The evolution of trade patterns and economic development ina dynamic Ricardian framework
Colin Davis
Doshisha University∗
Yasunobu Tomoda
Osaka University of Economics, Hiroshima University†
August 2024
Abstract
Does the international division of labor based on comparative advantage
lead to persistent income inequality between developed and developing countries? Does specialization in primary commodities hinder the economic development of low-income countries? We construct a North-South model in whichhuman capital determines labor productivity, and differences in the human capital elasticity of output across industries cause comparative advantage to evolve
dynamically as the countries invest in human capital accumulation, with theNorth (South) exporting the more (less) human capital elastic good. We findthat income inequality persists in the long run with both countries gaining fromtrade when trade patterns are characterized by complete specialization in production. The gains from trade vanish along with income inequality, however,when trade patterns feature incomplete specialization.
JEL Classifications: F63, O11, O41
Keywords: Comparative advantage; Human capital elasticity of output; Termsof trade; Persistent income inequality; Prebisch-Singer hypothesis∗

The Institute for the Liberal Arts, Doshisha University, Karasuma-Higashi-iru, Imadegawa-dori,
Kamigyo, Kyoto, Japan, 602-8580, cdavis@mail.doshisha.ac.jp.
†
Faculty of Economics, Osaka University of Economics, 2-2-8, Osumi, Higashiyodogawa, Osaka,
Japan, 533-8533, y.tomoda@osaka-ue.ac.jp.
Acknowledgements: We are grateful for helpful comments from Tetsugen Haruyama, Ken-ichi
Hashimoto, Noritsugu Nakanishi, Ken Tabata, Daishin Yasui, and participants of the Rokkodai
Macroeconomics Seminar at Kobe University. This research was financially supported by the Japan
Society for the Promotion of Science: Grants-in-Aid for Scientific Research 15K03444.

1

1Introduction

Since the introduction of David Ricardo’s principle of comparative advantage, there
has been persistent concern that low-income countries specializing in primary commodities are potentially hampered in their economic development by the international
division of labor. Indeed, it is often noted that development paths that depend chiefly
on the export of primary commodities may not be sufficient for low-income countries
trying to catch up with developed economies, potentially resulting in persistent income
inequality between the North and South. In fact, commodity exports tend to assume
an oversized role in many low-income countries. For example, the average share ofprimary commodities in total exports exceeded 70% in Least Developed Countries
between 1995 and 2009 (UNDP, 2011). Given the dependence of many low-income
countries on commodity exports, how then does North-South trade specialization influence economic development?
This paper offers a new perspective on the long standing question regarding therelationship between international trade, economic development and income inequality through the study of a novel dynamic Ricardian framework. In particular, we
introduce a North-South overlapping generations model in which labor is employedin the production of a primary good and an industrial good, both of which are consumed by households and used in the provision of education. Household investment
in education drives the creation of human capital. And, in each country, sector-level
labor productivity improves with human capital accumulation, linking development
and trade through the evolution of patterns of comparative advantage.
Our framework highlights the varying importance of human capital across different
employment types by focusing on the human capital elasticity of output associatedwith each sector. Consider, for instance, a relatively monotonous type of employmentthat requires more labor hours than it does labor skill. Replacing a worker withlow human capital for a worker with high human capital is unlikely to significantly2impact overall output. As such, this employment type could be described as having a
relatively low human capital elasticity of output. In contrast, for an employment typethat requires relatively more labor skill, replacing a worker with low human capitalfor a worker with high human capital is likely to generate a substantial increase in
output. This employment type could then be characterized as having a relatively highhuman capital elasticity of output.
In our model, the labor productivity coefficients associated with linear Ricardian
production technologies are set as functions of the country-specific stocks of humancapital, and the human capital elasticities of output are allowed to differ across the primary and industrial sectors. In this setting, we show that the human capital elasticity
of output plays a crucial role in how trade patterns based on comparative advantage
influence income inequality along the path of economic development. Assuming theNorth has a greater initial stock of human capital, the North has a comparative advantage in the industrial good, which has a larger human capital elasticity of output.
And, the South has a comparative advantage in the primary good, which has a smallerhuman capital elasticity of output.
Then, following the standard Ricardian model, a long-run trade pattern withcomplete specialization in production requires that national labor supplies are large
enough to meet the world demands for each good. In our framework, however, wederive the additional condition that industrial output must be human capital elastic
for complete specialization to persist in the long run. In particular, we find that when
the northern industrial sector exhibits increasing returns to scale, a falling price forthe industrial good results in a deterioration of the North’s terms of trade, preventing divergence in the dynamics of trade patterns towards incomplete specialization.
This result has important implications for the gains from trade. Specifically, if thehuman capital elasticity of output is inelastic in both industries, the utilities of thetwo countries converge to their autarky levels, and the gains from trade disappear.

3

Alternatively, if the industrial sector has an elastic human capital elasticity of output,
the North-South income gap persists in the steady state, and both countries gain fromtrade. In other words, an important result is that the North and South benefit fromtrade in the long run if and only if the North-South income gap endures as a resultof complete specialization.
In the prevailing international economics literature, the relationship between tradepatterns and cross-country income differentials (i.e., the North-South income gap) has
generally been analyzed using extensions of the Heckscher-Ohlin framework. In thestandard Heckscher-Ohlin model, capital and labor are commonly employed with homothetic production technologies. Accordingly, following the results of the standard
neoclassical growth model, the poorer a country is, the faster its income grows as
its capital accumulates, promoting convergence in cross-country income levels (Chuiet. al., 2002; Johnson and Pagageorgiou, 2020). Moreover, the factor price equalization that coincides with increased trade also tends to equalize per-capita income
across countries.1 Although the Heckscher-Ohlin model is known to provide a good
explanation of 19th century trade patterns (O’Rourke and Williamson, 1999), in recent years empirical support has been wavering (Trefler, 1993, 1995; Trefler and Zhu,
2000; Estevadeordal and Taylor, 2002). In contrast to rising skepticism regarding thepredictive power of the Heckscher-Ohlin model, the Ricardian model has garneredstrong empirical support (Golub and Hsieh, 2000).
The main message of the Ricardian model is that relative labor productivity acrossindustries is the key determinant of international trade patterns, rather than differences in resource endowments. In the modern era, with maturing industries and thecontinual advancement of information and communication technologies, labor produc1

A number of studies in the endogenous growth literature show that the North-South income gap
generates product cycles in which new products are produced in the North and then imitated in theSouth (Krugman 1979; Grossman and Helpman 1991). In addition, the new economic geography
literature considers how supply and demand externalities lead to the geographical concentration ofindustry, producing persistent income gaps between technologically advanced core countries and lessadvanced periphery countries (Baldwin et al. 2001).

