Marxian Value Theory
PROPERTY AND CONTRACT
THE CASE FOR
David P. Ellerman*
The One Commodity Model
A simple corn with labor input-output model [e.g., von Weizsacker 1971] is used in this chapter to present and analyze the modern treatment of the Marxian labor theory of value and exploitation. There is one produced good, corn, and homogeneous labor. The input-output technology for each enterprise is specified by:
A = number of bushels of seed corn per bushel of harvest corn, and
a = number of hours of labor needed per bushel harvest corn.
A firm’s gross output of X bushels of corn requires AX bushels of seed corn as an input and L = aX hours of labor.
For simplicity (not realism), it is assumed that no other scarce inputs are required. For the technology to be viable, it is assumed that A < 1, i.e., less than a bushel is needed to grow a bushel. It takes one time period, called a «year,» for the labor L = aX to produce the output X by using up the inputs AX. The inputs are required at the beginning of the year and the outputs are available at the end of the year.
Let p*= money price of corn per bushel, let w = money wage rate (at year-end), and let r = rate of interest for the year. Wages are paid at the end of the year while corn inputs are purchased at the beginning of the year. The capital outlay per unit output is p*A. If that capital is deposited in a savings account, it will compound to (1+r)p*A at the
end of the year. That is the passive use of capital. Alternatively, if the capital is used «actively» by being invested in production, then one unit of output will be produced (no uncertainty), sold for p*, and then Labor is paid the wage wa. In a perfectly competitive model with no uncertainty, capital can be switched freely between the passive and
active uses, so competitive arbitrage will enforce equality in equilibrium between the passive return of (1+r)p*A and the active return of p* – wa: Competitive Equilibrium Condition: p* = wa + (1+r)p*A.
Solving for the equilibrium price yields:
p* = wa[1–(1+r)A]-1.
Dividing through by the money wage rate w expresses the price p = p*/w in terms of the numeraire of labor:
p = p*/w = a[1–(1+r)A]-1.
Marxian Labor Theory of Value and Exploitation
After more than a century of analysis and interpretation, Marxian economics has arrived at a precise modern formulation of a Marxian labor theory of value and exploitation [e.g., Morishima and Seton, 1961; Okishio, 1963;
Morishima 1973; Wolfstetter, 1973]. It is this modern theory that is analyzed here. The theory stands by itself. We are not concerned here with the question of whether or not this «Marxian labor theory of value and exploitation» represents «what Marx really meant.»
There are several ways to state the definition of the Marxian labor value v of a unit of corn. From the neo-classical viewpoint, it is the equilibrium price of a unit of corn if the interest rate («rate of profit») is zero. The equilibrium price (in terms of labor) was
p(r) = a[1–(1+r)A]-1
so we have
Marxian Labor Value = v = p(0) = a[1–A]-1.
There is also the «net product» definition. Given the gross product X, the required seed corn was AX so the net product is defined as Y = X – AX (note how the definition subtracts seed corn from harvest corn «as if» there was no time difference). Then the Marxian labor value v could be defined as the labor necessary to produce one unit of net output. If net output = Y = 1 = X – AX, then the required gross product is
X = [1–A]-1
so the required labor is
v = a[1–A]-1.
The «historical» definition is based on the summation of all the labor directly and indirectly embodied in a unit of corn (assuming constant technology throughout the past). One unit of corn requires the direct labor of a units. It required the seed corn A which required the labor aA. The seed corn A also required the seed corn AA, which required the labor aAA, and so forth. Summing the labor
v = a + aA + aA2 + aA3 + … = a[1 + A + A2 + A3 + …] = a[1–A]-1
(using the formula for the geometric series to evaluate the sum for 0<A<1). Note that this definition adds together labor from different time periods «as if» there was no time difference. All the definitions of labor value are equivalent [see Wolfstetter 1973 for yet another equivalent definition].
Each unit of labor is paid the money wage w so the physical wage in terms of corn is the
Wage-Basket = z = w/p* = 1/p.
One could think of labor as being paid z bushels per unit of labor. When a worker expends one unit labor, the payment is the wage-basket z with the Marxian value (or «labor content») vz. This is called the Necessary Labor = Paid Labor = vz.
The remainder is called the Surplus Labor = Unpaid Labor = 1 – vz.
The ratio of surplus labor over necessary labor is the Marxian Rate of Exploitation = e = (1 – vz)/vz.
Morishima [1973, see Ellerman 1983 for a proof in the simple one commodity model] proves the: Fundamental Marxian Theorem (FMT). The rate of exploitation e is positive
if and only if the rate of interest r is positive.
