Comparative Advantage, Trade And Payments In A Ricardian Model With A Continuum Of Goods

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R. Dornbusch

COMPARATIVE ADVANTAGE,

TRADE AND PAYMENTS IN A RICARDIAN

MODEL WITH A CONTINUUM OF GOODS

S. Fischer P. A. Samuelson

Number 178 April 1976

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institute of

technology

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Cambridge, mass. 02139

COMPARATIVE ADVANTAGE,

TRADE AND PAYMENTS IN A RICARDIAN

MODEL WITH A CONTINUUM OF GOODS^

R. Dornbusch S. Fischer P. A. Samuelson

Number 178 April 1976

The views expressed in this paper are the authors' sole

responsibility and do not reflect those of the Department

of Economics, the Massachusetts Institute of Technology,

the Ford Foundation, or the National Science Foundation.

JUN 26 1976 I

Revised

April 1976

COMPARATIVE ADVANTAGE, TRADE AND PAYMENTS IN A RICARDIAN

MODEL WITH A CONTINUUM OF GOODS*

R. Dornbusch S. Fischer P. A. Samuelson

Massachusetts Institute of Technology

This paper discusses Ricardian trade and payments theory in the case

of a continuum of commodities. The analysis thus extends the development

of the many- commodity, two-country comparative advantage analysis as

presented, for example in Haberler (1937) and as historically reviewed

by Chipman (1965) . Perhaps surprisingly, the continuum assumption simplifies

the analysis in comparison with the discrete many-commodity case.

The distinguishing feature of the Ricardian approach emphasized in this

paper is the determination of the competitive margin in production between

imported and exported goods. The analysis advances the existing literature

by showing formally how tariffs and transport costs establish a range of

commodities that are not traded, and how the price-specie flow mechanism does

or does not give rise to movements in relative cost and price levels.

The formal real model is introduced in Part I. Its equilibrium

determines the relative wage and price structure and the efficient international

specialization pattern. Part II considers standard comparative

static questions of growth, demand shifts, technological change, and

transfers. Extensions of the model to nontraded goods, tariffs and

R. Dornbusch acknowledges gratefully a Ford Foundation Grant; S. Fischer,

NSF GS-41428; and P. Samuelson, NSF 75-04053.

0727632

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transport costs are then studied in Part III. Monetary considerations

are introduced in Part IV, which examines the price-specie mechanism under

stable parities, floating exchange rate regimes, and also questions of

unemployment under sticky money wages.

- 3 -

I. THE REAL MODEL

In this part we develop the basic real model and determine the

equilibrium relative wage and price structure along with the efficient

geographic pattern of specialization. Assumptions about technology are

specified in Section A. Section B deals with demand. In Section C the

equilibrium is constructed and some of its properties are explored.

Throughout this section we assume zero transport costs and no other impediments

to trade.

A. Technology and Efficient Geographic Specialization

The many-commodity Ricardian model assumes constant unit labor

requirements (aI,,...n, a ) and (a,I,..n.,a ) for the n commodities that can

be produced in the home and foreign countries, respectively. The commodities

are conveniently indexed so that relative unit labor requirements

are ranked in order of diminishing home-country comparative advantage,

* * *

a,1/a1, > ... > a.1/a1. > ... > a /a n n

where an asterisk denotes the foreign country.

In working with a continuum of goods, we similarly index commodities

on an interval, say[o,l], in accordance with diminishing home-country comparative

advantage. a commodity, z, is associated with each point on

the interval, and for each commodity there are unit labor requirements in

*

the two countries, a(z) and a (z) , with relative unit labor requirement

given by

*

(1) A(z) = 2_i£L . A . (z) < . a(z)

For a review of the Ricardian model, see Chipman (1965)

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The relative unit labor requirement function in (1) is by assumption

continuous and by construction (ranking or indexing of goods) nonincreasing

in z. We make the stronger assumption that A(z) is in fact

decreasing in z. The function A(z) is shown in Figure 1 as the downward

sloping schedule.

Consider now the range of commodities produced domestically and

those produced abroad, as well as the relative price structure associated

*

with given wages. For that purpose we define as w and w the domestic

and foreign wages measured in any (common!) unit. The home country will

efficiently produce all those commodities for which domestic unit labor

costs are less than or equal to foreign unit labor costs. Accordingly,

any commodity z will be produced at home if

* *

(2) a (z) w < a (z) w

or

(2') oj < A(z)

where

*

(3) co = w/w

is the ratio of our real wage to theirs (our "double-factorial terms of

trade"). It follows that for a given relative wage co the home country

will efficiently produce the range of commodities

(4) < z < z(w)

,

where, taking (2 1

) with equality defines the borderline commodity

(5) z" = A

_1

(oj) ,

A ( ) being the inverse function of A( )

.

- 5 -

By the same argument the foreign country will specialize in the

production of commodities in the range

(4') z(00) < z < 1.

The minimum cost condition determines the structure of relative prices.

The relative price of a commodity z in terms of any other commodity z',

with both goods produced in the home country, is equal to the ratio of home

unit labor costs:

(6) P(z)/P(z') = wa(z)/wa(z') = a(z)/a(z') ; z < z, z* < z.

The relative price of home-produced z in terms of a commodity z" produced

abroad by contrast is

(7) P(z)/P(z") = wa(z)/w a (z") = 0Ja(z)/a (z") ; z < z < z"

.

In summarizing the supply part of the model we note that any

specified relative real wage is associated with an efficient geographic

specialization pattern characterized by the borderline commodity z (to) as

well as a relative price structure. (The pattern is "efficient" in the

sense that the world is on, and not inside, its production-possibility

frontier.

)

B. Demand

On the demand side we impose a strong homothetic structure in the

form of J.S. Mill or Cobb-Douglas demand functions that associate with

each commodity (i) a constant expenditure share, b. . We further assume

identical tastes for the two countries or uniform homothetic demand.

6 -

By analogy with the many commodity case, which involves budget shares

n

b.1=1P.1C./Y ; lb. =1 ; b. = b. 1 11

we therefore have for the continuum case a given b(z) profile:

b(z)dz = 1 ; b(z) = b* (z)

where Y denotes income, C demand for and P the price of commodity z.

Next we define the fraction of income spent (anywhere) on those goods

in which the home country has a comparative advantage:

z

(9) 0(z) = b(z)dz ; 9'(z) = b(z) >

where again (0,z) denotes the range of commodities for which the home

country enjoys a comparative advantage. With a fraction 6 of each country's

income, and therefore of world income, spent on domestically produced goods

it follows that the fraction of income spent on foreign produced commodities

is

(9') 1 - 9(z) = b(z)dz.

C. Equilibrium Relative Wages and Specialization

To derive the equilibrium relative wage and price structure and the

associated pattern of efficient geographic specialization we turn next to

the condition of market equilibrium. Consider the home country's labor

market, or equivalently the market for domestically produced commodities.

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With z denoting the hypothetical dividing line between domestically and

foreign produced commodities, equilibrium in the market for home produced

goods requires that domestic income, wL, equals world spending on domestically

produced goods:

* *

(10) wL = 8(z) (wL + w L )

.

*

Equation (10) associates with each z a value of the relative wage w/w

such that market equilibrium obtains. This schedule is drawn in Figure 1

as the upward sloping locus and is obtained from (10) by rewriting the

equation in the form:

6 ( z) * *

(10') 03 = ^— (L /L) = B(z; L /L)

...