Wednesday, September 17, 2014

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)
to Frontier
Sample Size78
Std. Dev.0.08998
Coeff. of Var.1.30187
1st Quartile0.01915
3rd Quartile0.08330
Interquartile Range/Median1.63703


Magpie said...


If I understand, Figure 1 would be, then, an empirically-determined version of Vernengo's equation (5) -- see in the post linked to below --, only with the axis changed, right?

Most curves in the figure do present a curvature (i.e. are concave up), although the one for China seems basically straight (note that it crosses the Hong Kong curve). Is that meaningful?

I also find noteworthy a couple of things.

Firstly, that in most cases the sum of the proportions do not add 1. For instance, in China's case the observation, although it corresponds almost exactly to the curve, shows about 30% as rate of profit, and midway between 0.45 and 0.5.

But, shouldn't this add up to 1 or thereabouts? If so, what would account for the difference?

On second thoughts, the second observation (the intersection of the curve for Japan with the horizontal axis, for instance, exceeding the 100% mark), is probably trivial: I suppose this could be just an artefact of the estimation procedure.

Magpie said...

Whoops, above it should read

"Firstly, that in most cases the sum of the proportions do not equal 1. For instance, in China's case the observation, although it corresponds almost exactly to the curve, shows about 30% as rate of profit, and midway between 0.45 and 0.5 as wage rate".

Robert Vienneau said...

Yes, the curves are more or less, Matias' Equation 5 in that post.

The profit share and wage share should add to 1. But the X axis in my graphs is the RATE of profits, not the profit SHARE.

The curvature in most cases, as I understand it, results from capital goods being produced by more capital-intensive processes and consumer goods being produced by more labor-intensive process, given any particular rate of profits.