4

tivity hinges on the education, knowledge and skill of the labor force. It is natural,
therefore, to set labor productivity as a function of human capital. Trade patterns
then depend on how human capital influences labor productivity, linking trade patterns with human capital endowments. Importantly, connecting labor productivity
with human capital allows for an analysis of the link between trade and North-South
income inequality in a framework where human capital accumulation is the engine of
economic growth.
Our paper is closely related to a branch of the development economics literature
examining the Prebisch-Singer hypothesis (Prebisch, 1950; Singer, 1950). The main
premise is that primary goods tend to have low Engel coefficients, suggesting that
relative demand falls as income rises and implying that the terms of trade deteriorate
for countries that are dependent on primary commodity exports. The result is that
low-income countries may not gain from trade in the long run. While the conclusion
is that international specialization dampens the development of the South and leads
to persistent North-South income inequality, the empirical evidence for the PrebischSinger hypothesis has generally been mixed (Lutz, 1999; Baffes and Etienne, 2016;
Di Iorio and Fachin, 2018). Moreover, with an emphasis on the static gains fromtrade, the classical trade literature has been relatively silent on the long-run dynamic
gains from trade. In contrast, our analysis pays particular attention to this issue. We
show that although a decrease in the expenditure share of primary goods leads to a
deterioration in the South’s terms of trade in the short run, once long run adjustmentsin the human capital stocks are complete, the South experiences an overall terms oftrade improvement. The stark difference in the short-run and long-run effects of a fallin the spending share of primary goods underscores the importance of analyzing thelinks between trade and economic development in a dynamic framework.
Van de Klundert and Smulders (1996) and Devereux (1997) also consider production technologies for which labor productivity is contingent on the stock of human5capital.2 In these frameworks, however, it is technology spillovers and learning by
doing that fostor human capital accumulation as a by-product of production, rather
than investment in education. In particular, van de Klundert and Smulders (1996)
investigate the role of knowledge spillovers in the convergence of the North-South income gap. With learning by doing, the larger market shares of northern firms facilitatea faster rate of human capital accumulation, leading to divergence in the North-Southtechnology gap. Technology spillovers from the North to the South are determined
as an increasing function of the North-South technology gap and promote technology
convergence. Ultimately, the strength of knowledge spillovers determine whether or
not the cross-country income gap converges to a stable level in the long run.
In contrast, in our framework the international price mechanism governs the longrun stability of the North-South income gap. Specifically, the sector-specific humancapital elasticities of output determine trade patterns and cross-country income inequality, but the weighted-average of the human capital elasticities of output regulates
the aggregate dynamics of human capital accumulation. Moreover, the fact that the
production technology of the industrial good is an increasing function of the humancapital stock implies that the rise in output that coincides with human capital accumulation induces a fall in the relative price of the industrial good, putting a break on
the expansion of the North-South income gap as the developed North’s terms of trade
deteriorate. As such, in our framework it is the international price mechanism thatstabilizes the income gap in the long run, when the trade pattern features complete
specialization.
The remainder of the paper is organized as follows. We introduce our overlapping
generations framework in Section 2. In Section 3, we then discuss the basic featuresof the model through an examination of the closed economy equilibrium. Section 4
2

There is a broad literature extending the Ricardian framework to consider the effects of tradeliberalization on economic growth and national welfare. See, for example, Taylor (1993), Dinopoulos
and Syropoulos (1997), and Naito (2012). However, these studies do not model labor productivity
as a function of human capital.

6

investigates the open economy equilibrium. Section 5 provides concluding remarks.

2

The ModelThis section introduces our two-country overlapping generations model of comparative advantage. We consider a northern (N) developed country, and a southern (S)developing country. Each country is endowed with a labor force Li , where the superscript i ∈ {N, S} indicates variables associated with the North and the South. Thereis no population growth.
Time flows discretely, and all individuals live for two periods. In the first period,
children develop human capital through education. In the second period, adults earnlabor income, and incur expenditures on consumption and investment in the educationof their children. All household decisions are made by the adult generation. Withineach country, a representative adult has a human capital stock of hit , where the subscript t denotes the discrete time period. We assume that initially the representativeS
stock of human capital is greater in the North than in the South (hN
0 > h0 ).

Each country potentially employs labor in a primary sector that produces good
1 and an industrial sector that produces good 2. These two sectors are competitive,
with free market entry and exit, and a large number of firms. We adopt the primary
good 1 as the model numeraire, setting its price equal to one, and then denote therelative price of the industrial good 2 by pt . The production technology employed bya worker with a stock of human capital hit in country i and sector j ∈ {1, 2} isxij,t = Aj (hit )αj ,(1)where Aj > 1 and 0 < α1 < α2 < ∞. A key feature of our framework is the humancapital elasticity of output αj = (∂xij,t /∂hit )(hit /xij,t ), which we allow to differ between
sectors. Importantly, in this paper we consider two cases. In the first, the human7Figure 1: Human Capital and Sector Output
xij,t
x2

xN
2,t
xS2,t

x1

xN
1,t
xS1,t

hSt

0

hN
t

hit

Human capital and output for α1 < 1 < α2 and 1 < hSt < hNt .

capital elasticity of output is inelastic for both sectors (α1 < α2 < 1). And, in thesecond, the human capital elasticity of output is inelastic for the primary sector, butelastic for the industrial sector (α1 < 1 < α2 ).
In the Ricardian framework, relative labor productivity generates a pattern of comparative advantage that in turn guides international trade patterns. In our model, thelabor productivities associated with sectors 1 and 2 depend on northern and southernstocks of human capital. For example, Figure 1 illustrates the relationship between
human capital stock and sector output for α1 < 1 < α2 . With a smaller human capitalS
elasticity of output in the primary sector, the human-capital differential (hNt − ht )S
generates a relatively small output gap across countries (xN
1,t − x1,t ). In comparison,

the larger human capital elasticity of output in the industrial sector ensures that thehuman-capital differential leads to a relatively large output gap between the NorthS
and the South (xN2,t − x2,t ).

As introduced above, adults invest in the education of their children. Accordingly,
an individual born in country i at time t develops a human capital stock of hit+1 attime t + 1. Adapting the development framework of Glomm and Rivikumar (1992),

8

we assume that the education technology takes the following form in country i:
hit+1 = B(ei1,t )β1 (ei2,t )β2 ,(2)where B > 0, βj ∈ (0, 1), and β1 + β2 = 1.3 The parents of each household set
expenditures ei1,t and ei2,t on goods 1 and 2 in the education of their children.
Household preferences are symmetric across countries. Specifically, the per-period
utility of a household residing in country i is
U = (ci1,t )δ1 (ci2,t )δ2 (hit+1 )σ ,(3)

where cij,t is household consumption of good j ∈ {1, 2}, δj ∈ (0, 1), σ ∈ (0, 1), andthe utility function is assumed to be homothetic, with δ1 + δ2 + σ = 1. Under thisformulation for preferences, households derive altruistic utility from the education(i.e., future human capital) of their children. The household budget constraint is
wti = (ci1,t + ei1,t ) + pt (ci2,t + ei2,t ),(4)

with wti denoting the labor income earned by a representative household.
Adults set their expenditures on consumption and education with the aim of maximizing household utility (3) subject to the budget constraint (4). The solution to theutility maximization problem yields the following household demands for the primary
good 1 and the industrial good 2 in consumption and education in country i:
ci1,t = δ1 wti ,ci2,t =

δ2 wti,pt

ei1,t = σβ1 wti ,ei2,t =

σβ2 wti.
pt(5)Summing households demands (5) across the populations of each country (i.e., Di,t =3

For simplification, we do not include an externality from the human capital stock of the child’s
parents in the education technology. This assumption does not qualitatively affect our results.

9

N
N
S
S
S
(cN
i,t + ei,t )L + (ci,t + ei,t )L ), we derive the world demands for the primary good andthe industrial good as

D1,t = φ1 It ,D2,t =

φ 2 It,pt(6)where It = wtN LN + wtS LS is world income, and we express the relative expenditure
ratio for good j using φj = δj + σβj . Free trade ensures that all households face thesame product prices, regardless of their country of residence. Given the parameterassumptions for the education technology (β1 + β2 = 1) and household preferences(δ1 + δ2 + σ = 1), the relative expenditure ratios sum to one; that is, φ1 + φ2 = 1.
i
The total output of sector j in country i is Xj,t= xij,t Lij,t = Aj (hit )αj Lij,t , where

Lij,t denotes sector-level employment. Our Ricardian framework therefore features thestandard property that sector output is a linear function of labor employed. National
labor market clearing requires Li = Li1,t + Li2,t .

