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Growth or Decline of Comparative Advantage

Alan V. Deardorff

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Growth or Decline of Comparative Advantage

Alan V. Deardorff

The University of Michigan

Paper presented at DEGIT XVII

September 14, 2012, Milan

August 13, 2013

Paper: DEG.docx

ABSTRACT

Growth or Decline of Comparative Advantage

Alan V. Deardorff

The University of Michigan

The question here is whether the dynamic effects of opening to trade will increase or

decrease comparative advantage. When comparative advantage is based on the abundance or

scarcity of something that is costly to acquire, one expects rational behavior to respond to a

change in prices by increasing that abundance or scarcity. To explore this issue in simple

theoretical terms, this paper examines two types of simple growth model – a Solow Model with

proportional saving, and a Ramsey model with optimal saving – to see whether this reasoning is

born out. In the Ramsey model, it is. In the Solow model, on the other hand, results vary, but

this should not be surprising, as the proportional saving assumption does not embody optimizing

behavior. To the extent that we believe that aggregate saving behavior is indeed based on

rational and fully informed optimization, we should therefore expect that the dynamic effects of

trade do indeed operate in the direction of increasing comparative advantage over time.

Keywords: Comparative Advantage Correspondence:

Dynamic trade models

Trade and growth Alan V. Deardorff

Ford School of Public Policy

JEL Subject Code: University of Michigan

F11 Neoclassical Trade Models Ann Arbor, MI 48109-3091

Tel. 734-764-6817

Fax. 734-763-9181

E-mail: alandear@umich.edu

http://www-personal.umich.edu/~alandear/

1

August 13, 2013

Growth or Decline of Comparative Advantage *

Alan V. Deardorff

The University of Michigan

I. Introduction

In this paper I will explore, with simple theoretical models of trade and growth, how

trade liberalization may be expected to alter comparative advantage over time.

Specifically, I will ask how trade liberalization will change the dynamics of economic

growth, and then how these dynamics will alter a country’s comparative advantage (i.e.,

its relative autarky prices) over time. The simple question is: Does trade liberalization

cause comparative advantage to increase or decrease over time?

I was motivated to ask this question for two reasons. The first is that I was invited

to contribute to the DEGIT XVII conference in September 2012 in Milan, with the

specific intent of honoring Bjarne Jensen on the occasion of his 70th

birthday. Bjarne was

one of the original organizers of this series of conferences and his efforts have sustained

it over more than a decade and a half. Bjarne has always been generous to me in both his

invitations to these conferences and in his writings on trade and growth, which both

complimented and complemented my own contributions on that subject. I wanted now

write something along those same lines.

Second, I came across two recent papers that touched on this subject and that

stimulated my interest. The first was by my colleagues Levchenko and Zhang (2011).

* I have benefited from the questions and comments raised by the participants in the DEGIT XVII

conference.

2

They used a large computational model with an Eaton and Kortum (2002) framework to

measure comparative advantages of multiple countries over time. They found that

comparative advantage tended to converge among rich countries and among poor

countries, but that it did not converge between these two groups of countries. I found that

a fascinating result, and unfortunately it is not one that I will be able to produce, as it

would presumably require a model with more countries than I will be able to handle here.

The second paper was co-authored by my former student, Paoli Chang – Chang and

Huang (2012) – and was also presented at the DEGIT conference. This paper presented a

model in which an endogenous education system responds to trade by increasing

comparative advantage based on type of education.

Together with the work that I had done decades ago on trade and growth,1 these

observations suggested a basic insight that I thought might hold across a variety of

theoretical contexts: Suppose that comparative advantage (C-A) is based upon the

relative abundance of something, which might be a factor endowment, a level of

expertise, or anything else that can be acquired through effort. Then liberalizing trade, by

raising the domestic price of the good in which a country has C-A, will also raise the

price of that determinant of C-A and thus the return to acquiring it. As a result, trade will

simulate the acquisition of that determinant of C-A, and hence also increase C-A over

time.

