HIAS-E-49

Globalisation and Inequality in a Dynamic Economy:
An Axiomatic Analysis of Unequal Exchange
Roberto Veneziani
School of Economics and Finance, Queen Mary University of London, Mile End Road, London E1 4NS, UK

Naoki Yoshihara
University of Massachusetts Amherst, Crotty Hall, 412 North Pleasant Street, Amherst, MA 01002, USA
The Institute of Economic Research, Hitotsubashi University, Naka 2-1, Kuni- tachi, Tokyo 186-8603, Japan
School of Management, Kochi University of Technology, Tosayamada, Kami-city, Kochi 782-8502, Japan

February 2017

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
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Globalisation and Inequality in a Dynamic Economy:
An Axiomatic Analysis of Unequal Exchange1
Roberto Veneziani2

Naoki Yoshihara3

February 24, 2017

1 Paper prepared for the special issue in honour of John Roemer.

We thank Jon Cogliano,
Amitava Dutt, Gary Dymski, Peter H. Matthews, Rajiv Sethi, Gil Skillman, Peter Skott, theEditors of this journal, two anonymous referees, and audiences in London, New York, Chicago,
Amherst, and Sendai, for useful comments and suggestions. The usual disclaimer applies.
2 School of Economics and Finance, Queen Mary University of London, Mile End Road, London
E1 4NS, UK. E-mail: r.veneziani@qmul.ac.uk
3 (Corresponding author) Department of Economics, University of Massachusetts Amherst,
Crotty Hall, 412 North Pleasant Street, Amherst, MA 01003, USA; The Institute of Economic
Research, Hitotsubashi University, Kunitachi, Tokyo 186-0004 Japan; and School of Management,
Kochi University of Technology, Kochi 782-8502, Japan. E-mail: nyoshihara@econs.umass.edu

Abstract
An axiomatic analysis of the concept of unequal exchange (UE) between countries is developed in a dynamic general equilibrium model that generalises Roemer’s [22] economy with a
global capital market. The class of UE deﬁnitions that satisfy three fundamental properties including a correspondence between wealth, class and UE exploitation status - is completely
characterised. It is shown that this class is nonempty and a deﬁnition of UE exploitation between countries is proposed, which is theoretically robust and ﬁrmly anchored to empiricallyobservable data. The full class and UE exploitation structure of the international economy
is derived in equilibrium.
JEL classiﬁcation: D63; F02; B51.
Keywords: Exploitation, classes, unequal exchange, international economy.

1

Introduction

Egalitarians - and more speciﬁcally, socialists - have long questioned the structure of international relations, pointing in particular at the large inequalities in income and standardof living among countries. An inﬂuential analysis within this tradition has been developedby the so-called dependence school, according to which such inequalities are the product of
exploitative relations between rich and poor nations.1 In his classic work, for example, Emmanuel [9] has argued that the core-periphery structure of international relations generates
an unequal exchange (UE) between rich and poor nations. According to Emmanuel, given
institutionalised wage dierentials between developed and less developed nations, capital
mobility across borders and the international trade of commodities cause a transfer of surplus labour from poor nations with low capital-labour ratios to wealthy nations with highcapital-labour ratios, which results in the impoverishment of the former to the advantage ofthe latter.
UE theory has generated substantial controversy on both empirical and theoretical grounds.
Empirically, some studies emphasise the positive eects of international exchanges and provide evidence suggesting that the “movement toward free trade may actually have just the
opposite eect [than predicted by UE theory], leading to a reduction in income disparity
among countries” (Ben-David [4], p.653). UE theory is deemed theoretically doubtful, because it is based on the comparison between the amounts of labour embodied in the goods
traded, and the notion of labour embodied is widely considered to be metaphysical and at best - of very limited validity, as it can only be deﬁned in rather special cases.2 In thispaper, we examine the theoretical foundations of UE theory in order to determine whether
a general, consistent, and empirically grounded deﬁnition of UE can be identiﬁed.3
The starting point of our investigation is a seminal contribution by Roemer [22], which1

The literature is too vast for a comprehensive list of references. For a discussion of the classic contributions, we refer the reader to the excellent reviews by Bacha [3] and Gri!n and Gurley [13].
2
UE theory is also criticised because it contradicts the principle of comparative advantage, according towhich proﬁt equalisation and capital ﬂows from rich to poor countries have growth-inducing and inequalityreducing eects. See, for example, the debate between Paul Samuelson and Arghiri Emmanuel in The Journalof International Economics in 1978.
3
Empirically, we simply note that recent studies have provided evidence supporting the idea that international inequalities have indeed increased. See, for example, Slaughter [26].

1provides an elegant analysis of UE between countries, in a static model with revenuemaximising countries and a Leontief technology (see also Roemer [21]). Roemer [22] departs
from the standard UE approach in two key respects. First, he adopts a rigorous deﬁnition of
UE exploitation that is unrelated to the labour theory of value and that does not presume
global markets to be non-competitive. Second, in order to analyse the normative foundations of UE theory, he focuses on a model in which “all dierences between countries [areeliminated] save one: the capital-labor ratio” (Roemer [22], p.35). This allows him to showthat class relations and UE exploitation emerge in the international economy, even assuming
perfectly competitive global commodity and credit markets, and even if all countries haveaccess to the same technology and share the same preferences. In equilibrium, the global
economy is characterised both by mutual gains from trade and by asymmetric international
relations because the economic development of less developed countries is crucially dependent on capital exports from developed countries, and surplus is transferred from the former
to the latter via international capital markets. “Unequal exchange does not preempt mutual
gains from trade” (ibid.).4 Major heterogeneities in institutions, technology or preferences
and market imperfections are unessential for these results: unequal and asymmetric relations between countries “can be entirely explained by dierential capital-labor ratios acrosscountries” (Roemer [22], p.34).
As insightful as Roemer’s [21, 22] contributions are, it is still an open question whether
his key insights are robust. It is not clear, for example, whether Roemer’s conclusions
hold under more general assumptions concerning preferences and technology, and outside ofstatic models. In later contributions, Roemer himself has raised doubts on the generalityof UE theory,5 and on the possibility of identifying a rigorous deﬁnition that captures thefundamental normative intuitions of UE exploitation.
In this paper, we investigate the robustness of UE theory and extend Roemer’s [21,22] analysis by adopting a more general formal framework and a dierent methodologicalapproach. Formally, instead of restricting attention to static, one-period economies, we set up
4This insight is compatible with the classical Marxian theory of exploitation, as Marx ([15], chapter 20,(e)) notes that “a richer country exploits a poorer one, even when the latter beneﬁts from the exchange.”5
For example, Roemer [23]. See Veneziani [28] for a thorough discussion.

2

a dynamic general equilibrium model of the global economy in which intertemporal decisions
are explicitly considered and the concept of UE exploitation is analysed at general dynamic
equilibria. Further, the model incorporates more general assumptions on preferences - byassuming that countries’ welfare depends both on consumption and on leisure - and on
technology - by allowing for convex production sets.
Methodologically, instead of focusing on a speciﬁc deﬁnition of UE exploitation, we adopt
the axiomatic approach pioneered by Yoshihara and Veneziani [33] and signiﬁcantly extend it
by considering dierent properties, and by requiring our axioms to hold at general dynamic
equilibria. We examine three fundamental properties. The ﬁrst one, called Labour Exploitation (LE), is a domain axiom that captures some basic intuitions deﬁning the core of UE
theory that all admissible deﬁnitions should satisfy. It is a dynamic generalisation of similar
properties formulated in the literature (see, e.g., [31, 29, 30]), and it reduces to the latterin the special case of a static, one-period economy. Intuitively, according to LE, exploitative international relations are characterised by systematic dierences between the labour
performed by agents in a country and the amount of labour ‘contained’ in some reference
commodity bundles that capture their consumption possibilities.6
The other two properties, called the Class-Exploitation Correspondence Principle (CECP)
and the Wealth-Exploitation Correspondence Principle (WECP) capture axiomatically two
intuitions of UE theory originally introduced by Roemer [21, 22], who proved them to holdin certain economies, under speciﬁc UE deﬁnitions. According to CECP, a correspondence
should exist between a country’s position in the global capital market and its exploitation
status: in equilibrium, nations that optimise by lending capital abroad should emerge as
UE exploiters, whereas nations that optimise by borrowing capital should be UE exploited.
CECP has been ﬁrst formulated axiomatically by Yoshihara [31] in a static context; we extend it here to hold at general dynamic equilibria. WECP is a novel axiom in the literature:
it states that the exploitation status of countries in the international arena should be determined by their level of development (proxied by the value of their productive endowments).
6
A rigorous statement of all three axioms is in section 4. LE is conceptually related to the classic theories
of unequal exchange (Emmanuel [9]) and underdevelopment (Amin [2], Frank [11]).

3

We completely characterise the class of UE deﬁnitions that satisfy all three axioms at
the general dynamic equilibria of the international economy. Then, we prove that this classis nonempty: a deﬁnition recently proposed by Yoshihara and Veneziani [31, 33, 29] basedon the ‘New Interpretation’ (Duménil [6, 7]; Foley [12]; Duménil et al [8]) satisﬁes all threeproperties. Contrary to the received view, a rigorous and logically consistent deﬁnition ofunequal, exploitative exchange exists, which is ﬁrmly anchored to empirically observabledata and extends the key insights of UE theory to general, dynamic international economies.
Our analysis here continues a research programme in axiomatic exploitation theory
started in Yoshihara and Veneziani [33] (see also [31, 29, 30]) and signiﬁcantly extends
it to the dynamic context, and to deal with dierent intuitions of UE theory, formalised in adierent set of axioms. We analyse general intertemporal economies because this allows us
to evaluate the robustness of Roemer’s [21, 22] key insights, and the generality of the ‘New
Interpretation’, but also because they raise some important conceptual and formal issues inaxiomatic UE theory. In fact, as shown below, in a dynamic context, the appropriate definition of UE exploitation depends on the normative relevance of agents’ consumption andsavings decisions, and this requires a number of adjustments in the conceptual framework,in the axiomatic system, and in the formal arguments.
To be sure, this paper does not fully answer the question of the normative relevance ofUE and the wrongfulness of exploitative international relations. Yet the rigorous, axiomaticcharacterisation of a nonempty class of deﬁnitions that preserve some key insights of UE
theory is a crucial ﬁrst step in order to address that question.7

2

The Model

The economy consists of a set N = {1, ..., N} of countries, with generic element D, in which asequence of nonoverlapping generations exist, each living for T periods,8 and indexed by the7

We discuss some normative implications of UE exploitation theory in Veneziani [27, 28] and Yoshihara
and Veneziani [33].
8
We specify the framework in the case with a ﬁnite T in order to highlight the similarity with Roemer’s[21, 22] economies. However, the notation and deﬁnitions can be extended in a straightforward way to thecase with one inﬁnitely-lived generation, and all of our results hold both if T is ﬁnite and if it is inﬁnite.

