Conclusion

The Solow model is a key model in the theory of economic growth. Although it is rooted in older models and has theoretical and empirical weaknesses, this model provides an extremely useful, relatively simple, and flexible framework for the analysis of the process of economic growth.

However, the process of physical capital accumulation (which is the main engine of economic growth in the Solow model) cannot fully explain either the long-run growth of output per worker that has been observed in developed economies or the large differences in output per worker between developed and less developed economies. In fact, only a small part of these phenomena can be explained by the accumulation of physical capital. To explain the rest, one has to rely on total factor productivity, the efficiency of labor, and technical progress, which are considered exogenous in the Solow model.

In this sense, the Solow model (and any model that makes similar assumptions about technology and technical progress) shows us how to overcome its weaknesses and try to explain total factor productivity, labor efficiency, and technical progress. Extensions of the Solow model to address such weaknesses are examined in chapter 8.

Another theoretical weakness of the Solow model is the assumption that the savings rate is exogenous. At the time that the Solow model first appeared, this was a widely accepted assumption. However, the assumption is not satisfactory, as it does not take into account the underlying determinants of household savings behavior.

In the next two chapters, we examine two alternative classes of dynamic general equilibrium models of savings behavior, where savings result from the optimal intertemporal behavior of households that have access to competitive capital markets. These two classes of models, which are essential building blocks of modern intertemporal macroeconomics, are the representative household and the overlapping generations classes of models.

1. This model is often referred to as the “Solow-Swan model,” as a similar analysis was published in the same year by Swan [1956].

2. See Inada [1964].

3. In appendix A, we discuss a more general functional form, the CES (constant elasticity of substitution) production function, put forward by Arrow et al. [1961]. It is shown that the CES production function encompasses the Cobb-Douglas, the Leontieff, and the linear production function as special cases but does not necessarily satisfy the Inada conditions.

4. Insection 3.8, we also analyze the Solow model in discrete time, where t = 0, 1, 2, … is an integer that refers to discrete time periods, such as years, months, weeks, or days.

5. Technically, (3.7) and (3.8) are first-order linear differential equations, whose solution is given by (3.5) and (3.6), respectively. For an introduction to ordinary differential equations, see appendix C.

6. An impulse response refers to the response of any dynamic system to some external impulse. The impulse response function describes the reaction of the system as a function of time (or possibly as a function of some other independent variable that parameterizes the dynamic behavior of the system). Impulse response functions are plotted for most of the dynamic models analyzed in this book.

7. The term “asymptotic” means that a variable approaches its steady state value arbitrarily closely as t tends to infinity.

8. The significance of the golden rule for the Solow growth model and growth theory was first highlighted by Phelps [1961], although the concept was previously alluded to by John von Neumann and Maurice Allais.

9. This discussion highlights the difficulties arising in ad hoc dynamic models that do not address the optimizing behavior of households. There is nothing in the Solow model that would help us assess the welfare of consumers, as we do not know the utility function of the representative household.

10. In fact, as we saw in chapter 1, this only applies since the Industrial Revolution and the early nineteenth century.

11. Jones and Romer [2010] have recently codified some additional stylized facts that a satisfactory theory of economic growth must be able to account for. We shall examine these additional stylized facts in chapter 8.

12. Kaldor, who was quite critical of neoclassical theory, considered the Solow model to be incompatible with at least some of the stylized facts that he identified, mainly stylized facts 1, 2, and 6. He was critical because the Solow model is compatible with these facts only when one assumes exogenous technical progress, which drives the efficiency of labor, per capita income, per capita consumption, and real wages along the balanced growth path. Without the assumption of exogenous technical progress, the Solow growth model cannot account for all the Kaldor stylized facts.

13. Mankiw et al. [1992] have generalized the Solow model, attributing differences in the efficiency of labor to investment in human capital (education of the labor force). However, they retain the assumption that total factor productivity increases at an exogenous rate g. The generalized Solow model that they put forward is analyzed in chapter 8, and it seems to explain the growth experience of 98 non-oil producing countries after 1960 fairly well. See also Jones [2002] and chapter 8 for other generalized models for economic growth that rely on investment in both physical and human capital.

14. This solution method is analyzed in Chiang [1974] and is also discussed in Jones [2002] and Barro and Sala-i Martin [2004].

15. Note that (3.53) is a nonlinear first-order difference equation that can be analyzed diagrammatically.