3

Closed Economy

In order to provide a benchmark, we first consider the autarky equilibrium. Here we
set the superscript as i = A to facilitate a comparison with the open economy model
introduced in later sections. The autarky population is normalized to one (LA = 1).
In each sector, free market entry and exit drives profit to zero. Noting that theA AAA αj A
profit of a representative firm in sector j is πj,t
= pAj,t Aj (ht ) Lj,t − wt Lj,t , we set

profit equal to zero to obtain the equilibrium wage rate:
α1
A α2
wtA = A1 (hA= pAt )t A2 (ht ) .

(7)

Subsequently, the zero profit conditions determine the relative price of the industrial

10good aspAt =A1 A α1 −α2
(h )
.
A2 t

(8)

The greater human capital elasticity of output in the industrial sector (i.e., α2 > α1 )ensures that the relative price falls with a rise in the stock of human capital.
A
Equating the product demands (6) with the sector supplies (Xj,t), we derive themarket clearing conditions for the primary good 1 and the industrial good 2:
φ2 wtA
α2 A= A2 (hAt ) L2,t ,
pA
t

α1 A
φ1 wtA = A1 (hAt ) L1,t ,(9)A
where we have used It = wtA LA
t = wt . Together (8) and (9) give the ratio of laborAemployed in sectors 1 and 2 as LA
2,t /L1,t = φ2 /φ1 . In our framework, the Cobb-Douglas

formulation for household preferences naturally leads to fixed expenditure shares foreach good, yielding constant labor employment shares for each sector.
The dynamics of the closed economy are described by combining the education
technology (2), household utility (3), demands (5), the relative price (8), and thewage rate (7) to obtain the following differential equations for the evolutions of thestock of human capital, the relative price and indirect utility:4
β1 β2
A ω
hA
t+1 = A1 A2 (ht ) ,1−α2 α1 −1pA
A2pA
t+1 = A1t

(10.1)
ω

,

A
Ut+1
= (δ1δ1 δ2δ2 )(1−ω) Aη11 Aη22 (UtA )ω ,(10.2)(10.3)

where we have set B ≡ σ −1 β1−β1 β2−β2 , η1 ≡ α2 β1 +(1−α2 )φ1 , and η2 ≡ α1 β2 +(1−α1 )φ2
to simplify notation, and ω ≡ α1 β1 + α2 β2 describes the efficiency of human capital4

We can also obtain a differential equation to describe wage rate dynamics using (7), (8), andA(10.1): wt+1
= (Aβ1 1 Aβ2 2 )α1 (wtA )ω .

11

accumulation.5 Note that Aj > 1 allows for the possibility that the economy converges
to a long-run equilibrium with a positive rate of economic growth.
Referencing (10.1), the dynamics of the system depend on whether the efficiencyof human capital accumulation (ω) is greater than, less than, or equal to unity. We
briefly discuss each of these cases. First, for ω > 1, the system is unstable and does
not converge to a long-run equilibrium. Henceforth, we exclude this case from theanalysis. Second, for ω < 1, the system converges to the following long-run values forthe stock of human capital, the relative price, and indirect utility:
1

 1−ω
,
hA∗ = Aβ1 1 Aβ2 2(11.1)

 1
2
pA∗ = A1−αAα2 1 −1 1−ω ,1
1∗

U A = δ1δ1 δ2δ2 (Aη11 Aη22 ) 1−ω .

(11.2)(11.3)

From (8), we find that if the stock of human capital is increasing over time (ht < hA∗ ),then pt > pA∗ and the relative price of the industrial good falls over time. Intuitively,
the larger human capital elasticity of output in the industrial sector (α2 > α1 ) causes
labor productivity to improve and supply to expand in the industrial sector at fasterrates than in the primary sector. Consequently, as the expenditure shares of each good
are fixed by our assumption of Cobb-Douglas preferences, the relative price decreases
gradually towards its steady-state value.
In the third case, for a unitary efficiency of human capital accumulation (ω = 1),
the long-run equilibrium features endogenous growth with the per-capita stock ofhuman capital growing at a constant rate. Note that ω ≡ α1 β1 + α2 β2 = 1 requiresthat the output of the industrial sector be human capital elastic (α2 > 1), under ourparameter restriction β1 + β2 = 1. Accordingly, the long-run rates of change in the5

The efficiency of human capital accumulation (ω) is calculated as the inner product of vectors(α1 , α2 ) and (β1 , β2 ) and describes the process through which human capital is employed in the
production of the primary and industrial goods in period t, with a share of the output of these goods
subsequently invested in the formulation of human capital for period t + 1.

12

stock of human capital, the relative price, and indirect utility are derived ashA
t+1= Aβ1 1 Aβ2 2 ,
hA
t
A
pt+1
2
= A1−αAα2 1 −1 ,
1
A
pt
A
Ut+1
= Aη11 Aη22 .
UtA(12.1)(12.2)(12.3)In this paper, we have assumed that ω ≤ 1 to guarantee convergence to a stable longrun equilibrium that potentially exhibits long-run growth. Then, with endogenous
growth, the relative price declines at a constant rate.
The following proposition summarizes the results for the autarky equilibrium.
Proposition 1 In a closed economy, the efficiency of human capital accumulationω ≡ α1 β1 + α2 β2 determines the dynamics of each variable. (1) For ω < 1, the stockof human capital, the relative price, and the indirect utility of households respectively
converge to the steady-state values hA∗ , pA∗ , and U A∗ . (2) For ω = 1, the economy exhibits endogenous growth with the stock of human capital growing at the rate
β1 β2
1−α2 α1 −1
A
A
A
A2
<
hA
t+1 /ht = A1 A2 > 1, the relative price falling at the rate pt+1 /pt = A1
A1, and indirect utility improving at the rate Ut+1
/UtA = Aη11 Aη22 > 1.

Because the closed economy must produce both goods, an increase in the stockof human capital improves labor productivity and expands the output of each sector,
but at different rates depending on the human capital elasticity of output in eachsector (αj ). Following from the education technology (2), the contribution of eachgood to human capital formation is regulated by the input intensity of each good in
education (βj ). As a result, in our framework the dynamics of human capital are
determined not by the sector-specific human capital elasticities of output, but rather
by the weighted-average of the human capital elasticity of output across sectors as
captured by the efficiency of human capital accumulation (ω).
13

4Open Economy

We now turn to a study of the trade patterns that arise in an open economy wheneach country imports and exports according to its pattern of comparative advantage.
The following subsections characterize the open economy equilibrium and then study
the model dynamics that arise under decreasing and constant returns.
4.1

Market Equilibrium

Referencing the production technology (1), the unit production cost in sector j is
wti /(Aj (hit )αj ) for a given stock of human capital in country i. Thus, presuming thatS
the North has a larger stock of human capital than the South (hNt > ht ), a cross-country comparison of the relative unit production costs between the primary andindustrial sectors yields the following inequality to describe the pattern of comparativeadvantage:
α1A1 (hSt )α1A1 (hNt )<,α2A2 (hNA2 (hSt )α2
t )

