Naoki Yoshihara - Reexamination of Marxian Explotation Theory

8/13/2019 Naoki Yoshihara - Reexamination of Marxian Explotation Theory

1/35

Reexamination of the Marxian ExploitationTheory

Naoki Yoshihara

The Institute of Economic Research, Hitotsubashi University,Naka 2-1, Kunitachi, Tokyo 186-0004, Japan.

First: October 2004, This version October 2006

Abstract

In this paper, we reexamine the mathematical analysis of Marx-ian exploitation theory. First, we reexamine the validity of the twotypes of Marxian labor exploitation, Morishimas (1974) type and Roe-mers type (1982), in the argument of Fundamental Marxian Theorem(FMT). We show that the FMT does not hold true if we adopt the

Roemer exploitation, and equilibrium notions are the reproduciblesolution [Roemer (1980)]. Also, we show that the FMT does nothold true for the Morishima exploitation if there exist heterogeneousdemand functions among workers. Second, we reexamine the Class-Exploitation Correspondence Principle (CECP) [Roemer (1982)]. Weshow that the CECP does not hold true in the general convex coneeconomy even if we adopt the Roemer exploitation. Finally, we pro-pose a new definition of Marxian labor exploitation, and show that allof the above difficulties can be resolved under this new definition.

JEL Classification Numbers: B24, B51, D31, D46.

Keywords: reproducible solutions; the Fundamental Marxian The-orem; the Class-Exploitation Correspondence Principle; the Roemer(1982) definition of labor exploitation; the Morishima (1974) definitionof labor exploitation.

The author appreciates John Roemer for his comments.Phone: (81)-42-580-8354, Fax: (81)-42-580-8333. e-mail: [email protected]

1


2/35

1 Introduction

During the 1970s and 1980s, there were remarkable developments in thedebate about the exploitation of labor in Marxian economic theory. TheFundamental Marxian Theorem (FMT) was originally proved by Okishio(1963) and later named as such by Morishima (1973). The FMT showed acorrespondence between the existence of positive profit and the existence ofexploitation. It gives us a useful characterization for non-trivial equilibria,where a trivial equilibrium is one such that its social production point iszero.1 After the seminal work by Morishima (1973), there were many gener-alizations and discussions of the FMT. While the original FMT is discussedin the simple Leontief economy with homogeneous labor, the generalizationof the FMT to the Leontief economy with heterogeneous labor was made byFujimori (1982), Krause (1982), etc. The problem of generalizing the FMTto the von Neumann economy was discussed by Steedman (1977) and onesolution was proposed by Morishima (1974). Furthermore, Roemer (1980)generalized the theorem to a convex cone economy. These arguments mayreflect the robustness of the FMT.

There has also been a remarkable development in works on Marxian eco-nomic theory. This is the General Theory of Exploitation and Class pro-moted by Roemer (1982, 1986). It argues that in a capitalist economy ifthe labor supplied by agents is inelastic with respect to their wealth (that

is, the value of their own capital), then the Class-Exploitation Correspon-dence Principle (CECP), the Class-Wealth Correspondence Principle, andthe Wealth-Exploitation Correspondence Principle can be proven. These the-orems imply that under identical preferences of agents, class and exploitationstatus in the capitalist economy accurately reflect inequality in the distribu-tion of wealth if the labor supply by any agent is inelastic with respect to hiswealth at equilibrium prices. This argument was criticized by some Marxiantheorists,2 since it assumed a standard neoclassical labor market, which was

1Note that the FMT was originally considered to prove the classical Marxian argumentthat the exploitation of labor is the source of positive profits in the capitalist economy.

However, it does not follow from the FMT that the exploitation of labor is the uniquesource of positive profits. The reason is that any commodity can be shown to be exploitedin a system with positive profits whenever the exploitation of labor exists. This observationwas pointed out by Brody (1970), Bowles and Gintis (1981), Samuelson (1982), and wasnamed the Generalized Commodity Exploitation Theorem (GCET) by Roemer (1982).

2For instance, Bowles and Gintis (1990) and Devine and Dymski (1991, 1992).

2


3/35

regarded as not areal, but anidealmodel of the capitalist economy by these

critics. However, as Yoshihara (1998) showed, those theorems essentially holdtrue even if the neoclassical labor market is replaced by a non-neoclassicallabor market with efficiency wage contracts, which was interpreted as a morerealistic aspect of the capitalist economy by those same critics.

In this paper, we reexamine the above theorems in Marxian exploitationtheory. We introduce two types of Marxian exploitation of labor: one isMorishimas (1974) type and the other is Roemers type (1982). Then, first,we show that the FMT does not hold for the Roemer type of labor exploita-tion if equilibrium notions are the reproducible solution [Roemer (1980)].We also show that the FMT does not hold for the Morishima type of labor

exploitation if there exist heterogeneous demand functions among workers.Second, we show that the CECP no longer holds true in a convex cone econ-omy, whichever of the Roemer and the Morishima types of exploitation weassume. Finally, we propose a new definition of Marxian exploitation of la-bor, and show that all of these difficulties can be resolved under this newdefinition.

In the following paper, section 2 defines a basic economic model withconvex cone production technology, and also introduces the two types ofMarxian labor exploitation. Section 3 discusses the failure of the FMT byusing the two types of Marxian labor exploitation respectively. Section 4discusses the failure of the CECP by using the two types of exploitationrespectively. Finally, section 5 introduces the new definition of exploitation,and examines its performance in terms of the FMT and the CECP.

2 The Basic Model

Let P be the production set, which is assumed to be a convex cone. Phas elements of the form = (0,,) where 0 R+, Rm+, and Rm+. Thus, elements ofPare vectors inR

2m+1. The first component,0, is the direct labor input of the process ; and the next m components,

, are the inputs of goods used in the process; and the last m components,, are the outputs of the m goods from the process. We denote the netoutput vector arising from asb . We assume thatP is a closedconvex cone containing the origin inR2m+1. Moreover, it is assumed that:

A 1. P s.t. 0 0and = 0,[ 0 0 > 0]; and

3


4/35

A 2. commoditym vectorc Rm+, P s.t.b = c.Given such P, we will sometimes use the notations like P(0= 1) andbP(0= 1), where

P(0= 1) {(0,,) P |0= 1}

and

bP(0 = 1) {b Rm | = (1,,) P s.t. b} .Given a market economy, any price system is denoted by p Rm+, which

is a price vector ofm commodities. Moreover, a subsistent vector of com-moditiesb Rm+ is also necessary in order to supply one unit of labor perday. We assume that the nominal wage rate is normalized to unity when itpurchases the subsistent consumption vector only. By assumption,pb = 1holds.

2.1 Two Definitions for Marxian Exploitation of Labor

To define Marxian exploitation of labor in terms of surplus value, we mustfirst define the labor value of a vector of commodities. In the economicmodel with joint production such as the von Neumann model and the convex

cone model, there have been two candidates for the definition of Marxianexploitation of labor: one is of Morishima (1974) and Roemer (1981), andthe other is Roemer (1982). The difference between them comes from thedifferent definitions of labor value between them.

