The Rexsf Model Of The Us Economy
Since US economic growth for the past century, at least up to 2000, can be explained with considerable accuracy by three factors, K, L, U, it is not unreasonable to expect that future growth for some time to come  several decades, at least, will be explained quite well by simulated extrapolations of these variables, plus a growing contribution from information and communications technology (ICT). A powerful qualitative argument for this approach is that, no matter which direction the causality runs between useful work performed and growth (and we believe it runs both ways), it is hard to believe that a model that has high explanatory power for a century will suddenly spring a leak at the end of that time. But we are getting ahead of ourselves.
The simplest method of extrapolation of labor, capital and exergy consumption and conversion efficiency would be to do an econometric fit of each variable against a suitable mathematical function, such as an exponential. The simplest procedure would be to extrapolate output Y by fitting an exponential to past economic growth, and assuming future growth will continue at the same average rate r. This is, essentially, what most economists actually do in practice.6 The next simplest procedure is to extrapolate aggregate labor supply L, capital stock K, exergy intensity (E/Y) or exergy/capital (E/K). We can then calculate the aggregate thermodynamic conversion efficiency (U/E), by a similar technique.
However, our basic mental model of the feedback process which drives growth suggests that, while GDP is indeed a function of capital, labor and useful work (which is a product of exergy inputs times conversion efficiency), all of these driving variables, except (arguably) labor supply, are also functions of past GDP. In short, the model must be recursive.
The solution we propose differs fundamentally from other forecasting models discussed in the literature. It is called the REXSF (Resource Exergy Services Forecasting) model. The model is based on the explanatory model (for growth in the US and Japan) developed in Chapters 6 and 7. But in contrast to the historical explanatory model (REXSH), the forecasting version REXSF treats the form and the parameters of the LINEX production function as given.7 It focuses on forecasting capital and useful work output, not by simple extrapolation (as illustrated above), but by means of wellknown techniques from the field of 'systems dynamics', originally pioneered by Forrester and his followers (Forrester 1961). Briefly, systems dynamic models differ structurally from most simple economic models in that the variables are not divided into dichotomous categories, independent or dependent. On the contrary, systems dynamic models assume mutual dependence: each of the variables may influence several of the others, while simultaneously being affected by the others, through feedback loops. Thence causality is always mutual, rather than unidirectional. To be more specific, it is no longer assumed that GDP is a dependent variable, causally determined by labor, capital and/or exergy services. On the contrary, each of those variables is also dependent on previous, or even current, values of GDP. Similarly, future values of the capital, exergy and useful work variables are partly dependent on future values of the GDP, with a time lag.
We extend the model into the future by introducing two explicit learning processes. In the first, production experience drives down the energy and materials intensity of output. (Recall that experience models were discussed in Chapters 1 and 2, especially Section 2.7). In the second, experience gained in supplying energy to the economy acts to increase the efficiency with which energy services (useful work) are supplied to the economy. The REXSF model formally consists of four distinct linked modules, namely (1) capital accumulation, (2) population growth, (3) resource consumption and (4) technological change dynamics, all of which are linked together by the production function derived previously for the explanatory model developed in Chapters 6 and 7.
The labor supply module of REXSF operates like a birth and death process, where births are considered equivalent to hires and deaths to retirements or layoffs. This simple formulation bypasses the need to model population growth, male and female labor force participation, length of active working life, and so forth, even though a more sophisticated model
Year
Figure 8.1 Simulated labor hire and fire rate (USA, 19002000)
Year
Figure 8.1 Simulated labor hire and fire rate (USA, 19002000)
would have to include these considerations. Nevertheless, our simple formulation is sufficient to create a simulated time series for labor supply. First order fractional entry and departure rate parameters were assumed to be constants, although they were not. To correctly reproduce the historical time series, it was necessary to allow each of these parameter values to change only once (discontinuously) over the entire 100year period. Standard optimization methods were used to identify the years when the constant parameter values should change. In 1920, the fractional retirement rate shifted from 0.10 to 0.12. In 1959, the fractional entry rate increased from 0.124 to 0.135. These independent shifts generate three identifiable periods of relatively constant labor force dynamics, from 190020, 192059 and 1959 to the present day. The empirical and simulated results are presented in Figures 8.1 and 8.2.