(13)

where we continue to assume that α2 > α1 . From this expression, we find that thehuman capital elasticity of output transforms the North-South human capital gap intoa pattern of comparative advantage. Specifically, as the North has a relatively low
production cost in the industrial sector, and the South has a relative low production
cost in the primary sector, the North has a comparative advantage in the industrial
good and the South has a comparative advantage in the primary good.
The equilibrium trade pattern is characterized by one of three regimes, contingent upon the relative price. First, a pattern of incomplete specialization occurs
α1
N α2
S α1
S α2
in Regime (i) for pt = (A1 (hNt ) )/(A2 (ht ) ) < (A1 (ht ) )/(A2 (ht ) ), with theSouth specializing in the production of the primary good and the North producing both goods. Second, a pattern of complete specialization arises in Regime (ii)
14

α1
N α2
S α1
S α2
for (A1 (hNt ) )/(A2 (ht ) ) < pt < (A1 (ht ) )/(A2 (ht ) ), with the South produc-

ing the primary good and the North producing the industrial good. Third, a patα1
N α2
tern of incomplete specialization occurs in Regime (iii) for (A1 (hNt ) )/(A2 (ht ) ) <(A1 (hSt )α1 )/(A2 (hSt )α2 ) = pt , with the North specializing in the production of the industrial good and the South producing both goods. Hereafter, we indicate variables
associated with each trade regime using the superscripts (i), (ii), and (iii).
Next, we consider equilibrium in the goods markets for each regime. For example,
in Regime (i), with both the North and the South producing the primary good, we set
(i)

S,(i)

N,(i)

supply equal to demand to obtain D1,t = X1,t + X1,t . And, because only the North
(i)

N,(i)

produces the industrial good, equating supply with demand yields D2,t = X2,t .
Utilizing similar conditions for each regime, we match supply (1) with demand (6) torewrite the market clearing conditions for each of the three equilibrium regimes as

Regime (i) −→Regime (ii) −→Regime (iii) −→
α1 N,(i)φ1 It(i) = A1 (hSt )α1 LS + A1 (hNt ) L1,t(14.1)(i)It
α2 N,(i) φ2(i)= A2 (hNt ) L2,tpt
φ1 It(ii) = A1 (hSt )α1 LS(14.2)

(ii)

It

α2 N
 φ2(ii)= A2 (hN
t ) L
pt



φ1 It(iii) = A1 (hSt )α1 LS,(iii)1,t(14.3)(iii)
S,(iii)Itα2 N φ2(iii)= A2 (hSt )α2 L2,t + A2 (hN
t ) L

pt

The zero-profit conditions are combined to obtain the equilibrium wage rates associated with each regime. In Regime (i), for example, the profits of firms operS,(i)ating in the primary sector in the South are π1,tS,(i)= A1 (hSt )α1 − wt

= 0, andthe profits of firms operating in the primary and industrial sectors in the North are
N,(i)

π1,t

N,(i)

α1
= A1 (hN
− wt
t )

N,(i)

= 0 and π2,t

(i)

N,(i)

α2
= pt A2 (hN
− wt
t )

= 0. Using simi-

lar zero profit conditions for each regime, we calculate the equilibrium wages for the15

North and South as follows:

Regime (i) −→Regime (ii) −→Regime (iii) −→



(i)

α1
α2
wtN,(i) = A1 (hN= pt A2 (hN
t )
t )



wtS,(i) = A1 (hSt )α1w N,(ii) = p(ii) A (hN )α2
t

2

t

t



wtS,(ii) = A1 (hSt )α1α2wtN,(iii) = pt(iii) A2 (hNt )(15.1)(15.2)(15.3)


wtS,(iii) = A1 (hSt )α1 = pt(iii) A2 (hSt )α2
The equilibrium wage levels (15.1) and (15.3) establish the relative price levels
associated with the patterns of incomplete specialization that arise in Regimes (i) and(iii). And, the product market equilibrium conditions (14.2) determine the relative
price level associated with the pattern of complete specialization that coincides with
Regime (ii). The regime-specific relative prices are

Regime (i) −→Regime (ii) −→Regime (iii) −→

pt = A12 (htN )α2



pt



pt

A (hN )α1(i)

(16.1)t

A (hS )α1(ii)

= Φ A21(hNt )α2(16.2)A (hS )α1(16.3)

t(iii)= A12 (htS )α2t

Referencing (14.2), (15.2) and (16.2), we find that the relative price associatedwith the pattern of complete specialization that arises in Regime (ii) depends closelyon the ratio of the expenditures faced by workers in the North and the South:
φ 2 ItΦ≡ NL



φ 1 It,LS(17)where φ1 and φ2 capture the shares of world income allocated to expenditure on goods

16Figure 2: Relative Supply for the Industrial Goodpt
Regime (iii)bα1A1 (hSt )α2A2 (hS)tRegime (ii)α1A1 (hNt )α2A2 (hN)taRegime (i)

0

x2

α2 N
A2 (hNt ) L

1 and 2, with the North producing the industrial good, and the South producing the
primary good. The numerator describes the demand per northern worker, and thedenominator the demand per southern worker. As such, we refer to Φ as the effective
demand gap in Regime (ii). It is both time invariant and determined independentlyof the human capital gap. Intuitively, from (16.2), we observe that a larger effective
demand gap corresponds to a higher relative price. More concretely, an increase inΦ indicates a rise in the expenditure share faced by northern workers, leading to ahigher relative price, as a fixed northern labor force is employed to meet the expansion
in demand for the industrial good. Importantly, from (15.2) and (16.2), the effective
N,(ii)

demand gap equals the income gap (Φ = wt

S,(ii)/wt), providing a measure of cross-country income inequality in Regime (ii).6
Following from the Ricardian structure of our model, the balance of supply anddemand in each of the goods markets determines the equilibrium pattern of trade. Figure 2 provides an illustration of the kinked supply curve for the industrial good. Theintersection of the supply curve with the demand curve (6) determines the equilibrium
6

The labor market equilibrium associated with incomplete specialization is derived in a similarmanner. From (14.1) and (16.1), in Regime (i) the number of northern workers in the primary sector
N,(i)α1is L1,t = φ1 LN − φ2 LS (hSt /hNt ) . And, from (14.3) and (16.3) in Regime (iii) the number ofS,(iii)southern workers employed in the industrial sector is L2,t

17

S α2
= φ2 LS − φ1 LN (hNt /ht ) .

Figure 3: Regime Thresholds for Φ > 1
Regime (iii)1

Regime (ii)Φ

1/α2

Regime (i)
Φ

gt

1/α1

S
regime. Defining the cross-country human capital gap as gt ≡ hN
t /ht , we obtain thefollowing proposition to characterize the equilibrium trade pattern through a comparison of the human capital gap and the income gap between the North and theSouth:
Proposition 2 The equilibrium trade pattern is characterized by complete specialization under Regime (ii) for Φ1/α2 < gt < Φ1/α1 , but features incomplete specialization
under Regime (i) for Φ1/α1 < gt or Regime (iii) for gt < Φ1/α2 .
Proof: See the Appendix.
Figure 3 provides an illustration of the results of Proposition 2. The larger thehuman capital gap (g) between the North and the South, the greater the differencein labor productivity. Thus, for larger values of g, the North has a higher relative
productivity in the primary sector, and is more likely to produce both goods within
Regime (i). Alternatively, for smaller values of g, the North has a lower relative
productivity, and the South is more likely to produce both goods within Regime (iii).
Note that because 1 ≤ g, Regime (i) is the only possible equilibrium outcome if Φ ≤ 1.
More specifically, referencing (17) we find that the effective demand gap (Φ) is small
when the demand for for the industrial good, which the North has a comparativeadvantage in, is low even though the labor supply of the North is large. In this case,a pattern of complete specialization does not occur as the North continues to produce
both goods.
The equilibrium trade regime governs the evolution of state variables in each country. Combining the education technology (2) and household demands (5), with theequilibrium wage rates (15.1-15.3) and relative prices (16.1-16.3), we obtain the fol18

lowing differential equations to describe human capital accumulation in the North andSouth and the relative price dynamics in each trade regime:


