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# WRP curves in 49 industries, USA 2007 and 2014

Up until now, we have estimated the WRP curves of the economy using as numeraire the vector of the economy’s gross output, which we found to be quite different from the standard output. In similar fashion, the vector of employment coefficients was found to be quite different from the l.h.s.

eigenvector of the matrix of the vertically integrated input—output coefficients. Consequently, we cannot attribute the observed near linearity in the WRP curves to the alleged closeness of the output and employment vectors to their respective counterparts.

In this section, using as our numeraire not the gross output vector of the total economy, but rather each industry’s output, we get as many different WRP curves as the number of our effective 49 industries in the effort to shed further light on developments taking place in each individual industry in connection with the rest of the economy. We experiment first within the more popular circulating capital model and subsequently with the more realistic fixed capital model. By selecting two years quite distant from each other, such as 2007 and 2014, we make meaningful comparisons with respect to technological change.

## WRP curves, circulating capital model

By assuming away the matrices of depreciation and indirect tax coefficients, we get from Equation 5.3 the following dot division:

which gives two row vectors on which we apply an element-by-element division, ./, to get the WRP curves of each of the 49 industries of the U.S. economy. We say 49 industries, because, as we explained in Chapter 4, some similar industries are simply repeated in the WIOD (2016) database. In the interest of simplicity and clarity of presentation, we select two years, 2007 and 2014, which are quite apart from each other. A technological change, even though might be significant, does not appear immediately in the input—output data because only a relatively few establishments adopt the new technique in one or just a few industries. The estimated techniques, as these are reflected in the WRP curves, are average techniques. Consequently, only after the passage of a sufficiently long time, we start to distinguish differences in productivity and the techniques in use inasmuch the majority of establishments have adopted or have been affected by the new technology. We believe that a seven-year period is enough to allow these processes to take place.

In Figure 5.9, we present the WRP curves of those industries using a circulating capital model, which displays the expected or usual shape. These curves represent techniques, which are chosen from the hypothetical book of blueprints available in the U.S. economy. The graphs below, as Ochoa (1984) admits, are very difficult to explicate. In our view, each of the graphs below shows WRP curves of individual industries, whose connections with the others is depicted by the input—output coefficients in Equation 5.5. The equation might be described as the unit cost of production, which is expected to decrease with the passage of time because, other things constant, the technological change increases labor productivity and reduces the unit cost of production. If the relative rate of profit is zero, we get the unit labor values per industry, whose reciprocal is equal to the (gross) productivity of labor or the maximum real wage, the intercept of the vertical axis. If the relative rate of profit increases, it follows that the unit cost increases as well until the attainment of the standard ratio (or “productivity” of capital approximately equal to the maximum rate of profit) at a relative rate of profit equal to one. In short, the further out to the right from the origin of the WRP curve of industry, the more efficient the technology in use, the higher the productivity and the lower the unit cost of production.

Figure 5.9 WRP curves, circulating capital, USA 2007 and 2014.

Figure 5.9 (Continued)

The industries whose WRP curves of the two years cross each other and, therefore, are characterized by switching points are displayed in a separate set of graphs in Figure 5.10.

The outer dotted WRP curve stands for the year 2014, while the inner solid line represents the year 2007 in all of the 39 graphs in Figure 5.9. The

Figure 5.10 Industries displaying switching, circulating capital model, USA 2007 and 2014.

estimations are in constant prices (2010). A cursory look at the graphs reveals that the system indeed behaves more “irregularly” rather than “regularly” in Schefold’s usage of the terms. During these seven years, the U.S. economy experienced technological change and, therefore, improvements in productivity as this stands out in the shape of the 39 WRP curves. The effect of technological change is reflected in the difference of the size of the area under the two curves. In defining technological change by the difference in the areas under the WRP curves, we observe industries experiencing significant technological change, for example, industry 1 (crop and animal production, hunting, etc.), industry 5 (manufacture of food, beverages and tobacco products) and industry 31 (land transport and transport via pipelines), among others. However, we also notice industries that are only marginally affected by technological change, for instance, industry 32 (water transport), industry 34 (warehousing and support activities for transportation) and industry 42 (insurance, reinsurance and pension funding, except social security), among others. For the nomenclature of the other industries, see Table 4.2.

In the next set of graphs in Figure 5.10, we display the industries in which crossing in their WRP curves has taken place. We observe ten occurrences of switching, that is, 10/49=20.41% of the cases and none of the double switching. A careful examination of all the graphs shows the near-linear nature of the WRP curves that rules out the case of the many different curvatures and more than one switch points. The shape and location and the trajectories of WRP curves reveal the absence of any significant technological change in these ten industries. We have essentially, on average, the same technology in use during this period of seven years, and naturally, one expects that there will be crossings in quite many cases. If, however, we repeat the experiment comparing the year 2014 with the more distant 2000, it is very unlikely to find industries with the productivity of labor higher than that of the year 2014. In other words, the technology of the year 2014 will dominate absolutely in all industries of such distant years.

In examining each and every one of these graphs in Figure 5.10, it is important to bear in mind the following: for the year 2007, the equilibrium relative rate of profit is p = 31.69% and the standard ratio (maximum rate of profit) R = 1.06. By taking into account the equilibrium relative rate of profit in the case of the circulating capital model, we observe that only in industry 37 (publishing industries and software) crossing takes place at a relative rate of profit p = 30%, that is, at a point near the equilibrium. In all other cases, crossing takes place at rates of profit too high (near the maximum), or too low (near the minimum) and rarely around the equilibrium relative rate of profit. It is also interesting to note that by construction, the intercept on the horizontal axis is the same for the two techniques. The difference between the two curves is negligible in terms of the maximum rate of profit since for the year 2014, the maximum rate of profit R = 1.07 is slightly higher than that of the year 2007. By contrast, the maximum real wage or the productivity of labor of the year 2014 is usually much higher than that of the year 2007 in all industries except three (industries 4, 15 and 37).

Figure 5.1i Differences in WRP curves with switching, circulating capital model, USA 2007 and 2014.

In Figure 5.11, we place together all the more “regularly” (in the mathematical sense of the term) behaving WRP curves by taking their difference for the two years. If the difference is on the positive (negative) side, it means that the WRP curve of the year 2014 is above (below) that of the year 2007.

We observe that there is at most one crossing and, therefore, the sign alternates from positive to negative or vice versa, only once. This is equivalent to saying that the case of double-switching does not appear like a realistic possibility in the actual data of the U.S. economy in these two years. Of course, we do not exclude a priori the cases of double switching; we only ascertain the near-linear nature of the WRP curves rendering the case of re-switching a remote possibility. It is important to note that we also used the actual maximum rate of profit, R, for each year and not the maximum relative rate of profit, p. The results were, as expected, the same; that is, the signs changed only once. In constructing the WRP curves, we decided to use the relative rate of profit because the maximum rates of profit of the years 2007 and 2014, as we have already noted, although appear to be different, their difference is only in the second decimal, 1.06 vs. 1.07.

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