Data By Country On Gross And Net Investment?

My article demonstrating the empirical soundness of a simple labor theory of value needs updating. In particular, I should calculate the rate of profits on total capital. So I need data on both constant and circulating capital, not just circulating capital.

Or, at least, I need data on depreciation expenses by some consistent set of conventions. In other words, I need data on gross and net investment. Perhaps it would be sufficient for empirical approximations to have this data on the country level for every country or region in the world. I do not expect to find such data broken down for each country by industry.

Does anybody have suggestions or comments on where to find such data?

Income Distribution And A Simple Labor Theory Of Value

I have a new paper available on the Social Science Research Network:

Title: Income Distribution And A Simple Labor Theory Of Value: Empirical Results From Comprehensive International Data

Abstract: This paper presents the results of an empirical exploration, with data from countries worldwide, of Sraffian, Marxian, and classical political economy. Income distribution, as associated with systems of prices of production, fails to describe many economies. Economies in most countries or regions lie near their wage-rate of profits frontier, when the frontier is drawn with a numeraire in proportions of observed final demands. Labor values predict market prices better than prices of production do. Labor values also predict market prices better than they predict prices of production. In short, a simple labor theory of value is a surprisingly accurate price theory for economies around the world.

On And Off The Wage-Rate Of Profits Frontier

Figure 1: Wage-Rate of Profits Frontier for Seven Countries

This post reports on the analysis of wage-rate of profits frontiers drawn for each of 87 countries or regions. The input-output tables used for this analysis are 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. Figure 1, above, presents seven examples of such frontiers. Figure 1 also shows two points:

• The observed wage share and rate of profits as a point, typically off the frontier.
• The nearest point on the frontier, in some sense, to the observed point.

The wage-rate of profits frontiers is a decreasing function relating the wage to the rate of profits. The wage, in this case, is expressed as a proportion of the output of the unit output of the industry producing the numeraire commodity basket. I take the numeraire to be in the same proportions as observed net outputs (also known as final demands) in the data. The numeraire-producing industry is conceptually scaled to a level such that the system that produces it employs one unit labor. Since different countries produce commodities in different proportions, the wage is measured for a different numeraire for each wage-rate of profits frontier on my graphs.

The wage-rate of profits frontier is drawn based on several assumptions. First, one assumes the existence of steady state prices. That is, relative prices are the same for inputs and outputs. Under this assumption, the same rate of profits is earned in all industries in a country or region. I also assume wages are paid out of the output at the end of the year, not advanced at the beginning of the year. Prices, with the distribution of income under these assumptions, are known as prices of production.

One might expect the curvature of empirically-developed wage-rate of profits frontiers to deviate from a straight line, with the convexity even being different for different parts of a frontier. Such curvature arises from variations in capital-intensities, so to speak, between net output and the intermediate goods used in producing net output.

The observed wage and rate of profits might be off the frontier for a number of reasons. Wages are paid throughout the year, so even if prices of production prevailed, the assumptions with which I am drawing the frontiers are not exact. But points will also lie off the frontier because prices of production cannot be expected to prevail. Entrepreneurs will have different expectations. Some of these expectations will be disappointed, and some will not be optimistic enough. I also wonder about the importance of foreign trade. If a country is thoroughly integrated in the global economy, might its rate of profits be somewhat independent of the system formed by domestic production?

Anyways, this data allows one to explore the empirical adequacy of the theory of prices of production. How far away do the countries or regions, as described by this dataset, lie from the wage-rate of profits frontier? In the data, nine countries or regions had an actual rate of profits exceeding the theoretical maximum: the Philippines, Sri Lanka, the Rest of North America, Uruguay, Austria, Belgium, Croatia, Cyprus, and the Rest of Middle East. These countries are excluded from the histogram and the statistics given below.

Using the observed rate of profits, one can predict the wage from the wage-rate of profits frontier. Figure 1 shows the distribution of the absolute error in such predictions, while Table 1 provides descriptive statistics for this distribution. Uganda, Singapore, Vietnam, Hong Kong, Luxembourg, and Central America are the countries or regions with the wage on the frontier, at the observed rate of profits, furthest from the observed wage. I find encouraging how the countries or regions that stick out as most anomalous are, mostly, either regions that, for purposes of data collection, consist of disparate countries aggregated together; small countries that presumably have economies that cannot be regarded as systems separate from the economies of their neighbors; or countries and ports that are notable for heavy involvement in international trade.

It seems that most countries lie close to the wage-rate of profits frontier constructed from their observed input-output relations and produced commodities.

Figure 2: Distribution of Distance to Wage-Rate of Profits Frontier

Table 1: Descriptive Statistics for Wages (Four Countries Removed)

Statistic	Distance to Frontier
Sample Size	78
Mean	0.06912
Std. Dev.	0.08998
Coeff. of Var.	1.30187
Skewness	2.59744
Kurtosis	6.75223
Minimum	0.00025
1st Quartile	0.01915
Median	0.03919
3rd Quartile	0.08330
Maximum	0.42903
Interquartile Range/Median	1.63703

Failing to Empirically Render Visible What Was Hidden

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 labor-intensive or capital-intensive (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 input-output matrix A and the vector a₀ of direct labor coefficients. The gross output, q, of the standard system is a (right hand) eigenvector of the Leontief input-output 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:

a₀ 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 self-reproducing 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) + a₀ 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 wage-rate 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 = a₀ [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 (non-basic) 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 wage-rate of profits function for the standard system. I have looked at wage-rate 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 wage-rate 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 off-shore 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 input-output 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.

Estimates Based On Labor Values More Precise Than Those Based On Direct Labor Coefficients

Table 1: Variations Across Countries

1.0 Introduction

This post is an empirical exploration of a simple labor theory of value as a theory of price. The precision of estimates of labor values is compared with the precision of estimates based on direct labor coefficients. The question of the accuracy of the labor theory of value is left to later posts.

I think of precision and accuracy in terms of darts. Suppose all your dart throws cluster together. Then they are precise, even if that cluster is not near the bulls eye. But if they are also in the bulls eye, then your throws are accurate, as well.

2.0 Direct Labor Coefficients and Labor Values

Labor values are calculated in the manner I find most straightforward, from a pure circulating capital model. Each industry in a modeled country, in the year in which the country is observed, produces a flow of a single commodity. Inputs for each industry consist of labor power and a flow of commodity inputs. The quantity of labor directly used, per unit output of the industry, constitutes the direct labor coefficient for that industry.

