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Lecture 1
Class, exploitation, and inequality
John E. Roemer
Yale University
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I will cover
four topics:
1. The microfoundations of exploitation and class
a. definition of embodied labor time
b. definition of exploitation
c. definition of class
d. The Class Exploitation Correspondence Principle
(CECP)
e. The Class-Wealth Correspondence Principle
(CWCP)
2.
What is ethically wrong with Marxian exploitation?
3.
The ‘socialist’ allocation in a simple economy: the
Proportional Solution
4.
Inequality vs. exploitation: social democracy vs.
socialism
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I.
Exploitation and class
(J. Roemer, A General Theory of Exploitation and Class,
Harvard UP, 1982)
1.
Production: n goods, Leontief production (A,L) .
Thus to produce a vector of final demands y requires a
vector of inputs Ay and labor in amount Ly.
2. The vector of embodied labor times or labor values is
given by   L(I  A)1 .
Proof 1:
L(I  A)1 y  Ly  LAy  LA2 y  ...
Proof 2: x  Ax  f  x  (I  A)1 f  Lx  L(I  A)1 f .
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3.
Each individual must consume a ‘subsistence bundle’
b. Necessary labor time in the sense of Marx is thus b .
4. Endowments: individual i is endowed with an input
vector of  i °
n

and one unit of labor time.
5. At a price vector ( p, w) °
n1

, the cost of a commodity
vector y is py .
6. Preferences: Each agent wishes to minimize the labor
she must expend to be able to purchase b, while not
reducing the value of her initial endowment.
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7.
The economy. At a price vector (p,w) each agent
chooses three things:
a. the vector of activities, y, to produce by hirinig
labor to work on his own capital stock 
b. the vector of activities to run, using his own labor,
and capital stock
c. the quantity of labor, z, to sell to others.
Thus the optimization problem for an agent i is:
choose (x i , yi , z i ) to
min Lx i  z i
s.t.
A(x i  y i )   i
wz i  p(x i  y i )  wLy i  pb  pA(x i  y i )
These are the constraints of feasibility and reproducibility,
resp.
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8. Equilibrium.
A reproducible solution is a price
vector (p,w) and a set of choices (x i , yi , z i ), i  1,..., N such
that:
a. A(x  y)  
b. (I  A)(x  y)  Nb   .
Proposition 1.
indecomposable.
Assume that the technology (A,L) is
Then at a reproducible solution (p,w)
there is a number   0 such that
p  (1  )( pA  wL) .
 is the uniform rate of profit.
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9.
Exploitation.
An agent i is exploited at a
reproducible solution if
Lx i  z i  b ;
he is an exploiter if
Lx i  z i  b .
Thus a person’s status as exploited/er is endogenous at the
equilibrium.
10.
Class status.
At a reproducible solution, every agent belongs to one of
the following five classes
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(x, y, z)
(0, , 0) pure capitalist
(+,+,0) petty capitalist
(+,0,0) independent artisan
(+,0,+) semi-proletarian
(0,0,+)
proletarian
Theorem
A. (CECP) Suppose (p,w) is a reproducible solution with
>0.
Any agent who must hire labor to optimize
(i.e.,
a pure or petty capitalist) is an exploiter at equilibrium;
any agent who must sell labor to optimize is exploited.
B. (CWCP) Class membership, as defined above, is
monotone decreasing in wealth ( p   i ).
This theorem provides the microfoundations of several
important Marxian claims: namely, that those who sell their
labor power are exploited, that those who hire labor are
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exploiters, and that exploitation status is decreasing in
wealth.
11.
How does the theorem generalize when we admit
arbitrary preferences instead of subsistence preferences?
It is then necessary to provide a definition of exploitation
that does not rely on the ‘subsistence bundle.’
Definition.
Let the revenues of an agent be denoted Ri.
Let his activities at equilibrium be (x i , yi , z i ) .
that an agent is exploited if
max{  X | p  X  Ri }  Lx i  z i .
X
An agent is an exploiter if
min{  X | p  X  Ri }  Lx i  z i .
X
We say
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The result is: The CECP remains true for general
preferences.
But the CWCP does not remain generally
true.
12. The truth of the CWCP depends upon the elasticity of
labor supply with respect to wealth.
Theorem If labor supply is elastic w.r.t. wealth at the
equilibrium, then the CWCP is true.
if
I.e.,
d(Lx i  z i ) p i
 0.
d( p i ) Lx i  z i
Intuitively, consider a land-poor old woman who hires a
land-richer young man to work on her land; she makes a
profit from his labor, and he agrees to the contract to
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increase his income. She will exploit him, although he has
greater wealth than she.
13.
A final comment
Note that, if we view the capital assets as land, we give the
following names to the five classes:
(x, y, z)
(0, , 0) landlord
(+,+,0) rich peasant
(+,0,0) middle peasant
(+,0,+) poor peasant
(0,0,+)
landless laborer
Indeed, it is precisely these definitions that Lenin made in
his book The development of capitalism in Russia (1899)
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and that Mao Zedong used in Analysis of classes in Chinese
society (1926). So our analysis shows that Lenin and Mao
indeed got it right.
II. What’s wrong with exploitation?
1. What is wrong with exploitation, defined in its
technical Marxian sense – that b  labor expended?
The neoclassical economist will say ‘The worker is paying
rent to the capitalist for access to the means of production
which make his labor more productive.’
This is true.
So the question becomes: What is the ethical status of
inequality in the distribution of capital assets?
2. Marx’s analysis of the ‘primitive accumulation’ of
capital in England in Part VIII of Capital , volume 1.
Robbery, enclosure, pillage.
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3. But what if capital accumulation comes about
‘cleanly?’ It is then not so obvious that exploitation is
ethically bad. One has to then ask: Did everyone have the
opportunity to accumulate capital –- what is our evaluation
of the moral status of those opportunities?
4. Exploitation comes about if three conditions hold:
a. private and unequal ownership of capital stock (or
land)
b. competitive markets
c. a scarcity of labor relative to capital stock
available to employ it.
This is so even if everyone has the same skills.
If
skills are differentially distributed, then unequal
