Most economists are still using versions of a theory of growth developed for a single-sector model half a century ago by Robert Solow, who was awarded a Nobel Prize for his accomplishment (Solow 1956, 1957); a very similar model was set forth at about the same time by Trevor Swan (Swan 1956). The theory was developed further by Meade, another Nobel laureate (Meade 1961). The key feature of the Solow-Swan model was to
express the logarithmic time derivative of output (growth rate) as the sum of output elasticities with respect to capital, labor and time multiplied by their corresponding growth rates.6
The growth rate for labor is normally taken to be equal to the population growth rate, although some models incorporate more detailed considerations (with regard to gender, retirement age, years in school, etc.), while the growth rate of capital is defined as the rate of savings (investment) less depreciation. The output Y is a function of capital stock K and labor employment L.7 If the factor shares happen to be constants, they can be interpreted as output elasticities and the differential expression can be integrated to yield the familiar and convenient Cobb-Douglas form with an exogenous multiplier A(t) depending only on time.
Solow did not specify a particular mathematical form for the production function in his 1956 paper, but in his 1957 paper he specified the Cobb-Douglas form (Solow 1956, 1957). Since then most economic models have utilized either the well-known Cobb-Douglas form, or the so-called 'constant elasticity of substitution' (CES) model (Arrow et al. 1961). One implication of the Solow-Swan model, or any production function model, is that capital and labor are perfectly substitutable for each other. Adding a third or fourth factor of production does not change this requirement for mutual substitutability.
In equilibrium, assuming many price-taking firms in equilibrium producing a single composite product, constant returns to scale, integrabil-ity, and factor substitutability, it can be proved that the elasticities of the factors are equal to factor cost shares. The formal proof of this theorem is given in Appendix A. The reasoning behind this argument is spelled out in many economic textbooks (for example, Mankiw 1997, pp. 50-55). It goes like this: imagine an economy consisting of a large number of firms making a single all-purpose product (call it bread!). They all use capital and labor as inputs. They all must hire labor and rent capital equipment - from an agency outside the economic system - to stay in business. In a competitive economy in equilibrium, the wages paid to labor must be equal to the marginal productivity of that labor, and similarly, the rents for capital equipment and infrastructure must be equal to the marginal productivity of the capital, which is proportional to the corresponding elasticity. In this idealized economy all workers are paid the same, so marginal productivity is less than average productivity, the difference being profit. The payments to labor and capital together exhaust the total of all payments which, in turn, equals the total output of the economy. Q.E.D. We discuss this issue further in the next section.
The origins of physical production in the neoclassical paradigm remain unexplained, since the only explanatory variables are abstract labor and abstract immaterial capital. The realism of the core assumption (that only labor force expansion and/or capital accumulation drives growth) was sharply challenged in the early 1950s. Research based on reconstructions of historical time series of the supposed factors of production (labor and capital) drastically reduced the apparent role of capital accumulation (Abramovitz 1952, 1956; Fabricant 1954). For example, Fabricant estimated that capital accumulation accounted for only 10 percent of US economic growth since the middle of the 19th century. The need for a time-dependent multiplier A(t) arises from the observation that the GDP has grown faster than either capital K or labor L or any combination of the two that satisfies the requirement of constant returns to scale (Euler condition); namely that the production function must be homogeneous of the first order.
The neoclassical paradigm does not allow any role for 'real' material flows, except as consequences, but not causes, of economic activity. It considers the economy as a closed system in which production and consumption are linked only by flows of money (wages flowing to labor and expenditures flowing to production). The goods and services produced and consumed are supposedly measured in real terms, though in practice they are measured only in monetary terms. Of course, the simplest version of this model is too simple for serious analysis, since it presumes that a part of the composite product is diverted to producing more capital. The simple model is normally modified and extended to include an investment component that produces capital. A still more elaborate version of the basic model can incorporate extraction and waste flows, but it is still only an abstraction without physical properties.
Another implication of the Solow-Swan model is that technological progress is not created by capital or labor. Otherwise a 'sector' would have to exist, converting capital and labor into technological progress which, in turn, becomes an input to other sector(s). In other words, Solow's use of the single-sector assumption requires technological progress to be exogenous. Some economists have called it 'manna from heaven'. The analogy is apt.
The multiplier A(t) is usually expressed, in practice, as an exponential function of time which increases at a constant average rate based on past history. The multiplier is now called 'total factor productivity' (TFP). Of course, naming a disease is not the same as explaining it. Nevertheless, thanks to the miracle of differential calculus, it is standard practice to speak of the productivity of labor, the productivity of capital and (in some circles) the productivity of resources. Productivity estimation and explanation has become a mini-industry (Kendrick 1956, 1961, 1973; Gollop and Jorgenson 1980; Kendrick and Grossman 1980; Hogan and Jorgenson 1991). Some economists, such as Denison, have made careers of decomposing observed productivity in terms of other variables (for example, Denison 1962, 1967, 1974, 1985). More recently the emphasis has been on international comparisons to explain differences in growth rates in terms of policy-related variables (for example, Barro 1991; Barro and Sala-I-Martin 1995; Sala-I-Martin 1996, 1997; Easterly and Levine 2001; OECD 2003). This activity is called 'growth accounting'. In some respects, our work, reported hereafter, can be regarded as a small contribution to this literature.
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