The labor value embodied in a commodity consists of all labor directly or indirectly used as an input for producing it. In the model, all inputs into production can be reduced to an infinitely long, dated stream of labor inputs. For example, the input into the industry for wearing apparel includes labor directly employed in the given year, as well as some labor directly employed in the textile industry in the previous year. (In calculating such dated labor inputs, one abstracts from changes from technology, at least in the approach that I am using. The same technique is assumed to have been used forever in the past.) Inputs directly used in the textile industry include outputs of the industry for wool and silk worm cocoons. Thus, the labor inputs into the industry for wearing apparel include some labor directly employed in that industry two years ago, as well as some labor employed three years ago in the industry for bovine cattle, sheep and goats, and horses. Given that the technique for the economy is viable, the sum of the infinite sequence of labor inputs constructed in the way outlined converges to a finite sum. I know that the techniques for all countries that I am considering are viable, based on previous empirical work.

3.0 Source of the Data

Labor values are found, for each of one of 87 countries or regions, as calculated from a Leontief matrix and vector of direct labor coefficients for a country. Each Leontief matrix was derived from a transaction table. The transactions tables, in turn, are 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.)

Quantities of each commodities, including labor power, are measured such that a unit of each commodity can be purchased with one billion dollars at prices observed when the data was taken. With this choice of units, and the adoption of one billion dollars as the numeraire, observed market prices are unity for each produced commodity.

4.0 Results and Discussion

Figures 2 and 3 show direct labor coefficients and labor values, as calculated from the data. Each point in, say, Figure 2, represents the direct labor coefficient in a specific country for the industry with the label on the X axis. Many points are plotted for each industry, since that industry exists in many countries.

Table 2: Direct Labor Coefficients By Industry

Table 3: Labor Values By Industry

The labor value for each industry, in a given country, exceeds the corresponding direct labor coefficient. I was surprised to see that any direct labor coefficients or labor values exceed unity. The largest labor coefficient and labor value is for the industry producing oil seeds in Greece. Looking at the transactions tables, I see value added includes rows for a value-added tax, as well as income for labor, returns to capital, and rents on land. In Greece, the value-added tax for oil seeds is negative. Perhaps the government of Greece has decided that, for example, the olive oil industry is important to them for cultural reasons. And they subsidize it. So this most extreme point on my graph points to something of economic interest.

The labor values, for example, for a specific industry constitute a sample, with each country contributing a sample point. For the labor values for that industry, one can calculate various statistics, including the sample size, the mean, the standard deviation, skewness, and kurtosis. The sample size will never exceed 87, since Leontief matrices were calculated, in the analysis reported here, for 87 countries.

The coefficient of variation is a dimensionless number. It is defined as the quotient of the standard deviation to the mean. Since the coefficient of variation is dimensionless, it does not depend on the choice of physical units in which to measure the quantities of the various commodities.

Figure 1, at the top of this post, shows the distributions of the coefficient of variation, for labor values and direct labor coefficients, across countries. The variation in labor values tends to be smaller and more clustered than the variation in direct labor coefficients. Consider two theories, where one states that prices in a country tend to be proportional to labor values. The other theory is that prices tend to be proportional to direct labor coefficients. This post is an empirical demonstration that the first theory is more precise.

Even If The Workers Could Live On Air

The Maximum Rate Of Growth Around The World

Consider a model of an economy in which all commodities are produced from inputs of labor and previously produced commodities. And suppose the commodities needed as inputs in the production of commodities are described through a Leontief input-output matrix in which no commodity can be produced with (unassisted) direct labor alone. Consider the special case in which wages are zero. In a sense, this special case can be seen as a description of a futuristic economy in which all production is automated, and robots are used to produce robots.

In the theory, the input-output relations determine a finite maximum rate of profits, corresponding to the maximum eigenvalue of the Leontief matrix. This maximum rate of profits is also the maximum rate of growth that arises in the Von Neumann growth model. A composite commodity, proportional to the associated eigenvector, arises from the Leontief matrix. Along the Von Neumann ray, the output of the economy each year consists of an evenly expanding output of this standard commodity, as Piero Sraffa called it. The standard commodity, in some sense, is a generalization of "corn" in David Ricardo's corn model (which was expounded in his 1815 Essay on the Influence of a Low Price of Corn on the Profits of Stock). The commodities with positive quantities in the standard commodity are known as basic commodities, once again in Sraffa's terminology.

As this post demonstrates, this is an operational model. The graph above is based on an eigenvector decomposition of Leontief matrices. Each Leontief matrix was derived from a transaction table for a country or region. The transactions tables, in turn, are 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. Quantities of each commodities are measured such that a unit of each commodity can be purchased with one billion dollars at prices observed when the data was taken.

The graph above and the table below show the maximum rate of profits or growth for each country or region for the snapshot yielding the data. The actual rate of profits for prices that allow for the smooth reproduction of the economy falls below the maximum, sometimes considerably, because the workers do not live on air. The larger the proportion of the net output of the economy paid out in wages, the lower the corresponding rate of profits. At any rate, prices of production fall out, given some information on the distribution of income and production conditions.

Along with calculating the maximum rate of profits, I found the standard commodity and identified which commodities are basic for each country or region. For example, the commodities produced by the following industries are basic commodities in the United States: Cereal Grains; Vegetables, Fruits, Nuts; Crops; Bovine Cattle, Sheep and Goats, Horses; Animal Products; Raw Milk; Coal; Oil; Minerals; Bovine Meat Products; Meat Products; Dairy Products; Sugar; Food Products; Beverages and Tobacco Products; Textiles; Wearing Apparel; Wood Products; Paper Products; Publishing; Petroleum, Coal Products; Chemical, Rubber, Plastic Products; Mineral Products; Ferrous Metals; Metals; Metal Products; Motor Vehicles and Parts; Transport Equipment; Electronic Equipment; Machinery and Equipment; Manufactures; Electricity; Gas Manufacture, Distribution; Water; Construction; Trade; Transport; Water Transport; Air Transport; Communication; Financial Services; Insurance; Business Services; Recreational and Other Services; and Public Administration, Defense, Education, Health. Which commodities are basic varies among countries, and I typically found a few non-basic commodities in each country.

I think this data is fairly comprehensive, and I hope that I can do further believable analyses with it.