ownership of capital stock will emerge dynamically, even if
everyone has the same preferences.
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Marx was right that exploitation is a phenomenon that
exists even with competitive markets. Important point.
This distinguishes Marxian exploitation from neoclassical
exploitation, which is a phenomenon of non-competitive
markets – when workers are not paid their marginal product.
5. The Marx-Lenin-Mao solution was to socialize capital
assets, that is, to end private ownership.
But Lenin and
Mao also replaced competitive markets with central
allocation.
The proposal of market socialism (originally: Oscar
Lange) is to socialize capital assets but to retain
competitive markets:
exactly how to do so is an
interesting question.
The purpose of this would be to
stop exploitation but to retain the good allocative properties
of competitive markets.
(See J. Roemer, A future for
socialism (1994), there are two Chinese translations.)
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6.
On the other hand, social democracy maintains that it
is inequality that is ethically bad, not exploitation.
The
social-democratic solution is to retain unequal ownership
of capital but to redistribute income through two processes:
a.
wage differential reduction (solidaristic wage)
b.
taxation and redistribution.
This pair of institutions has made the five Nordic countries
the most egalitarian in the world, in terms of after-tax
income and consumption.
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III. The socialist allocation
(J. Roemer and J. Silvestre, 1993. “The proportional
solution in economies with public and private ownership,”
J. Econ. Theory)
1. The socialist allocation is: “From each according to his
ability, to each according to his work.” In contrast to
‘communism’ which is “….to each according to his need.”
2.
Let us define such an allocation for a general economy
where agents possess different skill levels.
We work with a simple economy.
There is one good produced from labor:
x  F(l ), l is labor measured in efficiency units.
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Agent i has skill level si.
Thus,
  s i Li .
Think of F as the aggregate production function,
where all capital is efficiently employed.
Thus,
F(l )  G(K,l ) , K is total capital stock.
3.
If G exhibits CRS, the F will exhibit decreasing
marginal productivity w.r.t. labor.
4. Preferences: agent i has a utility function u i (x, L) . i
=1,…,N.
5. Thus we have an economy defined by {u, s, F}.
6.
We say that an allocation   ((x1 , L1 ),...,(x N , LN )) is a
proportional solution if
a. there is a number k such that
for all i,
xi  ksi Li , and
b.  is Pareto efficient.
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I claim this is correct definition of the socialist allocation.
No exploitation, because output is distributed in proportion
to labor expended.
Note that this is a price-independent
concept.
7.
Theorem.
Under weak conditions on (F,u), a
proportional solution exists.
(Roemer and Silvestre (1993); Roemer (2005).)
Proportional solutions are locally unique: like Walrasian
equilibria.
There are a finite number (often just one) for
a typical economic environment.
8. There is no simple way to implement the proportional
solution with a price mechanism. There are game forms
which can implement it, but they are not simple.
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Nevertheless, the PS provides a normative benchmark
for what the ‘socialist’ should want.
9. The tragedy of the commons.
There is a lake, used
by fishermen or ‘fishers’ of different skills, or with
different size boats. Output of fish is F( s i Li ) .
Suppose the lake is commonly owned: each can fish as
much as she pleases. Fish will be caught in proportion to
the amount of efficiency units of labor a person expends in
fishing.
Thus we have a game: the fish caught by each
fisher is a function of the labor expended by all fishers,
because of decreasing marginal productivity of labor on the
lake.
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An equilibrium under common ownership is a Nash
equilibrium of this game, where the strategy of each fisher
is her labor supply.
It is a well-known fact that the equilibrium under
common ownership is Pareto inefficient.
There is ‘over-
supply’ of labor: everyone would be better off, were all to
reduce their labor supply a little bit. This is called the
tragedy of the commons.
The proportional solution is the resolution to the
tragedy of the commons.
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10. Kantian equilibrium.
Consider the following general problem of cooperative
production.
Each agent can expend an effort and the
utility of each agent depends upon the efforts of all: thus
we write the utility of agent i as vi (L1 , L2 ,..., LN ) .
fishing problem,
Definition.
In the
vi is a decreasing function of Li .
A Kantian equilibrium is a vector of
efforts ( L̂1,..., L̂N ) such that:
for all r  0, for all i  1,..., N v(rL̂)  v( L̂) .
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Thus, nobody would like that everybody change his labor
supply by the same factor – for any factor r.
Theorem The proportional solutions in the problem
(u,F,s) are precisely the Kantian equilibria. (Roemer;
1996, 2005)
We also have, for completely general cooperative
enterprises {vi } :
Theorem Kantian equilibria are Pareto efficient.
In other words, the cooperative counterfactual of the
Kantian equilibrium solves the inefficiency associated with
the Nash counterfactual.
The first theorem tells us that we have a natural
identification of the ‘socialist’ allocation with a cooperative
counterfactual.
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IV. Inequality vs. exploitation
1. It is important to note that the income distribution will
in general be unequal at the proportional solution for an
economy (u,F,s). This is due both to differential skills and
to differential preferences.
2. In this section I ask: How much inequality in modern
economies is due to exploitation, and how much to these
other factors (u,s)?
I attempt to make this question
precise by asking: How much taxation would be needed, in
the US, to produce an after-tax distribution of income with
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the same Gini coefficient as the US economy would have in
its proportional solution?
I.e., when would ‘social democracy’ produce the same
degree of income inequality as ‘socialism?’
3.
A stylized version of the US economy:
Single output x  K 1 L .
Distribution of skills s, where s : F on support