4

date of birth kT , k = 0, 1, 2, ... In every period t, countries consume n produced commodities,
and leisure. Technology is freely available to all countries: in every t, capitalists in eachcountry can operate any activity in the production set P  R2n+1 , with generic elementk = (kl , k, k), where kl 5 R+ is the direct labour input; k 5 Rn+ are the inputs of the ngoods; and k 5 Rn+ are the outputs of the n goods. The net output vector arising from k isdenoted as kb  k  k. Let 0 be the null vector. The set P is assumed to be a closed convex
cone containing the origin in R2n+1 , and to satisfy the following standard properties.9

Assumption 1 (A1). For all k 5 P , if k  0 then kl > 0 and k  0.
b = c.
Assumption 2 (A2). For all c 5 Rn+ , there exists k 5 P such that k

Assumption 3 (A3). For all k 5 P , and for all (k0 , k0 ) 5 Rn3 × Rn+ , if (k0 , k0 ) 5(k, k) then (kl , k0 , k0 ) 5 P .
A1 implies that labour and some capital are indispensable to produce any output; A2 statesthat any non-negative commodity bundle is producible as net output; A3 is a standard freedisposal condition. The set of e!cient production activities is YP = {k 5 P | @k0 5 P such that k0 > k}Commodities and capital can freely migrate across borders, while labour is immobile. In
every t, (pt , rt ) is the 1 × (n + 1) international price vector, where pt denotes the prices of
the n commodities and rt is the interest rate that prevails in competitive capital markets. In
order to focus on international inequalities, agents are assumed to be identical within eachcountry; thus, the superscript D denotes both a country and its representative agent.
Following Roemer [21, 22], we explicitly model the time structure of exchange and production. Each production period t is divided into two stages: the capital market and themarket for productive assets operate at the beginning of t, where goods are exchanged at theprices pt31 ruling at the end of t  1/beginning of t. At the beginning of t, /tD is the vectorD
of productive assets owned by D, - where /kT
denotes the endowments inherited when bornin kT , - and the market value of D’s endowments, D’s wealth, is WtD = pt31 /tD .
At the beginning of every t, each D 5 N can borrow an amount pt31 q Dt on the international
9Vector inequalities: for all x, y 5 Rm , x = y if and only if xi = yi (i = 1, . . . , m); x  y if and only if
x = y and x 6= y; x > y if and only if xi > yi (i = 1, . . . , m).

5

³
´
D
credit market to purchase q Dt in order to operate production activity qtD = qltD , q Dt , q t 5

P . Otherwise, it can use its wealth WtD either to purchase capital goods kDt to operate activity

ktD = (kltD , kDt , kDt ) 5 P ; or to buy commodities BtD 5 Rn+ to be stored and sold at the endof the period; or to lend capital ztD 5 R+ abroad.
Because production takes time, output is exchanged on the ﬁnal goods market at the endof t, at end-of-period prices pt . For each country D 5 N , proceedings from production are³´Dgiven by pt kDt + q t and the return to lending ztD is (1 + rt ) ztD , thus gross national income³´DDat the end of t is pt kt + q t +(1 + rt ) ztD from which the rental cost of the borrowed capital(1 + rt ) pt31 q Dt must be paid. The rest of D’s income can be used to purchase consumption
D
5 Rn+ .
goods cDt 5 Rn+ and to ﬁnance accumulation /t+1

Given production decisions (ktD , qtD ), in every t, the total amount of labour performed byagents in D 5 N is given by \Dt = kltD + qltD and it cannot exceed the labour endowment, L,
which is assumed to be the same for all countries without loss of generality. Therefore foreach D 5 N , leisure enjoyed at t is ltD = L  \Dt , and we assume that country D’s welfare att can be represented by a function u : Rn+ × [0, L] $ R+ : u(ct , lt ) can be interpreted either
as a standard utility function or as an objectivist index of well-being.10
In order to characterise the structure of international relations and the dynamic patternof exploitation and classes, it is necessary to impose some structure on the function u. We
assume that u(ct , lt ) =  (lt ) + v(ct ), where v : Rn+ $ R and  : [0, L] $ R are strictlyincreasing and twice dierentiable. Further, in order to avoid a number of unnecessary
technicalities, we assume that v is strictly quasi-concave and homogeneous of degree one,while  is strictly concave with liml<L 0 (l) = 0 and liml<0 0 (l) = 4.
These assumptions signiﬁcantly generalise the canonical models of exploitation theory
by Roemer [21, 22]. They are appropriate from a normative perspective (see, for example,
Silvestre [25]), and standard in international economics - and speciﬁcally, in the literature on
Heckscher-Ohlin models (see, for example, Chen [5]). The assumptions on , for example,
rule out implausible equilibria with countries performing zero labour, or enjoying no leisure10

For a discussion of subjective and objective principles, see Roemer and Veneziani [24] and, in the context
of exploitation theory, Yoshihara and Veneziani [34].

6

at all. It is worth stressing, however, that the restrictions on u are imposed mostly fortechnical convenience, and the main results of this paper can be derived under more generalassumptions, albeit at the cost of a signiﬁcant increase in technicalities.11(k+1)T 31

Let cD = {cDt }t=kT

be D’s lifetime consumption plan; and likewise for kD , q D , z D , B D ,© D ª(k+1)T 31
and \D , and let / D = /t+1
be D’s lifetime accumulation plan. Let (p, r) =t=kT(k+1)T 31{(pt , rt )}t=kT

be the path of international price vectors during the lifetime of a gen-eration. Let 1 D = (kD , q D , z D , B D , cD , /D ) denote a generic intertemporal plan for D. Let
0 < 4 5 1 be the time preference factor. Each D is assumed to choose 1 D to maximise welfare
subject to the constraint that in every t, (1) gross national income is su!cient for consumption and accumulation; (2) wealth is su!cient for production and lending; (3) production
activities are technologically feasible. Finally, (4) we follow Roemer [21, 22] and assume thateach agent optimises subject to the requirement that he does “not run down the value of his
assets, which is the reproducibility requirement” (Roemer [21], p.63). Unlike in Roemer’s
static economies, however, the reproducibility condition (4) does not hold in every period.
In any given t, agents can use their wealth to ﬁnance consumption but every generation k
is constrained to bequeath at the end of its life at least as much wealth as they inherited.
Formally, given (p, r), each D solves programme MP D .12

DMP D : V (/kT
) = maxD1(k+1)T 31

X

t=kT

4t [ (L  \Dt ) + v(cDt )] ,subject to: for every t = kT , . . . ,(k + 1) T  1,
pt kDt +h D
iDD
pt q t  (1 + rt ) pt31 q t + (1 + rt ) ztD + pt BtD = pt cDt + pt /t+1,
pt31 (kDt + BtD ) + ztD = pt31 /tD ,ktD , qtD 5 P , \Dt 5 L,DDp(k+1)T 31 /(k+1)T= p(k+1)T 31 /kT
.

(1)
(2)
(3)
(4)

11
For example, it is possible to allow for heterogeneous preferences over consumption goods withu (ct , lt ) = ! (L  t ) + v (ct ); a weakly concave !; v being homogeneous of degree k < 1; and so on.
12Constraints (1)-(2) are written as equalities without loss of generality, given the monotonicity of u.

7

MP D is a suitable way of modelling country D’s decision problem, given the representativeagent assumption, and it generalises Roemer’s [21, 22] static models in which countries
maximise national income.
In order to capture the role of ﬁnancial markets in exploitative international relations,only short-term credit contracts are considered as in Roemer [21, 22]: within each period,
countries can operate on the international capital market to ﬁnance their production plans,
but contracts do not extend over time and credit plays a limited role in fostering accumulation. Consumption, debt, and savings must be ﬁnanced out of current revenue. Due to thepossibility of saving, and noting that net savings are allowed to be negative, however, Roemer’s [21, 22] static models are generalised by allowing for intertemporal trade-os within acountry, consistently with a dynamic setting in which agents live for more than one period.
For all D 5 N , let OD (p, r) be the set of vectors 1 D that solve MP D at prices (p, r). Let
¡ 1
¢
2
N
lkT = /kT, /kT, ..., /kT. Let E (P, N , u, 4, lkT ), or as a shorthand notation E(lkT ), denotethe international economy with technology P , countries N , welfare function u with discountPfactor 4, and productive endowments lkT . Let ct = DMN cDt ; and likewise for all othervariables. For the sake of simplicity, let “for all t” stand for “for all t = kT, . . . , (k +1)T 1”.
Following Roemer [21, 22], the equilibrium concept can now be deﬁned.13
Deﬁnition 1: A reproducible solution (RS) for E(lkT ) is a price vector (p, r) and an

associated proﬁle of actions (1 D )DMN such that:(i) 1 D 5 OD (p, r) for all D 5 N ;(ii) kt + q t + Bt = ct + /t+1 with kt + q t  0 for all t;(iii) kt + q t + Bt 5 /t for all t;(iv) pt31 q t = zt for all t;(v) /(k+1)T = /kT .
In other words, at a RS, (i) every country optimises. Conditions (ii) and (iii) are standard
excess demand conditions: in the markets for ﬁnal goods and capital goods, respectively,
aggregate demand should not exceed aggregate supply in any period. Condition (ii) also
13

The existence of a reproducible solution is proved in the Addendum.

8

requires the RS to be non-trivial in that some production takes place in every period. Condition (iv) requires that the international credit market clears in every period. Finally, thereproducibility condition (v) is a generalisation of analogous equilibrium conditions imposedby Roemer [20, 21, 22]: it requires that every generation leaves to the following at least
as many resources as they inherited. As Roemer ([20], p.507) has argued, the notion of
reproducibility is one of the distinctive features of the concept of equilibrium in the formal
literature on exploitation theory (compared to a more standard concept of general equilibrium). It is also a standard condition in Ramsey-type growth models with a ﬁnite horizon(see, e.g., Morishima’s [16] classic model) and it is quite natural given that countries - rather
than individuals - are the focus of analysis. For, although each generation dies, the country
itself lives on, and so its capital stock should not be depleted.
In the rest of this section, we derive some preliminary results that describe the characteristics of the equilibria of the international economy. First, the strict monotonicity ofv implies that at any RS, it must be pt > 0 for all t. Next, given (p, r), at any t, lett )pt31 k
wtmax = maxkMP pt k3(1+r: by the assumptions on P , wtmax is well-deﬁned. Hence letklnopt k3(1+rt )pt31 kw
max
Pt (p, r) = k 5 P | wt =. Proposition 1 proves that only processes withkl

the highest return to labour are activated and, as is well-known in international economics,
even without an international labour market, wages are equalised in all countries at all t.
¢¡
Proposition 1: Let (p, r) , (1 D )DMN be a RS for E(lkT ). Then wtmax > 0 for all t, andktD , qtD 5 Ptw (p, r) for all D and all t.