This is the insight that I will look for here in a series of simple models. It should

be noted that, just as the abundance of something can create comparative advantage, its

scarcity can do the opposite: create comparative disadvantage. Therefore, the same

mechanism by which trade may cause comparative advantage to be enhanced over time

1 See for example Deardorff (1973, 1974) and Deardorff and Hanson (1978).

3

will also cause an increase in comparative disadvantage. For example, if a country starts

out capital-scarce and therefore has comparative disadvantage in capital-intensive goods,

one of the effects of trade will be to make capital even more scarce than it was before.

In what follows I will use simple growth models to look for such effects of

opening to trade on C-A. Although I believe that the mechanism would hold for

determinants of C-A other than factor endowments, I will confine my attention to

neoclassical growth models in which only the capital-labor endowment ratio plays any

role. I will look first at the simple growth model of Solow (1956), if necessary extended

as in Deardorff (1973) to two sectors so as to allow for trade. And I will look second at a

model of the same sort, but with saving based on optimization over time, as in Ramsey

(1928).

In each case there are two questions to consider:

1. How will trade alter the path to an unchanged steady state?

2. How will trade alter the steady state itself?

The first of these questions is most interesting without the second, so I will start with that.

That is, I will initially abstract from any effect of trade on the steady state itself, and just

ask how trade alters the path to an unchanged steady state. I will then address the second

question, also in isolation, assuming that a country starts in its initial steady state which is

then altered by opening to trade.

II. Case 1: Autarky Steady State Same as the World

The question can be addressed most simply within the context of the one-sector Solow

model of Figure 1. Country A and the Rest of World, R, share the same aggregate per-

4

worker production function . They also share the same savings

propensity, s, and the same population growth rate, n.2 Therefore, country A’s

investment (equals saving) per unit labor in autarky is the same function as that of the

world, and it shares the same steady-state capital-labor ratio: .

Away from steady state the change in the capital-labor ratio, , is

the vertical distance between the I/L curve and the nk line. Therefore, starting at

below steady state, the economy’s K/L ratio rises over time at a rate proportional to that

distance, in the limit approaching .

Now suppose that country A opens instead to free trade. With its initial capital-

labor ratio different from that of the world, country A will trade and gain from trade, as

shown by the upward shift of the production function to the dashed curve. Since a

fraction of that increase is saved and invested, investment rises and the country’s rate of

growth increases. Since both trade and the gains from trade disappear when country A’s

capital-labor ratio reaches that of the world, there is no change in the steady state. But

the country benefits from trade during the approach to steady state, which occurs more

rapidly than it would have in autarky.

This effect of trade on growth is familiar from Baldwin (1992). Strictly speaking,

as noted by Mazumdar (1994), the effect occurs only if the country is able to import the

investment good, a distinction that cannot be captured in this one sector model. But the

idea that the gains from trade can translate into a faster rate of growth, due to increased

saving and investment, is familiar.

2 For simplicity I assume that capital does not depreciate and that the labor force grows at the same rate as

population.

5

What may not have been stressed, however, is that this means that trade causes

the country’s comparative advantage – which would have disappeared gradually over

time even in autarky – to disappear more rapidly. Although comparative advantage

cannot be read directly from Figure 1, it is implicit in a comparison of the tangent to the y

curves at and . The slope of this tangent is the rental price of capital, its vertical

intercept is the wage, and therefore its horizontal intercept is the wage-rental ratio, w/r,

which determines relative prices of goods. As grows from to , all of

these properties of the tangent converge to those of the rest of world.

All of this yields the following result:

Proposition 1-A: In a Solow Model (i.e., with proportional saving) of a country whose

autarky steady state is the same as that of the world

Comparative advantage exists only out of steady state.

Without trade, comparative advantage will disappear over time, as the steady state

is approached.

With trade

o The country’s rate of growth will increase.

o Comparative advantage will diminish more rapidly over time than under

autarky.

Is this result generally true? No. A small departure from this result occurs if the

country’s comparative advantage is in a good or goods that are used for investment. As

noted by Mazumdar (1994), in that case while trade increases income in units of

consumption goods, it does not increase income in units of investment goods, and

6

proportional saving yields no increase in investment. Therefore the time path of

comparative advantage is the same as it would have been in autarky.