Maximum Rate Of Growth By Country

Country	Rate of Growth (Percent)
Peru	144.8
Turkey	132.5
Rest of Southeast Asia	127.4
Albania	125.3
Uganda	122.5
Zambia	121.6
Rest of Southern Africa Development Community	120.7
Mozambique	119.3
Greece	117.8
Mexico	117.3
Argentina	116.9
Columbia	115.0
France	109.8
Sri Lanka	109.6
Chile	108.8
United States	107.1
Bangladesh	106.4
Zimbabwe	106.3
Rest of Sub-Saharan Africa	105.6
Spain	104.4
Madagascar	104.1
Rest of South Asia	104.1
Venezula	104.1
Japan	103.4
Switzerland	102.5
Italy	101.0
Botswana	98.2
United Kingdom	97.7
India	93.8
Indonesia	93.0
Rest of Free Trade Area of the Americas	92.8
Rest of EFTA	92.0
Portugal	91.4
Canada	90.7
Malta	89.3
Rest of the Caribean	88.9
Denmark	88.7
Rest of Central America	88.4
Tanzania	87.7
Rest of South America	87.1
Australia	87.1
Rest of Europe	86.2
Brazil	85.6
Sweden	85.2
Rest of North Africa	84.3
South Africa	83.5
New Zealand	82.7
Rest of Middle East	82.5
Tunisia	82.3
Taiwan	82.1
Netherlands	81.7
Finland	80.0
Poland	76.8
Germany	76.6
Latvia	75.8
Rest of South African Customs Union	75.0
Malawi	72.9
South Korea	72.9
Hungary	72.4
Austria	70.3
Luxembourg	67.6
Romania	66.6
Russia	66.2
Lithuania	66.0
Rest of East Asia	64.4
Philippines	64.2
Estonia	62.3
Thailand	61.6
Malaysia	60.5
Vietnam	60.0
Rest of Oceania	57.6
Ireland	55.2
Central America	54.3
Slovakia	54.0
China	53.8
Slovenia	53.7
Croatia	51.1
Czech Republic	47.1
Belguim	46.9
Morocco	45.9
Hong Kong	40.6
Singapore	35.0
Cypress	24.6
Rest of Former Soviet Union	12.7
Uruguay	12.4
Bulgaria	8.6
Rest of North America	4.7

Dominance of Financial Capital in the United States

Profits by Selected Industry as Percent of Total Profits in the United States

The data for the graph are taken from Use tables for the United States. I am thinking of trying my hand again at some empirical exploration of input-output tables. When graphs for such start looking like those I know how to generate from Java code, I will have begun to start to make some progress. I have found a tool, Apache POI, for Java programs to read Excel spreadsheets.

Components Of United States GDP

Table 1: Components, As Percentage Of GDP

I thought I'd expand on a recent graph. I was curious to see how state and federal government spending break down in the United States. To draw the graphs in this post I performed some aggregation from the data:

• Consumption: Listed as "Personal consumption expenditures".
• Investment: Combines "Private fixed investment" and "Changes in private inventories".
• Trade deficit: Combines "Exports of goods and services" and "Imports of goods and services".
• Federal Government: Combines "National defense: Consumption expenditures", "National defense: Gross investment", "Nondefense: Consumption expenditures", and "Nondefense: Gross investment".
• State Government: Combines "State and local government consumption expenditures" and "State and local government gross investment".

I suppose I could find a price index, and plot absolute amounts, rather than percentages of GDP. Then you could see, for example, that GDP in 2009 is actually lower than the 2008 value, as a result of the global crash. Does the breakdown of government spending into consumption and gross investment components reflect the influence of Robert Eisner?

Table 2: Selected Components, As Percentage Of GDP

Size of Government in USA

I thought that Krugman had a post about Paul Ryan stating, incorrectly, that Obama had increased the size of the government. And he wondered why conservatives make factual statements that can be easily shown to be wrong. But I cannot find such a post. I can find this one on Rand Paul making a different incorrect factual claim. I am fairly sure I am thinking of something more recent than this post about Rand Paul being confused in 2012.

(By the way, Paul Krugman is wrong about what heterodox economists believe about marginal productivity theory. If he reads this, though, I would rather read his comments about the empirical correlation between increased government size and increased equality.)

Trends in Hardware and Software Costs as an Example of Structural Economics Dynamics

Empirical Trends in Costs for Computer Systems

1.0 Introduction¹

Over time, the proportion of the cost of computer systems consumed by software has tended to rise. Figure 1, originally in Boehm (1973) illustrates. In this post, I offer a theoretical explanation of this empirical observation. One might take this post as an illustration of an empirical use of the Labor Theory of Value.

2.0 The Model

Assume a computer system consists of equal amounts of hardware and software, both measured in some standard units^{1, 2}. Earlier computer systems delivered less units, while current computer systems deliver more. Next, assume that both hardware and software are produced directly from labor³.

2.1 Definitions and Assumptions

Let l_h be the staff-hours needed to produce a unit of hardware. Define ρ_h to be the rate of growth of labor productivity in the hardware industry:

ρ_h = - (1/l_h)(dl_h/dt)

Similarly, let l_s be the staff-hours needed to produce a unit of software, and define ρ_s to be the rate of growth of labor productivity in the software industry:

ρ_s = - (1/l_s)(dl_s/dt)

The last assumption is that the rate of growth of productivity is higher in producing hardware:

0 < ρ_s < ρ_h

One last variable must be defined. Let p be the ratio of labor costs to total system costs for a software system:

p = l_s/(l_h + l_s)

This completes the exposition of the model assumptions and variable definitions.

2.2 The Solution of the Model

Some algebraic manipulations with the above definitions yields the following differential equation:

(1/p) (dp/dt) = Δ(1 - p),

where Δ is the difference in the growth rates of labor productivities in hardware and software productivity:

Δ = ρ_h - ρ_s

This differential equation expresses the rate of growth of software cost, as a proportion of total system cost. The solution to this differential equation is:

p(t) = 1/[1 + c exp(-Δ t)]

where c is a constant determined by an initial value:

c = [1/p(0)] - 1

2.3 Numerical Values

Calibrating the model is the last step in the analysis presented here. Suppose 20% of the cost of a system is software in 1960, and that 80% of the cost of a system is software in 1995. The rate of growth of labor productivity is then 8% more in hardware than in software.

Δ = (1/35)[ln(4) - ln(1/4)] ≈ 7.9 %

The integrating constant for the initial value is:

c = 4/exp(-1960 Δ) ≈ 1.1 x 10⁶⁸

Figure 2 shows the relative proportion of system costs, as generated by the model with these parameters. Notice how closely Figure 2 resembles Figure 1. The model provides an explanation of the empirical observations.