Morishimas definition of labor value of a vector of commodities is givenindependently of the particular equilibrium the economy is in. Letc Rm+be a vector of produced commodities. Let

(c) { P |

b = c} ,

which is the set of the production points which produce, as net output vectors,at leastc. Then:

Definition 1: The Morishima (1974) labor value of commodity vector c,l.v. (c), is given by

l.v. (c) min {0 | = (0,,) (c)} .

4


5/35

Given the consumption vector,c Rm+, of the worker which is purchasedby his wage revenue per day, the Morishima rate of labor exploitation isdefined as follows:

Definition 2: The Morishima (1974) rate of labor exploitation at the con-sumption bundlec Rm+ is

e (c) 1 l.v. (c)

l.v. (c) .

It is easy to see that (c)is non-empty by A2. Also,

{0| = (0;;) (c)}

is bounded from below by 0, by the assumption 0 P and A1. Thus,l.v. (c)is well-defined sincePis compact. Moreover, by A1,l.v. (c)is positivewheneverc 6= 0, so thate (c)is well-defined.

In contrast to the Morishima (1974) labor value, the definition of laborvalue in Roemer (1982) depends, in part, on the particular equilibrium theeconomy is in. Given a price vectorp Rm+ and P, let (p;)

p

b0

p+0.

Given the consumption bundle c Rm+ and a price vector p R

m+ , let

P(p) Pbe the set of profit-maximizing production points when the pricesystem is(p, 1).3 Then, let

(c;p) P(p)|b = c ,

which is the set of those profit-rate-maximizing actions which produce, asnet output vectors, at least c. Then:

Definition 3: The Roemer (1982) labor value of commodity vector c,l.v. (c;p),is given by

l.v. (c;p) min {0| = (0,,) (c;p)} .

3That is, P(p) arg max { (p;)| P}.

5


6/35

Definition 4: The Roemer (1982) rate of labor exploitation at the consump-

tion bundlec Rm+ is

e (c;p) 1 l.v. (c;p)

l.v. (c;p) .

It is easy to verify that l.v. (c;p) is well-defined, and has a positive valuewhenever c 6= 0, so that e (c;p) is well-defined. Also, l.v. (c;p) l.v. (c)which indicates that e (c;p)> 0 impliese (c)> 0.

In the following discussion, we will introduce two types of Marxian eco-nomic models; one is for discussing the Fundamental Marxian Theorem

(FMT) (Roemer (1981; Chapter 2)), and the other is for discussing the ClassExploitation Correspondence Principle (Roemer (1982; Chapter 5)). We will,then, examine the viability of Morishimas (1974) and Roemers (1982) de-finitions of labor exploitation respectively in both of the Marxian economicmodels.

3 Failure of the Fundamental Marxian The-

orem

3.1 The Model for the Fundamental Marxian TheoremIn the model for the FMT, it is assumed that there are two classes, theworking class and the capitalist class. The working class is characterizedby agents, each of who has no initial endowment of input goods, while thecapitalist class is made up of agents having some non-negative and non-zeroamount of input goods. Moreover, the consumption of any agent in theworking class is given exogenously by the subsistent consumption bundle b,while any capitalist is implicitly assumed to save all his revenue for investingin the next production period.

There are |N| capitalists; the -th one is endowed with a vector of pro-

duced commodity endowments . Workers have no endowments of producedcommodities; they have only one unit of labor endowment, which is homoge-nous among the workers. We assume that every worker has the same level oflabor skill.

Under the assumption ofstationary expectationson prices [Roemer (1980,1981; Chapter 2)], capitalist s program is given, facing a market price vector

6


7/35

pand the wage rate1, by:

choose Pto maximize p (p + 0 )

s.t. p + 0 p.

The set of the production processes which are the optimal solutions of theabove problem is denoted by A (p, 1). Given this, we are ready to discussthe definition of equilibrium in this economic system:

Definition 5 [Roemer (1980, 1981; Chapter 2)]: A reproducible solution(RS) for the economy specified above is a pair

p, {}N

, wherep Rm+

and P, such that:

(a) N,

A

(p, 1)(profi

t maximization);(b)b = 0b (reproducibility),

whereb PN( )and0 PN0 ;(c)pb= 1(subsistent wage); and

(d) + 0b 5 (social feasibility),where

PN

and P

N.

The three parts except (a) need some comments. Part (b) says that netoutputs should at least replace employed workers total consumption. This isequivalent to requiring that the vector of social endowments does not decrease

in terms of components, because (b) is equivalent to (+ 0b) + = ,where the right hand side is the social stocks at the beginning of this period,the left hand side is the stocks at the beginning of the next period. Part (d)says that intermediate inputs and workers consumption must be availablefrom current stocks. Here, we assume that wage goods are dispensed at thebeginning of each production period, therefore stocks must be sufficient toaccommodate them as well. Finally, Part (c) says that the equilibrium wagerate is equal to the subsistent level. It implies that workers are plentifulrelative to the social endowments of capital stocks, so that a percentageof workers should be unemployed (industrial reserve army), by which the

equilibrium wage rate is driven down to the subsistent level.The existence of the reproducible solution is guaranteed as the followingproposition shows:

Proposition 1 [Roemer (1980; 1981)]: Let b Rm++ . Under A1, A2, andstationary expectation of prices, areproducible solution (RS)of Definition 5exists for the economy specified above.

7


8/35

3.2 Failure of the FMT, using Roemers Definition

Morishima (1974) showed that, in the balanced growth equilibrium of the vonNeumann economy, the warranted rate of profit is positive, if and only if theMorishima (1974) rate of labor exploitation is positive. This theorem (FMT)is robust, even if the von Neumann system of production technology containssome inferior production processes. In contrast, Roemer (1981) showed thatthe equivalent relationship between the positive rate of Morishimas (1974)labor exploitation and the positive profit rate no longer holds true, if the pro-duction technology contains an inferior process. This difference comes fromthe different equilibrium notions between them: Roemer (1981) discussed theprofit rate which prevails under the reproducible solution defined in Defini-

tion 5 of this paper. Furthermore, Roemer (1981) gave us a condition, underwhich Morishimas (1974) labor exploitation is equivalent to profit-makingeven in the reproducible solution of a general convex production economy.This condition requires non-existence of an inferior production process, andis calledIndependence of Production.

In this subsection, we also discuss the validity of FMT when the equi-librium notion of the convex cone production economy is the reproduciblesolution. Here, we examine the case of Roemers (1982) exploitation of labor.We set up the same situation as Roemer (1981; Chapter 2) to discuss thisproblem. Up to this point, it is not obvious at all, even under the assumption

of the independence of production, whether the positive profit rate entailsthe positive rate of the Roemer (1982) exploitation or not, since e (c) > 0does not necessarily imply e (c;p) > 0. Note that Roemer (1982; Chapter5, p. 158) already wrote that the FMT continued to hold using the Roemer(1982) exploitation. We will show that this conjecture of him is wrong.

At the first place, let us introduce the assumption of the independence ofproduction:

A 3. (Independence of Production)(0,,) P,

b = 0, and0 5 c

b, then (00,

0,0) P suchthat 0 0

=cand0

0


9/35

(i)e (b)> 0;

(ii) there exists a reproducible solution yielding positive total profits;(iii)all reproducible solutions yield positive total profits.

