## Numerical Results

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The ordinary least squares (OLS) fit can be done in two ways: either by using the log-variables and the ratios or, and alternatively, by using the year-to-year differences. As already noted, a simple two or three parameter production function, whether of the Cobb-Douglas or LINEX type, cannot be expected to explain short-run, year-to-year differences accurately. The fact that such a function has any short-run explanatory power at all is fairly remarkable. It is tempting, therefore, to do a fit with year-to-year differences instead of values per se. In the latter case, the model essentially forecasts the differences for the next period and uses them to adjust the current GDP for one period at a time. At each step, the actual GDP is used rather than the GDP calculated from the previous period. It turns out that the difference method can 'explain' the local variations in history extraordinarily well. As one might expect, the residual error is extremely small when the time series are differenced, except for the years of World War II. However, this method filters out the changes in the mean and hence cannot be used to forecast the future, or even to explain the major trends in the past, with any confidence. Hence we do not utilize the year-to-year difference approach hereafter.

Using the simpler two-period approximation with five independent parameters (two exponents for each period plus one for normalization at the beginning of the second period), the Cobb-Douglas model for the US yields results for GDP that are still not very good, especially after 1980, as shown in Figure 7.4a. There is quite a large and growing discrepancy between predicted and actual GDP after 1985. For Japan, the situation is slightly better. There is only one significant break in the C-D residual, again corresponding to World War II (1942-5). In this case, again, only five parameters are needed, two exponents for each period, plus a normalization for the second period. The resulting fit is also shown in Figure

Figure 7.4a Empirical and estimated GDP (USA, 1900-2005, excluding 1941-48)

7.4b. The parameter values needed to define the models are indicated on the graphs. The Cobb-Douglas residuals themselves are shown in Figure 7.5a for both countries.

The LINEX case for two periods is somewhat different, for both countries. In this case, again, there is only one significant break in the residuals, corresponding to World War II (1942-5). The LINEX residuals are shown in Figure 7.5b. The LINEX fits for the US and Japan were shown in Figures 7.4a and 7.4b. The fits are obviously very close. However, it will be recalled that Kummel's generic LINEX model (Equation 6.18) included two time-dependent parameters, a(t) and b(t). The optimal choices for a(t) and b(t), corresponding to Figure 7.5, are graphed in Figures 7.6a and 7.6b. Time-averaging these functions in each of the two periods yields a simpler parametric form of the production function, with only four independent parameters, two in each period. The resulting fit (not shown) is only slightly less good than Figure 7.5.

In the US case, we note that the LINEX function provides a significantly better fit than Cobb-Douglas from the beginning of the century until the break in 1942, and again after 1945 to 1992 or so. But it underestimates economic growth significantly thereafter. We suspect that the

Figure 7.4b Empirical and estimated GDP (Japan, 1900-2005, excluding 1941-48)

underestimate may be due to either or both of two different factors. The first is the increasing importance of information and computer technology (ICT). The second is the increasing US trade deficit in recent years, which results in an underestimation of the role of domestic exergy services (useful work) in propelling growth in GDP. The point here is that the ratio of value-added to useful work-added to imports is significantly greater than the corresponding ratio for domestic production. This is because most of the useful work is done in extraction and primary processing, increasingly done abroad, rather than in later stages of the production chain carried out in the US. As a partial confirmation of our conjecture, it is noteworthy that in Japan the gap between model and GDP data for recent years is reversed in sign. In fact, Japan exports a significant amount of exergy (and useful work) embodied in the automobiles, machinery and other products that leave its shores. Clearly, these conjectures must be tested statistically at a later time.

In the case of Japan, the LINEX model is very slightly inferior to the C-D model in terms of residual error before 1926, again 1939-43 and again 1995-8. But the LINEX model provides a better fit for the rest of the time, and overall throughout the century. The statistics are summarized in Tables 7.1 and 7.2.

40 30 20

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Year

Figure 7.5 a Cobb-Douglas residuals (USA and Japan, 1900-2005, excluding 1941-48)

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Year

Figure 7.5 b LINEX residuals (USA and Japan, 1900-2005, excluding 1941-48)

Elasticities of output are constant, by assumption, for C-D models, although fitted values are not necessarily positive in all periods. In fact, fitting the Cobb-Douglas model for the US seems to imply negative elasticities for both labor and useful work since 1984, probably due to the fact

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 ■ ■-■ Japan ■ ■ r ■ — 4 ♦ V- - / ■ ■■ ♦♦♦ ♦

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Year

USA Japan

 f^^ - a(t) left axis 1 \ b(t) right axis 1--- _ ^ *** '" —— -

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Year

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Year

Figure 7.6a Parameters of the LINEX function (USA, 1900-2005)

0.65 0.6 0.55 0.5 0.45 ' 0.4 0.35 0.3 0.25 0.2 0.15

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Year

Figure 7.6b Parameters of the LINEX function (Japan, 1900-2005)

noted earlier that the OLS regressions in this case are spurious. However, constant elasticities over long periods of time are unrealistic, in any case. Hence we prefer to concentrate on the LINEX production function hereafter. As a condition of the fitting procedure (mentioned earlier), the fitted