N,(i)

ω

ht+1 = Aβ1 1 Aβ2 2 (hN

t )



S,(i)
ω−α1
Regime (i) −→ ht+1
= Aβ1 1 Aβ2 2 (hSt )α1 (hN
t )






1−α2 α1 −1 (i) ωp(i)A2 (pt )t+1 = A1N,(ii)
α2 β2

ht+1 = Aβ1 1 Aβ2 2 Φβ1 (hSt )α1 β1 (hN

t )



S,(ii)
α2 β2Regime (ii) −→ ht+1
= Aβ1 1 Aβ2 2 Φ−β2 (hSt )α1 β1 (hN
t )






1−α2 α1 −1 (α1 −1)(α2 −1) (ii) ωp(ii)A2 Φ(pt )t+1 = A1N,(iii)α2ht+1 = Aβ1 1 Aβ2 2 (hSt )ω−α2 (hN

t )



Regime (iii) −→ hS,(iii)= Aβ1 1 Aβ2 2 (hSt )ω
t+1





(iii)
(iii)

2
pt+1
= A1−αAα2 1 −1 (pt )ω
1(18.1)(18.2)(18.3)

Noting the results of Proposition 2, we then use the the cross-country gap in humancapital (g) to reduce the dynamic system as follows:(gt )α1 ,gt+1 = Φ,






(gt )α2 ,if Φ1/α1 < gt (Regime (i))
if Φ1/α2 < gt < Φ1/α1 (Regime (ii))(19)

if gt < Φ1/α2 (Regime (iii))

Thus, we find that the balance between the human capital gap (g) and the effective
demand gap (Φ) determines the system dynamics.
Figure 4 illustrates the evolution of the human capital gap when the effective
demand gap is greater than one (Φ > 1). Following from (19), the economy always
lies on the bold curve, with the two kinks delineating the trade regimes that arise fordifferent levels of the human capital gap. Crucially, the human capital elasticity of19

Figure 4: Dynamics of Human Capital Accumulation
gt+1

gt+1
45◦ line

gtα2

45◦ line

gtα2
gtα1

gtα1
Φ

1

Φ1/α2 Φ1/α1

Φ

gt

1

(a) α1 < α2 < 1

Φ1/α2

Φ1/α1

gt

(b) α1 < 1 < α2output (αj ) determines the dynamics of the human capital gap. Figure 4(a) presents
the case for which α1 < α2 < 1, with the human capital gap vanishing as the economy
moves to a long-run equilibrium at g ∗ = 1. In this case, the long-run trade patternis characterized by Regime (iii). Alternatively, Figure 4(b) depicts the case for which
α1 < 1 < α2 . The economy converges to g ∗ = Φ > 1 in finite time, and the long-run
equilibrium trade pattern is described by Regime (ii), exhibiting a persistent income
gap.7
Importantly, a fall in the effective demand gap (Φ) expands the range over which
long-run trade patterns are described by Regime (i). Specifically, a decrease in theeffective demand gap shifts the horizontal line gt+1 = Φ downwards, contracting themarket for the industrial good relative to the market for the primary good. As a result,the North is less likely to specialize completely in the production of the industrial good
given the current human capital gap (gt ) and is more likely to produce both goods.
If the effective demand gap falls below one (Φ < 1), from Proposition 2 we find that
Regime (i) arises in equilibrium for all values of gt . In this case, the human capital7

In the last case, for which 1 < α1 < α2 , the human capital gap diverges. Because we haveassumed that ω ≤ 1 and β1 + β2 = 1, however, this final case is eliminated from the analysis.

20

gap always vanishes because α1 < 1. These results are summarized in the following
proposition.
Proposition 3 (1) When Φ ≤ 1, Regime (i) arises in the long-run with the Northproducing both goods and the human capital gap converging to one (g ∗ = 1). (2) WhenΦ > 1 and α1 < α2 < 1, Regime (iii) occurs in the long-run with the South producing
both goods and the human capital gap converges to one (g ∗ = 1). (3) When Φ > 1
and α1 < 1 < α2 , Regime (ii) arises in the long-run, with complete specialization in
production, and the human capital gap converges to the income gap (g ∗ = Φ).
Thus, we find that the human capital elasticity of output in the industrial sector(α2 ) and the effective demand gap (Φ) not only govern the static pattern of trade,from Proposition 2, but also regulate the evolution of the human capital gap (gt ) anddetermine the long-run trade regime. In addition, Proposition 3 implies that longrun trade patterns are determined independently of the efficiency of human capital
accumulation (ω ≡ α1 β1 + α2 β2 ). In the following sections, however, we show thatω plays a critical role in the economic development of both countries: Subsection 4.2
studies the case of decreasing returns (ω < 1) and Subsection 4.3 considers the caseof constant returns (ω = 1).
4.2

Dynamics of Decreasing Returns

The path of economic development depends on the equilibrium trade regime with
Proposition 3 summarizing the evolution of trade patterns. In this subsection, westudy the development paths associated with incomplete and complete specialization
under decreasing returns (ω < 1). In addition, we link the key features of our dynamic
framework with the Prebisch-Singer hypothesis.
When the equilibrium trade regime features incomplete specialization, with one
country producing both goods (i.e., Regimes (i) and (iii)), we find that it is the human21capital accumulation of the incompletely specialized country that drives the dynamics
of the system, as the economy converges to a long-run equilibrium with equality inboth income and stocks of human capital at their autarky levels. Consider, for example, Regime (i) in which the North continues to produce both goods and the South
specializes in the primary good. Referencing (18.1), we observe that northern humancapital accumulation proceeds independently of the South, ensuring that human capital stocks are driven to their autarky levels in both countries as the human capital gap
vanishes; that is, hS,(i)∗ = hN,(i)∗ = hA∗ = (Aβ1 1 Aβ2 2 )1/(1−ω) . Moreover, the open economy dynamics for the relative price (18.1) are identical to those for autarky (10.2),
1/(1−ω)2driving the relative price to its autarky value, p(i)∗ = pA∗ = A1−αAα2 1 −1, and1guaranteeing that household utility converges to its autarky level in both countries
in the steady state: U N,(i)∗ = U S,(i)∗ = U A∗ = δ1δ1 δ2δ2 (Aη11 Aη22 )1/(1−ω) . Similar results
are obtained for Regime (iii), allowing us to generally conclude that the pattern ofcomparative advantage and the gains from trade disappear in the long run when theequilibrium trade regime features incomplete specialization.
In contrast, when the equilibrium trade regime features a pattern of complete specialization (Regime (ii)), we find that development paths depend on a common rateof human capital accumulation for the North and the South, generating a constantN,(ii)

human capital gap that is equal to the cross-country income gap (htS,(ii)= Φht).