Now, let us examine the robustness of FMT, when the definition of laborexploitation is replaced by the Roemer (1982) type. The following theorem,however, implies that the answer is negative:

Theorem 1: Letb Rm++. Under A1, A2, A3, and stationary expectation ofprices, there exists an economy with convex cone production technology suchthat every reproducible solution

p, {}N

yields positive total profits and

zero rate of theRoemer (1982) exploitation e (b;p) = 0.

Proof. Letm = 2, #N = 1, and is the unique capitalist in this society.Also let us define an economy as follows: in the first place, when the workerprovides one unit of labor, then his subsistent consumption bundle is givenbyb = (1, 1). The social endowment of capital in the economy is given by = (2, 1), which is owned by the unique capitalist . Second, let us definethree production points:

1 =

10,

1,1

= (1, (2,1) , (2, 3)) ;

2

= 20,2,2= (1, (1, 0) , (3, 1)) ; and3 =

30,

3,3

= (1, (1,1) , (4, 1)) .

Now, letP be aclosed, convex conesubset ofR2m+1 such that1) 0 P; and2)co {1,2,3}= P(0= 1), wherecoXis the convex hull of a set X.

Given the above defined economy, we will show that there is a uniquereproducible solution (RS). Note that:

p 4m\ {(1, 0)} ,p< p (0 +b) (0 co

1,2,3

\

2

), (*)

where 4m is the simplex set. Also, for p = (1, 0), p = p (0 +b) forany 0 co {2,3}. Note that if (p,) is a RS, then + 0b by Definition 5(d). Thus, if (p,) is a RS with 6= 2, then 0 < 1.Moreover, by Definition 5(b), if (p,) is a RS, thenb = 0b. Let 12 121 + 1

22, and P(0= 1) be the set of upper-boundary of P(0 = 1).

Then, co {12,2} =0 P(0 = 1)|b0 = b. Thus, by linearity ofP,

9


10/35

if(p,) is a RS, then there exist t (0, 1] and0 co {12,2} such that

= t0

.Note that ifp =

13

, 23

, thenpb1 10 =pb220=pb000 > pb00000

for any 0 co {1,2} and any 00 co {2,3}. However, because of theproperty (*), the existence of capital constraint in the profit maximizationproblem implies that 2 is the unique profit maximizer at the pricep=

13

, 23

.

Note also that ifp =12

, 12

, thenpb220 = pb00000 > pb000 for any

0 co {1,2}and any 00 co {2,3}. Thus, by the same reasons as theabove paragraph, 2 is the unique profit maximizer at the price p =

12 ,

12

also.

Note that ifp has the property that 12

< p1 < 23

, 12

> p2 > 13

, then 2

is the unique profit maximizer at that price. Moreover, for any other pricep, there is no 0 co {12,2}such that for some appropriatet (0, 1],t0

constitutes a profit maximizer at that price.In summary, if there exists a RS, say (p,), then = 2 should hold.

In fact, for anyp 4m (2) p 4m | 1

25 p15 23 ,

12= p2= 13

,(p,2)

constitutes a RS. Moreover, in this case, (p;2)> 0.

Insert Figure 1 around here.

Finally, we can check that in (p,2), where p 4m (2), e (b;p) = 0holds, whereas its corresponding profit is positive. This is because P(p) ={t2 P |t R+} and (20b;p

) ={2}, which implies l.v. (20b;p) =20,

so thatl.v. (b;p) = 1.

The above proof constructs an economy such that for any reproduciblesolution (p,), is the unique profit maximizer at this price p. Moreover,the corresponding net outputb of this production point does not stronglydominate the subsistent consumption vector b:b b, as described in Figure1. In such a case, the minimum amount of direct labor to produce at leastbover the profit-maximizing production set is equal to the amount of directlabor at the reproducible solution.

Note that Roemer (1982; Chapter 5; footnote 6) pointed out that theassumption of independence of production must be made concerning the re-stricted production setP(p)so as to keep the FMT holding using the Roemer(1982) exploitation. The constructed economy in the proof of Theorem 1 doesnot meet this revised version of independence of production. But, it seemsthat such a restriction for possible types of production sets is uncompelling.

10


11/35

A3 in this paper implies the elimination of inferior production processes

from possible production sets, which is an acceptable restriction, whereasthe revised version of independence of production in Roemer (1982) doesnot necessarily have such an implication. Alternatively, the Roemer (1982)independence of production eliminates the types of production sets whosecorresponding net output possibility sets,bP(0= 1), are representable bystrictly concave and continuously differentiable functions. This is because insuch types of production sets, any reproducible solution has only the uniqueprofit-maximizing path, as described in Figure 1, and so the Roemer (1982)independence of production is violated. That seems to be too restricted.Moreover, we cannot identify whether or not the targeted economy satis-fi

es the Roemer (1982) independence of production in advance to discoverwhich of price vectors constitutes a reproducible solution in this economy.The failure of the FMT in the above theorem comes from the explicit

introduction of capital constraints into the equilibrium notion rather thanthe violation of the Roemer (1982) independence of production. In fact,if we define an equilibrium notion without the explicit capital constraint asin Definition 1(d), then the FMT continues to hold even under the Roemer(1982) exploitation of labor and without the Roemer (1982) independenceof production. To discuss this, let:

Definition 5*: A reproducible solution* (RS*) for the economy specified

above is a pair p, {}N, where p Rm+ and P, such that theconditions (b) and (c) of Definition 5 hold, and:

(a*) N, A (p, 1)(profit maximization),where A (p, 1) arg max

pb0 00b| 0 P and00 W andW

iss financial endowment;

(d*) 05 L(social feasibility of labor demand),where 0

PN

0 , and L

PNW

is the aggregate financial endow-ments for purchasing labor.

This equilibrium notion is based upon Roemer (1981; Chapter 2; Appen-dix 2).

Theorem 2: Let b Rm++. Under A1, A2, A3, and stationary expectationof prices, there exists a reproducible solution*

p, {}N

such that it yields

positive total profits if and only if e (b;p)> 0.

11


12/35

Proof. By Theorem 2.17 [Roemer (1981; Chapter 2; Appendix 2)], there

exists a RS*, p, {}N, so thatb = 0b. Note that in this RS*, 0 =Lby profit maximization, and p is an efficiency price which supportsb bP0 = L as an efficient production. In other words, pb = pb0 holds foranyb0 bP0=L. Note that by b Rm++ andb = 0b,b Rm++.Then, by A3, p Rm++ . Thus, since the total profits of this RS* is equal top (b 0b), it holds that

p (b 0b) Q 0 b 0b Q 0.If

b 0b = 0, then it is easy to see that l.v. (0b) = 0 by A3. Thus,

l.v. (0b;p)

=

0 since l.v. (

0b;p)

= l.v. (

0b), so that e (b;p)

50. Ifb 0b 0, then there exists P such that 0 = 0 andb 0b > 0 by

A3. Let p Rm++ be a price vector which supportsb bP0=L asan efficient production. Then, we can constitute a new RS*

p, {}N

such that

PN

=. Since the only constraint for each capitalist is hisbudget for purchasing his employers, it is easy to find a division {}Nof such that A (p, 1) and

PN

0 = L. In this new RS*,

sinceb 0b > 0, it holds p (b 0b) > 0. Moreover, by A3, thereexists P such that

b 0b= 0 and0 < 0. In particular, since

b 0b >0, we can choose

=t with t (0, 1) by the cone propertyof P. In this case, P(p) so that l.v. (0b;p

) < 0. This impliese (b;p)> 0.

