1.0 IntroductionPiero Sraffa, in his critique of neoclassical theory, described a system of prices in which capitalist earn the same rate of profits in every industry. One can derive, in the pure circulatingcapital version of this system, a tradeoff between wages and the rate of profits.
The shape of this wagerate of profits curve depends on the (possibly composite) commodity chosen for the numeraire. It is a straight line when Sraffa's
standard commodity is used for the numeraire. If the wagerate of profits curve were a straight line for all other numeraires, the labor theory of value would be be exactly true as a theory of relative prices, abstracting from deviations between market prices and prices of production and from the theory of joint production. This theorem of mathematical economics raises an empirical question. How far does the wagerate of profits curve vary from a straight line for various numeraires?
P. Petrovic ("Shape of a WageProfit Curve, Some Methodology and Empirical Evidence",
Metroeconomica, V. 42, N. 2 (1991): pp. 93112) explored this question for 1976 and 1978 data from Yugoslavia. Petrovic found that the empirical wagerate of profits curve never deviated much from a straight line, no matter what numeraire was chosen.
I was only able to partially replicate Petrovic's results with 2005 USA data. The 2005 USA wagerate of profits curve drawn with a numeraire in the proportions of net output is indeed quite close to a straight line. But the 2005 USA wagerate of profits curve can be quite convex or quite concave, depending on the choice of the numeraire. My methodology differed from Petrovic's in that I introduced a normalization of the numeraire quantity to fix the maximum wage at unity for each numeraire.
My estimate of the rate of profits in the USA is higher than I expected. I am beginning to think that my approach is too simple. Perhaps I need to account for depreciation, fixed capital, and the distinction between productive and unproductive labor. I may post more analyses in this series before revisiting my past results.
2.0 Derivation of the WageRate of Profits CurveConsider an economy in which each of
n commodities are produced from labor and inputs of those
n commodities. Let
a_{0, j} be the personyears of labor used in producing one unit of the jth commodity. Let
a_{i, j} be the physical units of the ith commodity used in producing one unit of the jth commodity. The direct labor coefficients are elements of the
nelement row vector
a_{0}. The remaining inputoutput coefficients are the entries in the
nx
n Leontief inputoutput matrix
A, which is assumed to satisfy the HawkinsSimon condition.
The Sraffa prices equations, in which wages are paid out of the surplus, are:
p A (1 + r) + a_{0} w = p
where
p is a row vector of prices,
w is the wage, and
r is the rate of profits. After some manipulation, one has:
a_{0} [ I  A (1 + r)]^{1} w = p
The HawkinsSimon conditions guarantees the existence of the matrix inverse for rates of profits between zero and some maximum rate of profits. Let
e be a column matrix representing the numeraire. Multiply on the right by the numeraire:
a_{0} [ I  A (1 + r)]^{1} e w = p e = 1
The wagerate of profits curve is then:
w = 1/(a_{0} [ I  A (1 + r)]^{1} e)
3.0 Empirical ResultsFigure 1 shows the range of convexities, depending on the numeraire, of the wagerate of profits curve in the USA in 2005. The straightline wagerate of profits curve is constructed using Sraffa's standard commodity as the numeraire.

Figure 1: WageRate Of Profits Curve For Selected Numeraires 
I examined a numeraire for each of the 65 industries in the 2005 Use Table. The numeraire corresponding to each industry consists solely of the output of that industry; the output of all other industries is zero in this noncomposite numeraire commodity. The quantity of the selected numeraire commodity is set to ensure the maximum wage, corresponding to a rate of profits of zero, is unity. In other words, the numeraire quantity is normalized such that its embodied labor value is one thousand personyears, the unit in which the BEA measures labor.
Figure 1 shows wagerate of profits curves for two of these 65 numeraires. The wagerate of profits curve for the numeraire consisting solely of output of
Warehousing and Storage industry has the highest positive displacement from the straightline wagerate of profits curve. The wagerate of profits curve for the numeraire consisting solely of output of the
Petroleum and Coal Products industry is the furthest below the straightline wage rateof profits curves. The wagesrate of profits curves for all other numeraires are closer to the straightline wagerate of profits curve.
The remaining wagerate of profits curve shown in Figure is drawn for a numeraire in the proportions of positive quantities in the net output (final demand) quantities in the 2005 Use Table. (Final demand quantities are net of the circulating capital goods replaced out of gross output; they still include, however, depreciation charges against fixed capital.) Among the components of final demand, imports and nondefense consumption expenditures from the Federal government can be negative. Thus, the final demand for the output of an industry can be negative, if, for example, more of that industry's output is imported than exported. The following industries have negative quantities in final demand:
 Forestry, fishing, and related activities
 Oil and gas extraction
 Wood products
 Nonmetallic mineral products
 Primary metals
Finally, Figure 1 shows a point for the year 2005. Wages, in numeraire units, are calculated from data on employee compensation, full time equivalent employees, and net output. The data on full time equivalent employees is included with data on gross output and was used to calculate labor values. I did not make any correction here for negative quantities in final demand.
Compensation of employees is a component in Value Added in the Use Table. The other two components of Value Added are
Taxes on production and imports, less subsubsidies and
Gross operating surplus. The actual wage is 0.575 of the net output of a thousand personyears. The corresponding rate of profits is 53.3%. The wage, when net output is used as the numeraire lies 0.0766 numeraire units above the straight line wagerate of profits curve, close to the maximum difference along these two curves of 0.0773 numeraire units.
Theoretically, the wagerate of profits curve for numeraires other than the standard commodity can be of any convexity. Furthermore, the convexity can differ over different ranges of the rate of profits. One might find surprising how close the wagerate of profits curve is to a straight line when net output is chosen as a numeraire. The rate of profits can be increased by an increase in productivity, which moves the wagerate of profits curve outward. The rate of profits can also be increased by a decrease in the wage, that is, by increasing the exploitation of the workers.