.
We assume that capital ownership is distributed according
to a power function, so that a worker of type s owns sd
amount of capital.
Thus per capita capital and labor in
the economy are:
K    s d dF(s), L   sdF(s) .
1
Preferences: u(x, L)  x   L
1

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Thus, the economy is e  (F,  , ,d,  , ).
Defn. 1 A socialist allocation for e is a pair of functions
(x(s), L(s)) and a number c such that:
(1) for all s, x(s)  csL(s)
(2) K
1
 sL(s)dF(s)   x(s)dF(s), and

(3) (x(), L()) is Pareto efficient.
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Defn. 2 A social democratic equilibrium at an income tax
rate t is : a wage-interest rate pair (w,r), a labor-supply
%K%)
function L̂(s) , firm demands for labor and capital ( L,
such that:
%K%) solves the firm’s profit-max program:
(1) ( L,
max K 1 L  wL  rK
L,K
(2) L(s) solves worker s’s utility maximizing program:
max(1 t)wsL   L11/
L
(3) Markets clear:
 sL(s)dF(s)  L%,
d

s
 dF(s)  K%.
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4.
I calibrate the model to US data;
the tax rate is
t 0  0.31. We have   0.74,   0.035. It turns out that
d=3.77, or d = 4.0 from two ways of estimating the
distribution of capital ownership. I fit the distribution F of
skills to a lognormal distribution.
5.
Thus, at any tax rate t,
equilibrium of the model.
I get a good fit.
can compute the unique
Define:
y pre (s;t)  w(t)sL(s;t)   r(t)s d
y
post
(s;t)  (1 t)y (s;t)  tK
pre
1
% 
L(t)
.
Thus, can compute the Gini coefficient.
6.
I compute that the Gini coefficient in the proportional
solution is 0.263.
A tax rate of between 0.35 and 0.36 gives the same
Gini coefficient in the social-democratic equilibrium!
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Thus, an increase in the size of the state sector by only
5 percentage points would generate the same degree of
income inequality as ‘socialism.’
7. Conclusion from this exercise: if we are concerned with
income inequality, then (at least in the US) social
democracy seems a better solution than ‘socialism.’
I do
not know what the comparable calculation would produce
for China.
8.
Are there reasons to prefer ‘socialism’ to social
democracy, other than income inequality?
 concentration of capital ownership may engender
concentration
undesirable
of
political
power
which
is
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 induced preferences for public bads: pollution,
private autos vs. public transport, poor working
conditions, wars to secure natural resources
 social ethic of greed vs. one of cooperation.
These are large, open questions.
References
Roemer, J. 1982. A general theory of exploitation and
class, Harvard UP
Roemer, J. 1986. Value, exploitation, and class.
Harwood Academic Publishers
Roemer, J. 1985. “Should Marxists be interested in
exploitation?” Philosophy and Public Affairs 14,, 30-65
Roemer, J. 1994. A future for socialism, Harvard UP
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Roemer, J. and J. Silvestre, 1993. “The proportional
solution for economies with both public and private
ownership,” Journal of economic theory 59, 426-444
Roemer, J. 1996. Theories of distributive justice,
Harvard University Press
Roemer, J. 2005. “Kantian solutions”
Roemer, J. 2006 . “ Socialism vs. social democracy
as income-equalizing institutions”