Conversely, ife (b;p) > 0, then e (b) > 0. Thus, by Proposition 2, allRS*s yield positive total profits.

Although the above theorem seems to show the validity of the Marxiantheory of labor exploitation and profits using the Roemer (1982) definition, itcannot be regarded as such: the theorem permits the situation that althoughthe workers continue to supply a constant amount of labor and receive aconstant amount of real wage goods, they are exploited or not according tothe aggregate production points chosen in equilibrium. For instance, given

0 > 0and0b >0, ifb0b >0 in a reproducible solution, say,(p, (b,0)),then they should be exploited at(p, (b,0)), while ifb00b 0in anotherreproducible solution, say,

p0,

b0,0, then they might not be exploitedatp0,b0,0; in fact, they are not exploited at p0, b0,0ifbP(0= 1)

has such an upper-boundary as representable by a strictly concave and con-tinuously differentiable function, because P(p0) consists only of the type of

12


13/35

tb0, t0 where t R++. This is unacceptable, since the status of workersin terms of exploitation has to be identified only based upon their objective

conditions of labor like 0 and0b. But, the above example indicates thatthe choice of net output components may influence the exploitation status ofworkers even under the same labor condition, which is inappropriate.

3.3 The case of Workers Heterogenous ConsumptionDemands

In this subsection, we go back to Morishima (1974) type of labor exploita-tion, and examine the validity of FMT in reproducible solutions with hetero-

geneous consumption demands among workers. We can see that the FMTholds true in the Leontief economy even if workers have heterogeneous con-sumption demands. That is, in that economy, the positive rate of profitprevails in the reproducible solution, if and only if the average rate of ex-ploitation of all workers is positive [Roemer (1981)]. Moreover, the lattercondition holds, if and only if each and every workers rate of exploitation ispositive. Thus, in the economic model with Leontief production technologyand heterogeneous consumption demands, the scenario of Marxian exploita-tion of labor is completely consistent. Such a result, however, can no longerhold once we discuss it in a general convex cone economy with heterogeneous

consumption demands.LetIbe the finite set of types of the workers with heterogeneous consump-tion demands with a generic element . LetF() [0, 1] be the fraction ofthe-type workers. By definition,

PIF() = 1. Let

minI{F()},and let us assume that the aggregate labor endowments of the-type work-ers is normalized to one. Thus, for any type I, the aggregate laborendowments of this type of workers is given by F()

F(). Then, let

P0()

0 R+|

= (0 , ,) P s.t. 0

F()

F()

be the set of feasible production points by the -type workers.

Given a price system p Rm+ , let us denote the consumption demandof the -type worker per one unit of income by d (p) Rm+ . The demandfunctiond ()is assumed to be derived from a continuous, strictly monotonic,quasi-concave, and homothetic utility function of the -type worker, andpd (p) = 1for anyp Rm+ normalized to

Pmj=1pj = 1.

13


14/35

Given a price system p Rm+ and a production plan = (0,,)

P, let the aggregate labor demand 0 consist of (0 )I, where 0 is theemployed labor amount of the -type workers, and

PI

0 =0. Then, the

average consumption demand of the employed workers is defined by:

dp; (0 )I

PI

0d

(p)

0.

Note thatpdp; (0 )I

= 1by definition.

An economy is specified by a listE=

P; N; ()N; I; (F())I; (d ())I

.

Now, we are ready to define reproducible solutions with heterogeneous con-sumption demands among workers:

Definition 6: A reproducible solution(RS) for the economy with heteroge-neous consumption demands among workersEis a pair

p, {}N

, where

p Rm+ and P, such that:

(a) N, A (p, 1)(profit maximization);

(b)b = 0d p; (0 )I (reproducibility),whereb PN( )and0 PN0 =PI0 ;

(c) I,pd (p) = 1and0 P0(); and

(d) + 0dp; (0 )I

5 (social feasibility),

where PN and PN

.

This definition is essentially the same as Definition 5, except the point thatthe aggregate consumption demands of employed workers are endogenouslygiven by each employed workers demand function.

Proposition 3: Under A1, A2, and stationary expectation of prices, a re-producible solution (RS) of Definition 6 exists for the economy E.

Theorem 3: Under A1, A2, A3, and stationary expectation of prices, letp, {}N

be thereproducible solution (RS)with the average consumption

demand of the employed workers, d

p; (0 )I

, for the economy E. Then,

the RSyields positive total profits if and only if e d p; (0 )I> 0.Proof. (): Let

p, {}N

be a RS with a positive total profit. Thus,

p

XN

( )

!XN

0 = p

XN

( )XI

0d (p)

!= p

b 0d p; (0 )I> 0.14


15/35

Since p Rm+ andb = 0d p; (0 )I by Definition 6(b), the last strictinequality impliesb 0d p; (0 )I andb 6= 0d p; (0 )I. Thus, by

A3,l.v.0d

p; (0 )I

< 0. By linearity,l.v.

dp; (0 )I

< 1, which

implies e

dp; (0 )I

> 0.

(): Since there is no RS with a negative total profit, it suffices todiscuss only the case of zero profit. Let

p, {}N

be a RS with a zero

total profit. Thus,pb 0d p; (0 )I = 0. By Definition 6(b),b=

0dp; (0 )I

. If for some commodityj,bj 0dj p; (0 )I> 0, then

it follows that pj = 0. However, since every worker has a strictly monotonicpreference on Rm+, pj = 0 implies dj

p; (0 )I

= +, a contradiction.

Thus,

b= 0d

p; (0 )I.

Suppose l.v. 0d p; (0 )I < 0. Then, l.v. (b) < 0, which im-plies thatb / bP(0), where bP(0) is the set of upper-boundary ofbP(0) {b Rm | 0 = (0,0,0) P s.t. 0 0 b}. This alsoimplies / P, so that there exists at least one capitalist N such that / A (p, 1), a desired contradiction. Thus, l.v.

0d

p; (0 )I

= 0,

so thate

dp; (0 )I

= 0.

Let bP(0 = 1)is the set of upper-boundary ofbP(0 = 1), and letbP (0= 1) b

P(0= 1) \

bP(0 = 1).

Theorem 4: Under A1, A2, A3, and stationary expectation of prices, letp, {}Nbe thereproducible solution (RS)with the average consumptiondemand of the employed workers, d

p; (0 )I

, for the economy E. Then,

the following two statements are equivalent:(I) the RS yields positive total profits if and only if e (d (p)) > 0 for any I;(II)d (p) bP (0 = 1) for any I.Proof. On [(II)(I)]. Let d (p) bP (0= 1) hold for any I. Thisimplies for any I, there exists

b

bP(0 = 1) such that

b > d (p).

Then, since P is a convex cone with A 2, there exists P with 0 0for any I. Conversely, let us suppose there exists 0 I such thatd

0

(p) /bP (0 = 1). Then, for any0 P with 00 = 1, eitherb0 =d0 (p)orb0 d0 (p) holds for this 0 I. This implies e d0 (p) 0 for this0 I. Thus, the condition (II) holds if and only ife (d (p))>0 holds forany I.

