The Solowswan Model Of Economic Growth
Until the 1950s growth theory remained primitive and qualitative because it lacked any empirical base. (Some will argue that it is still primitive.) However, thanks to the development of the system of national accounts (SNA) in the 1930s and 1940s, it became possible for the first time to construct historical GDP figures for the US and some other countries for a number of prior decades. Economists had previously assumed that economic growth was determined by the accumulation of capital stock per worker. The availability of SNA data and quantitative estimates of historical GDP enabled economists to test this assumption for the first time.
Capital stock in the neoclassical paradigm is measured strictly in monetary terms.3 Capital stock is normally estimated  'constructed' might be a better word  by a procedure called the 'perpetual inventory method' or
PIM. In brief, net investment is accumulated from a convenient historical starting point. Net investment in a period can be estimated as the product of total investment expenditure (often equated with savings) allocated among capital types  as given in the system of national accounts  times useful service life. Or it can be equated with gross expenditure for capital less depreciation. Service lives can be determined by survey, or using a mortality function. Depreciation rates can be determined by tax rules, company accounts or surveys.4 It is important to note that there is no adjustment in the PIM method for increasing productivity (or quality) of capital in use.
Using the PIM construct, it was discovered in the early 1950s that historical growth of the US economy could not be explained by the accumulation of capital stock, or the increase in capital per worker, as most economists had previously assumed (for example, Fabricant 1954; Abramovitz 1956). The key innovation in growth theory at that time was the explicit use of an aggregate production function of capital and labor services which enabled economists to account for the relative importance of the two factors of production and sources of productivity growth (Solow 1956, 1957; Swan 1956). Though not all economists are happy with the use of production functions, their limitations have been relegated in recent years to footnotes or ignored altogether.
It has also been convenient, although somewhat inconsistent with observed scale economies at the microscale, to assume constant 'returns to scale' at the macroscale. This is tantamount to assuming that if the inputs of capital and labor are doubled (or multiplied by any constant), then the output (GDP) will be larger by the same factor. Mathematically, this implies that the production function should be a homogeneous firstorder function of the input variables (the socalled Euler condition), together with a timedependent multiplier.5 With this analytic machinery it is easy to calculate the marginal productivities of each input factor, namely as the respective logarithmic partial derivatives of the production function with respect to the input variables. The simplest functional form satisfying the Euler condition is the socalled CobbDouglas function, which is widely used in growth models. (However that function also implies that the marginal productivities are constants, independent of time, which is not necessarily realistic.) It also seemed natural, based on a simple theory of income allocation, to equate these calculated marginal productivities with corresponding payment shares in the national accounts, as Solow did (Solow 1956). Thus, returns to capital stock can then be equated to payments to capital (interest, dividends, rents and royalties) in the national accounts. Similarly, returns to labor can be equated with payments to labor, consisting of wages and salaries. Solow observed that the capital share of payments in the SNA had indeed remained relatively constant at about 30 percent throughout the period covered by his analysis (190949), with the labor share relatively constant at about 70 percent. This appears to justify the choice of CobbDouglas production functions. However, we reconsider the use of production functions later in this book.
Solow was surprised to discover that the capital/labor ratio, as determined by the perpetual inventory method (PIM), could not account for nearly 90 percent of observed growth in US GDP, per capita, between those same years, 190949 (Solow 1957). The difference had to be explained by 'something else'. That something could have been timedependent multipliers of capital and/or labor, respectively (interpreted as quality improvements), or a 'neutral' timedependent multiplier for the capitallabor combination as a whole. Statistical tests, admittedly not conclusive, originally suggested that the latter scheme was best. Solow called this overall multiplier 'technological progress', although he admitted that it was simply 'a measure of our ignorance'. Others have called this multiplier the 'Solow residual'. More recently the annual increments of Solow's progress multiplier have been termed as increases in total factor productivity (TFP). One of our objectives in this book is to offer a plausible explanation of TFP in terms of measurable changes in real technology as related to the use of energy (or, to be more precise, exergy).
Of course, thanks to technological change, older capital is normally less productive than more recent vintage capital. Similarly, labor becomes more productive, thanks to education and training. Hence timedependent augmentation multipliers can be introduced to explain part of the Solow residual, mentioned above. But, in this case, the apparent returns to capital and labor inputs, as such, are reduced by the inverse of the augmentation factors. Neither augmented capital stock nor returns to capital can be measured independently of the other. The same is true for labor. This has been a source of controversy and confusion. Indeed, some have argued that aggregate capital cannot logically be measured independently of its rate of return, and  for this and other reasons  that the concept of production function itself is faulty (Robinson 19534; Pasinetti 1959; Sraffa 1960; Sylos Labini 1995).
On the other hand, there is a statistical way out of the difficulty, if one is willing to assume that the augmentation functions are smooth and mathematically tractable; for example, simple exponentials. Inserting such functions into the production function previously introduced  commonly of the CobbDouglas type  it is possible to carry out a statistical fitting procedure to determine the 'best fit' parameters of the augmentation functions. In principle, this might eliminate the TFP multiplier, though in practice it does not appear to do so.6
Given a population of perfectly competitive producers of a single allpurpose good in a simple singlesector model of income allocation, in equilibrium, it follows that the demand for capital and labor services will be proportional to their respective marginal productivities.7 The two factor CobbDouglas production function with constant returns is particularly convenient because it provides an immediate economic interpretation for the parameters of the function, which (as noted above) are set equal to the marginal productivities.
The annual increments of total factor productivity or TFP tend to fluctuate around a longterm trend. The fluctuations have some regularities. Enormous effort has been expended on identifying 'business cycles' with various periodicities, from four years to 50 years and attempting to explain them. Productivity calculations and projections have become a miniindustry. Nevertheless, it is important to realize that the (presumed) trend itself is assumed to be exogenously determined. The socalled 'endogenous theory' introduced by Romer and others (discussed in Chapter 5) offers various qualitative explanations, but nothing quantitative.
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