Modeled Trends in Costs for Computer Systems

3.0 Conclusion

This post has presented a model, with its attendant idealizations. And that model shows how the empirical observation that productivity increases faster in hardware than software can account for the empirical observation that the cost of computer systems have become mostly software costs. Hardware costs, as a proportion of total system costs have been declining for decades.

Footnotes

1. This post draws on work I did elsewhere decades ago.
2. Floating Point Operations per Second (FLOPs) is a common measure of output in hardware. I suppose one should also specify the power at which these FLOPs are generated.
3. Source Lines Of Code (SLOC) is a common measure of software size. I have heard the analogy that measuring software in SLOC is like measuring the size of a house by the number of nails used in its construction. I guess one could always use Function Points (FPs) as a measure of software.
4. A natural extension would be to assume both hardware and software are produced solely from inputs of labor, hardware, and software. I am not sure if I ever stepped through such a model in this context.

References

• Barry W. Boehm (May 1973). "Software and Its Impact: A Quantitative Assessment", Datamation.
• Luigi L. Pasinetti (1993). Structural Economic Dynamics: A Theory of the Economic Consequences of Human Learning, Cambridge University Press.

Peak And Off-Peak Electricity As A Joint Product (Continued)

Figure 1: A Photo Probably From The Same Week I visited A Joint Production Process

1.0 Intrduction

I thought I would continue thinking about the joint production example in the previous post. I want to consider the price equations for three processes that might be operated with this apparatus in the course of a full day. In the first process, labor operates the main generator and pump for 12 off-peak hours. The second and third processes execute in parallel during 12 peak hours. Labor operates the main generator alone in the second process. And, in the third process, labor operates the secondary generator alone.

Assume this electric company takes the wage, the costs of operating the main and secondary generators, and the cost of operating the pump as given. What rate of profits and relative prices of peak and off-peak hours of electricity justifies the utility in operating these processes (when this apparatus is new and no quasi-rent is being charged)? This post gives an incomplete outline answering this accounting question.

2.0 Assumptions And Price Equations

Some definitions follow:

• p₁ = cost of operating main generator for 12 hours.
• p₂ = cost of operating pump for 12 hours.
• p₃ = cost of operating secondary generator for 12 hours.
• p₄ = 1 = price of a unit of non-peak hours of electricity.
• p₅ = price of a unit of peak hours of electricity.
• p₆ = price of a unit of pumped and stored water.
• w = the wage.
• r = the rate of profits (for a 12 hour period).
• b₄₁ = Units of off-peak hours of electricity produced in 12 hours when the pump is operating
• b₅₃ = Units of peak-hours of electricity produced in 12 hours by the secondary generator.
• a₀₁ = Person-hours of labor needed to operate the main generator and pump for 12 hours
• a₀₂ = Person-hours of labor needed to operate the main generator alone.
• a₀₃ = Person-hours of labor needed to operate the secondary generator alone.

I have taken a unit of non-peak hours of electricity as the numeraire. Assume that electricity is measured in units such that the output of the main generator operating alone is one unit of electricity. Since the pump is operating during the production of off-peak hours of electricity, the electricity generated during this period is less than one-unit:

0 < b₄₁ < 1.

Measure pumped and stored water in units such that the amount pumped in 12 hours is a unit. The second law of thermodynamics implies the following additional constraint:

0 < b₄₁ + b₅₃ < 1.

Finally, I assume that less labor is required to operate the main generator alone than is required to operate it with the pump:

0 < a₀₂ < a₀₁.

These assumptions allow one to specify the following price equations:

(p₁ + p₂)(1 + r) + a₀₁w = b₄₁ + p₆

(p₁)(1 + r) + a₀₂w = p₅

(p₃ + p₆)(1 + r) + a₀₃w = b₅₃p₅

The price equations show that wages are paid out of the surplus, not advanced. The price equation for the first process shows that it produces a joint product.

3.0 The Solution Prices

The solution prices are:

w = [(b₅₃p₁ - p₃ + b₄₁)(1 + r) - (p₁ + p₂)(1 + r)²]/[a₀₁(1 + r) - a₀₂b₅₃ + a₀₃]

p₅ = (p₁)(1 + r) + a₀₂w

p₆ = [(b₅₃p₅ - a₀₃w)/(1 + r)] - p₃

In a more thorough analysis, one would consider when the wage-rate of profits curve is downward sloping, when the price of peak-hours electricity is positive, and when the price of pumped and stored water is positive. As is typical in price theory, prices depend on the distribution of income. The analysis uncovers the accounting price for pumped and stored water. Since this is a long-period model, consumer demand enters only in determining the scale at which this facility is constructed. Prices can be found without ever considering consumer demand schedules.

4.0 Discussion and Conclusions

A fuller development would look at the depreciation of the pump and generators. If one were to look at the economy as a whole, instead of just this electric company, one would want to include processes for producing pumps and generators, perhaps with inputs that include electricity. And one could add further complications. Anyways, I think I have justified, in this post, the (unoriginal) claim that Sraffa's book has empirical implications.

Peak And Off-Peak Electricity As A Joint Product

Figure 1: A Hydro-Electric Facility

The appearance and effects of joint production are sometimes hard to see, and they often require a degree of abstraction to understand¹. For example, suppose only one process exists in the Leontief input-output matrix to produce a certain pair of joint products. And no other process produces either one of them alone. It does not necessarily follow that the Leontief matrix is non-square. It could be that two processes exist for producing another product, but with different ratios of inputs. The inputs in this pair of processes consist, among others of the pair of joint products. And so prices of production still can be explained without specifying demand schedules for consumers. The net output of the economy can vary in some range of proportions with the same prices of production. Demand and supply remain asymmetrical.

But I want to concentrate in this post in describing a specific combination of processes for making a joint product. The joint products, at some level of abstraction in this case, are peak-time and off peak-time electricity². The apparatus illustrated in the figure above produces these joint products.

The dam has an associated generator. During off-peak hours, some of the resulting electricity is used to pump water up the hill and into the storage area. Only some of the off peak-time electricity is delivered to the grid.

On the other had, during peak hours, two generators are operated, and all of the generated electricity is delivered to the grid. The underground pipe to the storage area flows backwards from how it flows during off-peak hours. This water flowing downwards is used to operate one of the generators, the one not operating during off-peak hours.

It seems to me these are not fixed coefficient processes. I imagine more off-peak hours electricity can be delivered to the grid if not as much water is pumped up to the storage area. So peak and off-peak electricity can be traded off to some extent, but not one for one. Some of the off-peak electricity would be lost to operating the pump and necessary³ inefficiencies in operating the generators. So one unit of off-peak hours electricity would be sacrificed for less than one-unit of peak hours electricity. But the configuration of the apparatus, I gather, sets a limit to maximum amount of electricity that can be generated.