15


16/35

Thus, if the condition (II) holds, thene d p; (0 )I> 0which impliesthat the RS yields positive total profits by Theorem 3. If the RS yields posi-

tive total profits, then givend (p) bP (0 = 1)for any I,e (d (p))> 0trivially holds for any I.

On [(I)(II)]. Suppose (II) does not hold. Let the RS yield positive totalprofits. Since the condition (II) is equivalent toe (d (p))>0 holds for any I, there exists 0 I such that e

d

0

(p) 0 in spite of the RS with

positive total profits. Thus, (I) is violated.

By the above theorem, we derive the following situation:

Corollary: Under A1, A2, A3, and stationary expectation of prices, there

exists an economy with convex cone technology and heterogeneous consump-tion demands among workers, such that there exists a reproducible solu-tion (RS),

p, {}N

, with the average consumption demand of the em-

ployed workers, dp; (0 )I

, which yields positive total profits and a neg-

ative rate of the Morishima (1974) exploitation of some -type workers,e

d

(p)

< 0.

Proof. Letm = 2, #N = 1, and is the unique capitalist in this society.Also let us define an economy as follows: at the first place, there are two typesof workers, the type and the type: the-type (resp. -type) workers are

characterized by their common consumption demand functionsd

() (resp.d ()). Assume that each of these two demand functions is respectivelyderived from a continuous, monotonic, quasi-concave, and homothetic utilityfunction, and that

d

(p) = (0.5, 1.25) andd (p) = (2.5, 0.25) ifp=

1

3,2

3

.

Moreover, F() = 0.5 = F(). The social endowment of capital in theeconomy is given by = (2.5, 0.75), which is owned by the unique capitalist. Second, let us define four production points:

1 =10,

1,1

= (1, (1, 0) , (2.5, 1)) ;

2 =20,

2,2

= (1, (0,1) , (2.5, 1.5)) ;

3 =30,

3,3

= (1, (1.5, 0) , (1.5, 1.01)) ; and

4 =40,

4,4

= (1, (0,1.5) , (2.6, 1.5)) .

16


17/35

Now, we are ready to define a production possibility set of this economy. Let

P be aclosed, convex conesubset ofR2m+1 such that1) 0 P;2)co {1,2,3,4}= P(0= 1);

3) the net output possibility set at one unit of labor input ofP,bP(0 = 1),is defined by: bP(0= 1) =co b1,b2,b3,b4, 0 .

Given the above defined economy, we will show the existence of one re-producible solution (RS). Let us consider (p,) =

13

, 23

,1

. We will

show it is a RS. Since d

(p) = (0.5, 1.25)andd (p) = (2.5, 0.25), we candefine the aggregate consumption demand of all the employed workers by

d p; 0 ,0 = F() d (p) +F() d (p) = (1.5, 0.75), assuming theratio of employed workers between the two types are equal. Thus, at (p,),we haveb = (1.5, 1) (1.5, 0.75) =0 d p; 0 ,0 , which implies thatDefinition 6(b) holds. Also, +0 d

p;

0 ,0

= (1, 0)+(1.5, 0.75) =

(2.5, 0.75) =, which implies that Definition 6(d) holds. Moreover, by con-struction, we can check that p is an efficiency price for . Thus, sincep +0 = p

< p+ 0 for any co {1,2,3,4} \ {1}, it fol-lows that A (p, 1), Definition 6(a) holds. Finally, by construction,Definition 6(c) holds.


Note that this RS yields a positive profit: p (p + 0) = 16

> 0.Also, e

dp;

0 ,0

> 0, since l.v.

dp;

0 ,0

< 1. The last

inequality follows from l.v. (b) = 1,b d p; 0 ,0 , and A3. Atthe same time, in this RS, the -type employed workers are not exploitedin the Morishima (1974) sense, since e

d

(p)

< 0. The last inequal-

ity follows from d

(p) >b3, l.v. b3 = 1, and A3. In other words,d

(p) /

bP(0 = 1) = co {0, (0, 1.01) , (1.5, 1) , (2.5, 0.5) , (2.6, 0)} implies

l.v.

d

(p)

> 1.

The above theorem implies that once we consider a more general modelthan the Leontief economy, the Marxian theory of labor exploitation can nolonger be a consistent argument for explaining the source of positive profitseven in the case of the Morishima (1974) definition of labor exploitation.This is because all workers are homogeneous in their labor endowments and

17


18/35

their labor skills: they engage in the same working hours with the same

labor skill and the same wage rate. They are mutually identical exceptfor their consumption demands when they are faced with the same budgetconstraint. In this situation, we would expect that every worker is in thesame position with respect to labor exploitation, since the exploitation statusseems to reflect only workersobjectivelabor conditions. However, the abovetheorem shows that some types of workers are not exploited due to theirsubjectiveconsumption demands, even if they are under the same objectivelabor condition as others who are exploited.

Although the above theorem assumes the notion of reproducible solutionsproposed by Roemer (1981), the same type of conclusion could hold true

even if the positive profi

t is not associated with the reproducible solution,but rather associated with the balanced growth equilibrium as in Morishima(1974). Thus, the above negative conclusion is not based on the choice ofequilibrium notions.

4 Failure of the Class Exploitation Correspon-

dence Principle in Convex Cone Economies

4.1 The Model for the Class Exploitation Correspon-

dence PrincipleIn the model for the CECP, a fixed class structure like the model for the FMTis not assumed in advance of economic analysis. For the sake of simplicity, letus follow the same setting as that in Roemer (1982; Chapter 5). That is, ourschematic model of a capitalist economy is that all agents are accumulatorswho seek to expand the value of their endowments as rapidly as possible.Let us denote the set of agents by N with generic element . All agentshave access to the same technology P, but they differ in their bundles ofendowments. An agent Ncan engage in three types of economic activity:he can sell his labor power 0 , he can hire the labor powers of others to

operate = 0 ,, P, or he can work for himself to operate = (0 ,

,) P. His constraint is that he must be able to affordto lay out the operating costs in advance for the activities he chooses tooperate, either with his own labor or hired labor, funded by the value of hisendowment. He can choose the activity level of each of,, and0 under

18


19/35

the constraints of his capital and labor endowments. Thus, given (p, w),

where w is a nominal wage rate, his program is:

max( ; ; 0 )PPR+

[p ( )] +hp

w0

i+ [w0 ]

such that

p +p 5 p W,0 +

0 5 1.

Given(p, w), letA (p, w)be the set of actions(; ; 0 ) P PR+which solve s program at prices(p, w).

The equilibrium notion of this model is given as follows:

Definition 7 [Roemer (1982; Chapter 5)]: A reproducible solution(RS) forthe economy specified above is a pair

(p, w) , {(; ; 0 )}N

, where

p Rm+, w = pb= 1, and(; ; 0 ) PP R+, such that:

(a) N,(; ; 0 ) A (p, w)(profit maximization);

(b) + 5 (social feasibility),where

PN

,

PN

, and

PN

;

(c)05 0 (labor market equilibrium)where 0 P

N0 and0

PN

0 ; and

(d)b +b= 0b+ 0b (reproducibility),whereb PN( ),bPN( ), and0 PN0 .