Returning to (18.2), the northern and southern stocks of human capital can be solved
for as hN,(ii)∗ = (Φ(1−α1 )β1 Aβ1 1 Aβ2 2 )1/(1−ω) and hS,(ii)∗ = (Φ(α2 −1)β2 Aβ1 1 Aβ2 2 )1/(1−ω) . Accordingly, a straightforward comparison with the autarky equilibrium shows that completely specialized trade leads to increases in human capital stocks: hA∗ < hS,(ii)∗ <hN,(ii)∗ . In addition, comparing the open economy price dynamics (18.2) with those
for autarky (10.2), we establish that the relative price declines at a faster rate with
(ii)

(ii)

completely specialized trade; that is, pt+1 /pt

22A(α1 −1)(α2 −1)< pA< 1 given
t+1 /pt since Φthat α1 < 1 < α2 is required for Regime (ii). The open economy relative price,p(ii)∗ = Φ(α1 −1)(α2 −1)/(1−ω) pA∗ < pA∗ ,(20)

then settles at a level that is lower than that for autarky.
The increase in human capital and the decrease in the relative price have important
implications for the gains from trade. Substituting household demands (5), the relative
price (16.2) and the income gap (17) into utility (3) yields
ω
N,(ii),Aη11 Aη22 Φ(1−α1 )[(1−α2 )δ1 +{α2 (1−σ)+σ}β1 ] Ut
1−ω η1 η2 (α2 −1)[(1−α1 )δ2 +{α1 (1−σ)+σ}β2 ]  S,(ii) ωS,(ii)
Ut+1 = δ1δ1 δ2δ2
A1 A2 Φ,Ut
N,(ii)

Ut+1= δ1δ1 δ2δ2

1−ω(21.1)(21.2)

where α2 > 1 for Regime (ii). Figure 5 plots the northern and southern welfare paths
associated Regime (ii). The autarky welfare (10.3) path is also illustrated for purposesof comparison. The steady-state welfare levels are obtain as
U N,(ii)∗ = U A∗ Φ
U S,(ii)∗ = U A∗ Φ

(1−α1 )[(1−α2 )δ1 +{α2 (1−σ)+σ}β1 ]1−ω
(α2 −1)[(1−α1 )δ2 +{α1 (1−σ)+σ}β2 ]1−ω

,(22.1).

(22.2)

Thus, the key observation is that completely specialized trade leads to welfare improvements in both countries along the development path, with the North experiencing thelargest increase in utility: U A∗ < U S,(ii)∗ < U N,(ii)∗ .
The results for the gains from trade that arise under decreasing returns (ω < 1)
are summarized in the following proposition.
Proposition 4 Suppose ω < 1. (1) When the long-run trade regime features incomplete specialization, human capital stocks, the relative price, and welfare all converge
to their autarky levels. (2) When the long-run trade regime is characterized by complete specialization, trade increases the northern and southern stocks of human capital,
23

Figure 5: Dynamics of National Welfare with Complete Specialization
Ut+1
45◦ line
UtN
UtS
UtA

1

U A∗

U S,(ii)∗ U N,(ii)∗Utlowers the relative price, and improves welfare in both countries.
Table 1 summarizes the long-run trade regimes and gains from trade that arisefor different values of the effective demand gap and the human capital elasticity ofoutput in the industrial sector under decreasing returns (ω < 1). As discussed earlier,if either the effective demand gap is less than unity (Φ < 1), and or the humancapital elasticity of output is inelastic in both sectors (α1 < α2 < 1), the stock ofhuman capital converges to the autarky level in each country. Then, the two countries
become identical as the cross-country gap in human capital vanishes and income levels
converge, leaving no role for international trade. In other words, the degeneration of
the pattern of comparative advantage eliminates the gains from trade.
Alternatively, if the effective demand gap exceeds unity (Φ ≥ 1), and the industrial
sector is human capital elastic (α1 < 1 < α2 ), the dynamics of the open economy
differ markedly from those of the autarky economy, with the economy converging to apattern of complete specialization, sustained by a persistent cross-country gap in thestock of human capital. Importantly, even though specialization in the production of
the human capital elastic industrial good supports a faster rate of human accumulationin the North, the human-capital gap does not diverge. Specifically, under a pattern24

Table 1: Gains from Trade under Decreasing Returns (ω < 1)
α1 < α2 < 1

α1 < 1 < α2

There is no international trade: U N ∗ = U S∗ = U A∗ .

Closed EconomyΦ≤1Open Economy
Φ>1Regime (i) with incomplete trade specialization and no gains
from trade: U N,(i)∗ = U S,(i)∗ = U A∗
Regime (iii) with incomplete Regime (ii) with complete
Bothtrade specialization and no trade specialization.
countries gain from tradegains from trade:
U N,(ii)∗ > U S,(ii)∗ > U A∗ .
U N,(iii)∗ = U S,(iii)∗ = U A∗ .

of complete specialization the improvement in industrial labor productivity causes
the international price of the industrial good to fall over time following (18.2). The
continuous deterioration in the northern terms of trade slows the rate of human capital
accumulation, pushing the northern stock of human capital to its steady-state level.
Simultaneously, the continuous improvement in the southern terms of trade facilitates
greater imports of the industrial good at a lower price, implying that the southernstock of human capital expands with international trade. The North ultimately has
a greater stock of human capital in the steady state, however, because it specializes
in the sector with the greater output elasticity of human capital.
As we have seen, an effective demand gap (Φ > 1) is required for the economy toconverge to a long-run trade pattern characterized by complete specialization. The
North-South income gap (17) is determined by population sizes and the shares of world
income allocated to expenditure on each good, rather than national stocks of humancapital. These results contrast with van de Klundert and Smulders (1996) where theNorth-South income gap is driven by technological spillovers and learning by doing.
In our framework it is the mechanism through which the increasing marginal productof human capital in the northern industrial sector drives adjustments in the terms oftrade that explains the long-run persistence of a North-South income gap.
The North-South income gap (Φ) also plays an important role in determining the25cross-country gap in steady-state welfare. From (18.2), (22.1), and (22.2), we have
(α2 −1)[(1−α1 )δ2 +{α1 (1−σ)+σ}β2 ]U S,(ii)∗
1−ω
> 1,
=
Φ
U A∗

U N,(ii)∗= Φ > 1.
U S,(ii)∗(23)

These expressions demonstrate that although a larger North-South income gap leads
to a greater steady-state international welfare gap, it also raises the gains from trade
enjoyed by the South. In our model, the only fundamental difference between thetwo countries are the population sizes and the world demands that the two countries
face when complete specialization is achieved. Reconsidering (17), the income gap(Φ) may therefore also be perceived as measuring the degree of heterogeneity between
the North and the South. The greater the heterogeneity between the two countries,
the larger the income gap between the North and the South, but also the larger thesouthern gains from trade.
We now assess the implications of our framework for the Prebisch-Singer hypothesis, which suggests that the southern terms of trade deteriorate with economic development. Specifically, we complete a comparative static analysis using the demandparameters of the utility function (3) to clarify the effects of a fall in the expenditure
share of the primary good. Maintaining a constant value for σ, consider a decreasein δ1 and an increase in δ2 that satisfies δ1 + δ2 + σ = 1. Since international trade
only affects the long-run equilibrium in Regime (ii), we restrict our analysis to thecase where α1 < 1 < α2 and Φ > 1. First, recalling that φj = δj + σβj , from (17) wefind that a fall in the expenditure share of the primary good (δ1 ) raises the effective
demand gap; that is, dΦ/dδ1 < 0. Thus, from (17) and (19) the decline in the shareof spending on the primary good leads to a larger human capital gap and a greater
income gap between the North and South.
To analyze the effects of changes in expenditure shares on the terms of trade,
we take the total derivative of the equilibrium relative price (16.2) associated with26