Note however how this Solow Model contrasts with the reasoning embodied in

the basic insight that I discussed in the introduction. In the Solow model, accumulation

of capital is not responsive to any incentive, such as the rate of return to capital. Instead

it just depends on income and the fixed fraction of income that is saved. Therefore it

should not be surprising that trade has no permanent effect on capital accumulation or

comparative advantage.

This suggests therefore that we look at a Ramsey model, in which savings and

investment are responsive to the rate of return on capital, which serves as the incentive to

trade and invest.

In a Ramsey model, savings and therefore capital accumulation respond to the

difference between the rate of return on capital, r, and the sum of the rate of population

growth, n, and the discount rate, , at which consumers discount future consumption. If

), then capital accumulates relative to labor, and vice versa. The implications

under autarky can be seen easily in Figure 1. Let r* be the return to capital in the rest of

the world in steady state, thus the slope of the y curve at : . If country

A has the same production function, same population growth rate, and same discount rate

as the rest of the world, then the fact that the slope of the y curve is greater than r*, means

that it too will accumulate capital up to the point where . Thus the implication

under autarky is the same as in the Solow Model, even though the mechanism is

different.

7

But now consider the opening to trade. We know from the Heckscher-Ohlin

model that the per-labor production function does not just shift up wherever the country’s

capital-labor ratio differs from that of the world; it does so in a particular way. The

implication of Samuelson’s (1948, 1949) Factor-Price-Equalization (FPE) Theorem is

that, as long as the capital-labor ratio is not so different from as to force complete

specialization, country A will have the same factor prices, including r*, as the rest of the

world. Thus the shift of the production function in Country A due to free trade is not as

in Figure 1, but as in Figure 2. That is, the function becomes a straight line for capital-

labor ratios between a lower bound and an upper bound that define the FPE set or

diversification cone.

The implications of trade then depend on how far below the steady-state capital-

labor ratio the country starts. If it starts below , then opening to trade causes it to

specialize completely in the labor-intensive good and to grow toward the FPE set. But

when it reaches the FPE set, growth stops, and it continues indefinitely to specialize in

and export that labor-intensive good.

If on the other hand Country A start with a somewhat larger capital-labor ratio

that is inside the FPE set, then opening to trade changes its return to capital immediately

to the steady-state value r*, and further capital accumulation stops. Thus the country’s

pattern of trade and comparative advantage stops evolving and become frozen at

whatever level it happened to be when trade was opened.

8

To conclude, we have the following:

Proposition 1-B: In a Ramsey Model (i.e., with saving motivated by the return to capital

relative to a discount rate) of a country whose autarky steady state is the same as that of

the world

Without trade, comparative advantage will disappear over time, as the steady state

is approached.

With trade, what happens depends on the initial capital-labor ratio:

o If the country starts below the FPE set, then growth continues until that set

is reached, after which comparative advantage and trade remain fixed over

time.

o If the country starts inside the FPE set, then free trade causes growth to

stop immediately. Comparative advantage and trade become fixed at their

initial values.

III. Case 2: Autarky Steady State Not Same as the World

Now consider the perhaps more interesting case in which the country of interest has a

different autarky steady state than that of the world with which it trades. I will look only

at the case of a country with a lower steady-state capital-labor ratio than the world, since

the opposite case is analogous. Depending on the model, this difference could be due to a

lower savings propensity or a higher discount rate than prevails abroad, or it could be due

to a difference in technology or population growth rate.3 It will suffice to examine the

case of a small two-sector economy that takes as given the relative prices in the world,

3 The latter is tricky, however, as it can lead to the country becoming large relative to the world, a case that

I will not consider here. See Deardorff (1994) for the interesting complications that can arise with

differences in population growth rates.

9

which are different from its own relative prices in autarky and therefore indicative of

comparative advantage. Because the country’s steady-state capital-labor ratio is below

that of the world, it will start with comparative advantage in the labor-intensive good, the

relative price of which will rise when it opens to free trade. The question will be, how

does its comparative advantage change over time after the country opens to trade.

Although this may seem like a well-defined single question, there are actually two

cases to consider. The reason is that, in a two-sector model where one of the goods is the

investment good that must be used for capital accumulation, it matters whether that good

is labor-intensive or capital-intensive relative to the consumption good. I will examine

both cases, using the geometric technique of Deardorff (1974).