The essential concept of the reproducible solution here is almost the same asthat for the model of the FMT. Each condition of profit maximization, socialfeasibility, and reproducibility in Definition 7 has the same implication aseach of those conditions in Definition 5, except for the following two points:First, rational agents have the three components of actions in Definition 7,

and secondly wages should be paid at the end of the current period. Theonly different factor from Definition 5 is (c), the condition of labor marketequilibrium. This condition allows strict inequality between labor demand0and labor supply0. If it holds in strict inequality, then the nominal wagerate is driven down to the subsistent wage w = pb = 1. If it holds in equality,then it might hold that w = pb= 1.

19


20/35

The existence of the reproducible solution in this definition is also guaran-

teed in a similar way to Proposition 1. LetP() {= (0,,) P | }and0() max {0| = (0,,) P()}. Then:

Proposition 4: Let b Rm++ and 0() 5 |N|. Under A1, A2, andstationary expectation of prices, areproducible solution (RS)of Definition 7exists for the economy specified above.

This proposition can be shown in a similar way to Proposition 1.

4.2 Failure of the CECP both in the Morishima and

the Roemer Definitions of ExploitationFollowing Roemer (1982; Chapter 5), let us define possible classes in themodel of section 4.1. At every RS in the model of section 4.1, differentproducers relate differently to the means of production. An individuallyoptimal solution for an agent at the RS consists of three vectors(;; 0 ).According to whether these vectors are either zero or nonzero at the RS,all producers are classified into the following four types: that is, (+, +, 0),(+, 0, 0), (+, 0, +), and (0, 0, +), where + means a nonzero vector in theappropriate place. Here, the notation(+, +, 0)implies, for instance, that anagent works for his own shop and hires others labor powers; while(+, 0, +)

implies that an agent works for his own shop and also sells his own laborpower to others, etc.

Let us define four disjoint classes as follows:

CH = { N | A (p, w) has a solution of the form (+, +, 0) \ (+, 0, 0)} ,

CPB = { N | A (p, w) has a solution of the form (+, 0, 0)} ,

CS = { N | A (p, w) has a solution of the form (+, 0, +) \ (+, 0, 0)} ,

CP = { N | A (p, w) has a solution of the form (0, 0, +)} .

We can see that the set of producers Ncan be divided into these four classes

at any RS.Given a price system(p, w), let

max (p, w) max

p (p +w0)

p | = (0,,) P

20


21/35

and

P(p, w) = (0,,) P |

p (p+w0)

p =max (p, w)

.

In the following discussion, let us restrict our attention to a RS with a positiveprofit. Then:

Proposition 5[Roemer (1982; Chapter 5)]: Let(p, w)be aRSwithmax (p, w)>0. Then,

CH W > maxP(p,w)

p

0 , CPB min

P(p,w)

p

0

5 W 5 max

P(p,w)

p

0

,

CS 0< W < minP(p,w)

p

0

,

CP W = 0.

Definition 8: Let(p, w) be a reproducible solution with net outputb+b.Afeasible assignment is any collection of nonnegative vectors (f)N suchthat:

(i)PNf =b+b;(ii) pf =max (p, w) W +w.

Let us denote the class of feasible assignments by F, and the set of bundlesassigned to inF byF. Then:

Definition 2*: A producer N is exploited in the Morishima (1974)senseif and only if:

maxfF

l.v. (f)< 1,

and he is an exploiter in the Morishima (1974) senseif and only if:

minfF

l.v. (f)> 1.

Definition 4*: Let(p, w) be a reproducible solution with full employmentof labor. A producerN is exploited in the Roemer (1982) sense if and

21


22/35

only if:

maxfF l.v. (f;p, w)< 1,4

and he is an exploiter in the Roemer (1982) senseif and only if:

minfF

l.v. (f;p, w)> 1.

Now, CECP, which is a principle we would like to verify, is introduced asfollows:

Class Exploitation Correspondence Principle (CECP):For any econ-omy defined as in section 4.1, and any reproducible solution with a positive

profit rate, it holds that:(A) every member of CH is an exploiter.(B) every member of CS CP is exploited.

We are ready to show that the CECP cannot hold if the Morishima (1974)definition of labor exploitation is used. Let us define:b c Rm+ | (c) :0 = 1& 0 is minimized over (c) .Then, given a RS(p, w), let

bec

bbe such thatp

bec = pcfor allc

b. Also, let

bc b be such thatpbc 5 pcfor allc b. We can check that is an exploiterin the Morishima (1974) sense if and only ifmax (p, w) W+ w > pbec. Also,is exploited in the Morishima (1974) sense if and only ifmax (p, w) W+w 0,in which there exists CH

who is not an exploiter in the Morishima (1974) sense.

Proof. let us define four production points:

1 =10,

1,1

= (1, (1, 0.5) , (1, 2.25)) ,b1 = (0, 1.75) ;2 =

20,

2,2

= (1, (1,1) , (2, 2.5)) ,b2 = (1, 1.5) ;3 =

30,

3,3

= (1, (3,4) , (5, 5)) ,b3 = (2, 1) ; and4 =

40,

4,4

= (1, (5,3) , (8, 3)) ,

b4 = (3, 0) .

Now, we are ready to define a production possibility set of this economy. Let

P be aclosed, convex conesubset ofR2m+1 such that1) 0 P;2)co {1,2,3,4} P;

3) the net output possibility set at one unit of labor input ofP,bP(0 = 1),is defined by: bP(0= 1) =co b1,b2,b3,b4, 0 .

23


24/35

LetN={, , 0}, and =0

= 0, and = (2, 2). Also, letb= (1, 1)

be the subsistent consumption bundle for any agent supplying one unit oflabor. Let us consider the situation that (p, w) = ((0.5, 0.5) , 1)and

;; 0

= ((0, 0, 0 ) ; (0, 0, 0);1) ,

(;; 0) = (((2, 2.5) , (1, 1) , 1);((2, 2.5) , (1, 1) , 1);0) , and

0

;0

; 0

0

= ((0, 0, 0 ) ; (0, 0, 0);0) .

This indicates that under the price system(p, w) = ((0.5, 0.5) , 1), the agent is an employed worker who sells his one unit of labor to the agent ;the agent operates his owned capital by his own working and buying one

unit of labor from the agent ; and the agent 0

is an unemployed worker.Thus,CH ={} andCP ={, 0}. We will show that this combination ofthe price system and the list of every agents action constitutes a RS withmax (p, w) > 0. Moreover, we will show that the agent cannot be anexploiter in the Morishima (1974) sense.

First, we will show that the above list of economic actions{(;; 0 )}Nis the list of optimal solutions for all agents at (p, w) = ((0.5, 0.5) , 1).Note that ((p, w) ;1) < 0, ((p, w) ;2) = 1

4, ((p, w) ;3) = 1

7, and

((p, w) ;4) = 18 , when (p, w) = ((0.5, 0.5) , 1). Thus,max (p, w) = ((p, w) ;2),

and so operating with only t2 up to the budget constraint is the opti-

mal solution for the agent with nonnegative endowment of capital

.Thus,{(;; 0 )}Nconstitutes the list of optimal solutions at (p, w) =((0.5, 0.5) , 1). Second, the conditions (b) and (c) of Definition 7 are sat-isfied by the list {(;; 0 )}N at that price system. Finally, 2b =0 +

0

b b +b = 2b2, which implies the condition (d) holds. Thus,

(p, w) , {(;; 0 )}N

is a RS withmax (p, w)> 0.