Proof: 1. Suppose, contrary to the statement, that wtmax 5 0 for some t. Then 1 + rt > 0Dand at the solution to MP D , it must be kDt = q t = 0 for all D. To see this, note that, sinceD

pt > 0, all t, if either kDt  0 or q t  0 for some D, then by A1 agent D can reduce labour
performed and increase revenues by setting kt0D = qt0D = 0 and zt0 = zt + pt31 kD . Therefore
kt = q t = 0 which contradicts Deﬁnition 1(b).
2. The second part of the statement follows immediately from MP D .
pit
At any RS, 1 + rt = maxi pit31
must also hold at all t. For if there were some good i suchpit
> 1 + rt , at some t, then noting that by Proposition 1 wtmax > 0, at the solutionthat pit31

9to MP D the gross revenue would have the form of wtmax qltD + pt BtD with pt31 BtD = pt31 /tD and
qtD 5 Ptw (p, r), where qltD > 0 follows from the assumptions of . Then, ktD = 0, ztD = 0, andpt31 q Dt > 0 by pt > 0 and A1, all D 5 N . This contradicts Deﬁnition 1(iv).
Lemma 1 proves a useful property of the set of solutions of MP D .
pit
for allLemma 1: Let (p, r) be a price vector such that wtmax > 0 and (1 + rt ) = maxi pit31

t. For all D 5 N , if (kD , q D , z D , B D , cD , /D ) solves MP D , then (k0D , q 0D , z 0D , B D , cD , /D ) also
D
D
0D
0D
solves MP D whenever kt0D + qt0D = ktD + qtD , zt0D + pt31 k0D
t = zt + pt31 kt , kt , qt 5 P , andzt0D = 0, all t.
Proof: By construction, the constraints of MP D are all satisﬁed. Furthermore, qlt0D + klt0D =qltD + kltD , at all t and c0D = cD . Hence (k0D , q 0D , z 0D , B D , cD , / D ) yields the same welfare.
By Lemma 1, and the convexity of P , we can consider solutions of MP D with kD = 0,
without loss of generality.
DD= p(k+1)T 31 /kT, all D. Hence, at a
For all k, at the solution to MP D , p(k+1)T 31 /(k+1)T
D
D
= /kT
is feasible and optimal for
RS, /(k+1)T = /kT since p(k+1)T 31 > 0. Moreover, /(k+1)TDD
all D. Therefore, without loss of generality, we can focus on solutions with /(k+1)T
= /kT
.
¡
¢
Hence, if (p, r) , (1 D )DMN is a RS for E(lkT ), then it is also a RS for E(l(k+1)T ), and inwhat follows generation k = 0 can be considered without loss of generality.

A subset of equilibria of particular interest are those where agents optimise at an interior
solution. Thus:
Deﬁnition 2: An interior RS (IRS) for E(lkT ) is a RS such that for all D, 1 D 5 OD (p, r)
is an interior solution to MP D with cDt > 0 for all t.
The next result proves a necessary condition for an IRS.
¢¡
Lemma 2: Let (p, r) , (1 D )DMN be an IRS for E(l0 ). Then for all D 5 N , \D = \W =0W)
0 (L3\W )(k+1)T 31t{\Wt }t=kT
where  (L3\
= 4(1 + rt+1 ) wmaxt+1 , for all t.
wmax
t

t+1

Proof: 1. By Proposition 1, for all D 5 N , at the solution to MP D , it must be (1 + rt ) pt31 /tD +D, all t. Then it is immediate to prove that, at an interior sowtmax \Dt = pt ct + pt /t+110

v0 (cWD )lution 1 D 5 OD (p, r) to MP D , for all t and all D 5 N , it must be v0i (ctWD ) = ppjtit and
j

0

t

WD

max vi (ct )for all i, j. By the linear homogeneity of v(.), this implies that at
 (L  \WDt ) = wt
pit
0

cWD

cW>

jt

jt

an interior solution to MP D , at all t, it must be citWD = citW> for all D, > 5 N , and therefore
W>
W>
0
0
WD
0
vi0 (cWDt ) = vi (ct ) and  (L  \t ) =  (L  \t ) for all D, > 5 N .

The ﬁrst part of thestatement then follows from the strict concavity of .
v 0 (cW )2. At any t, let cWt 5 Rn+ be such that v0i (ctW ) = ppjtit , for all i, j. Then by step 1, atjan IRS, it must be

cWD
t

=

ktD cWt

where

ktD

=

t

D
(1+rt )pt31 /tD +wtmax \Wt 3pt /t+1
pt cWt> 0, at all t andfor all D 5 N . Take any two adjacent periods t,t + 1, and consider D 5 N such thatDD
> 0. Consider a small one-period perturbation of / D such that dktD =  pt1cW pt d/t+1
pt /t+1
t

t+1 )
D
D
and dkt+1= (1+rp d/t+1. By the linear homogeneity of v, the resulting change in welfare
pt+1 cWt+1 t
t+1 )
D
D
D
is v(cWt )dktD + 4v(cWt+1 )dkt+1= v(cWt ) pt1cW pt d/t+1+ v(cWt+1 )4 (1+r
pt d/t+1.
pt+1 cW
t

t+1

0

WD

v (c )
3. By step 1, at an IRS, at all t,  (L  \Wt ) = wtmax i pitt for all i and all D 5 N . By the
0

WD

W

t

t

t )
t)
linear homogeneity of v, this implies that 0 (L  \Wt ) = wtmax v(c= wtmax v(c. Using thept cWD
pt cW

0
W)
D
t
latter expression, the change in welfare can be written equivalently as   (L3\pt d/t+1+
wtmax

4 (1 + rt+1 )0 (L3\Wt+1 )D
pt d/t+1. Therefore a necessary condition for 1 D 5 OD (p, r) to be an inmax
wt+1

¸

0 (L3\Wt+1 )0 (L3\Wt )DD
pt d/t+1
terior solution to MP is that  wmax + 4(1 + rt+1 ) wmax
5 0 for all d/t+1
,
D

t

t+1

which holds only if the expression in brackets is equal to zero.

3

The International Class Structure

In this section, we derive the international class structure, where “classes of countries can bedeﬁned with reference to the use of the credit market” (Roemer [22], p.54). Let (a1 , a2 , a3 )be a vector where ai 5 {+, 0}, i = 1, 3, a2 5 {+, 0}, and “+” means a non-zero vector in theappropriate place. We extend Roemer’s deﬁnition of classes within each period t.
¡¢
Deﬁnition 3: Let (p, r) , (1 D )DMN be a RS for E(l0 ). Country D is said to be a memberof class (a1 , a2 , a3 ) within period t, or W Pt class (a1 , a2 , a3 ), if there is a 1 0D 5 OD (p, r) with/ 0D = / D such that (kt0D , zt0D , qt0D ) has the form (a1 , a2 , a3 ) in t.
There are eight conceivable classes (a1 , a2 , a3 ), but only four of them are theoretically11¡¢
relevant, as argued by Roemer [21]. At a RS (p, r) , (1 D )DMN for E(l0 ), at all t, let KDt ={(kt0D , zt0D , qt0D ) | 1 0D 5 OD (p, r) with / 0D = /D }. We say that KDt has a solution of the form

(a1 , a2 , a3 ) \ (a01 , a02 , a03 ) to mean that KDt contains a vector (kt0D , zt0D , qt0D ) of the form (a1 , a2 , a3 )but not one of the form (a01 , a02 , a03 ). Then:
Ct1 = {D 5 N | KDt has a solution of the form (+, +, 0) \ (+, 0, 0)} ,Ct2 = {D 5 N | KDt has a solution of the form (+, 0, 0)} ,Ct3 = {D 5 N | KDt has a solution of the form (+, 0, +) \ (+, 0, 0)} ,Ct4 = {D 5 N | KDt has a solution of the form (0, 0, +)} .
At all t, countries in Ct1 are net lenders in the global market; countries in Ct2 can optimise
without using the capital market; countries in Ct3 must borrow foreign capital to optimise;countries in Ct4 must borrow all of their operating capital. This deﬁnition of classes based on
credit relations conveys the intuition that a country’s position in the capital market aects
its international status.
Below, it is proved that in equilibrium the set of countries N can indeed be partitioned
into these four W Pt classes at all t. First, Lemma 3 proves that W Pt classes (+, +, +) and(0, +, +) can be ignored.
¡¢
Lemma 3: Let (p, r) , (1 D )DMN be an IRS for E(l0 ). Let 1 D 5 OD (p, r) be such that D isa member of W Pt class (+, +, +) or (0, +, +) in t. Then:if zt0D > pt31 q 0D
for all (kt0D , zt0D , qt0D ) 5 KDt , then D 5 Ct1 ;tfor some (kt0D , zt0D , qt0D ) 5 KDt , then D 5 Ct2 ;if zt0D = pt31 q 0Dt
if zt0D < pt31 q 0D
for all (kt0D , zt0D , qt0D ) 5 KDt , then D 5 Ct3 .
t
Proof: 1. By the convexity of MP D , it follows that if zt0D < pt31 q 0D
for some (kt0D , zt0D , qt0D ) 5tfor some other (kt00D , zt00D , qt00D ) 5 KDt , then there exists (kt000D , zt000D , qt000D ) 5
KDt and zt00D > pt31 q 00Dt. Therefore, the three cases in the statement are mutually
KDt such that zt000D = pt31 q 000Dt

