
Figure 1: Wage Share versus Ratio of Rate of Profits 
1.0 Introduction
Consider the theory that Sraffa's standard system can be used to empirically predict distribution and prices in existing economies. Although individual commodities might be produced with extremely laborintensive or capitalintensive (at a given rate of profits?) processes, large bundles of commodities chosen for technical characteristics, such as net output or wage goods, would be expected to be of average labor intensity. And the standard commodity formalizes the idea of a commodity of average capital intensity.
The data I looked at rejected this theory as a universal description of economies around the world.
2.0 Theory
The standard system is here defined for a model of an economy in which all commodities are produced from labor and previously produced commodities. The technique in use is characterized by the Leontief inputoutput matrix A and the vector a_{0} of direct labor coefficients. The gross output, q, of the standard system is a (right hand) eigenvector of the Leontief inputoutput matrix, corresponding to the maximum eigenvalue of the matrix:
(1 + R) A q = q,
where R is the maximum rate of growth (also known as the maximum rate of profits). The maximum rate of profits is related to the maximum eigenvalue, λ_{m}, by the following equation:
R = (1λ_{m})  1
From previous empirical work, I know that the maximum rate of profits is positive for all countries or regions in my data. The standard system is defined to operate on a scale such that the labor employed in the standard system is a unit quantity of labor:
a_{0} q = 1
The standard commodity, y, is the net output of the standard system:
y = q  A q
In the standard system, such aggregates as gross output, the flow of capital goods consumed in producing the gross output, the net output, the commodities paid in wages, and the commodities consumed out of profits all consist of different amounts of a single commodity basket, fixed in relative proportions. Those proportions spring out of the technical conditions of production in the actual economy.
Prices of production represent a selfreproducing system in which tendencies for capitalists to disinvest in some industries and disproportionally invest in other industries do not exist. In some sense, they arise in an economy in which all industries are expanding so as to maintain the same proportions. Such prices can be represented by a row vector, p, satisfying the following equation:
p A(1 + r) + a_{0} w = p,
where
r is the rate of profits and
w is the wage paid out of the net product. The adoption of the standard commodity as numeraire yields the following equation:
p y = 1
One can derive an affine function for the wagerate of profits. (Hint: multiply both sides of the first equation above for prices of production above on the right by the standard commodity.) This relationship is:
w = 1  (r/R)
Prices of production in the standard system can easily be found for a known rate of profits.
p = a_{0} [I  (1 + r) A]^{1} [1  (r/R)]
If wages were zero, the rate of profits would be equal to its maximum in the standard system. If the rate of profits were zero, the wage would be equal to unity. The wage represents a proportion of the net output of the standard system. It declines linearly with an increased rate of profits.
The gross and net outputs of any actually existing capitalist economy cannot be expected to be in standard proportions, particularly since some (nonbasic) commodities are produced that do not enter into the standard commodity. But do conclusions that follow from the standard system hold empirically? in particular, the average rate of profits, the proportion of the net output paid out in wages, and market prices are observable. Given the average rate of profits for the economy as a whole, the proportion of the standard commodity paid out in wages can be calculated. Is this proportion approximately equal to the observed proportion of wages? Do the corresponding relative prices of production calculated with the standard commodity closely resemble actual relative market prices? This post answers the question about wages. The empirical adequacy of prices of production is left to a later post.
3.0 Results and Discussion
I looked at data on 87 countries or regions, derived from the GTAP 6 Data Base, compiled by the Global Trade Analysis Project at Purdue. (I had help extracting the database and putting it in a format that I can use.) GTAP 6 data is meant to cover the year 2001. The data covers up to 57 industries. (Not all industries exist in each country.)
For each country or region, I calculated:
 The observed proportion of the net output paid out on wages.
 The observed rate of profits, as the proportion of the difference between net output and wages to the total prices of intermediate inputs.
 The maximum rate of profits for the standard system.
 The ratio of the observed rate of profits to the maximum rate.
Figure 2 shows the distributions of the observed and maximum rate of profits.