By the above definition, it holds that P(p, w) = {t2 P |t R+}, = (p, w) ={2}, andb= co b1,b2 cob2,b3 cob3,b4. Then,max(p,w)p= p

2, so thatmaxp

bw=p

b2 1 = 1

4, while

bec=

b3,

which implies that 14

=pb2 1 = maxpbw < pbecw= pb3 w= 12 .Thus, the inequality (1) cannot be satisfied. In fact,max (p, w)p +w =1.5 = pbec, which implies that cannot be an exploiter in the Morishima(1974) sense.

The above theorem was pointed out by Roemer (1982; Chapter 5). Basedon this result, he criticized the Morishima (1974) definition of labor exploita-

24


25/35

tion, since the CECP is false under that definition. In contrast, as Roemer

(1982; Chapter 5; Theorem 5.3) showed, all agents in CH become exploiterswhenever the Roemer (1982) exploitation is used. However, we can also showthe existence of a RS, in which there could be an agent in CS CP who isnot exploited, even if the Roemer (1982) exploitation is used.

Given(p, w), let

c Rm+ | (c; (p, w)) :0 = 1& 0 is minimized over (c; (p, w))

.

Thus, is the collection of nonnegative consumption bundles whose laborvalues in the Roemer (1982) sense are unity. Then, given a RS (p, w), let

ec be such that pec = pcfor allc . Also, let c be such that p c5 pcfor allc . We can check thatis an exploiter in the Roemer (1982) senseif and only ifmax (p, w) W +w > pec. Also, is exploited in the Roemer(1982) sense if and only ifmax (p, w) W +w < pc

. Then:

Theorem 6: Under A1, A2, A3, and stationary expectation of prices, thereexists an economy with convex cone technology such that there exists a repro-ducible solution(p, w)withmax (p, w)> 0,in which there exists CSCP

who is not exploited in the Roemer (1982) sense.

Proof. Let us consider the same economic environment as in the proof of

Theorem 5. Then,P(p, w) ={t

2 P |t

R+}, and

=

(t1, 1.5) R2+ | t1 [0, 1]

(1, t2) R2+| t2 [0, 1.5]

.


Thus, c

for (p, w) = ((0.5, 0.5) , 1) is given by c

= (1, 0). Thus, p c

=12


26/35

5 A New Definition of Labor Exploitation

Roemer (1982) argued that the epistemological role of the CECP in ourunderstanding of the capitalist economy is as an axiom, although the formalversion of it emerges as a theorem. So, if we wish to verify the CECP, we mustseek an appropriate model which will preserve this principle as a theorem.By this reason, Roemer (1982) insisted that the Roemer (1982) definitionof labor exploitation is superior to the Morishima (1974) one. Based uponthis argument made by himself, however, the Roemer (1982) type of laborexploitation is unable to be justified, since the CECP fails to hold even in themodel with the Roemer (1982) exploitation as shown in Theorem 6. Thus,we seek further for a new definition of Marxian labor exploitation.

In this section, following Roemer (1982), we still adopt the definitionof labor value of commodities as in Definition 3. However, we refine thedefinition of labor exploitation from Roemers (1982). The definition of laborexploitation we propose is the difference between one unit of labor suppliedby an agent per day and the minimal amount of direct labor socially necessaryto provide the agent with his incomeper day.

Note that in Roemer (1982), labor exploitation was the difference be-tween the supplied labor and the minimal amount of labor which was so-cially necessary to produce a commodity vector as a net output, where thecommodity vector can be regarded as the subsistent consumption vector or

as an agents demand vector subject to his budget constraint. In this case,exploitation status of agents would be influenced by the characters of thesubsistent consumption vector or of agents demand vectors, as Theorems1, 4, and 6 indicated. This seems unconvincing from the Marxian point ofview, because the exploitation status of agents should reflect their objectiveconditions of labor only. Any agents who earn the same income by supplyingthe same amount of homogenous labor should be placed at the same status interms of exploitation, regardless of their consumption vectors. This problemcould be overcome by the new definition we propose.

The new definition is formally given by the following:

Definition 9: Let (p, w) be a reproducible solution. Then, an agent Nisexploitedif and only if:

minfF

l.v. (f;p, w)< 1.

26


27/35

In contrast, an agent N is anexploiter if and only if:

minfF

l.v. (f;p, w)> 1.


Note that if an agent N is a worker who purchases a subsistent con-sumption vectorb Rm+ by his wage income under a RS

(p, w) , {}N

,

then for anyf Rm+ such thatpf =pb = w,

l.v. (f;p, w) = min {0 | = (0,,) (f;p)} .

Thus,minfFl.v. (f;p, w)in Definition 9 implies the minimizer ofl.v. (f;p, w)

over the budget set

B(p, w)

f Rm+ |pf =pb = w

.

This implies that the labor value in Definition 9 is concerned not with anagentsconsumption vector,butrather with an agentsincomeearned. Thus,the new definition implies the following: Suppose an economy is under areproducible solution (p, w). Then, if the minimal amount of labor sociallynecessary to provide each agent with income max (p, w) W +w is less(resp. more) than unity, then is exploited (resp. exploiter).

Noting that this fact, we can show the following theorems:

Theorem 7:Under A1, A2, and stationary expectation of prices, letp, {}N

be thereproducible solution (RS)in the sense of Definition 5. Then, theRSyields positive total profits if and only if every worker is exploited in the senseof Definition 9.

Proof. (): Letp, {}N

be a RS with a positive total profit. Thus,

p XN

( )!XN

0 = p XN

( )XN

0b!= p(b 0b)> 0.

Since p Rm+ andb = 0b by Definition 5(b), the last strict inequalityimpliesb 0bandb6=0b. Letf Rm+ be such thatpf=pband0f=tb

27


28/35

for some 0 < t < 1. Then, by the convex cone property of the production

set, l.v. (0f;p, 1)5 l.v. (0b;p, 1), and l.v. (0f;p, 1)< l.v. (0b;p, 1) holdswhenever f 6= b. Thus, l.v. (0f;p, 1) < 0. By linearity, l.v. (f;p, 1) < 1,

which implies minbfFl.v.bf;p, 10. Note for

this RS p, {}N, any profit-maximizing production points0 P(p, 1)P(0 = 1)has the property that pb0 = 1 bymax (p, 1) = 0. Thus, for any0 P(p, 1) P(0 = 1), pb0 = pb0 = pb. This implies for anyf Rm+such that pf = pb, l.v. (f;p, 1) = 1 holds. Hence,minbfFl.v.

bf;p, 1 = 1,so that no worker is exploited in the sense of Definition 9.

Ifp 0, it may be the case thatb 0b andb 6= 0b. However, as p(b 0b) = 0and {}Nconstitutes a profit-maximizing production planatp,b bP(0= 1)holds. By the same argument as above, for anyf Rm+such thatpf=pb,l.v. (f;p, 1) = 1holds. Thus,minbfFl.v.

b

f;p, 1

= 1, so

that no worker is exploited in the sense of Definition 9.