Regime (ii):1 dp(ii)∗1 dΦ
α1 dhS,(ii)∗α2 dhN,(ii)∗=+−,p(ii)∗ dδ1
Φ dδ1 hS,(ii)∗ dδ1
hN,(ii)∗ dδ1
where the terms on the righthand side describe the effects of changes in the effective
demand gap (Φ), the southern stock of human capital (hS,(ii)∗ ), and the northern stockof human capital (hN,(ii)∗ ). First, consider the short-run impacts of a decrease in δ1 on
the South’s terms of trade, with the human capital stocks fixed at their current levels
in each country. Then, we have dp(ii)∗ /dδ1 = (p(ii)∗ /Φ)(dΦ/dδ1 ) < 0. In other words,
the decline in the share of spending on primary goods raises the effective demand
for the industrial good, improving the northern terms of trade, and deteriorating thesouthern terms of trade. In the short run, our framework therefore supports the
Prebisch-Singer hypothesis.
Next, allowing human capital stocks to adjust, the long-run terms of trade effectis now given by dp(ii)∗ /dδ1 = ((α1 − 1)(α2 − 1)/(1 − ω))(p(ii)∗ /Φ)(dΦ/dδ1 ) > 0, with
α1 < 1 < α2 and ω < 1. Consider, once again, a decrease in δ1 . Although the increasein the effective demand gap (Φ) directly improves the northern terms of trade, italso operates indirectly on the terms of trade through adjustment in human capitalstocks. Importantly, the lower share of spending on the primary good expands thehuman capital gap between the North and South and the faster rate of productivity
growth in the North lowers the relative price of the industrial good deteriorating thenorthern terms of trade. Overall, the effect of adjustments in human capital stocks
dominates, and a lower share of spending on the primary good improves the southernterms of trade in the long-run. Thus, in our framework the premise that primary
goods have low Engle coefficients offers support for the Prebisch-Singer hypothesis in
the short run, but not in the long run, once adjustments in human capital stocks havebeen taken into account.

27

Lastly, we consider the effect of changes in expenditure shares on economic welfare.
From (23), a lower share of spending on the primary good (δ1 ) increases the gap in
economic welfare between North and South. At the same time, the South experiences
a welfare improvement through the rise in the southern terms of trade. We summarize
the results associated with expenditure share adjustments in the following proposition.
Proposition 5 Suppose ω < 1. In the case of complete specialization of production, adecrease in the share of spending on the primary good expands the cross-country gaps
in human capital, income, and economic welfare. In addition, the South’s term oftrade deteriorate in the short run. In the long run, however, the decrease in spendingon the primary good raises the southern terms of trade and thus improves the economic
welfare of the South.
A key premise associated with the Prebisch-Singer hypothesis is that the relative
demand for primary goods tends to fall as income rises over time because primary
goods have low Engel coefficients. Investigating this premise through a comparative
static analysis, we show that changes in expenditure shares have opposing short-run
and long-run effects that may provide a basis for further understanding of the generally
mixed results for the Prebisch-Singer hypothesis presented in the empirical literature(Lutz, 1999; Baffes and Etienne, 2016; Di Iorio and Fachin, 2018).
4.3

Dynamics of Constant Returns

In this final subsection, we investigate the development paths associated with constantreturns (ω = 1). To begin, we once again note that under our parameter restriction
β1 + β2 = 1, a unitary efficiency of human capital accumulation (ω ≡ α1 β1 + α2 β2 = 1)
requires that the output of the industrial sector be human capital elastic (α2 > 1).
Consequently, returning to Proposition 3, we find that Regime (i) and Regime (ii) are
the only long-run trade regimes that are consistent with endogenous growth. Then,28

it is the effective demand gap (Φ) that determines which of these trade regimes isrealized in long-run equilibrium.
In the first case for Φ ≤ 1, the long-run trade regime features incomplete specialization, with the South specializing in the primary sector and the North producing both goods. From (18.1), the northern stock of human capital accumulates
N,(i)

N,(i)

β1 β2
A
= hA
t+1 /ht = A1 A2 . Similarly, refer-

at the constant autarky rate ht+1 /ht

encing (18.1) and (19), the evolution of human capital in the South is rewritten as
S,(i)

S,(i)

ht+1 /ht(i)(i)= Aβ1 1 Aβ2 2 (gt )1−α1 . Because gt≥ 1, we find that human capital ac(i)cumulates at a faster rate in the South than in the North, driving gt

to unity.

Therefore, southern human capital accumulation also approaches the autarky rate in
S,(i)

S,(i)

the long run: limt→∞ ht+1 /ht

A= hA
t+1 /ht . In addition, from (18.1), the relative
(i)

(i)

2
price of the industrial good declines at the autarky rate Pt+1 /Pt = A1−αAα2 1 −1 < 1.
1

Combining the dynamics outlined above, we establish that the rate of growth in indirect utility approaches the autarky rate in each country in the long run; that is,
S,(i)

S,(i)

limt→∞ Ut+1 /Ut

N,(i)

N,(i)

= Ut+1 /Ut

A= Ut+1/UtA = Aη11 Aη22 .

In the second case for Φ > 1, the long-run trade regime is characterized by(ii)complete specialization. Incorporating (18.2) with gt= Φ, we find that opening

up to trade accelerates the rate of human capital accumulation in both countries
N,(ii)

N,(ii)

ht+1 /ht

S,(ii)

S,(ii)= ht+1 /ht

A= Aβ1 1 Aβ2 2 Φ(1−α1 )β1 > hA
t+1 /ht , where we have refer-

enced (12.1). The relative price of the industrial good also decreases at a faster
(ii)

(ii)

rate: pt+1 /pt

1−α2 α1 −1
A= A1−α2 A2α1 −1 Φ(α1 −1)(α2 −1) < pA
A2
< 1. Referring to
t+1 /pt = A1

(22.1) and (22.2), we use ω = 1 to obtain the growth rate for indirect utility:
S,(ii)

N,(ii)

Ut+1

N,(ii)
Ut=

Ut+1

S,(ii)Ut= Aη11 Aη22 Φ(α2 −1)β2 (α1 φ1 +α2 φ2 ) .
N,(ii)

Therefore, maintaining the welfare gap Ut

S,(ii)/Ut(24)= Φ > 1, welfare continues toimprove in both countries at an increased rate with international trade. These results

29are summarized in the following proposition.
Proposition 6 Suppose ω = 1. (1) When the long-run trade regime features incomplete specialization, human capital accumulation, the decline in the relative price of
the industrial good, and the improvement of welfare all converge to their autarky rates.
(2) When the long-run trade regime is characterized by complete specialization, tradeleads to a faster rate of decline in the relative price of the industrial good, and greater
rates of growth in human capital and indirect utility in both countries.
Lastly, we briefly consider the implications of Proposition 6 for the relationshipbetween trade and the North-South income gap. In the autarky economy, underconstant returns (ω = 1) both countries realize equal rates of endogenous humancapital accumulation. Consequently, the higher initial northern stock of human capitalguarantees that the South can never catch up with the North. Opening up to trade,
if the long-run trade regime features complete specialization, the rate of growth insouthern welfare increases with trade, but the cross-country gap in income and welfare
persists. If the long-run trade regime is characterized by incomplete specialization,
however, a comparative advantage in the primary good facilitates a faster rate ofhuman capital accumulation that allows the South to catch up with the North in thelong run.