Figure 3 first provides a reminder of how the steady state appears in a two-sector

Solow growth model with the investment good labor-intensive. As explained in

Deardorff (1974), if we fix prices – in this case at their autarky steady-state levels – the

aggregate per-labor output function, y, now includes portions of two separate industry-

level production functions plus a straight-line segment tangent to both of them. Output

per person of the labor-intensive good rises along the first portion of this curve until

reaches this straight segment, after which it falls along a straight line to zero. As it is

falling, another straight line (not shown in Figure 3) gives the rising per person output of

the capital-intensive good, which then proceeds along the second curved portion of the

aggregate output curve for high k.

As in the one-sector Solow model, savings and investment are given by this same

aggregate output curve scaled down by the savings propensity. Steady state requires that

this per capita savings equal the amount required to match population growth, nk. In

10

addition, in autarky, this output must also be equal to what is produced. Hence the

autarky steady state has all three curves intersecting at k*.4 Had the investment good

been instead capital-intensive, this intersection would have to be instead with the upward-

sloping straight line for the per capita output of that good, not shown in Figure 3.

What now happens when the country opens to trade at prices different from its

steady-state autarky prices? In order to infer dynamics from changes in savings and

investment, we must fix our units of measurement in terms of the investment good. Thus,

while I stated above that the relative price of the labor-intensive good would rise with

trade, we must introduce this change as a fall in the price of the capital-intensive

consumption good. That is done in Figure 4.

The fall in the price of the capital-intensive good causes the right-hand portion of

the aggregate income curve, y, to shift down. The straight-line tangent to the two

industry curves rotates clockwise in order to remain tangent to both, as show by the

dashed curve. The downward sloping straight line showing per capital output of the

(labor-intensive) investment good also shifts to the right, but I have left this out of the

figure for simplicity. Most importantly, the downward shift of all but the left portion of

the aggregate income curve causes the per capita savings/investment curve also to shift

down, and this moves its intersection with nk to the left.

The result, then, is that opening to trade causes the steady-state capital-labor ratio

of the country to fall. When it first opens to trade, the country’s comparative advantage

in the labor-intensive good causes it to export that good, as usual. Then, over time,

4 The curves in Figure 3 are drawn for the prices that prevail in the autarky steady state. They cannot be

used directly to depict out-of-steady-state behavior in autarky, because autarky prices would have to change

with k in order to keep supply equal to demand.

11

because of the change in the steady state, its capital-labor ratio falls and its comparative

advantage in the labor-intensive good increases. So does its trade.5

Proposition 2-A-i: In a two-sector Solow Model of a country whose autarky steady-state

capital-labor ratio is below that of the world, the following occur due to trade if the

investment good is labor-intensive:

The country has comparative advantage in the labor-intensive good, both initially

and in the new steady state.

Opening to trade causes the capital-labor ratio to decline toward a new, lower,

steady state.

Thus trade causes the country’s comparative advantage (and trade) to increase

over time.

This result conforms to the expectation based on the basic insight above, even

though it is not the case that savings is being induced directly by the rate of return to

capital. Here, however, what has induced the decline in investment is a worsening of the

terms on which the country can convert a portion of its output into investment – the rise

in the price of the investment good – that can be said to have induced the fall in growth

and the increase in comparative advantage. Note, though, that while it appears in Figure

4 that the country has been hurt by trade, that is not really true. Had we measured income

in units of the consumption good, on which welfare is more correctly based, its income

would have gone up.

Consider now the opposite case, still in a Solow model of savings, of the

investment good being capital-intensive. This is shown in Figure 5. Now the opening to

5 As drawn, this change is not large. But with a larger change in prices, the country could be driven to

complete specialization in the labor-intensive good and therefore a more extreme response of trade, both

before and after the fall in the capital-labor ratio due to declining growth.

12

trade raises the price of the labor-intensive good, since it is the consumption good. Again

this shifts a large part, but not all, of the aggregate per capita income curve, this time

shifting up its left and middle portions. This in turn shifts the saving/investment curve up

as well, causing an increase in the steady-state capital-labor ratio. Over time, capital

accumulation causes the country’s comparative advantage in the labor-intensive

consumption good to decline. Trade shrinks, and may even reverse.