Table 7.1 Statistics of model fit

1900-1940

1947-2005

Cobb-Douglas

LINEX

Cobb-Douglas

LINEX

Durbin-Watson Dickey-Fuller R2 Japan Durbin-Watson Dickey-Fuller R2

1.72

0.03

3.540

0.997

0.11 1.451

0.999

0.15

2.306

0.999

1.10

Notes: Critical test values for the Dickey-Fuller unit-root test: *90%-1.606, **95%-1.950,

values of the elasticities of output remain positive throughout the century in the LINEX case, for both countries (Figures 7.7a and 7.7b). As expected, based on the Dickey-Fuller and other tests, also mentioned earlier, there are sharp breaks between 1942 and 1945 for both countries.

Since the national accounts do not distinguish payments for 'useful work' from other payments, it must be assumed that the payments for 'useful work' are accounted for indirectly as payments to capital and labor used in the production of useful work. However, even if all the payments to useful work are really attributable to labor, and none to capital, it can be seen from our results (Figures 7.7a and 7.7b) that the calculated labor share has fallen well below the traditional 70 percent of GDP and the capital share is much higher than the traditional 30 percent. This is disturbing. It suggests, at least, that our model overestimates the output elasticity for useful work and underestimates the output elasticity of labor.

Nevertheless, it is interesting to observe that for both countries the elasticity and hence the marginal productivity of labor falls throughout the century (except during World War II) and becomes very small at the terminal point (2004). Since labor still accounts for something like 70 percent of total costs (payments), the elasticity calculations suggest that marginal productivity of labor in both the US and Japan has been declining for a long time and is now quite low. In fact, the model results shown in Figures 7.7a and 7.7b suggest that adding a unit (man-hour) of labor, by itself, produces almost no added value in either country.

Numerical results for the US and Japan Table 7.2 Coefficients of production functions

Coefficients of Cobb-Douglas functions

USA 1900-1940 1947-2005 Japan 1900-1940 1947-1998

Capital (a) 0.33 ±0.064 0.78 ±0.037 Capital (a) 0.37 ±0.094 0.51 ±0.038

Labor (b) 0.31 ±0.038 -0.03 ±0.018 Labor (b) 0.44 ±0.033 0.34 ±0.009

 USA 1900-1940 k p q r a(t) 0.08 97.86 10.26 c(t) —4.12 80.85 63.04 2.6 USA 1947-2005 k p q r a(t) 0.19 107.6 11.50 c(t) — 0.27 53.44 89.10 0.47 Japan 1900-1940 k p q r a(t) 0.15 74.24 6.38 c(t) 9.5 9.5 9.5 9.5 Japan 1947-1998 k p q r a(t) 0.21 138.96 57.82 c(t) —0.35 19.03 83.99 1.26

Notes: Where a 11 ) = k/1 + exp c 11 ) = k/1 + exp ln 81

P ln 81

This is consistent with our observations (i) that output elasticity may not coincide with cost share in a real economy where the several conditions - equilibrium, profit maximization and constant returns - required for the proof (Appendix A) do not hold and (ii) that the elasticity calculations, based as they are on parameters determined by a non-linear fitting procedure, are not statistically robust.

Regarding the first possibility, there is evidence that the real economy is indeed quite far from equilibrium. The theoretical arguments against the equilibrium hypothesis have been discussed in the literature (Kaldor 1971, 1972, 1979; Kornai 1973; Day 1987). The mechanisms responsible (for example, 'lock-in' of sub-optimal technologies) have been analysed extensively by Arthur (1994). We certainly cannot rule out that possibility. As regards profit maximization, there is extensive empirical evidence that firms neglect profitable options (for example, Nelson 1989) and that the

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Year

Figure 7.7a Elasticities of factors of production - LINEX function (USA, 1900-2005, excluding 1941-47)

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Year

Figure 7.7a Elasticities of factors of production - LINEX function (USA, 1900-2005, excluding 1941-47)

SSMJ

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Year

Figure 7.7b Elasticities of factors of production - LINEX function (Japan, 1900-2005, excluding 1941-47)

real economy uses considerably more energy than the least-cost solution, due to a variety of regulatory and oligopoly barriers (Sant and Carhart 1981; Morris et al. 1990; Casten and Collins 2003). Finally, several of the so-called endogenous growth models actually postulate positive returns to scale (Young 1928; Romer 1986, 1987b).

From another perspective on the equilibrium question, natural capital is clearly being underpaid today. The earth's stock of natural capital - from forests to topsoil to mineral resources - is now being depleted without being 'paid' (or replaced) at all. In an equilibrium economy, depleted capital stocks would have to be replaced. As existing stocks of cheap petroleum are exhausted, new and higher cost resources will have to be exploited. Natural capital in the form of oil or gas in easy-to-reach geological formations will have to be replaced by man-made capital in the form of nuclear fission or fusion reactors, wind farms or large-scale photovoltaic facilities.