Hence the Marxian theory concludes that Labor is exploited if the rate of interest or rate of profit is positive.
Analysis of Marxian Exploitation Theory Some Misconceptions
There are a few frequent misinterpretations of Marxian exploitation theory which must be mentioned first. The Marxian theory is not a bargaining power theory; the setting is a perfectly competitive model. Marx wrote extensively about the inequalities of the marketplace, but he wanted to criticize capitalist production itself, not just monopolistic
imperfections. Hence he set out to expose exploitation in competitive capitalism, and the modern formulation preserves that competitive setting.
Marx also tried to relate the exploitation analysis to the workplace power relationship of the employer over the workers. In spite of the rhetoric which usually accompanies presentations of the modern theory, the role of power relations did not survive in the modern reformulation. No assumptions about power relations were made in the inputoutput model, yet the presence of Marxian exploitation can still be derived. Hence the modern result does not depend on power relationships. The veneer of rhetoric about «the capitalist forcing the workers to work longer than it takes to produce their labor-power» (wage-basket z) only obscures the real basis for the exploitation result.
Analysis of Marxian Value
The definition of Marxian value v systematically neglects the effect of time–an effect registered by the interest rate. Time puts a difference on commodities. As any farmer could testify, having corn available to plant at planting time is quite different from having the otherwise identical corn at harvest time. The seed corn and harvest corn are
economically distinct–like «apples and oranges.» One unit of a commodity at time t is equivalent to 1+r units at time t+1 in the sense that the market will trade one for the other (at constant prices). For example, the loan market trades $1 for $(1+r) a year from now. The Marxian value definition treats the units of labor (or corn) at different times as
being the same (so they can be meaningfully added together), and thus the definition implicitly treats the interest rate as being zero.
Consider the net product definition of Marxian value v. The definition of the net product y = X – AX assumes that the beginning-of-the-year inputs AX are commensurate with the end-of-the-year outputs X so that the former can be subtracted from the latter to arrive at the net product. However, the difference X – AX is as meaningful as the difference «4 apples minus 3 oranges.» The inputs AX are equivalent to (1+r)AX units at the end of the year, so the time-corrected net product in terms of commodities timed with outputs is:
y(r) = X – (1+r)AX.
When the corrected net product y(r) is equal to 1, the gross output is:
X = [1 – (1+r)A]-1.
Then the time-corrected Marxian value of a bushel of corn is:
v(r) = a[1 – (1+r)A]-1
in terms of beginning-of-the-year labor–which is precisely the price p(r) of a bushel of corn in terms of labor.
It is also possible to apply the time correction to the «historical» definition of Marxian value since that definition adds to labor performed in different time periods. If one bushel of corn is produced at year’s end, then all the past embodied labor can be transformed into the equivalent beginning-of-the-year labor before being summed. The labor aAn performed n years before the beginning of the current year is equivalent to a(1+r)nAn units of labor at the beginning of the year. Hence the corrected historical definition of Marxian value is:
a + a(1+r)A + a(1+r)2A2 + …= a[1 + (1+r)A + (1+r)2A2 + …]= a[1 – (1+r)A]-1= v(r) = p(r)
[assuming that (1+r)A<1] which is the same as the corrected net product definition.
What is the difference between the Marxian value v and the competitive market price p(r)? The difference is that Marxian value definition ignores time. Time is registered by the interest rate in the model. The uncorrected Marxian value v is p(0) the price when the interest rate is zero, and the corrected Marxian value v(r) is identical with the price
What happens to «exp loitation» under the time correction? The time-corrected necessary labor contained in the wage-basket z paid for one unit of labor is:
v(r)z = p(r)z = 1
so the surplus labor is 1 – v(r)z = 1 – 1 = 0. Hence the exploitation result vanishes under the time correction.
Marxian Value Theory as a Just-Price Theory
The Fundamental Marxian Theorem ( e > 0 if and only if r > 0) is often interpreted as showing the exploitation is the hidden inner meaning of the charging of interest. Our results indicate that precisely the opposite is the case; the charging of interest is the hidden inner meaning of «exploitation.»
The modern formulation of the Marxian labor theory of value and exploitation is in fact a just-price theory. It takes as a normative benchmark the time-saturated regime where the rate of interest (called the «rate of profit») is zero. It evaluates the transactions of the actual economic regime (where r is positive) at the benchmark prices. It finds that the workers receive less in the actual regime than they would in the benchmark regime; that difference is precisely the «exploitation.» Of course, Marx did not intend or desire the theory to be only a just-price theory. But that is one of the ways a theory might fail. When finally worked out in a detailed and consistent form, the theory might fall far short of the original expectations.