Proposition 2-A-ii: In a two-sector Solow Model of a country whose autarky steady-

state capital-labor ratio is below that of the world, the following occur due to trade if the

investment good is capital-intensive:

The country has comparative advantage in the labor-intensive good, initially but

not necessarily in the new steady state.

Opening to trade causes the capital-labor ratio to rise toward a new, higher, steady

state.

Thus trade causes the country’s comparative advantage (and trade) to decrease

over time, and perhaps even reverse.

In this case, then, the result is again contrary to the simple insight from the

introduction: Opening to trade causes a labor-abundant country to expand its capital-

labor ratio and thus diminish its comparative advantage and trade over time. The latter

could be seen more easily in Figure 5 if I had included the rightward shift of the upward

sloping straight line showing per capita output of the investment good. That line would

be necessarily steeper than the savings curve, and therefore the initial imports of the

investment good diminish as capital accumulates and may even reverse to become

exports.

13

Why is all this happening? It is true that the rate of return to capital (shown as the

slope of the aggregate income curve) has fallen due to trade, which might suggest that the

incentive to accumulate capital has reduced. But proportional savings a la Solow does

not respond to this incentive. Instead, what is driving accumulation here is the reduced

relative price of the investment good. Thus while the benefits of capital accumulation

have fallen, the costs have fallen even more, and the country accumulates capital.

This may seem a bit perverse, and indeed it is. Uzawa (1961) noted long ago in

his pioneering explication of the two-sector closed-economy growth model that the case

of a capital-intensive investment good can produce some odd results, precisely because a

fall in its price can induce capital accumulation that increases supply more than demand.

Thus there is a potential for dynamic instability.

Note too that, had we drawn Figure 5 in units of the consumption good instead of

the investment good, we would have seen the possibility that per capita consumption

could fall due to trade, even in steady state. That too should not be surprising in this

model of proportional saving, which without constraints may lead to greater capital

accumulation that dictated by the Phelps (1961) Golden Rule of Capital Accumulation.

In such a case, if trade leads to even more accumulation of capital, as here, then per

capita consumption will fall in both the short run and the long run.6 With saving (and

thus demand for the two goods) not being determined optimally, it should not be

surprising if the outcome violates the gains from trade.

All of which suggests that we examine also a Ramsey model for this question.

That is, suppose that savings is derived optimally over time and that the country has a

6 See Deardorff (1973).

14

lower steady-state capital-labor ratio in autarky than the world, due to a higher rate of

discount. What happens then when it opens to international trade?

In Figure 6, the gray curve shows per capita income of the rest of world, from

which its autarky steady state is determined as by equating its return to capital, , to

its sum of discount and population growth rates, . Country A, initially closed to

trade and in autarky, has a higher discount rate , and thus a lower steady-state capital-

labor ratio, .

When country A opens to free trade, it faces the autarky prices of the rest of

world, and thus, in a two-sector model, the aggregate per capita income curve shown as

. As already seen, because of the fixed (from the world) prices, this curve includes a

straight segment with slope equal to . Since , country A’s steady state is

necessarily below the FPE set. Thus, in the Ramsey Model with unequal autarky steady

states, the low-saving country necessarily moves due to trade to complete specialization

in the labor-intensive good.

Proposition 2-B: In a two-sector Ramsey Model of a country whose autarky steady-state

capital-labor ratio is below that of the world, the following occur due to trade regardless

of the relative factor intensities of the goods:

The country has comparative advantage in the labor-intensive good, initially and

also in the new steady state, in which it specializes completely in that good.

Opening to trade causes the capital-labor ratio to fall toward a new, lower, steady

state.

Thus trade causes the country’s comparative advantage (and trade) to increase

over time.

15

IV. Conclusion

The question here was whether the dynamic effects of opening to trade would increase or

decrease comparative advantage. When comparative advantage is based on the

abundance or scarcity of something that is costly to acquire, one expects rational behavior

to respond to a change in prices by increasing that abundance or scarcity. To explore this

issue in simple theoretical terms, this paper has examined two types of simple growth

model – a Solow Model with proportional saving, and a Ramsey model with optimal

saving – to see whether this reasoning is born out. In the Ramsey model, it is. In the

Solow model, on the other hand, results vary, but this should not be surprising as the

proportional saving assumption does not embody optimizing behavior. To the extent that

we believe that aggregate saving behavior is indeed based on rational and fully informed

optimization, we should therefore expect that the dynamic effects of trade do indeed

operate in the direction of increasing comparative advantage over time.