12

exclusive and they decompose the set of agents with WtD > 0 into disjoint sets.
2. Suppose zt0D > pt31 q 0D
for all (kt0D , zt0D , qt0D ) 5 KDt . Construct (kt0D , zt0D , qt0D ) such thatt
kt0D = ktD + qtD , zt0D = ztD  pt31 q Dt > 0, and qt0D = 0. By the convexity of P , kt0D 5 P ,
D
D
0D
0D
0D
D
klt0D = kltD + qltD and zt0D + pt31 k0D
t = zt + pt31 kt . Then by Lemma 1, (kt , zt , qt ) 5 Kt . Thus,

it remains to show that there is no solution in KDt of the form (+, 0, 0). Suppose, by way ofcontradiction, that KDt contains a solution of the form (+, 0, 0) at t. Construct (kt0D , zt0D , qt0D )such that kt0D = 0, zt0D = pt31 kDt > 0, and qt0D = ktD + qtD . Clearly, qt0D 5 P , qlt0D = kltD + qltD
D
D
0D
0D
0D
D
and zt0D + pt31 k0D
t = zt + pt31 kt . Then by Lemma 1, (kt , zt , qt ) 5 Kt , contradicting theassumption that zt0D > pt31 q 0D
for all (kt0D , zt0D , qt0D ) 5 KDt .
t
3. The other two cases are proved similarly.
Lemma 3 substantiates the claim that a country’s class status is determined by its positionin the capital market: net lenders form the ‘upper’ strata of the global economy, whereas
net borrowers occupy ‘lower’ positions. It is therefore natural to investigate whether classstatus (and so a country’s position in the credit market) is determined by its wealth.
min

kt
=
Consider an IRS. At all t, let ktmin be deﬁned as follows: ktmin 5 Ptw (p, r), pt31
kmin
lt
h
i
k
minkMPtw (p,r) pt31
, and kltmin = \Wt , where \Wt > 0 is deﬁned in Lemma 2. Similarly, ktmax
kl
h
i
kmax
pt31 ktw (p,r)is given by: ktmax 5 Ptw (p, r), pt31=
max
, and kltmax = \Wt . Note that
max
kMP
k
kl
t
lt

pt31 kmin
5 pt31 kmax
and that ktmin , ktmax are well-deﬁned. Theorem 1 generalises one of thettmain results of Roemer’s theory of classes: at an IRS, W Pt classes are pairwise disjoint andexhaustive, and W Pt class status depends on a country’s wealth.14¡¢
Theorem 1 (The Dependence School Theorem): Let (p, r) , (1 D )DMN be an IRS for E(l0 )pit
for all t. Then at all t:such that 1 + rt > maxi pit31(i) D 5 Ct1 / pt31 kmax
< WtD ;t
5 WtD 5 pt31 kmax;(ii) D 5 Ct2 / pt31 kmintt(iii) D 5 Ct3 / 0 < WtD < pt31 kmint ;(iv) D 5 Ct4 / WtD = 0.
pit
The condition 1 + rt > maxi pit1
ensures that undertaking production activities is better than storing
goods to be sold at the end of the period. In order to interpret this condition, note that at a stationary IRS
with pt = pt1 it reduces to the familiar requirement that rt > 0.
14

13

Proof: 1. By Lemma 2, at an IRS, \Dt = \Wt = kltD + qltD > 0, all D 5 N .
2. By step 1, it immediately follows that at any t, D 5 Ct4 if and only if WtD = 0.
5 WtD 5 pt31 kmax
. We show that KDt has a
3. Consider part (ii). Suppose pt31 kminttsolution of the form (+, 0, 0). By step 1, and noting that 1 + rt > 0, at an IRS it must
D
0D
0D
0D
0D
0D
w
be wtmax \Wt + (1 + rt ) WtD = pt cWD
t + pt /t+1 , and any (kt , zt , qt ) with kt , qt 5 Pt (p, r),0D
D
klt0D + qlt0D = \Wt , zt0D = 0, and pt31 k0D
t + zt = Wt is part of an optimal solution. But

then, since pt31 kmin
5 WtD 5 pt31 kmax
, by the convexity of P , it follows that there exists
t
t
WD
D
00D
W
some kt00D 5 Ptw (p, r), such that wtmax klt00D + (1 + rt ) pt31 k00D
t = pt ct + pt /t+1 , with klt = \tand pt31 k00D
= WtD . Conversely, if D 5 Ct2 , then there exists kt 5 Ptw (p, r) such thattDDW
(1 + rt ) pt31 kt + wtmax klt = pt cWD
t + pt /t+1 , with pt31 kt = Wt and klt = \t , which implies

pt31 kmin
5 WtD 5 pt31 kmax.
tt
4. Parts (i) and (iii) are proved similarly.
In order to clarify the normative relevance of Theorem 1, note that country D’s wealth,WtD , can be seen as the main proxy for its level of development. Theorem 1 implies that inequilibrium less developed countries (with a lower WtD ) are net borrowers, whereas developed
countries (with a higher WtD ) are net lenders: a nation’s wealth (and development) level
determines its class status. Given that international markets for commodities and capital are
perfectly competitive, in equilibrium all countries enjoy beneﬁts from trade, as international
capital ﬂows allow poor countries to improve their lot. Yet, the IRS is also characterisedby a four-class structure which reﬂects the wealth hierarchy and an asymmetric relationshipbetween countries. For the economic development of the countries in Ct3 ^ Ct4 is cruciallydependent on the existence of the rich countries in Ct1 which export their capital to the poor,
whereas the rich in Ct1 could realise a certain economic development with full employment
by themselves alone. In this sense, Theorem 1 captures some of the key insights of the‘dependence school’ discussed in the Introduction, and is named accordingly.
As noted by an anonymous referee, Theorem 1 echoes a standard result in neoclassicalmodels in which foreign capital ﬂows into underdeveloped countries because their endowmentof capital per capita is lower and therefore its marginal product is higher. At a closer look,14

however, some important dierences emerge. Although our assumptions on technology allow
for smooth production sets, we are not postulating the existence of a dierentiable aggregate
production function and therefore cannot derive any conclusions concerning the marginal
productivity of capital. The world interest rate is unique because of perfect competition in
the capital market but in principle there may be multiple equilibrium values of rt in any
period, none of which necessarily linked to a notion of marginal product of capital. Capitalﬂows are not driven by the search of the investment yielding the highest return: they are
determined by agents’ optimising choices on consumption, savings and leisure. Thus, forexample, in any t, a wealthy nation uses its capital to ﬁnance internal production up to
the point where its citizens are spending the optimal amount of labour. It then lends any
remaining capital - at the given international rate - in order to increase consumption, or
savings. (And a similar logic drives borrowing decisions in poor countries.)

4

Exploitative International Relations

Exploitation in international relations is conceived of as the unequal exchange of labour
between countries as in Roemer [21, 22]: exploitative international relations are characterised
by systematic dierences between the labour ‘contributed’ by agents in country D and thelabour ‘received’ by them via their national income. As intuitive as this deﬁnition may
seem, in general economies the notions of labour ‘contributed’ and labour ‘received’ are notobvious. Indeed, the very existence of a general, consistent deﬁnition which preserves the
key insights of UE theory has been put into doubt. In this section, we develop an axiomatic
analysis of UE exploitation and characterise a class of deﬁnitions that satisfy three important
properties. In the next section, we prove that the class is nonempty.

4.1

A domain axiom

In economies with homogeneous labour, the labour ‘contributed’ by D coincides with thelabour performed by workers in D, \Dt .15 Outside of static, two-class Leontief economies
15

For a generalisation to economies with heterogeneous labour, see Veneziani and Yoshihara [30, 29].

15

with subsistence wages, instead, many dierent deﬁnitions of the labour ‘received’ by D
can be, and have in fact been proposed, which incorporate dierent normative and positive
views.16 In recent work, Yoshihara and Veneziani [33, 31, 30, 29] have proposed an axiom
that identiﬁes the domain of admissible UE deﬁnitions: it imposes some weak restrictions on
the notion of labour received and all of the main approaches satisfy it in static economies.
In this subsection, we generalise it to the dynamic context.
At the most general level, in UE theory, the amount of labour ‘received’ by agent D isdetermined with reference to some bundles that belong to a normatively relevant budget set
and capture D’s consumption opportunities. In static models, all of the main UE approaches
usually (albeit sometimes implicitly) assume that in equilibrium agents at least replace any
wealth used up in production in every period and stipulate that the reference bundles are, or
can be purchased with agents’ actual net incomes at all t. In the general dynamic framework
considered here, the relevant notion of agents’ net income is not obvious and the choice of
the appropriate budget set is not unambiguous.
¢¡
Let (p, r) , (1 D )DMN be a RS for E(l0 ). The gross income of each D 5 N at t is givenby (1 + rt ) pt31 /tD + wtmax \Dt . In order to identify D’s ‘net’ income at t in this context, thefund for replenishing D’s wealth pt31 /tD in the next period should be deducted after adjustingfor the dierence in prices between t  1 and t. To do so, we deﬁne the inﬂation index at
pt /t, taking /t as the inﬂation basket. Given this index, D’s wealth pt31 /tD at
t, Rt  pt31
/tt  1 is evaluated as being equivalent to Rt pt31 /tD at t. Then, D’s ‘net’ income at t can bedeﬁned as (1 + rt ) pt31 /tD + wtmax \Dt  Rt pt31 /tD , and it identiﬁes the normatively relevant setof commodity bundles ‘received’ by D:©
ª
Bt ((p, r) ; pt31 /tD , \Dt )  c 5 Rn+ | pt c = (1 + rt ) pt31 /tD + wtmax \Dt  Rt pt31 /tD .
In other words, in a general dynamic context the set of commodity bundles ‘received’ by
D at t is deﬁned counterfactually by considering the net income that could be devoted toconsumption if D decided only to replace its wealth, i.e. to carry forward the real asset value of
16

See, for example, Morishima [17] and Roemer [21]. See Yoshihara [31, 32] and Veneziani and Yoshihara[29] for a thorough discussion.

16

WtD to the next period.17 The reason for this choice is threefold. First, countries are interested
in wealth, rather than in a speciﬁc vector of capital endowments. Second, from a normative
perspective, for a given gross income, in every t exploitation status should not depend on
speciﬁc saving and investment decisions, or on the speciﬁc vector of productive endowmentspurchased. According to Roemer ([22], p.53), for example, the appropriate notion of UE
exploitation should be preference-independent. Third, it is immediate to show that thefocus on bundles in Bt ((p, r) ; pt31 /tD , \Dt ) is a generalisation of the standard approach andit reduces to the latter at a RS with stationary prices and capital.
Let E denote the set of all economies E(P, N , u, 4, l0 ) that satisfy our assumptions ontechnology, agents, preferences and endowments. For all c 5 Rn+ , let  (c)  {k 5 P | k  c}be the set of production activities that can produce c as a portion of output. Given any
deﬁnition of exploitation, let Ntter  N and Ntted  N denote, respectively, the set of
exploiters at t, or W Pt exploiters, and the set of exploited agents at t, or W Pt exploitedagents, at a given allocation. Our domain axiom can now be formally introduced.
Labour Exploitation (LE): Consider any economy E(l0 ) 5 E. Let¡¢(p, r) , (1 D )DMN

be a RS for E(l0 ). Given any deﬁnition of UE exploitation, in each period t two subsets
Ntter  N and Ntted  N , Ntter _Ntted = B, constitute the set of W Pt exploiters and the set of
W Pt exploited agents if and only if for any D 5 N , there exist cDt , cDt 5 Bt ((p, r) ; pt31 /tD , \Dt )bct = pt cDt and kct 5  (cDt ) _ YP withsuch that there exist kct 5  (cDt ) _ YP with pt k
D

cD

D

D

cD

bct = pt cDt such that kl t = kl t , and the following condition holds,
pt k
D

cD

D 5 Ntter / \Dt < kl t ;cD

D 5 Ntted / \Dt > kl t .
LE requires UE exploitation status to be determined based on the labour contributed bycountries (the labour performed by their citizens) and on the labour received by them, where
the latter is determined in relation both to purchasing power, and to productive conditions.
17The set Bt ((p, r) ; pt1 $t , t ) does not necessarily contain ’s actual consumption bundle at t, as pt $t+1

may be dierent from Rt pt1 $t , in equilibrium.