Figure 2: Distribution of Actual Rate of Profits and Maximum in Standard System 
Four countries or regions in the data had an actual rate of profits exceeding the theoretical maximum rate of profits: The rest of North America, Uruguay, Belgium, and Cyprus. The rest of North America is a region consisting of Bermuda, Greenland, and Saint Pierre and Miquelon. The four countries and regions are excluded from the linear regression and statistics given below.
Figure 1 shows the results of a linear regression of the wage on the ratio of the rate of profits. If, for each country or region, the standard system were empirically applicable to that country or region the intercept of the regression line would be near one, and the slope would be approximately negative one. But the 99% confidence intervals of the intercept and slope do not include these values. In this sense, the theory is rejected by the data.
Figure 1 points out the twelve countries with the wage furthest away from the prediction from the standard system.
Why might the theory be off for these countries and the four excluded from the regression? Perhaps the net output is not near standard proportions. This possible variation of between the proportions of the standard commodity and the actual net output is abstracted from when plugs the observed rate of profits into the wagerate of profits function for the standard system. I have looked at wagerate of profits curves, drawn with the observed technique in use and the observed net output as numeraire. And countries far from the theory generally stick out as having wagerate of profits curves with extreme curvatures.
Another possibility is that the industries in an economy are not earning nearly the same rate of profits, not merely because of barriers to entry but because of the economy not being in equilibrium. Prices of production, for any numeraire do not prevail.
Another possibility is that the Leontief matrix and the vector of direct labor coefficients do not capture the economic potential of the country or region. For example, the calculation of the rate of profits abstracts from the existence of land and fixed capital. Most interestingly, suppose the country or region does not characterize an isolated economic system. A region in the data combines several countries for which data is difficult to get. And the above analysis highlights several of these regions: the rest of North America, Central America, and the rest of Middle East (which consist of all of the Middle East besides Turkey). Or the country under consideration might be small and heavily dependent on imports and exports. You might notice Hong Kong and Singapore, which are important international ports. Think also of small countries that provide offshore banking facilities. Recent events have alerted me to Cyprus serving this purpose for the countries that were formerly in the Soviet Union. I do not know much about Ireland, but recent discussion of how Apple shields its profits makes me wonder about the reported profits for its economy.
I do not know what to fully make of this analysis. The empirical use of the standard commodity seems to be more of a heuristic than the application of a claimed universal law. And the failure of its application seems to point out aspects of the deviating countries that seem of economic interest.
Appendix: Data Tables
Table 1: Descriptive Statistics for Rate of Profits (Four Countries Removed)
Statistic  Maximum Rate of Profits  Observed Rate of Profits  Ratio of Observed Rate To Maximum 
Sample Size  83  83  83 
Mean  84.852  48.623  0.591 
Std. Dev.  26.088  14.898  0.138 
Coeff. of Var.  0.307  0.306  0.234 
Skewness  0.374  0.044  0.623 
Kurtosis  0.326  0.591  0.134 
    
Minimum  8.623  5.495  0.356 
1st Quartile  66.195  39.947  0.476 
Median  86.242  47.385  0.575 
3rd Quartile  104.139  58.124  0.662 
Maximum  144.818  84.822  0.967 
Interquartile Range/Median  0.440  0.384  0.323 
Table 2: Descriptive Statistics for Wages (Four Countries Removed)
Statistic  Wage in Standard System  Observed Wage 
Sample Size  83  83 
Mean  0.409  0.431 
Std. Dev.  0.138  0.085 
Coeff. of Var.  0.338  0.198 
Skewness  0.623  0.397 
Kurtosis  0.134  0.597 
  
Minimum  0.033  0.246 
1st Quartile  0.338  0.360 
Median  0.425  0.453 
3rd Quartile  0.524  0.491 
Maximum  0.644  0.597 
Interquartile Range/Median  0.438  0.289 
Update (16 September 2014): The analysis reported above is based on Leontief inputoutput matrices which include investment as a sector. Apparently, it is common in Computational General Equilibrium (CGE) models to treat investment as endogenous, in some sense. I plan on redoing the analysis with this sector removed and with disaggregated investment included in final demands.