So we see here an application of Sraffian economics in energy economics.

Footnotes

1. Bertram Schefold has written much on this theme, including on applied problems.
2. Milk and gasoline are both measured in gallons. But nobody would say the ratio of the price of milk for delivery at one point of time to the price of gasoline at another point of time is an interest rate, despite what a superficial and mistaken dimensional analysis might say. Likewise, the ratio of the price of peak-time electricity to off-peak time electricity is not an interest rate.
3. See the second law of thermodynamics.

Numeraire-Free Tests Of The Labor Theory Of Value

A reminder to my self - the following article belongs on this list.

• Mariolis, Theodore and George Soklis (2011) "On Constructing Numeraire-Free Measures of Price-Value Deviation: A Note on the Steedman-Tomkins Distance" Cambridge Journal of Economic, V. 35, N. 3: 613-618.

Empirical Applications of Marxism - A Reading List

I've decided that if I want use data from the National Income and Products Account (NIPA) to explore Marxist and Sraffian economics, I need a more detailed understanding. I should read, sometimes again, at least these references, which are mostly Marxist:

• Applications
• Cockshott, W. Paul and A. F. Cottrell (1997) "Labour Time versus Alternative Value Bases: A Research Note," Cambridge Journal of Economics, Volume 21, Number 4, p. 545.
• Cockshott, W. Paul and Allin Cottrell (2003) "A Note on the Organic Composition of Capital and Profit Rates", Cambridge Journal of Economics, V. 27: 749-754.
• Cockshott, W. Paul and Allin Cottrell (2005) "Robust Correlations Between Prices and Labour Values: A Comment", Cambridge Journal of Economics, V. 29: 309-316
• Han, Z. and B. Schefold (2003). "An Empirical Investigation of Paradoxes: Reswitching and Reverse Capital Deepening in Capital Theory", Cambridge Journal of Economics, V. 30: 737-765.
• Izyumov, Alexei and Sofia Alterman (2005) "The General Rate of Profit in a New Market Economy: Conceptual Issues and Estimates", Review of Radical Political Economics, V. 37, N. 4 (Fall): 476-493.
• Kliman, Andrew J. (2002) "The Law of Value and Laws of Statistics: Sectoral Values and Prices in the US Economy, 1977-97", Cambridge Journal of Economics, V. 26: 299-311
• Kliman, Andrew J. (2005) "Reply to Cockshott and Cottrell", Cambridge Journal of Economics, V. 29: 317-323
• Mohun (2005) "On Measuring the Wealth of Nations: the US Economy, 1964-2001",Cambridge Journal of Economics, V. 29: 799-815
• Mohun (2006) "Distributive Shares in the US Economy, 1964-2001",Cambridge Journal of Economics, V. 30: 347-370
• Moseley, Fred (1988) "The Rate of Surplus Value, The Organic Composition, and the General Rate of Profit in the U.S. Economy, 1947-67: A Critique and Update of Wolff's Estimates", American Economic Review, V. 78, N. 1 (March): 298-303
• Ochoa, Edward M. (1989) "Values, Prices, and Wage-Profit Curves in the U. S. Economy" Cambridge Journal of Economics, V. 13, No. 3, September 1989, pp. 413-429.
• Petrovic, P. (1991) "Shape of a Wage-Profit Curve, Some Methodology and Empirical Evidence", Metroeconomica, V. 42, N. 2: 93-112.
• Podkaminer, Leon (2005) "A Note on the Statistical Verification of Marx: Comment on Cockshott and Cottrell", Cambridge Journal of Economics, V. 29: 657-658
• Shaikh, Anwar (1984) "The Transformation from Marx to Sraffa", in Ricardo, Marx, Sraffa (Edited by E. Mandel and A. Freeman), Verso
• Shaikh, Anwar M. and E. Ahmet Tonak (1994) Measuring the Wealth of Nations: The Political Economy of National Accounts, Cambridge University Press
• Venida, Victor S. (2007) "Marxian Categories Empirically Estimated: The Philippines, 1961- 1994", Review of Radical Political Economics, V. 39, N. 1 (Winter): 58-79.
• Weisskopf, Thomas E. (1979) "Marxian Crisis Theory and the Rate of Profit in the Postwar U.S. Economy", Cambridge Journal of Economics, V. 3 (December): 341-378
• Weisskopf, Thomas E. (1979) "Marxian Crisis Theory and the Rate of Profit in the Postwar U.S. Economy", Cambridge Journal of Economics, V. 3 (December): 341-378
• Wolff, Edward N. (1979) "The Rate of Surplus Value, The Organic Composition, and the General Rate of Profit in the U.S. Economy, 1947-67", American Economic Review, V. 69, N. 3 (June): 329-341
• Wolff, Edward N. (1988) "The Rate of Surplus Value, The Organic Composition, and the General Rate of Profit in the U.S. Economy, 1947-67: Reply", American Economic Review, V. 78, N. 1 (March): 304-306
• Methodology
• Pasinetti, Luigi L. (1973) "The Notion of Vertical Integration in Economic Analysis", Metroeconomica, V. 25: 1-29.
• Pasinetti, Luigi L. (1977) Lectures on the Theory of Production, Columbia University Press
• Raa, Thijs Ten (2005) The Economics of Input-Output Analysis, Cambridge University Press
• Steedman, I. and J. Tomkins (1998) "On Measuring the Deviation of Prices from Values", Cambridge Journal of Economics, V. 22: 379-385

The Wage As The Independent Variable

1.0 Introduction
Piero 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 circulating-capital version of this system, a trade-off between wages and the rate of profits.

The shape of this wage-rate 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 wage-rate 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 wage-rate of profits curve vary from a straight line for various numeraires?

P. Petrovic ("Shape of a Wage-Profit Curve, Some Methodology and Empirical Evidence", Metroeconomica, V. 42, N. 2 (1991): pp. 93-112) explored this question for 1976 and 1978 data from Yugoslavia. Petrovic found that the empirical wage-rate 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 wage-rate 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 wage-rate 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 Wage-Rate of Profits Curve
Consider an economy in which each of n commodities are produced from labor and inputs of those n commodities. Let a_{0, j} be the person-years 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 n-element row vector a₀. The remaining input-output coefficients are the entries in the nxn Leontief input-output matrix A, which is assumed to satisfy the Hawkins-Simon condition.