17

To be speciﬁc, by LE under any admissible deﬁnition, in equilibrium the sets Ntter ,Ntted are characterised in each t by identifying two (possibly identical) reference bundles
cDt , cDt 5 Rn+ for each D 5 N . The reference bundles must be aordable for any D who
simply replaces its wealth at t (cDt , cDt 5 Bt ((p, r) ; pt31 /tD , \Dt )) and must be technically
producible as a portion of the output of e!cient production activities while carrying forward
the real asset value of the commodity inputs of such production activities to the next periodbc = pt c, c = cDt , cDt ).18 The labour contained in cDt , cDt is equal to the(kc 5  (c) _ YP with pt kcD

cDamount of labour required to produce them as a portion of output: respectively, kl t , kl t .
h D DicD
cD
ccGiven kl t = kl t , the (possibly degenerate) interval kl t , kl t is the labour that D can receive

via its ‘net income’ at t and it determines D’s UE exploitation status at t, once comparedwith the labour contributed by D, \Dt . In equilibrium, at any t, D is a W Pt exploiter if

and only if D works less than the minimum amount of labour that D can receive via its ‘net
cD

income’, kl t ; whereas D is W Pt exploited if and only if D works more than the maximumcD

amount of labour that D can receive via its ‘net income’, kl t .
LE is a weak condition that captures some fundamental insights of UE theory sharedby all of the main approaches in the literature.19 It identiﬁes the domain of admissible
UE deﬁnitions, but it cannot discriminate among alternative deﬁnitions within this domain,which can be large indeed. For this purpose, some additional properties must be imposed.

4.2

Class, Wealth and Exploitation

A fundamental insight of UE theory is the existence of a relation between development or wealth, - exploitation status, and class position in the global economy. The existence ofsuch a relation is often proved as a result in a given economic environment, under certain
conditions. Yet its central relevance in UE theory is such that “its epistemological status
in our understanding is as a postulate. We seek a model which will make our postulated
18
Note that pt 
bc = pt c implies pt c 5 pt c  pt c, where the left hand side represents the real asset valueof the commodity inputs of production activity c .
19
In particular, it is worth noting that axiom LE does not require UE exploitation status to be deﬁned
based on imputing embodied labor magnitudes to exchanged commodity bundles as in standard approaches.
But nor does it rule out the possibility that the labour received by  corresponds to the labour embodied ina speciﬁc bundle. We are grateful to an anonymous referee for pointing this out.

18

belief true” (Roemer [21], p.152). In this subsection, we state this intuition axiomatically andformalise two properties that incorporate the relation between wealth, class, and exploitation
status. Then, we provide a characterisation of the class of deﬁnitions of UE exploitation thatsatisfy LE and both properties, in the dynamic international economies considered here.
The ﬁrst property captures the intuition that richer countries are UE exploiters whileless developed countries suer from UE exploitation:¡¢
Wealth-Exploitation Correspondence Principle (WECP): Let (p, r) , (1 D )DMN be an

IRS for E(P, N , u, 4, l0 ) such that 1+rt Rt > 0 for all t. For each t, there exist W t , W t > 0PPwith W t = W t such that for any E(P, N , u, 4, l00 ) 5 E with DMN /00D = DMN /0D and any¡¢
PP
IRS (p, r) , (1 0D )DMN for E(P, N , u, 4, l00 ) with DMN /t0D = DMN /tD , all t:
Wt0D > W t / D 5 Ntter ;Wt0D < W t / D 5 Ntted .

WECP states that, in equilibrium, in any given period there should be two (possibly
equal) threshold wealth levels, W t , W t , such that the set of W Pt exploiters (resp., W Pt
exploited) corresponds to the set of countries with wealth higher than W t (resp., lower thanW t ). The threshold levels may depend on equilibrium prices and aggregate endowments,
but not on the equilibrium wealth distribution.
The next Lemma characterises the set of deﬁnitions that satisfy WECP.
¡¢
Lemma 4 (WECP): Let (p, r) , (1 D )DMN be an IRS for E(P, N , u, 4, l0 ) such that 1 +rt  Rt > 0 for all t. Given any deﬁnition of UE exploitation satisfying LE, the following
statements are equivalent:(i) WECP holds;(ii) at all t, there exist W t , W t > 0 with W t = W t such that for any E(P, N , u, 4, l00 ) 5
¡
¢
P
P
0D
D
(p, r) , (1 0D )DMN for E(P, N , u, 4, l00 ) with
E withDMN /0 =DMN /0 and any IRS

19

P

0D
DMN /t =P

D
DMN /t , all t, and for each D 5 N ,0D
max ctpt c0Dklt  wt0D
0D;
Wt > W t / Wt >1 + rt  Rt
0D
max ctpt c0Dklt  wt0D
0D
Wt < W t / Wt <.
1 + rt  Rt¡¢
Proof: 1. Consider any economy E(P, N , u, 4, l00 ) 5 E and any IRS (p, r) , (1 0D )DMN

for E(P, N , u, 4, l00 ). Given a deﬁnition of exploitation satisfying LE, at any t, for each
0D

0D
0D
0D
ct
5  (c0D
D 5 N , there exist c0D
t , ct 5 Bt ((p, r) ; Wt , \t ) such that there exist kt ) _ YP
c0D

c0D

ct
t
with pt k
bct = pt c0D
5  (c0D
bct = pt c0D= kl t andt ) _ YP with pt kt such that klt and k
0D

0D

0D

c0D

c0D

ted
t
t
D 5 Ntter / \0D/ \0D
t < kl , and D 5 Nt
t > kl .

2. In order to prove the result, it is su!cient to show that for any economy E(P, N , u, 4, l00 ) 5¡¢
E and any IRS (p, r) , (1 0D )DMN for E(P, N , u, 4, l00 ), the following conditions hold at any
t and for each D 5 N ,0D
max ctkl
pt c0Dt  wtter
0D;
D 5 Nt / Wt >1 + rt  Rt
0D
max ctkl
pt c0Dt  wtted
0D
D 5 Nt / Wt <.
1 + rt  Rt

(5)
(6)

c0D

0D
0D
0D
t
Consider (5). By LE, D 5 Ntter / \0D
t < kl . Moreover, ct 5 Bt ((p, r) ; Wt , \t ) implies
c0D

c0D

0D

0D

pt ct 3(1+rt 3Rt )Wt0D
max 0Dtt\t . Therefore \0D
and thept c0Dt = (1 + rt  Rt ) Wt +wt
t < kl / kl >wmax
tdesired inequality follows by rearranging the latter expression and noting that 1+rt Rt > 0.
A similar argument proves that (6) also holds.
Theorem 1 and Lemma 4 provide two dierent partitions of the set of countries, according
to their UE exploitation or class status. Depending on the UE deﬁnition that one adopts,
the two partitions may or may not coincide. Yet, an important intuition of UE theory is theexistence of a robust relation between class and UE exploitation status. Based on Roemer[21], we formulate this intuition explicitly as follows:¢¡
Class-Exploitation Correspondence Principle (CECP): Let (p, r) , (1 D )DMN be an
20

IRS for E(l0 ) such that 1 + rt  Rt > 0 for all t. Then, at all t
D 5 Ct1 , D 5 Ntter ;D 5 Ct3 ^ Ct4 , D 5 Ntted .
CECP states that in equilibrium, countries in the upper classes in the credit market
should emerge as UE exploiters, while those in the lower classes should be UE exploited.
Theorem 2 provides necessary and su!cient conditions for a UE deﬁnition in the admissible
domain to satisfy both WECP and CECP:¢¡
Theorem 2 (CECP): Let (p, r) , (1 D )DMN be an IRS for E(P, N , u, 4, l0 ) such that 1 +pit
for all t. Given any deﬁnition of UE exploitation satisfying LE, the following
rt > maxi pit31

statements are equivalent:(i) WECP and CECP hold;
5 W t 5 W t 5 pt31 kmax
such that for(ii) at all t, there exist W t , W t > 0 with pt31 kmin
t
t
¡
¢
P
P
0D
D
0D
any E(P, N , u, 4, l00 ) 5 E with
for
DMN /0 =DMN /0 and any IRS (p, r) , (1 )DMN
PP
E(P, N , u, 4, l00 ) with DMN /t0D = DMN /tD , all t, and for each D 5 N ,0D
max ctkl
pt c0Dt  wt0D
0D;
Wt > W t / Wt >1 + rt  Rt
0D
max ctkl
pt c0Dt  wt0D
0D
Wt < W t / Wt <.
1 + rt  Rt¡¢
Proof: 1. Consider any economy E(P, N , u, 4, l00 ) 5 E and any IRS (p, r) , (1 0D )DMN forpit
for all t implies 1 + rt  Rt > 0 for all
E(P, N , u, 4, l00 ). Note that 1 + rt > maxi pit31

t. Given a deﬁnition of exploitation satisfying LE, at any t, for each D 5 N , there exist
0D
0D
0D
ct
c0D
5  (c0D
bct = pt c0D
t , ct 5 Bt ((p, r) ; Wt , \t ) such that there exist kt ) _ YP with pt k
t and
0D

c0D

0D

c0D

c0D

ter
t
t
t
kct 5  (c0D
bct = pt c0D
/ \0D
and
t ) _ YP with pt kt satisfying: kl = kl , and D 5 Nt
t < kl
0D

0D

c0D

t
D 5 Ntted / \0D
t > kl .

2. ((ii),(i)) Suppose that (ii) holds. Then by Lemma 4 and Theorem 1 it immediately
follows that WECP and CECP hold.