5

Conclusion

In the traditional Ricardian framework, each country specializes in the productionof the good for which it has a comparative advantage, and free trade makes theworld more efficient as all countries benefit from trade. This result is one of themost important conclusions in international economics. Concerns have been raised in
the economic development literature, however, that low-income countries specializingin the export of primary commodities may face deteriorating terms of trade that30

prevent them from catching up with developed economies. Furthermore, developing
countries often face challenges in accumulating the knowledge and technologies thatwould allow them to reduce their dependency on commodity exports. In this paper, westudy a North-South Ricardian model of trade in which human capital accumulation
drives economic growth and heterogeneous human capital elasticities of output acrossindustries play a key role in the evolution of patterns of comparative advantage. Our
novel dynamic Ricardian framework provides a new perspective on the long standing
question regarding the relationship between international trade, development, andincome inequality.
In our framework, the North has a larger initial stock of human capital and maintains a comparative advantage in an industrial sector with a relatively high humancapital elasticity of output, while the South has a comparative advantage in a primary industry with a relatively low human capital elasticity of output. We then findthat, on the one hand, when output is human capital inelastic in both sectors, theeconomic system converges to a pattern of incomplete specialization in which tradedoes not promote human capital accumulation. Although there is no cross-country
income inequality, neither the North or South experiences gains from trade. On theother hand, when the output of the northern industrial sector is human capital elastic,
the economic system converges to a pattern of complete specialization if and only if
the cross-country income gap persists in the long run. In other words, as some development economists have suspected, trade characterized by complete specialization
leads to a permanent gap in human capital and national welfare between the Northand the South.
Nevertheless, with complete specialization the human capital gap between the twocountries does not continue to widen in our framework. The sustained expansionof the northern industrial sector decreases the relative price of the human capitalelastic industrial good on the international market, lowering the price of imports and31generating gains from trade for the South. Stated differently, the southern terms oftrade improve, enhancing human capital accumulation in the South. Consequently,
the increasing returns associated with the northern industrial sector are balancedby the improvement in the southern terms of trade, holding the human capital gap
between the North and South constant. In comparison with the closed economy,
the steady-state economic welfare of the South improves, and the South continues tobenefit from free trade in the long run.
Our results are closely linked with the assumption of homothetic preferences. In
the economic development literature, however, a key theoretical assumption guiding
the Prebish-Singer hypothesis is that the declining share of expenditure on primarycommodities coinciding with the rise in world income has deteriorated the terms oftrade for low-income countries. As this feature of the demand for primary commodities
cannot be derived from a homothetic utility function, we leave the development of amodel with non-homothetic preferences as a topic for future research.

Appendix
This appendix provides a proof of Proposition 2. The intersection of the world supply
and demand curves indicates whether the equilibrium trade pattern is characterized
by Regime (i), (ii), or (iii). In the case where the supply curve is a horizontal line
α2 N
given by X2 = A2 (hNt ) L , the price of the industrial good 2 is determined by (16.2).

Figure 2 depicts the case where the supply curve is kinked at points a and b. On the
one hand, when the supply and demand curves do not intersect above point a, thefollowing inequality is satisfied:
(i)

pt =

α1
A1 (hSt )α1A1 (hN
(ii)
t )
>
p
=
Φ
.
t
α2
α2
A2 (hNA2 (hNt )t )32

This condition is rewritten as Φ1/α1 < gt , with the intersection of the supply anddemand curves positioned in the Regime (i) region of Figure 2. On the other hand,when the supply and demand curves do not intersect below point b, the following
inequality is satisfied:
(ii)

pt

=Φ

A1 (hSt )α1A1 (hSt )α1
(iii)
≥
p
=
Φ
.
t
α2
A2 (hNA2 (hSt )α2t )Similarly, this condition is rewritten as gt < Φ1/α2 , and the intersection of the supply
and demand curves lies in the Regime (iii) region of Figure 2. Finally, the equilibrium
trade pattern is characterized by Regime (ii) when
(i)

pt =

α1
A1 (hNA1 (hSt )α1A1 (hSt )α1
(ii)
(iii)
t )
≤
p
=
Φ
≤
p
=
.
t
t
α2
α2
A2 (hNA2 (hNA2 (hSt )α2t )t )This expression is rewritten as Φ1/α2 < gt < Φ1/α1 . Then, as illustrated in Figure 3,
(i)

(ii)

we have gt > gt

(iii)

> gt

, completing the proof of Proposition 2.

References[1] Baffes, J. and Etienne, X., 2016. Analysing food price trends in the context ofEngel’s Law and the Prebisch-Singer hypothesis. Oxford Economic Papers 68(3),688-713.
[2] Baldwin, R., Martin, P., and Ottaviano, G., 2001. Global income divergence,
trade, and industrialization: the geography of growth take-offs. Journal of Economic Growth 6, 5-37.
[3] Chui, M., Levine, P., Murshed, S.M. and Pearlman, J., 2002. North-South models
of growth and trade, Journal of Economic Surveys 16(2), 123-165.
[4] Devereux, M., 1997. Growth, specialization, and trade liberalization, International Economic Review 38(3), 565-585.
[5] Di Iorio, F. and Fachin, S., 2018. The Prebish-Singer hypothesis in the postcolonial era: Evidence from panel cointegration. Economics Letters 166, 86-89.
[6] Dinopoloulos, E. and Syropoulos, C., 1997. Tariffs and Schumpeterian growth,
Journal of International Economics 42(3-4), 425-452.

33

[7] Estevadeordal, A. and Taylor, A., 2002. A century of missing trade? American
Economic Review 92(1), 383-393.
[8] Glomm, G. and Ravikumar, B., 1992. Public versus private investment in humancapital: Endogenous growth and income inequality. Journal of Political Economy
100(4), 818-834.
[9] Golub, S. and Hsieh, C., 2000. Classical Ricardian theory of comparative advantage revisited, Review of International Economics 8(2), 221-234.
[10] Grossman, G., Helpman, E., 1991. Innovation and Growth in the Global Economy. Cambridge: MIT Press.
[11] Johnson, P. and Pagageorgiou, C., 2020. What remains of cross-country convergence? Journal of Economic Literature 58(1), 129-175.
[12] Krugman, P., 1979. A model of innovation, technology transfer, and the world
distribution of income, Journal of Political Economy 87(2), 253-266.
[13] Lutz, M., 1999. A general test of the Prebisch-Singer hypothesis. Review of Development Economics 3(1), 44-57.
[14] Naito, T., 2012. A Ricardian model of trade and growth with endogenous tradestatus, Journal of International Economics 87(1), 80-88.
[15] O’Rourke, K. and Williamson, J., 1999. The Heckscher-Ohlin model between
1400 and 2000: When it explained factor price convergence, when it did not, andwhy, NBER Working Paper Series 7411.
[16] Prebisch, R., 1950. The Economic Development of Latin America and Its Principal Problems, Late Success, NY: Economic Commission for Latin America.
United Nations. Department of Economic Affairs.
[17] Singer, H., 1950. The distribution of gains between investing and borrowing countries, American Economic Review 40(2), 473-485.
[18] Taylor, S., 1993. ‘Quality ladders’ and Ricardian trade, Journal of International
Economics 34(3-4), 225-243.
[19] Trefler, D., 1993. International factor price differences: Leontief was right! Journal of Political Economy 101(6), 961-987.
[20] Trefler, D., 1995. The case of the missing trade and other mysteries, American
Economic Review 85(5), 1029-1046.
[21] Trefler, D. and Zhu, S., 2000. Beyond the algebra of explanation: HOV for thetechnology age, American Economic Review 90(2), 145-149.

34

[22] UNDP, 2011. Towards Human Resilience: Sustaining MDG Progress in an Ageof Economic Uncertainty, United Nations Development Programme, Bureau forDevelopment Policy, New York.
[23] van de Klundert, T. and Smulders, S., 1996. North-South knowledge spillovers
and competition: Convergence versus divergence, Journal of Development Economics 50(2), 213-232.

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