The Sraffa prices equations, in which wages are paid out of the surplus, are:

p A (1 + r) + a₀ 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:

a₀ [ I - A (1 + r)]^-1 w = p

The Hawkins-Simon 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:

a₀ [ I - A (1 + r)]^-1 e w = p e = 1

The wage-rate of profits curve is then:

w = 1/(a₀ [ I - A (1 + r)]^-1 e)

3.0 Empirical Results

Figure 1 shows the range of convexities, depending on the numeraire, of the wage-rate of profits curve in the USA in 2005. The straight-line wage-rate of profits curve is constructed using Sraffa's standard commodity as the numeraire.

Figure 1: Wage-Rate 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 non-composite 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 person-years, the unit in which the BEA measures labor.

Figure 1 shows wage-rate of profits curves for two of these 65 numeraires. The wage-rate of profits curve for the numeraire consisting solely of output of Warehousing and Storage industry has the highest positive displacement from the straight-line wage-rate of profits curve. The wage-rate of profits curve for the numeraire consisting solely of output of the Petroleum and Coal Products industry is the furthest below the straight-line wage rate-of profits curves. The wages-rate of profits curves for all other numeraires are closer to the straight-line wage-rate of profits curve.

The remaining wage-rate 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 person-years. 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 wage-rate of profits curve, close to the maximum difference along these two curves of 0.0773 numeraire units.

Theoretically, the wage-rate 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 wage-rate 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 wage-rate 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.

If The Workers Were Able To Live On Air

"In so far as the development of productivity reduces the paid portion of the labour applied, it increases the surplus-value by lifting its rate; but in so far as it reduces the total quantity of labour applied by a given capital, it reduces the number by which the rate of surplus-value has to be multiplied in order to arrive at its mass. Two workers working for 12 hours a day could not supply the same surplus-value as 24 workers each working 2 hours, even if they were able to live on air and hence scarcely needed to work at all for themselves. In this connection, therefore, the compensation for the reduced number of workers provided by a rise in the level of exploitation of labour has certain limits that cannot be overstepped..." -- Karl Marx, Capital, Vol. 3, Part 3, Chap. 15, Sect. 2

Introduction
A maximum rate of profits arises in a model of the production of commodities by means of commodities. This maximum rate of profits is an upper limit on the rate of profits in any sublunary capitalist economy, where the workers produce commodities to consume and thereby reproduce their labor power.

This maximum rate of profits would be easily seen if the economy were a giant farm producing one commodity, corn, from inputs of labor and seed corn. The surplus would be the difference between harvested corn and the quantity of seed corn which needs to be set aside to continue production next year. The ratio of this surplus to the quantity of seed corn is the maximum rate of profits. The maximum rate of profits cannot be achieved because of the need to pay wages to the workers eats into this surplus.

Some of the simple lessons of the corn economy generalize to actual more-or-less capitalist economies, such as in the United States of America (USA). One can use the mathematics of eigenvalues and eigenvectors to set out the theory in this case.

2.0 The Standard System
Consider an economy in which each of n commodities are produced from labor and inputs of those n commodities. Let a_{0, j} be the person-years 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 n-element row vector a₀. The remaining input-output coefficients are the entries in the nxn Leontief input-output matrix A, which is assumed to satisfy the Hawkins-Simon condition.

This data determines Sraffa's standard system, in which the gross output, the net output, and capital goods have specific properties. Let q* be an n-element column vector denoting the gross quantities output in each industry, that is to say, the gross output in the standard system. The column vector A q* represents the physical quantities of capital goods needed to produce the gross output in the standard system. Let y* be an n-element column vector denoting the net quantities output in each industry in the standard commodity. The net output is available to be divided up between wages and profits after replacing the capital goods needed to reproduce the gross output. Net output and gross output, in any proportions, are related as follows:

y* = q* - A q* = (I - A) q*

In the standard system, the ratio between gross output and the quantity of capital goods needed to produce the gross output is invariant among commodities:

q* = (1 + R) A q*

Or:

A q* = [1/(1 + R)] q*

As a matter of fact, [1/(1 + R)] is the maximum eigenvalue of A. The standard system is scaled such that the amount of labor employed in the standard system is unity:

a₀ q* = a₀ (I - A)^-1 y* = 1

y* is the standard commodity.

One chooses the maximum eigenvalue to ensure, under the Hawkins-Simon condition, the existence of a standard commodity in which all components are non-negative and at least some components are strictly positive. The commodities which enter the standard commodities are called "basic". They enter directly or indirectly into the production of all commodities. Those commodities with zero entries in the standard commodity are called "non-basic". Either non-basic commodities do not enter into the production of any other commodity. Or they enter into the production only of non-basic commodities. For each non-basic commodity, there exist some commodity such that the non-basic commodity does not enter, either directly or indirectly, into the production of that commodity.

To explicate the concept of a commodity entering indirectly into the production of another commodity, consider the output of the Motor Vehicles, Bodies And Trailers, And Parts industry, one of the 65 industries in the 2005 Use Table for the USA available from the Bureau of Economic Analysis (BEA). 0.18 units of the output of the Primary Metals industry enter (directly) into the production of each unit produced by the Motor Vehicles, etc. industry. (A quantity unit of any industry is one hundredth of the quantity output of each industry in the year 2000, where the quantity unit in each year is a chain index.) 0.15 units of the output of the Mining, Except Oil And Gas, industry is an input into each unit produced by the Primary Metals industry. Since Mining, Except Oil And Gas, enters into Primary Metals, and Primary Metals enters into Motor Vehicles, etc., then Mining, Except Oil And Gas, enters indirectly into the production of Motor Vehicles, etc. (0.040 units of Mining, Except Oil And Gas, also enter directly into each unit output of Motor Vehicles, etc..) Any number of steps can separate the indirect production of one commodity by another.

Summary of Some Empirical Results
I've implemented the above mathematics with 2005 data for the USA. Sixty two industries in the USA in 2005 are basic and enter into the standard commodity with positive components. The three non-basic industries are

• Hospitals and Nursing and Residential Care Facilities
• Federal General Government
• State and Local General Government

I think the non-basic property of the general government industries is an accounting convention. The industries Federal Government Enterprises and State and Local Government Enterprises are basic and enter into the standard commodity with positive values.

The maximum rate of profits in the USA in 2005 was approximately 106.4%.

OCC Varies Less Among Vertically Integrated Industries (Part 2)

I gave a hostage to fortune in the first part. That part notes the empirical claim that the Organic Composition of Capital (OCC) varies less among vertically integrated industries, as compared to non-vertically integrated industries. But I did not demonstrate this claim with actual data. This part retrieves this hostage by presenting empirical results.