21

3. ((i),(ii)) Let WECP and CECP hold. By Lemma 4, it is su!cient to show that at
all t, W t 5 pt31 kmax
and pt31 kmin
5 W t.
tt, some t. We consider two cases.
Suppose, by way of contradiction, that W t > pt31 kmaxtCase 1: suppose that pt31 kmax
= pt31 /t0D , all D 5 N , for any E(P, N , u, 4, l00 ) 5 E with
t
¡
¢
P
P
P
0D
D
0D
0
0D
DMN /0 =DMN /0 and any IRS (p, r) , (1 )DMN for E(P, N , u, 4, l0 ) with
DMN /t =PD1max, by WECPDMN /t , all t. By Theorem 1, Ct = B. Moreover, because W t > pt31 kt
0

Ntter = B. But then, noting that the same holds for any W t = pt31 kmax
and that WECPt

does not require wealth thresholds to be unique, it is possible to set W t = pt31 kmaxt
P
Case 2: suppose that there exists an economy E(P, N , u, 4, l00 ) 5 E with DMN /00D =
¡
¢
P
P
P
D
0D
D
for E(P, N , u, 4, l00 ) with DMN /t0D =DMN /0 and an IRS (p, r) , (1 )DMN
DMN /t ,all t, such that pt31 kmax
< pt31 /t0D , for some D 5 N . If pt31 kmax
< pt31 /t0D 5 W t ,ttthen the desired contradiction follows from Theorem 1, CECP, and WECP. So, sup< W t < pt31 /t0D . Then by Lemma 2 it is immediate to show thatpose that pt31 kmaxt
P
P
D
there exists another economy E(P, N , u, 4, l000 ) 5 E with DMN /000D =DMN /0 and an
¡
¢
P
P
D
IRS (p, r) , (1 00D )DMN for E(P, N , u, 4, l000 ) with DMN /t00D =DMN /t , all t, such that
< pt31 /t00D 5 W t for some D 5 N , which yields the desired contradiction.
pt31 kmaxt
A similar argument can be used to prove that pt31 kmin
5 W t.
t
These results fully characterise exploitative international relations in the intertemporalmodel. Theorem 1 identiﬁes the structure of the global capital market, in which developed
countries emerge as net lenders and less developed countries as net borrowers. Lemma 4
derives necessary and su!cient conditions for exploitative international relations to map
inequalities in economic development (proxied by the value of productive endowments).
Finally, Theorem 2 provides necessary and su!cient conditions for class positions in the
global credit market to map wealth inequalities and exploitation status.
It is worth stressing at this point the generality of our results, which are derived inintertemporal economies, under rather general assumptions on preferences and technology,
and without restricting the analysis to steady states. Perhaps more importantly, they arederived without adopting any speciﬁc UE approach: they hold for every deﬁnition within

22the admissible domain identiﬁed by LE. The relation between wealth, class and exploitationis thus proved to hold for an entire (and potentially large) class of UE deﬁnitions.

5

A Deﬁnition of UE Exploitation

Section 4 provides a complete characterisation of the class of UE deﬁnitions that satisfy
LE, WECP and CECP. But are there any deﬁnitions that actually meet the conditions in
Theorem 2? This is not an idle question. Yoshihara [31] has shown that in static economies
with revenue-maximising agents, some of the received deﬁnitions - including Morishima’s [17]and Roemer’s [21] - satisfy LE but not CECP. Roemer [21, 23] himself has raised doubtson the robustness of the relation between wealth, exploitation, and class. In this section,
we show that the class of deﬁnitions identiﬁed by Theorem 2 is nonempty. To this end,
we introduce a dynamic generalisation of a deﬁnition recently proposed by Yoshihara and
Veneziani [33, 31, 29] and show that it satisﬁes LE and preserves WECP and CECP, in
the international economies considered in this paper.
¢¡
Let (p, r) , (1 D )DMN be a RS for E(l0 ) and let ktp,r + qtp,r denote the aggregate equilib³´
rium production activity at t. For any c 5 Rn+ , such that pt c 5 pt k
btp,r + qbtp,r , the labour³´
btp,r + qbtp,r =content of c is equal to tc (kltp,r + qltp,r ), where tc 5 [0, 1] is such that tc pt k
pt c.20 Thus, the labour contained in any bundle c (whose value does not exceed global in-of social labour, c (kltp,r + qltp,r ), necessary to produce a³´
fraction of aggregate net output, c k
btp,r + qbtp,r , that has the same value as c. We denotecome) is equal to the fraction

cthis as l.v. (c; (p, r) , ktp,r + qtp,r ): it is the labour value of c at t, at a RS with prices (p, r)
and aggregate production, ktp,r + qtp,r . Then:¡¢
Deﬁnition 4: Consider any economy E(l0 ) 5 E. Let (p, r) , (1 D )DMN be a RS forE(l0 ).

At all t, country D 5 N , which supplies \Dt , is W Pt -exploited if and only if\Dt > l.v. (ecDt ; (p, r) , ktp,r + qtp,r ) for any ecDt 5 Bt ((p, r) ; pt31 /tD , \Dt ), and a W Pt -exploiter

cDt ; (p, r) , ktp,r + qtp,r ) for any ecDt 5 Bt ((p, r) ; pt31 /tD , \Dt ).
if and only if \Dt < l.v. (e20³´If pt btp,r + btp,r = 0, we set tc = 0 by deﬁnition.

23

Deﬁnition 4 generalises the deﬁnition proposed by Yoshihara and Veneziani [33, 31, 29]
in a static context. In static economies, exploitation status is measured with reference toconsumption bundles that agents can purchase with their actual net income. As argued in
section 4.1 above, however, in the general dynamic framework considered here, actual net
income is not necessarily a normatively appropriate variable to determine agents’ exploitation status and we propose to focus instead on the net income that could be devoted toconsumption if an agent decided only to replace its wealth.
Deﬁnition 4 is conceptually related to the ‘New Interpretation’ (Duménil [6, 7]; Foley [12];Duménil et al [8]). In fact,hcDt

is D’s reference share of world income, and sohcDt(kltp,r + qltp,r )cDt . Then,is the share of total social labour that D receives by earning national income pte
as in the New Interpretation, country D is UE exploited if and only if the amount of social
labour it receives is less than the amount of labour expended by its workers, \Dt .
Several attractive features of Deﬁnition 4 are worth noting. First, it does not rely on thelabour theory of value and is not restricted to economies with a linear technology. Second,
unlike in the standard approach, exploitation is not a merely technological phenomenon andsocial relations play a central role. For in Deﬁnition 4 the deﬁnition of UE exploitation
requires knowledge of equilibrium prices and of the social reproduction point, and it isrelated to the production and distribution of global income and social labour. Third, UE
exploitation is identiﬁed as a feature of the competitive allocation of social labour rather thanas the result of productive ine!ciencies, or labour market imperfections. Fourth, Deﬁnition 4transparently captures the key intuitions of UE theory. For it identiﬁes exploitation status bycomparing the labour contributed by each country D and the share of aggregate social labour
received by D via its national income. Moreover, Yoshihara and Veneziani [33] have shownthat in a rich domain of (static) convex economies, Deﬁnition 4 is the only UE deﬁnition
that satisﬁes a small set of formally weak and theoretically desirable properties.
Theorem 3 proves that if Deﬁnition 4 is adopted then both WECP and CECP hold.
¢¡pit
for all
Theorem 3: Let (p, r) , (1 D )DMN be an IRS for E(l0 ) such that 1 + rt > maxi pit31

t. Then, under Deﬁnition 4, WECP and CECP hold.

24

¡
¢
Proof: 1. First, we show that Deﬁnition 4 satisﬁes LE at an IRS. Since (p, r) , (1 D )DMN

= /t , at all t.
is an IRS for E(l0 ), it follows that (ktp,r + qtp,r ) 5 Ptw (p, r) and ktp,r + q p,rt

Further, by Lemma 2, \Dt = \Wt for all t and all D 5 N . At all t, let (wtD )DMN 5 [0, 1]N be suchP
that DMN wtD = 1 and pt31 /tD = wtD pt31 /t for each D 5 N . Then, at all t, and for all D 5 N(1 + rt ) pt31 /tD + wtmax \Wt  Rt pt31 /tD= (1 + rt  Rt ) wtD pt31 /t + wtmax \Wt
i
h ³
´
p,r
p,r
p,r
p,r
D
max
b
= wt pt k
bt + qt
 wt (klt + qlt ) + wtmax \Wt .

Then, because 1 + rt  Rt > 0 and wtmax > 0, all t, in each period t, there exists ( tD )DMN 5
P(0, 1)N such that DMN tD = 1 and
D
t pt

³
´
h ³
´
i
k
btp,r + qbtp,r = wtD pt k
btp,r + qbtp,r  wtmax (kltp,r + qltp,r ) + wtmax \Wt .

According to Deﬁnition 4, at all t, D 5 Ntter if and only if \Wt < tD (kltp,r + qltp,r ); and D 5 Ntted
´
³
pt (k
ep,r +qep,r )p,rif and only if \Wt > tD (kltp,r + qltp,r ). By taking cDt = cDt = tD p ktp,r +qtp,r ktp,r + q t
and
t( t
t )
D
D
kct = kct = tD (ktp,r + qtp,r ), for all D 5 N , we can see that Deﬁnition 4 satisﬁes LE.
2. By step 1, it su!ces to show that under Deﬁnition 4, statement (ii) of Theorem 2¡¢
pit
holds. Let (p, r) , (1 D )DMN be an IRS for E(P, N , u, 4, l0 ) such that 1 + rt > maxi pit31

for all t. We show that at all t, W t = W t = WtW  N1 pt31 /t > 0 satisﬁes all conditions instatement (ii).
P
P
D
First of all, note that for any E(P, N , u, 4, l00 ) 5 E with DMN /00D =DMN /0 and¡¢PP
any IRS (p, r) , (1 0D )DMN for E(P, N , u, 4, l00 ) with DMN /t0D = DMN /tD , all t, WtW is welldeﬁned, unique and invariant.

Further, for any E(P, N , u, 4, l00 ) 5 E with

P

25

0D
DMN /0 =P

D
DMN /0 and any IRS¡¢(p, r) , (1 0D )DMN

for E(P, N , u, 4, l00 ) with
c0D
t

b
pt k

P

0D
DMN /t =P

D
DMN /t , all t, for each D 5 N , we haveh ³´
i
¡ 0p,r0p,r0p,r0p,r ¢
maxb
pt k
bt + qt
 wt
klt + qlt
h
³
´
³
´i
0p,r
0p,r
0p,r
+
q

p
k
+
q
= t0D (1 + rt ) pt31 k0p,r
t
t
t
t
t

c0D
 wtmax kl t =0Dt= t0D (1 + rt  Rt ) pt31 /t .

where the ﬁrst equality follows from step 1, the second equality follows from the fact that¢
¡ 0p,rkt + qt0p,r 5 Ptw (p, r) at a RS, and the last equality follows from the deﬁnition of Rt
0D

0D

noting that at a RS k0p,r+ q 0p,r= /t . Then, since pt k
bct = pt c0D
bct = pt c0Dt
t = pt k
t and
t
P
P
0D
0D
kct = kct by step 1, it follows that for any E(P, N , u, 4, l00 ) 5 E with DMN /00D = DMN /0D¡¢
PP
and any IRS (p, r) , (1 0D )DMN for E(P, N , u, 4, l00 ) with DMN /t0D = DMN /tD , all t, foreach D 5 N ,0D
max ctpt c0Dklt  wt0D
W
0D;
Wt > Wt / Wt >1 + rt  Rt
0D
max ctkl
pt c0Dt  wt0D
W
0D
Wt < Wt / Wt <.
1 + rt  Rt

Finally, since WtW = N1 pt31 /t = pt31
5 WtW 5 pt31 kmax.
pt31 kmintt³ k0p,r +q 0p,r ´ ¡
¢
0p,r
0p,r
t
t
,
k
+
q
5 Ptw (p, r) implies thatttN

In summary, statement (ii) of Theorem 2 holds under Deﬁnition 4 and therefore WECP
and CECP hold under Deﬁnition 4.
Theorem 3 implies that the set of deﬁnitions identiﬁed in Theorem 2 is nonempty. If Deﬁnition 4 is adopted, then both Roemer’s [21] Class-Exploitation Correspondence Principle
and the Wealth-Exploitation Correspondence Principle can be extended to general, dynamic
equilibrium paths of international economies with general convex technologies and welfare
functions. Theorems 1 and 3 allow us to identify the structure of dependent and exploitativeinternational relations emerging between developed and less developed countries as the equilibrium outcome of a perfectly competitive international economy. Mutual beneﬁts from freeinternational trade of commodities and capital coexist with an international stratiﬁcation of26

countries in the credit market and with unequal ﬂows of revenue and labour.