16

References

Baldwin, Richard E. 1992 "Measurable Dynamic Gains from Trade," Journal of Political

Economy 100, pp. 162-174.

Chang, Pao-Li and Fali Huang 2012 “Trade Divergence in Education Systems,” presented

at Asia-Pacific Trade Seminar 2012, Singapore, June 22.

Deardorff, Alan V. 1973 “The Gains from Trade In and Out of Steady-State Growth,”

Oxford Economic Papers 25, (July), pp. 173-191.

Deardorff, Alan V. 1974 “A Geometry of Growth and Trade,” Canadian Journal of

Economics 7, (May), pp. 295-306.

Deardorff, Alan V. 1994 “Growth and International Investment with Diverging

Populations.” Oxford Economic Papers 46 (July), pp. 477-491.

Deardorff, Alan V. and James E. Hanson 1978 "Accumulation and a Long-Run

Heckscher-Ohlin Theorem," Economic Inquiry 16, (April), pp. 288-292.

Eaton, Jonathan and Samuel Kortum 2002 “Technology, Geography, and Trade,”

Econometrica 70(5), (September), pp. 1741-1779.

Levchenko, Andrei and Jing Zhang. 2011 “The Evolution of Comparative Advantage:

Measurement and Welfare Implications,” Discussion Paper #610, Research

Seminar in International Economics, University of Michigan, September 29.

Mazumdar, Joy 1994 "Do Static Gains from Trade Lead to Medium-Run Growth?"

Journal of Political Economy 104, December, pp. 1328-1337.

Phelps, Edmund S. 1961 “The Golden Rule of Capital Accumulation," American

Economic Review 51: pp. 638–643.

Ramsey, Frank P. 1928 “A Mathematical Theory of Saving,” Economic Journal 38, pp.

543-559.

Samuelson, Paul A. 1948 "International Trade and the Equalisation of Factor Prices,"

Economic Journal 58, (June), pp. 163-184.

Samuelson, Paul A. 1949 "International Factor-Price Equalisation Once Again,"

Economic Journal 59, (June), pp. 181-197.

Solow, Robert 1956 “A Contribution to the Theory of Economic Growth,” Quarterly

Journal of Economics 70, pp. 65-94.

Uzawa, Hirofumi 1961 “On a Two-Sector Model of Economic Growth,” Review of

Economic Studies 29, pp. 40-47.

17

Figure 1

Solow model, same savings propensity as rest of world, R.

Starting below steady state, with trade country A gains and grows faster than in autarky.

k =K / Lk*R = k*Ak0

A

y =Y / L

I / L = S / L = sy

nk

Gain from trade

y etc.

r*=ρ*+n*

18

Figure 2

Ramsey model, same discount rate as rest of world, R.

Starting far below autarky steady state, with trade country A specializes and grows to edge of FPE set.

Starting inside FPE set, trade stops growth and comparative advantage stays fixed.

k2

k =K / Lk*R = k*Ak0

A

y =Y / L

r *+n*

r

FPE

k1

19

Figure 3

Autarky steady state in a 2-sector Solow model.

Investment good is labor-intensive

k =K / Lk *

y =Y / L

I / L = S / L = sy

nk

y etc. Output/L of K-int good

Output/L of L-int good

20

Figure 4

Opening to trade from autarky steady-state with labor-intensive investment good.

Price of capital-intensive consumption good falls.

k =K / Lk *

y =Y / L

I / L = S / L = sy

nk

k ¢*

y etc.

21

Figure 5

Opening to trade from autarky steady-state with capital-intensive investment good.

Price of labor-intensive consumption good rises.

k =K / Lk *

y =Y / L

I / L = S / L = sy

nk

k ¢*

y etc.

22

Figure 6

Opening to trade in a Ramsey Model for a country with a higher discount rate than the world.

k =K / Lk *Rk0

A

r *+n*

FPE

rA +nA

y etc.yA

yR

rR