The first part explained how to calculate the OCC for both non-vertically and vertically integrated industries, given Leontief Input Output tables. I performed these calculations with the Leontief Input Output table obtainable from the 2005 Use Table and other data available from the Bureau of Economic Analysis (BEA). Figure 1 shows these distributions of the OCC among the 65 industries aggregated by the BEA. Notice that the OCC does indeed seem to be more dispersed for non-vertically integrated industries.

Figure 1: Distribution of Ratio of OCC to Sum of Unity and Rate of Exploitation

In both cases, the distributions seem to be skewed and from a non-Gaussian distribution. Taking common logarithms yields the distributions shown in Figure 2. Table 1 presents summary statistics for these distributions. The absolute value of the coefficient of variation in the distribution of the OCC is indeed decreased by vertical integration. So these results replicate, for 2005 United States of America (USA) data, Shaikh’s and Petrovic’s earlier results for the USA in 1947 and Yugoslavia in 1976 & 1978, respectively.

Figure 2: Distribution of Common Logarithm of Ratio of OCC to Sum of Unity and Rate of Exploitation

Table 1: Logarithm of Ratio of OCC to Sum of
Unity and Rate of Exploitation

Statistic	Non-Vertically Integrated Industries	Vertically Integrated Industries
Number Industries	65	65
Mean	-0.0326576	-0.0770277
Standard Deviation	0.396980	0.225224
Coefficient of Variation (Absolute Value)	12.16	2.924

I suppose this analysis could be improved by performing formal statistical tests. In particular, one might use the Kolmogorov-Smirnov goodness of fit test to determine if the distributions of the OCC after a logarithmic transformation are Gaussian. I don’t know how to formally test for a change in the coefficient of variation. But, since the mean is so close to zero anyway, one might use an F test to contrast the variance in the distributions of the (transformed) OCC. I don’t plan on pursuing this line soon, though.

OCC Varies Less Among Vertically-Integrated Industries (Part 1)

1.0 Introduction
Anwar Shaikh claims that one can expect the Organic Composition of Capital (OCC) to vary less among vertically integrated industries than among non-vertically integrated industries. Shaikh shows his claim holds for the United States of America in 1947. Petrovic demonstrates the claim for Yugoslavia in 1976 and 1978.

This post lays out the theory formulating this empirical claim. Results replicating Shaikh's and Petrovic's test of the theory in new data are left for Part 2. I have yet to test the theory, and Part 2 remains unwritten for now.

2.0 Vertical Integration

Consider an economy in which each of n commodities are produced from labor and inputs of those n commodities. Let a_{0, j} be the person-years 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 n-element row vector a₀. The remaining input-output coefficients are the entries in the nxn Leontief input-output matrix A, which is assumed to satisfy the Hawkins-Simon condition. The challenge is to express an empirical claim about the variability of the OCC in terms of this empirically-observable data.

Let q be an n-element column vector denoting the gross quantities output in each industry. The column vector A q represents the physical quantities of capital goods needed to produce this gross output. Let y be an n-element column vector denoting the net quantities output in each industry. The net output is available to be divided up between wages and profits after replacing the capital goods needed to reproduce the gross output. Net output and gross output are related as follows:

y = q - A q = (I - A) q

Or:

q = (I - A)^-1 y

The jth column of (I - A)^-1 represents the gross output in a vertically integrated industry producing a net output of one unit of the jth commodity. This interpretation becomes apparent when one considers a net output vector consisting of one unit of the jth commodity:

y = e_j

where e_j is the jth column of the nxn identity matrix.

The above analysis of vertically integrated industries allows one to specify the amount of labor and the capital goods employed in each vertically integrated industry. Consider the n-element row vector v defined as:

v = a₀ (I - A)^-1

The jth element of v represents the labor (in person-years) employed in a vertically integrated industry producing one unit of the jth commodity net. This element is the labor directly and indirectly embodied in one unit of the jth commodity. v is the vector of labor values for this economy. The capital goods used in producing any gross output vector is found by pre-multiplying that vector by the Leontief input-output matrix A. Define the matrix H such that each column is the product of A and the gross output of a vertically integrated industry producing a net output of one unit of the corresponding commodity:

H = A (I - A)^-1

Luigi Pasinetti calls the columns of H "the vertically integrated units of productive capacities." A column "expresses in a consolidated way the series of heterogeneous physical quantities of commodities which are directly and indirectly required as stocks, in the whole economic system, in order to obtain one physical unit of [the corresponding commodity] as a final good."

3.0 The Organic Composition of Capital
According to Karl Marx, the labor value of a commodity is the sum of the labor embodied in the capital goods used in the production of that commodity, the labor value of the labor power used in the production of that commodity, and the surplus value:

v_j = c_j + w_j + s_j

where

• c_j is constant capital expended in producing one unit of the jth commodity
• w_j is variable capital (that is, the labor value of capital spent on the wages of workers) expended in producing one unit of the jth commodity
• s_j is the surplus value obtained in producing one unit of the jth commodity.

For Marx, the labor value of constant capital appears unchanged in the product. The source of profits is the appropriation by the capitalists of surplus value produced throughout a capitalist economy.

The OCC is defined to be the ratio of constant capital and variable capital, both expressed in labor values:

occ_j = c_j/w_j

where occ_j is the OCC for the jth industry. Marxist economics would be much less problematic if the OCC were invariant across industries. The rate of exploitation e is another important parameter in Marxist economics. The rate of exploitation is the ratio of surplus value to variable capital in each industry:

e = s_j/w_j

The equality of the rate of exploitation across industries follows from an assumption of competitive labor markets, inasmuch as workers are free to transition among industries in seeking work. The OCC in each industry can be expressed as a function of the rate of exploitation and the ratio of constant capital to the remaining labor value of the product:

occ_j = (e + 1) c_j/(w_j + s_j)

The rate of exploitation can be treated as a nuisance parameter in exploring the empirical question raised in this post.

Consider non-vertically integrated industries, each producing a gross output of one unit of each commodity. The jth industry in this case employs a_{0, j} person-years of labor. That is, the labor value of the product from newly applied labor is merely the corresponding direct labor coefficient:

w_j + s_j = a_{0, j}

The columns of A represent the capital goods needed in each of these industries. The labor embodied in the capital goods for the jth industry is the dot product of the row vector expressing the labor values of a unit of each commodity and the column vector denoting the quantities of these capital goods. Thus, one has:

c = v A

On the other hand, consider vertically integrated industries, each producing a net output of one unit of each commodity. The amount of labor directly employed in the jth vertically integrated industry is v_j. The labor value c^*_j embodied in the capital goods for the jth vertically integrated industry are easily found:

c^* = v H

where the elements of c^* are the desired labor values.

