Abstract
We present in this chapter the first growth model, introduced almost simultaneously by R. Solow and S. Swan in two different papers published in 1956. In fact, as we will see, the assumptions embedded in this model imply that, in the long run, and in the absence of technological growth, economies do not grow in per-capita terms. The possibility of aggregate growth arises only from either population growth or growth in factor productivity. Since neither factor is supposed to depend on the decisions of economic agents, this is known as an exogenous growth model. There are model economies for which there are steady-states with constant, non-zero growth rates determined by some decisions made by economic agents, like the level of education, or by some policy choices, like a given tax rate. These are known as endogenous growth models and will be studied in later chapters.
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Notes
- 1.
In fact, a steady-state is defined by constant rates of growth of appropriately chosen ratios of variables. In this introductory discussion, it is convenient to define it in terms of per capita variables, although in a later section of this same chapter we need to define it differently.
- 2.
- 3.
That would be the case, for instance, if leisure does not enter as an argument in their utility function, which we will not specify in this chapter.
- 4.
Consumers are able to work from the moment they are born.
- 5.
It is also sometimes known as neutral in the sense defined by Harrod.
- 6.
The argument in Sect. 2.2, suggests that, under our maintained assumption of decreasing returns to scale, the ratios of physical capital and output per unit of effective labour will experience zero growth in steady-state. In turn, that would imply that per-capita variables like \(\frac{K_{t}} {N_{t}}\) or \(\frac{Y _{t}} {N_{t}} = \Gamma _{t}f(k_{t})\) will grow in steady-state at a rate γ A . These results are shown in the next section.
- 7.
The equation has another root: k ss = 0. This would be a steady-state with zero capital, output and consumption.
- 8.
Notice that this is a result on absolute changes in the stock of capital per unit of effective labor, while the result above was on its rate of growth.
- 9.
It is clear that, being a constant, the correction on physical capital data would not need to be done to estimate the regression, so long as we are careful when interpreting the estimated intercept, although estimates of (2.26) would have a more direct interpretation.
- 10.
The following argument rests on utility comparisons, and we have not specified consumer preferences in this chapter. It is nevertheless interesting as an introduction to the type of normative analysis that is done in subsequent chapters. In fact, we will address again the suboptimality of the Golden Rule in Chap. 3.
- 11.
Substitution of the proposed solution yields, \(b\mu e^{\mu t} = Da + Dbe^{\mu t} - Dk_{\mathit{ss}}\) which can hold only if \(\mu = D,a = k_{\mathit{ss}}\). Hence, we have: \(k_{t} = k_{\mathit{ss}} + be^{Dt}\). To determine the value of the constant b we use the initial condition: \(k_{0} = k_{\mathit{ss}} + b\), so that: \(b = k_{0} - k_{\mathit{ss}}\).
- 12.
This assumption is not a proper element of the Solow–Swan model, which does not leave any role for a profit maximizing behavior on the part of producers of the single good in the economy.
- 13.
Even though in some European countries, tax incentives have recently been introduced in an attempt to increase the birthrate.
- 14.
The reader should not have much problem thinking about an economy which starts outside steady-state and changes its savings rate to s GR .
- 15.
Notice the different analytical representation for growth rates, relative to the exponential functions used in the continuous-time version of the model.
- 16.
This is not necessary for the exercise, as the reader may see by changing the value of either the savings rate or the output share of capital.
- 17.
Which are obtained by adding the depreciation loss to the need to provide new workers with the same stock of capital than the older ones.
- 18.
It would be simple to incorporate it into the simulation, but it would not change the qualitative aspects of the discussion.
- 19.
The reader can copy the spreadsheet and use the random number generator to write a different realization on top of the old one. All the calculations in the spreadsheet will change, providing a different set of time series for all the variables in the economy. We need to be careful about the fact that EXCEL does not automatically update the regression results.
References
Barro, R., and X. Sala-i-Martin. 2003. Economic growth, 2nd ed. Cambridge: MIT.
Solow, R.M. 1956. A contribution to the theory of economic growth. Quarterly Journal of Economics 70(1): 65–94.
Swan, T.W. 1956. Economic growth and capital accumulation. Economic Record 32: 334–361.
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Novales, A., Fernández, E., Ruiz, J. (2014). The Neoclassical Growth Model Under a Constant Savings Rate. In: Economic Growth. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54950-2_2
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