In the case of capital stock at any moment, annual increments can be crudely equated with savings, which can be assumed roughly proportional to GDP, with some adjustments, such as a declining savings rate, as wealth grows, minus losses due to depreciation. The rate of depreciation is usually taken to be a constant, based on the useful life of the capital good, but adjustments may be needed to reflect a changing mix of capital goods (more computers with a life of four years, fewer bridges, etc.) There is at least a secondorder relationship between depreciation and investment: the
Year
Figure 8.2 Simulated and empirical labor (USA, 19002000)
Year
Figure 8.2 Simulated and empirical labor (USA, 19002000)
greater the latter, the faster the obsolescence rate, which accounts for part of the depreciation.
There is also a secondorder relationship between investment and population cohort aging, because individuals save little or nothing during their first working years with young children, and they are likely to consume previous savings after retirement. Maximum savings are generated by age groups in their 40s and 50s. This phenomenon has been related to capital investment by Sanderson et al. (Sanderson 2006). Again, these relationships tend to change slowly, which means that the historical data can be used for parametric selection.
For REXSF model purposes, time series of total fixed capital were taken from standard published sources, as calculated by the socalled perpetual inventory method (Maddison 1995). We assume that future investment is a percentage of gross output, proportional to a savings rate, allocated among capital types (for example, infrastructure, equipment, structures, etc.). In principle, capital stock, by type, is depreciated based on appropriate estimates of useful lifetime. We assume that the mix of longlived capital (infrastructure and structures) and shortlived capital (for example, vehicles and computers) has shifted significantly in favor of the latter, and that aggregate depreciation was about 3 percent per annum in 1900 compared to about 8 percent per annum today. This assumption fits the
historical capital stock series with a savings (investment) rate of about 6 percent of GDP as shown in Figure 8.3. For forecasting purposes we extrapolated the fitted parameters for recent decades into the future. To construct more detailed scenarios it is possible to vary the future rates of investment and depreciation as well.
In the case of useful work (exergy services), there are two components. One is exergy consumption, which is almost proportional to GDP, except with a slight annual reduction in the EIGDP ratio. This decline arises from annual efficiency improvements and the structural shift away from exergyintensive manufacturing and processing activities and towards services. In this case, a straightforward linear extrapolation of the EIGDP ratio may be appropriate, for a few decades, at least.
Existing models, as far as we know, consider only the commercial fuel exergy (energy) inputs, E. In several models, the energy intensity of capital (the EIK ratio) is also assumed to be a monotonically decreasing function of time. However, using either the usual definition of E (commercial fuels) or the broader definition that includes biomass, Figure 8.4 shows the actual exergy intensity of capital for the two definitions of exergy (EIK) for the US (19002000). Evidently, the actual curves are not smooth or monotonic at all.
Start of Great End of World Depression War II
Start of Great End of World Depression War II
1900 1910 1920 1930 1940 1950 1
Year
P = total primary exergy supply (energy carriers plus metals, minerals and phytomass exergy) F = total fuel exergy supply (energy carriers only) K/ = capital index (1900 = 1)
Figure 8.4 Energy intensity of capital (USA, 19002000)
1970 1980 1990 2000
1900 1910 1920 1930 1940 1950 1
Year
P = total primary exergy supply (energy carriers plus metals, minerals and phytomass exergy) F = total fuel exergy supply (energy carriers only) K/ = capital index (1900 = 1)
Figure 8.4 Energy intensity of capital (USA, 19002000)
1970 1980 1990 2000
The alternative is to use the energy intensity of GDP (E/Y) ratio. Then, for each year  starting with the present  knowing Y we can calculate the probable exergy input E for the next year from d a D
whence, rearranging terms
where the energy intensity Y/E and its average rate of change D( Y/E) are both determined from the energy intensity graph Figure 8.5. Note the two definitions of energy (exergy) E. The narrower definition, corresponding to the lower curve, includes only commercial fuels, and the historical use pattern is an inverted U, with a peak in the mid1920s (typical of the socalled environmental Kuznets curve). Breaks in the slope in the years 1930, 1940 and 1970 are clear evidence of the sensitivity of structural shifts
Year
Figure 8.5 Energy intensity of GDP (USA, 19002000)
Year
Figure 8.5 Energy intensity of GDP (USA, 19002000)
in the economy. The upper curve, which includes all forms of exergy from biomass, is also somewhat smoother and more nearly monotonic.
Of course, DY/Y is the average historical economic growth rate r, also extrapolated. Then, knowing E and f = U/E we can calculate U. This is sufficient to calculate Y for the next year from the LINEX production function (Equation 6.32). However, if any of the four input variables (K, L, E, U) or their ratios are departing from a smooth historical trajectory, the calculated Y for the next year will differ from the simple extrapolation assuming growth rate DY/Y = r or a comparable rate of change for E/K.