The above observations can be brought together to summarize the empirical claim of interest here. The OCC in each non-vertically integrated industry is proportional to the ratio of the labor value of the capital goods used in that industry and the labor directly employed in that industry:

occ_j/(e + 1) = c_j/a_{0, j}

The OCC in each vertically integrated industry is also proportional to the ratio of the labor value of the capital goods used in that industry and the labor employed in that industry:

occ^*_j/(e + 1) = c^*_j/v_j

where occ^*_j is the OCC in the jth vertically integrated industry. The proportionality constant is the same function of the rate of exploitation in the above pair of equations. The empirical claim is that the expression on the right hand side varies less among industries for vertically integrated industries than among non-vertically industries. That is, the coefficient of variation is less among vertically integrated industries. Perhaps one should take a variance-stabilizing transformation, such as natural logarithms, before calculating the coefficient of variation.

References

• Luigi L. Pasinetti (1973) "The Notion of Vertical Integration in Economic Analysis", Metroeconomica, V. 25: pp. 1-29 (Republished in Pasinetti 1980)
• Luigi L. Pasinetti (Editor) (1980) Essays on the Theory of Joint Production, Columbia University Press
• P. Petrovic (1991) "Shape of a Wage-Profit Curve, Some Methodology and Empirical Evidence", Metroeconomica, V. 42, N. 2: pp. 93-112
• Anwar Shaikh (1984) "The Transformation from Marx to Sraffa", in Ricardo, Marx, Sraffa (Ed. by E. Mandel and A. Freeman), Verso

2005 USA Labor Values

Figure 1 and Table 1 list industries in order of declining labor values, as of 2005. Industries are as aggregated in the North American Industry Classification System (NAICS), and labor values are in units of person-years per $1,000 output. Table 1 also shows direct labor coefficients for each industry. Direct labor coefficients are the full time equivalent staff hired in each industry. Labor values for each industry are the sum of direct labor values and the labor embodied in the inputs purchased by that industry. For example, the labor embodied in the commodities produced by "Food Services and Drinking Places" includes the labor embodied in commodities purchased by such establishements from the Construction; Real Estate; Miscellaneous Professional, Scientific and Technical Services; and Wholesale Trade industries.

I can think of other analyses to do with this and related data, such as some measure of average wage in an appropriate numeraire. For example, one might examine the rate of exploitation, the variation in the organic composition of capital by industry, and the differences between prices and labor values.

Figure 1: 2005 Embodied Labor Values

Industry	Embodied Labor Values (Person-years per Thousand $ Output)	Direct Labor Coefficient (Person-years per Thousand $ Output)
Social Assistance	0.0219	0.0185
Food Services and DrinkingPlaces	0.0200	0.0157
Forestry, Fishing, and Related Activities	0.0192	0.0103
Transit and Ground Passenger Transportation	0.0174	0.0138
Administrative and Support Services	0.0172	0.0136
Educational Services	0.0169	0.0134
Amusements, Gambling, and Recreation Industries	0.0152	0.0122
Other Services, Except Government	0.0152	0.0112
Hospitals and Nursing and Residental Care Facilities	0.0146	0.0107
Warehousing and Storage	0.0141	0.0127
Wood Products	0.0141	0.00537
Apparel and Leather and Allied Products	0.0139	0.00840
Retail Trade	0.0137	0.0106
Accomodation	0.0133	0.00986
State and Local General Government	0.0129	0.00971
Furniture and Related Products	0.0128	0.00651
Printing and Related Support Activities	0.0119	0.00719
Textile Mills and Textile Product Mills	0.0119	0.00548
Other Transportation and Support Activities	0.0111	0.00908
Construction	0.0110	0.00623
Federal Government Enterprises	0.0105	0.00803
Ambulatory Health Care Services	0.0103	0.00727
Fabricated Metal Products	0.0102	0.00555
State and Local Government Enterprises	0.00996	0.00525
Truck Transportation	0.00989	0.00539
Motor Vehicles, Bodies and Trailers, and Parts	0.00963	0.00226
Waste Management and Remediation Services	0.00923	0.00492
Plastics and Rubber Products	0.0922	0.00410
Machinery	0.00920	0.00399
Misc. Professional, Scientific and Technical Services	0.00914	0.00501
Miscellaneous Manufacturing	0.00903	0.00450
Nonmetallic Mineral Products	0.00898	0.00444
Paper Products	0.00895	0.00302
Food and Beverage and Tobacco Products	0.00890	0.00247
Computer Systems Design and Related Services	0.00888	0.00627
Computer and Electronic Products	0.00880	0.00340
Electrical Equipment, Appliances, and Components	0.00880	0.00393
Information and Data Processing Services	0.00869	0.00339
Other Transportation Equipment	0.00847	0.00349
Air Transportation	0.00841	0.00352
Performing Arts, Spectator Sports, Museums, Etc.	0.00831	0.00512
Motion Picture and Sound Recording Industries	0.00821	0.00371
Federal General Government	0.00812	0.00430
Wholesale Trade	0.00811	0.00526
Insurance Carriers and Related Activities	0.00805	0.00372
Rental and Leasing Services and Lessors of Intangible Assets	0.00784	0.00243
Water Transportation	0.00780	0.00159
Farms	0.00778	0.00264
Legal Services	0.00758	0.00512
Management of Companies and Enterprises	0.00757	0.00469
Publishing Industries (Includes Software)	0.00751	0.00317
Primary Metals	0.00736	0.00237
Rail Transportation	0.00686	0.00326
Support Activities for Mining	0.00656	0.00263
Fed. Reserve Banks, Credit Intermediation, Etc.	0.00635	0.00408
Mining, Except Oil and Gas	0.00623	0.00329
Chemical Products	0.00621	0.00160
Broadcasting and Telecommunications	0.00598	0.00188
Securities, Commodity Contracts, and Investments	0.00542	0.00246
Pipeline Transportation	0.00524	0.000922
Funds, Trusts, and Other Financial Vehicles	0.00514	0.000918
Real Estate	0.00326	0.000687
Petroleum and Coal Products	0.00325	0.000274
Utilities	0.00294	0.00133
Oil and Gas Extraction	0.00250	0.000507