6

Conclusion

This paper develops an axiomatic analysis of the concept of unequal exchange between
countries in a dynamic general equilibrium model, which generalises Roemer’s [21, 22] economy with a global capital market. The class of deﬁnitions that preserve three fundamental
properties of UE exploitation theory - including the existence of a correspondence between
wealth, class and exploitation status, - in general dynamic equilibria is completely characterised. This class is shown to be nonempty: there exists a UE deﬁnition that satisﬁes abasic domain axiom and both the Wealth-Exploitation Correspondence Principle and theClass-Exploitation Correspondence Principle. This deﬁnition is conceptually related to theso-called ‘New Interpretation’ (Duménil [6, 7]; Foley [12]; Duménil et al [8]). It is logicallyconsistent, general, and ﬁrmly anchored to empirically observed data.
Based on this deﬁnition, unequal international relations are fully characterised and Roemer’s [21, 22] results generalised. In equilibrium, countries are partitioned based on their
UE exploitation status and on their position in the capital market: advanced countries are
net lenders and exploiters, less developed countries are net borrowers and suer from UE
exploitation. Mutual gains from trade and UE exploitation coexist in competitive markets. The exploitative nature of international relations is the product of capital ﬂows, whichtransfer surplus from less developed to more developed countries.
It may be objected that in reality capital ﬂows do not move in the direction predicted in
Theorem 1: as Lucas [14] famously argued, capital does not ﬂow to underdeveloped countries.
Two points should be made here that suggest that the so-called “Lucas paradox” does notpose a major challenge to our conclusions. Firstly, there is a growing empirical literaturesuggesting that there may be no paradox after all: the actual pattern of credit ﬂows may
be explained by some fundamental violations of the basic assumptions of the neoclassicalmodel which are also shared by Roemer’s [21, 22] model and our extension of it, such as
dierences in fundamentals, including institutional quality (Alfaro et al [1]), and in credit27risk (Reinhart and Rogo [18]). Actually, Reinhardt et al ([19], pp.235-6) have shown that“the prediction of the standard neoclassical theory holds only when taking into account thedegree of capital account openness, conditional on a set of fundamentals. Among countries
with an open capital account, richer countries tend to experience net capital outﬂows, while
poorer countries tend to experience net capital inﬂows. In contrast, in countries with closed
capital account, there appears to be no systematic relationship between the level of economic
development and net capital ﬂows.” According to them, the phenomenon observed by Lucas[14] is due to the relatively high degree of capital controls in the period he considered.
The paradox, however, disappeared in the 1990s when capital controls were lifted and “thisliberalization process was associated with signiﬁcant changes in the patterns of capital ﬂows
across countries at dierent income levels” (Reinhardt et al [19], p.236).21
Secondly, and perhaps more importantly, ours is not an investigation of the determinants of the size and direction of capital ﬂows in the current global economy. This paper
develops a normative, axiomatic analysis of the concept of exploitation and for this purpose
we have followed Roemer ([22], p.58) in using Occam’s razor and abstracting from “manyelements of friction or noncompetitiveness or ‘imperfections’”. These simplifying assumptions are theoretically appropriate from a normative perspective, as they allow us to deriveprecise conclusions on the concept of UE in an abstract and general, but at the same time
well-deﬁned context, focusing on a benchmark competitive economy. Our main axioms arethus restricted to hold at the competitive equilibria of the intertemporal economy with unrestricted capital ﬂows and no labour mobility - the “classical [UE] environment” (Roemer[22], p.53). The appropriate extension of our axiomatic analysis to noncompetitive settingsand disequilibrium allocations is an interesting and open question.
Our analysis thus provides a normative benchmark to evaluate international relations
under globalisation. For, inequalities in wealth and development among countries are atleast partly due to past “robbery and plunder” - especially during the colonial period 21

One may argue that such movements are still smaller than predicted by the standard neoclassical model
based on international dierences in the marginal product of capital. As Lucas ([14], p.92) put it, “one would
expect no investment to occur in the wealthy countries in the face of return dierentials of this magnitude.”This issue is not really relevant in our framework, especially given that, as noted in section 3, international
capital ﬂows are not determined by dierences in the marginal productivity of capital.

28

which makes them, and the unequal exchanges and exploitative relations resulting fromthem hardly justiﬁable, as argued by Ferguson and Veneziani [10]. To be sure, the radical
change in ownership relations in the world economy necessary to eliminate UE exploitation
may be considered politically infeasible. This does not make the concept of UE exploitation
any less relevant. For it is essential to establish a robust normative benchmark against whichto evaluate international relations, and even if it is not possible to eliminate UE exploitationin one stroke, there may be a number of measures to reduce it via international transfers
and redistribution. An interesting question from this perspective concerns the developmentof a measure of the degree or intensity of UE exploitation of each country, and an index of
aggregate UE exploitation in the international economy that goes beyond the rather coarse
classiﬁcation into UE exploiting and UE exploited nations. We leave this issue for further
research.

References[1] Alfaro, L., Kalemli-Ozcan, S., and V. Volosovych (2008): “Why Doesn’t Capital Flowfrom Rich to Poor Countries? An Empirical Investigation,” The Review of Economics
and Statistics 90, 347-368.
[2] Amin, S. (1976): Unequal Development: An Essay on the Social Formations of Peripheral Capitalism, Monthly Review, New York.
[3] Bacha, E.L. (1978): “An Interpretation of Unequal Exchange from Prebisch-Singer to
Emmanuel,” Journal of Development Economics 5, 319-330.
[4] Ben-David, D. (1993): “Equalizing Exchange: Trade Liberalization and Income Convergence,” Quarterly Journal of Economics 108, 653-679.
[5] Chen, Z. (1992): “Long-run Equilibria in a Dynamic Heckscher-Ohlin Model,” Canadian
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29[7] Duménil, G. (1984): “The So-Called ‘Transformation Problem’ Revisited: A Brief Comment,” Journal of Economic Theory 33, 340-348.
[8] Duménil, G., Foley, D.K., and D. Lévy (2009): “A Note on the Formal Treatment of
Exploitation in a Model with Heterogeneous Labor,” Metroeconomica 60, 560-567.
[9] Emmanuel, A. (1972): Unequal exchange, Monthly Review Press, New York.
[10] Ferguson, B. and R. Veneziani (2017): “Territorial Wrongs and Colonial Rights,” mimeo,
VU Amsterdam and Queen Mary University of London.
[11] Frank, A.G. (1978): Dependent accumulation and underdevelopment, Monthly Review
Press, New York.
[12] Foley, D.K. (1982): “The Value of Money, the Value of Labor Power, and the Marxian
Transformation Problem,” Review of Radical Political Economics 14, 37-47.
[13] Gri!n, K., and J. Gurley (1985): “Radical Analyses of Imperialism, The Third World,
and the Transition to Socialism: A Survey Article,” Journal of Economic Literature 23,1089-1143.
[14] Lucas, R. (1990): “Why Doesn’t Capital Flow from Rich to Poor Countries?” American
Economic Review 80, 92-96.
[15] Marx, K. (1968): Theories of Surplus Value, Vol.III, Lawrence & Wishart, London.
[16] Morishima, M. (1969): Theory of Economic Growth, Oxford University Press, Oxford.
[17] Morishima, M. (1974): “Marx in the Light of Modern Economic Theory,” Econometrica
42, 611-632.
[18] Reinhart, C.M. and K.S. Rogo (2004): “Serial Default and the ‘Paradox’ of Rich-toPoor Capital Flows,” American Economic Review 94, 53-58.
[19] Reinhardt, D., Ricci, L.A., and T. Tressel (2013): “International capital ﬂows anddevelopment: Financial openness matters,” Journal of International Economics 91,
235-251.
30

[20] Roemer, J.E. (1980): “A General Equilibrium Approach to Marxian Economics,”
Econometrica 48, 505-530.
[21] Roemer, J.E. (1982): A General Theory of Exploitation and Class, Harvard University
Press, Cambridge.
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Flows: A Theoretical Synthesis,” in: Desai, P. (Ed.), Marxism, Central Planning and
the Soviet Economy, MIT Press, Cambridge.
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New York.
[24] Roemer, J.E. and R. Veneziani (2004): “What we owe our children, they their children. . . ,” Journal of Public Economic Theory 6, 637-654.
[25] Silvestre, J. (2007): “Intergenerational equity and human development,” in: Roemer,
J. E. and K. Suzumura (Eds.), Intergenerational Equity and Sustainability, MacMillan,
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[26] Slaughter, M.J. (2001): “Trade Liberalization and per capita Income Convergence: aDierence-in-dierences Analysis,” Journal of International Economics 55, 203-228.
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[28] Veneziani, R. (2013): “Exploitation, Inequality, and Power,” Journal of Theoretical
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32

Existence of an Interior Reproducible
Solutions: Addendum to “Globalisation andInequality in a Dynamic Economy: an
Axiomatic Analysis of Unequal Exchange”
Roberto Venezianiand Naoki Yoshihara†
February 25, 2017

Abstract
This addendum proves the existence of an equilibrium for the
economies analysed in the paper. The proof shows the consistencyof the formal framework and establishes that the axioms in the paper
do not hold vacuously.


School of Economics and Finance, Queen Mary University of London, Mile End Road,
London E1 4NS, UK. E-mail: r.veneziani@qmul.ac.uk†(Corresponding author) Department of Economics, University of Massachusetts
Amherst, Crotty Hall, 412 North Pleasant Street, Amherst, MA 01003, USA; The Institute of Economic Research, Hitotsubashi University, Kunitachi, Tokyo 186-0004, Japan;
and School of Management, Kochi University of Technology, Kami-city, Kochi 782-8502,
Japan. E-mail: n_yoshihara_1967@yahoo.co.jp

1

1

Existence of an Interior Reproducible Solution

In this Addendum, we prove the existence of an IRS in the economy E(P, N , u, 4, l0 )with T = +4. Similar arguments hold in the case with T < +4.
To begin with, consider the following social planner’s optimisation programme MP sp :MP sp :

max{(ct ,\t ,B t ,kt ,/ t+1 )}+"
t=0

+"
X
t=0

4t (v (ct ) + N (L  \t ))subject to\t
N\t
kt + Bt
kt + Bt(klt , kt , kt )
/0

5
=
5
=
5

[0, L] , (;t) ;klt , (;t) ;/t , (;t) ;ct + / t+1 , (;t) ;P , (;t) ,X/ D0 .
DMN

To solve MP sp , we deﬁne the feasibility constraint correspondence [P :Rn+ ³ Rn+ as:[P (/t )ª©
 / t+1 5 Rn+ | < (c, kt ) 5 Rn+ × P : klt 5 NL, kt 5 /t , kt + (/t  kt ) = c + /t+1 .