In the REXSF model, we use the more general exergy/GDP (E/Y) ratio  including biomass  to define future exergy requirements. They are assumed to be proportional to GDP, but adjusted by a gradually decreasing exponential function of time. Based on data for the past century, the average rate of decline is 1.2 percent per annum. This assumption serves two purposes. First, it is simple. Second, it avoids the need to assume a constant capitalexergy relationship, for which there is little or no evidence. In the REXSF model, the rate of change of the E/Y decline is, as in other models, exogenously determined. Its value can be changed to reflect alternative 'dematerialization' policy efforts. In future versions of the model, we could envisage further developing the model to endogenize this aspect of technological progress, using a learning process controlled by production experience or R&D efforts.
The other component of useful work is the conversion efficiency itself, which reflects partly the mix of resource inputs and partly the state of the conversion technology per se. The former refers to structural shifts, for example, away from inefficient working animals to ICEpowered tractors on farms. The latter measures 'pure' technological improvements, for example, increasing thermal efficiency of electric power generation. Most of the primary exergy input to the economy is wasted due to an inefficient conversion process to physical work. Only the exergy services (useful work) delivered at the point of use can be considered productive. The lost fraction, at least its material component, is potentially harmful to the environment and can even hinder growth. As noted previously, the aggregate thermodynamic efficiency of exergy conversion f is a measure of the ratio of work (exergy service) delivered per unit of primary exergy consumed. This measure is a monotonically increasing function of time, as was shown in Figure 7.3.
Actually, we would expect the efficiency trend to have an elongated Sshape, rising slowly at first, then more rapidly as the mechanisms of technological advancement feed on themselves, but finally slowing down as the efficiency of conversion asymptotically approaches its theoretical maximum value (which is unity). As discussed in Chapter 1, the socalled logistic form of the elongated Sshaped progress or adoptiondiffusion curve is observed in a wide variety of phenomena (for example, Fisher and Pry 1971; Marchetti and Nakicenovic 1979; Marchetti 1981). The logistic curve is the general solution of a simple differential equation, of the form f 5 kA 1  f (8.3)
It happens to be symmetric around a point in time. Over the years, a wide variety of other nonsymmetric functional forms have been suggested and analysed (for example, Martino 1983, chapter 4; Skiadas 1985). More recently, the notion of multiple logistic (or other) curves has been suggested (for example, Watanabe et al. 2001).
A variety of algorithms for extrapolating the thermodynamic efficiency curve can be envisioned. The simplest is to fit the historical data to a simple twoparameter logistic curve, plotting technical efficiency (f) against cumulative production, a surrogate for experience. However, the data provide some indication that a bilogistic curve could fit better (Figure 8.6). Indeed the bilogistic model gave a better fit (rms error = 0.001017), and successfully captures the trend of increasing efficiency. The bilogistic model was
Cumulative primary exergy production (eJ)
Figure 8.6 Logistic and bilogistic Scurve fits to the trend of aggregate technical efficiency in the USA, 19002000
Cumulative primary exergy production (eJ)
Figure 8.6 Logistic and bilogistic Scurve fits to the trend of aggregate technical efficiency in the USA, 19002000
Parameters 
1st logistic* 
Parameters 
2nd logistic 
K, 
0.135 
K2 
0.2 
ml 
1560 
1 m2 
12326 
DTj 
4540 
dt2 
10000 
SSE 
0.004** 
*nested within function for bilogistic model. "corresponding to the fit of the single logistic alone. Notes: *nested within function for bilogistic model. "corresponding to the fit of the single logistic alone. used to provide the forecasts that follow. Model parameters are shown in Table 8.1. A tenyear moving average of the derivative of f versus cumulative GDP (Figure 8.7) reveals two peaks in 1962 and 1987, a valley in 1980 and another decline from 1987 to 1998. (We have not updated this particular graph.) The main conclusion is that technical progress is not as smooth as a firstorder view, such as Figure 8.6, suggests. This irregularity results from 1962 10 year moving average 10 15 20 25 Cumulative production (1900 = 1) 30 35 Figure 8.7 Rate of change of aggregate technical efficiency of primary exergy conversion (USA, 190098) 1987 1998 the combination of revolutionary and incremental engineering improvements, together with investment and behavioral changes. We do not propose any specific interpretation of the dates. However, this plot does provide some justification for the use of a bilogistic Scurve to forecast technical efficiency growth into the future.8 
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