Let r be the positive interest rate in the actual regime, so p(r) is the actual price of a bushel of corn in terms of the numeraire of labor. Since the price of labor in terms of labor is always unity, p(r)z = 1 so the real wage basket z = z(r) = 1/p(r) is also a function of r.
Price-Wage Tradeoff p(r)z(r) = 1
In the benchmark regime where r = 0, the just-price of corn is v = p(0) the Marxian value of a unit of corn. The just wage, which «represents the real wage rate that would prevail if there was no exploitation» [Morishima 1973, p. 54], is:
Just Wage = z* = z(0) = 1/p(0) = 1/v = [1–A]/a = Net Product per unit Labor.
As the interest rate r moves from zero to a positive value, the price of labor, the real wage z(r), decreases so laborsellers are worse off in the actual regime in comparison with the benchmark regime. How much worse off? In selling the labor L = aX, the workers would receive z*L in the benchmark regime and they receive zL in the actual regime.
The difference is:
[z(0)–z(r)]L = z*L – zL = ([1–A]/a – z)aX = X–AX – zL
= Net Product – Wage Corn = Surplus Product.
Thus the so-called «surplus product» is just the difference between the «just corn wages» and the actual corn wages for the labor L. And the benchmark value of that wage differential in terms of labor is:
p(0)[z(0) – z(r)]L = (1–vz)L = v(X–AX) – vzL = Total Surplus Labor.
In the (hypothetical) transition from the benchmark to the actual regime, the economic position of labor-sellers worsened, and that is precisely the «Marxian exploitation.» The difference in the wage-bill is the «surplus product» in terms of corn and the difference is the «surplus labor» in terms of labor.
The same sort of «exploitation» analysis could be applied to any price change. Here is an apple-selling example.
Suppose in the benchmark situation, Benchmark Prices: 10 Apples = 1 Bushel of Corn.
But in the actual situation, the price of apples dropped relative to corn.
Actual Prices: 15 Apples = 1 Bushel of Corn.
Suppose the apple-owner sells 300 apples in return for 300/15 = 20 bushels of corn. Let us «pierce the veil» of this competitive market transaction to «reveal its inner nature.» In return for the 20 bushels , the apple seller first gives up 200 apples. The 200 apples have the same «value» as the 20 bushels (i.e., «value» = benchmark prices). Everything seems fair and square. The 200 apples were «paid for» by the 20 bushels. But then the apple-seller is «forced to alienate» an additional 100 apples which is «appropriated as a surplus» by the corn-owner without any further corn payment in return. These extra 100 apples are the «unpaid» apples. In terms of corn, the corn-owner gave up 20 bushels to receive the «value» of 30 bushels so the surplus appropriated by the corn-owner represented 10 bushels of corn.
Apple-Corn Trades at Benchmark Prices
( 10 Apples = 1 Bushel of Corn)
Apple-Corn Trades after Price Change
(15 Apples = 1 Bushel of Corn)
«Exploitation» Analysis of Apple Price Change
The ratio of the unpaid apples to the paid apples is 100/200 = .5 so there is a 50% rate of exploitation. «Beneath the facade» of the market transaction, we have revealed the «exploitation» of the apple-seller by the «forced alienation» of the surplus apples.
Marxian exploitation theory applies this same methodology to the labor contract. It clearly has nothing to do with workplace power relations. The time-saturated benchmark regime defines the «just prices.» The just interest rate is zero, the just corn price per bushel is v, and the just wage is z* = 1/v. In the hypothetical transition from the zerointerest
benchmark model to the actual model, time enters as a scarce resource commanding a positive price (the positive interest rate). The price p(r) of corn in terms of labor is a monotonic increasing function of r:
p(r) = a[1 – (1+r)A]-1 = a[1 + (1+r)A + (1+r)2A2 + …]
so there is more labor for the same corn, i.e., less corn for the same labor, than in the benchmark model. Where, say, 200 units of labor may have previously traded for 20 bushels, it now takes 300 units of labor to buy 20 bushels of corn. The first 200 units of labor have the same «value» (= benchmark price) as 20 bushels of corn, so the last 100
units of labor represent «unpaid labor» (at benchmark prices). Thus there is «exploitation» if and only if the interest rate is positive. That «Fundamental Marxian Theorem» may «be considered as the heart and soul of Marxian philosophy…» [Morishima 1973, p. 6].
The modern Marxian labor theory of value and exploitation has nothing to do with workplace power relations, with wage labor, or even with capitalist property relations. It turned out to be a pre-liberal Aristotle-Aquinas «interest grumble» dressed up in Marxian garb.