national income in5terms of good 2 when the country produces at point P is w̃1 (L; pT1 )L, which isgreater than when it produces at point V . In Figure 3 (b) the extra gains from
specializing in good 1 is depicted by the vertically shaded rectangle.
When there are multiple equilibria, in general it is not clear which one islikely to prevail, though some sort of stability argument may helps the equilibrium selection.11 Myerson (2009, p. 1111) points out that the existence ofmultiple equilibria is “a pervasive fact of life that needs be appreciated andunderstood, not ignored by economists.”123

Discussion

Using the above illustration confers several advantages. First, since the labormarket diagram emphasizes differences in wage rates between sectors in the
short run, it is easier to understand the process of factor movements from theworkers’ perspectives. Second, it is intuitive to see at first the trade gains interms of labor income. This is later reinforced by the traditional separation of
trade gains into consumption and production gains. Third, the idea of multiple
equilibria and the stability considerations can be illustrated in a simple way.
Another major advantage relates to the sequencing of lectures of a typicalcourse of International Economics (or International Trade). Since both thespecific-factor model and the analysis of international factor movements makeextensive use of the graphs of labor-market (dis)equilibrium, it might be helpful
to provide a graph of the labor market in advance. The inclusion of labor market
graphs in the Ricardian Model under external economies of scale lecture lays auseful groundwork for the later lectures.
We recognize that there are many alternative ways to show Ricardian tradegains under external economies of scale. Still, we believe that the way presented
here provides some helpful tool for understanding (and teaching) Ricardian trademodels under external economies of scale.

References[1] Bowen, Harry P., Abraham Hollander and Jean-Marie Viaene (1998) Applied International Trade Analysis, London: Mcmillan.
[2] Caves, Richard E., Jeffrey A. Frankel, and Ronald W. Jones (2002) World
Trade and Payments: An Introduction, Ninth Edition, NY: Harper Collins.

11 Note that Proposition 1 implies that when when pT is lower than pA the country will not
1
1
be “driven” to the (unstable) equilibrium point P , by the usual stability argument, as illustrated by the arrows in Figure 3. Only an active industrial policy can lead to this equilibrium(e.g. by coordinating expectations)
12 Multiplicity of equilibria occurs in nature as well. As Stephen Jay Gould (1993, p. 28)
noted, “places with apparently identical vegetation, moisture, and temperature might harborshells of maximally different form.”6[3] Dixit, Avinash. K. and Victor Norman (1980) Theory of International
Trade, Cambridge: Cambridge University Press.
[4] Ethier, Wilfred J. (1982) “Decreasing Costs in International Trade andFrank Graham’s Argument for Protection,” Econometrica, Vol. 50, pp.
1243–1268.
[5] Francois, Joseph F., and Douglas Nelson (2002) “A Geometry of Specialization,” Economic Journal, Vol. 112, pp. 649–678.
[6] Gould, Stephen J. (1993), Eight Little Piggies: Reflections in Natural History, W.W. Norton, New York.
[7] Helpman, Elhanan and Paul Krugman (1985) Market Structure and Foreign
Trade, Cambridge, MA: MIT Press.
[8] Herberg, Horst and Murray C. Kemp (1969) “Some Implications of Variable
Returns to Scale,” Canadian Journal of Economics, Vol. 3, pp. 403–415.
[9] Jones, Ronald W. (1968) “Variable Returns of Scale in General Equilibrium
Theory,” International Economic Review, Vol. 9, pp. 261–272.
[10] Kemp, Murray C. (1969) The Pure Theory of International Trade and Investment, Englewood Cliffs, NJ: Prentice-Hall.
[11] Kikuchi, Toru and Ngo Van Long (2010) “A Decomposition of Ricardian
Trade Gains,” Discussion Paper No. 0922, Graduate School of Economics,
Kobe University.
[12] Krugman, Paul and Maurice Obstfeld (2008) International Economics:
Theory and Policy, Eighth Edition. Addison-Wesley.
[13] Markusen, James and James Melvin (1984) “The Gains from Trade Theorem with Increasing Returns to Scale,” in H. Kierzkowski (ed.) Monopolistic
Competition and International Trade, NY: Oxford University Press.
[14] Melvin, James (1969) “Increasing Returns to Scale as a Determinant ofTrade,” Canadian Journal of Economics, Vol. 2, pp. 389–402.
[15] Myerson, Roger B. (2009), “Learning from Schelling’s Strategy of Conflict,”
Journal of Economic Literature 47(4):1109-1125.
[16] Nishimura, Kazuo, and Koji Shimomura (2002), “Trade and Indeterminacy
in a Dynamic General Equilibrium Model,” Journal of Economic Theory,
Vol. 105, pp. 244-260.
[17] Panagariya, Arvind (1981) “Variable Returns to Scale in Production and
Patterns of Specialization,” American Economic Review, Vol. 71, pp. 221230.

7

Figure 1(b)(a)
p1/aL1

good 2

p2/aL2~w(L1, p1^A)

p2^A/aL2

p1^A/aL1(L1)V
A
P

good 1O1
sector 1 employment

L1

O2
sector 2 employment

Figure 2
(b)

(a)
D

p1/aL1

good 2Newequilibrium

p1^T/aL1(L)~w(L1, p1^T)

p1^T/aL1(L1)p1^A/aL1(L1)

C

V’
V

InitialequilibriumAP
B

p2/aL2

good 1O1
sector 1 employment

L1

O2
sector 2 employment

Figure 3(b)(a)
p1/aL1

good 2

p2/aL2Newequilibrium

p1^T/aL1(L)p2^A/aL2

p1^A/aL1(L1)V
B

p1^T/aL1(L1)
Initialequilibrium

C
A
P

good 1O1
sector 1 employment

L1

O2
sector 2 employment