Note that when showing Theorem 7, A3 is no longer indispensable.

Theorem 8:Under A1, A2, and stationary expectation of prices, letp, {}N

be thereproducible solution (RS)in the sense of Definition 6, with the aver-age consumption demand of the employed workers, d

p; (0 )I

. Then, the

RSyields positive total profits if and only if every type of worker is exploitedin the sense of Definition 9.

Proof. By Theorem 3, if it is shown that every individual has a commonexploitation rate in the sense of Definition 9, we can show the desired re-

sult. By definition, I, pd (p) = pd p; (0 )I = 1. Given the RSp, {}N

, letb PN( ) and0 PN0 . Letb0 b0 .

Then, as shown in the proof of Theorem 7, there exists some t (0, 1] suchthat I, minfvFvl.v. (fv ;p, 1) = l.v.

tb0;p, 1. This implies that all

individuals have a common exploitation rate in the sense of Definition 9.

28


29/35

We can check that is an exploiter in the sense of Definition 9 if and

only ifmax (p, w) W +w > pec. In addition, is exploited in the sense ofDefinition 9 if and only ifmax (p, w) W +w < pec. Then:Theorem 9: Under A1, A2, A3, and stationary expectation of prices, let(p, w) be the reproducible solution (RS) in the sense of Definition 7, withmax (p, w)> 0. Then, it holds that:

(A) every member of CH is an exploiterin the sense of Definition 9.(B) every member of CS CP isexploitedin the sense of Definition 9.

Proof. The part (A) is already shown by Roemer (1982). Thus, we onlyshow the part (B). To verify CS CP consisted of exploited agents in thesense of Definition 9, we have to show

pecwmax (p, w)

= minP(p,w)

p

0

. (1)

SinceP(p, w)is a cone, we can normalize the right hand side of the inequality(1) by taking P(p, w) for which 0 = 1. Thus, (1) can be reduced to

p

ecw

max (p, w)= min

(p,w)p, (2)

as in the case for inequality (2). Note that

min(p,w)

p 5min

p (3),

as in the case for the inequality (3). Then, it is sufficient to show:

pecwmax (p, w)

=min

p. (4)

Taking max (p, w)p pb w for any (p, w), the inequality (4) is

equivalent to:pecw =min

pbw. (5)

Note thatb for any . Thus, for anyb such that ,pec pb, which implies (5) holds true. This implies the part (B) is shown.

29


30/35

As shown in the above theorems in this section, the new definition of labor

exploitation performs well in terms of both the FMT and the CECP. However,the new definition has exclusively distinct characteristics in comparison withthe previous definitions, which may give us new insights on the Marxiantheory of labor exploitation and the theory of labor value.

First, Roemer (1982) claimed that prices should emerge logically prior tolabor values so as to preserve the CECP as a theorem in a general convexcone economy. According to Theorems 7, 8, and 9, we would also follow theabove claim of Roemer (1982) in order to verify not only the CECP, but alsothe FMT in more general economic models.

Secondly, in the orthodox Marxian argument, labor exploitation was ex-

plained by using the concept of the labor value of labor power. The laborvalue of labor power could be defined in the Morishima (1974) frameworkas the minimal amount of direct labor necessary to produce the subsistentconsumption vector as a net output. This could be accepted by the orthodoxMarxist as the formulation of the socially necessary labor time to reproducelabor power. Here, the subsistent consumption vector plays a crucial role inthe formulation of the labor value of labor power. In the new definition ofthis section, however, the labor value of labor power might be defined as theminimal amount of direct labor socially necessary to provide workers withthe income by which they can respectively purchase at least the subsistentconsumption vector. In this formulation, the subsistent consumption vectoris used, at most indirectly, to define the labor value of labor power. Thus,even the labor value of labor power no longer emerges logically prior to theprice of labor power (wage income). Hence, the concept of labor value inthis new definition is completely irrelevant to theories ofexchange values ofcommodities and labor power.

In spite of such a significant difference of this new definition from the or-thodox Marxian notion of labor exploitation, it would be justified, accordingto the scenario Roemer (1982) offered, since both of the FMT and the CECPhold true for this new definition.

6 References

Bowles, S., and Gintis, H. (1981): Structure and Practice in the LaborTheory of Value,Review of Radical Political Economics12, pp. 1-26.

Bowles, S., and Gintis, H. (1990): Contested Exchange: New Microfounda-

30


31/35

tions for the Political Economy of Capitalism, Politics & Society 18, pp.

165-222.Brody, A. (1970): Properties, Prices and Planning, New York: AmericanElsevier.

Devine, J., and Dymski, G. (1991): Roemers General Theory of Exploita-tion is a Special Case: The Limits of Walrasian Marxism, Economics andPhilosophy7, pp. 235-275.

Devine, J., and Dymski, G. (1992): Walrasian Marxism Once Again: AReply to John Roemer, Economics and Philosophy 8, pp. 157-162.

Fujimori, Y. (1982): Modern Analysis of Value Theory, Springer-Verlag,Berlin.

Krause, U. (1982): Money and Abstract Labor, New Left Books, London.

Morishima, M. (1973): Marxs Economics, Cambridge University Press, Cam-bridge.

Morishima, M. (1974): Marx in the Light of Modern Economic Theory,Econometrica42, pp. 611-632.

Okishio, N. (1963): A Mathematical Note on Marxian Theorems,Weltwirtschaft-sliches Archiv.

Roemer, J. E. (1980): A general equilibrium approach to Marxian eco-nomics, Econometrica 48, pp. 505-530.

Roemer, J. E. (1981): Analytical Foundations of Marxian Economic Theory,Cambridge University Press, Cambridge.

Roemer, J. E. (1982): A General Theory of Exploitation and Class, HarvardUniversity Press, Cambridge.

Steedman, I. (1977): Marx after Sraffa, New Left Books, London.

Samuelson, P. (1982): The Normative and Positive Inferiority of MarxsValues Paradigm,Southern Economic Journal 49, pp. 11-18.

Yoshihara, N. (1998): Wealth, Exploitation, and Labor Discipline in theContemporary Capitalist Economy,Metroeconomica49, pp. 23-61.

31


32/35

Figure1 Proof of Theorem1

2

( )2m

net output setcorresponding to

net output setcorresponding to

1

( )0 1p =

3

0 1

1

2

2 3

( )P p

( ),b p

b


33/35

3

1

2

4

( )* *

d p

3

1

2

4

(0,1.01)

(1.5,1)

(2.5,1)

(2.6,0)

=

=

=

=* 1 2,

3 3p

=

1.251.01

1

0.75

0.5

0.25

0.5 1 1.5 2.5 2.6

Figure2

( )( )** 0 , 0;d p

( )*d p


34/35

Figure3 Proof of Theorems 5 and 6

Net output set

corresponding to

1.75

1.5

1

0 1 2 3

1

2

3

4

b

( ),P p w

( )0 1P =


35/35

Figure4 Definition9

( )0 1P =

( )0 1P

Documents

Naoki Yoshihara - Reexamination of Marxian Explotation Theory