This correspondence is non-empty, compact-valued, and continuous. Let©
ªxP = (/t , / t+1 ) 5 Rn+ × Rn+ | / t+1 5 [P (/ t )be the graph of [P . For any given / 0 , the set of feasible sequences is©
ª
F P (/ 0 ) = (/ 0 , / 1 , . . . , /t , . . .) | / t+1 5 [P (/t ) for all t .
2This correspondence is called the plan correspondence and it is non-empty.
As in Aliprantis and Camera (2009, Theorems 2 and 3), the plan correspondence F P can be proved to be non-empty, compact-valued, graph-closed, andcontinuous. For each t, for each (/ t , /t+1 ) 5 xP , let the one-period return
function F : xP $ R be
F (/ t , / t+1 ) max(ct ,kt )MRn+ ×P

v (ct ) + N (L  \t )subject to N\t = klt 5 NL, kt 5 / t , kt + (/ t  kt ) = ct + / t+1 .
By Berge’s Maximum Theorem, F is continuous, as {/ t }×[P (/ t ) is compactvalued and continuous. Then, MP sp reduces to the following programme
MP sprd :+"X
max
4t F (/t , / t+1 ).
(/ 0 ,/ 1 ,...,/ t ,/t+1 ,...)MF P (/0 )t=0Following the arguments in Aliprantis and Camera (2009, Lemma 7(a)),
thesocial planner’s
programme MP sprd has an optimal solution¡ reduced¢WWWP/ 0 , /1 , . . . , / t , / t+1 , . . . 5 F (/ 0 ). Then, correspondingly, there exists an©¡¢ª+"optimal solution cWt , \Wt , B Wt , kWt , / Wt+1 t=0 to the programme MP sp . Let theset of optimal solutions to the programme MP sp be denoted by O(/ 0 ).
Given the deﬁnition of IRS, we consider the following class of optimal
solutions to programme MP sp :Deﬁnition A.1: An non-trivial
to programme©¡ interior optimal¢ªsolution+"
MP sp is an optimal solution cWt , \Wt , B Wt , kWt , / Wt+1 t=0 5 O(/ 0 ) such thatcWt > 0, 0 < \Wt < L, and kWt  0 for all t.

By the assumptions on preferences and technology, it is not di!cult to showthat the set O(/ 0 ) contains non-trivial interior optimal solutions.1©¡¢ª+"Let cWt , \Wt , B Wt , kWt , / Wt+1 t=0 be a non-trivial interior solution to programme MP sp . Given that the production possibility set P is convex and
For instance, to ensure t  0 for all t, assume on the contrary that t = 0 for somet. Then, lt = 0 = t and t = 0 must hold for the optimality. Then, ct + $ t+1 = $ t .
However, by A2 and the cone property of P , there exists 0 5 P such that 0l > 0 is
su!ciently small, 0 5 $ t , and 0 + ($ t  0 ) > ct + $ t+1 . Then, by the property of
the function !, for a su!ciently small positive labour supply 0 = 0l , it follows thatv (c0 ) + ! (0 ) > v (ct ) + ! (t ) for c0  0 + ($ t  0 )  $ t+1 , which is a contradiction.
Therefore, t  0 must hold for all t.
13

closed, we can deﬁne an implicit production function KP : R+ ×Rn+ × Rn+ $R as:(kl , k, k) 5 P / KP (kl , k, k) 5 0; and(kl , k, k) 5 YP / KP (kl , k, k) = 0.
This function is continuous
convex. Then,¢ª+" it is easy to show the existence©¡ 0 and0000of a feasible sequence ct , \t , B t , kt , / t+1 t=0 such that for each t, k0t + B 0t </0t , k0t + B 0t > c0t + / 0t+1 , and KP (k0lt , k0t , k0t ) < 0, and therefore we can apply
the Kuhn-Tucker Theorem. Thus,³ there exists´ a proﬁle of non-zero and nonnegative Lagrange multipliers >lt , >t , >t ,  t 5 R+ × Rn+ × Rn+ × R++ for¢ª+"©¡each t such that the non-trivial interior solution cWt , \Wt , B Wt , kWt , / Wt+1 t=0
to MP sp is an optimal solution to the following maximisation programme
MP L :
max{(ct ,\t ,B t ,kt ,/ t+1 )}+"
t=0

+"
X
t=0

4t [v (ct ) + N (L  \t )  >lt (klt  N\t )  >t (kt + B t  / t )+>t (kt + B t  ct  /t+1 )   t KP (klt , kt , kt )].
P
Note that  t > 0³must hold.
´ Then, since K is convex, there exists a suitablenon-zero vector qlt , qt , q t for each t such that

qlt kWlt + qt kWt + q t kWt = 0; andqlt klt + q t kt + qt kt 5 0 for all (kl , k, k) 5 P ,and moreover the following ﬁrst order condition holds forfor each t,
uv (cWt )  >t0 (L  \Wt ) + >lt
>lt   t qlt
>t   t q t>t   t q t
>t + 4>t+1³´
>t + >t B Wt
4©¡ W W W W W ¢ª+"ct , \t , B t , kt , / t+1 t=0 :

0,
0,
0,
0,
0,
0,

(A1)
(A2)
(A3)
(A4)
(A5)
(A6)

= 0.

(A7)

=
=
=
=
=
=

From (A1), >t = uv (cWt ) > 0. From (A2), >lt = 0 (L  \Wt ) > 0. Then, by(A3) and  t > 0, qlt < 0. From (A5) and >t > 0, it follows that q t > 0.
From (A6) and >t > 0, it follows that >t+1 > 0. Then, by (A4), q t+1 < 0.
Moreover, from (A6) and (A4), >t = 4 t+1 q t+1 , which in turn implies by(A5) that  t qt = 4 t+1 qt+1 .
Let (wtW , pWt )  (qlt , q t ) for each t. Then,  t pWt = 4 t+1 q t+1 . Let
¡
¢
¢ W
¡
t
W
W
431 . Then, qt+1 = 1 + rt+1
  t+1
pt holds. In this case,
1 + rt+1
¡
¡
¢ W
¢
W
W
W
. From (A1), t pt =  t+1 4 1 + rt+1 pt holds, so that  t =  t+1 4 1 + rt+1
vi0 (cWt )0 (L3\Wt )(A2), (A3), and (A5),  t = wW
= W holds for each commodity i and¡ t W ¢ pit
each t. Therefore,  t =  t+1 4 1 + rt+1 implies the Euler equation
¡
¢
0
W
¡
¢
L

\

0 (L  \Wt )t+1
W= 4 1 + rt+1
for each t.
W
wtW
wt+1

Finally, in (A7), note that >t +>t =  t (1 + rtW ) pWt31 + t pWt . To make B Wt = 0
for each t the optimal solution to the Lagrangian maximization programme
MP L ,  (1 + rtW ) pWt31 + pWt 5 0 must hold. Thus, the non-arbitrage condition(1 + rt ) pWt31 = pWt holds for each t.
In summary, we get a sequence of price vectors (pW , wW , rW )  {(pWt , wtW , rtW )}+"
t=0
such that pWt > 0, wtW > 0, andpWt kWt = (1 + rtW ) pWt31 kWt + wtW kWlt ; andpWt kt 5 (1 + rtW ) pWt31 kt + wtW klt for any kt 5 P©¡¢ª+"
hold for each t. Then, given (pW , wW , rW ), cWt , \Wt , B Wt , kWt , pWt31 / Wt , / Wt+1 t=0
constitutes an optimal solution to the following programme:
max

+"
X

{(ct ,\t ,B t ,q t ,zt ,/ t+1 )}+"
t=0 t=04t (v (ct ) + N (L  \t ))subject to: for each t,hipWt q t  (1 + rtW ) pWt31 q t + (1 + rtW ) zt + pWt B t = pWt ct + pWt / t+1 ;zt + pWt31 B t = pWt31 /t ;pWt31 q t = zt ;q t 5 P , \t 5 L.

5

We can now show that (pW , rW ) can support an IRS. To see this, it is¢ª+"©¡suﬁcient to note that cWt , \Wt , B Wt , kWt , pWt31 / Wt , / Wt+1 t=0 is consistent witheach country’s programme MP D given the price (pW , rW ). Given l0 , let wD 
pW31 /D0
for each D 5 N . Then, for each t, let
pW /0
31

£

¤

£

1
pW kW  (1 + rtW ) pWt31 kWt + wD (1 + rtW ) pWt31 / t  pWt /t+1
DN £ t t¤ £¤t 
pWt kWt  (1 + rtW ) pWt31 kWt + (1 + rtW ) pWt31 / t  pWt / t+1

By deﬁnition, for each t,

P

¤

> 0.

D
DMN  t = 1 holds. Then, for each t, let¶µW¡ WD WD WD WD WD WD ¢D W
D W
W D W kt
W D W
ct , \t , B t , q t , zt , / t+1   t ct , \t , w B t , , w pt31 / t , w /t+1
N
©¡
¢ª+"
WD
WD WD
WD
WD
for each D 5 N . Then, for each D 5 N , 1 D  cWD
t , \t , B t , q t , zt , / t+1
t=0
constitutes an optimal solution to the following programme MP D given (pW , rW ):MP D :

max{(

cDt ,\Dt ,B Dt ,q Dt ,ztD ,/Dt+1

+"

+"
X

)}t=0 t=0

4t (v (cDt ) +  (L  \Dt ))subject to: for every t,ih D
pWt q t  (1 + rtW ) pWt31 q Dt + (1 + rtW ) ztD + pWt B Dt = pWt cDt + pWt / Dt+1 ;ztD + pWt31 B Dt = pWt31 /Dt ;q Dt 5 P , \Dt = q Dlt 5 L.

Thus, 1 D 5 OD (pW , rW ) holds for eachN. P
P D 5 WDP
WD
by the deﬁnition,= P/WD
t+1 and
DMN q tDMN ct +DMN P
PP Moreover,
WD
WD
WD
W
WD
q
=
/
hold
for
every
t.
Finally,
p
q
=
DMN t
DMN t
DMN t31 t
DMN ztD
holds for every t. In summary, (pW , rW ) associated with (1 )DMN is an IRS.

References[1] Aliprantis and Camera (2009): “A Correspondence-Theoretic Approachto Dynamic Optimization”, Macroeconomic Dynamics 13 (Supplement1), 97-117.

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