Solow's Model of Growth (With Diagram)
We can then write the equation for the evolution of the capital stock as follows:. Once we have specified the function f , we can follow the evolution of the capital stock over time.
Generally, the path of the capital stock over time has two important properties:. To be more specific, the steady state level of capital solves the following equation:. At the steady state, the amount of capital lost by depreciation is exactly offset by saving. This means that at the steady state, net investment is exactly zero. If two countries share the same technology A and the same production function [ f k ], then over time these two countries will eventually have the same stock of capital per worker.
If there are differences in the technology or the production function, then there is no reason for the two countries to converge to the same level of capital stock per worker.
Chapter 5: The Solow Growth Model
In this presentation, we explain the balanced-growth path of the economy and prove some of the claims made in the text. The model takes as given exogenous the investment rate; the depreciation rate; and the growth rates of the workforce, human capital, and technology. The endogenous variables are output and physical capital stock. The notation for the presentation is given in Table There are two key ingredients to the model: the aggregate production function and the equation for capital accumulation.
The production function we use is the Cobb-Douglas production function :. If we apply the rules of growth rates to Equation When we impose this condition on our equation for the growth rate of output Equation This equation simplifies to.
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The growth in output on a balanced-growth path depends on the growth rates of the workforce, human capital, and technology. Using this, we can rewrite Equation The actual growth rate in output is an average of the balanced-growth rate of output and the growth rate of the capital stock.
The second piece of our model is the capital accumulation equation. The growth rate of the capital stock is given by.
The growth rate of the capital stock depends positively on the investment rate and negatively on the depreciation rate. It also depends negatively on the current capital-output ratio. Now rearrange Equation We can also substitute in our balanced-growth expression for g Y B G Equation The proof that economies will converge to the balanced-growth ratio of capital to GDP is relatively straightforward. First, go back to Equation Subtract both sides from the growth rate of capital:. Now compare the general expression for ratio of capital to GDP with its balanced growth value:.
If we want to examine the growth in output per worker rather than total output, we take the per-worker production function Equation With balanced growth, the first term is equal to zero, so. In this analysis, we made the assumption from the Solow model that the investment rate is constant. The essential arguments that we have made still apply if the investment rate is higher when the marginal product of capital is higher.
This increases the growth rate of capital and causes an economy to converge more quickly to its balanced-growth path. Previous Section. Table of Contents.
Solow Growth Model
Then system 25 is equivalent to the following system of ordinary differential equations:. In the remainder of this section, we will follow the ideas and the same notations as in Hassard et al. Noticing that W is real if u t is real, we consider only real solution. Writing the Taylor expansion, we have. Comparing the coefficients of 42 with those in 40 , we find. Hence, in the sequel, we will compute them. From 25 and 37 , we have. From 47 , 48 , and the definition of A ,. Similarly, again from 47 and 48 , we can derive. In the following, we will seek for E 1 and E 2.
From the definition of A and 46 , we have. From 44 and 45 , we have.
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Substituting 50 and 54 into 52 and noticing that. Similarly, from 51 and 53 , we can get. These show that E 1 and E 2 can be determined. Based on the above analysis, each g ij is computed. Therefore, we can calculate the following quantities:. In this section, we study how the long-run dynamics of the dynamical system 3 change when the time delay parameter varies. The time series of capital stock according to different values of the time delay are shown in Figure 2.
In particular, Figure 2 b describes endogenous oscillations not driven by stochastic shocks typical of real world economic variables. We have analysed the dynamical properties of a Solow model with bounded technological progress and time-to-build technology. We have shown that the introduction of these two components drastically changes the results of the classical models with exponential growth of technological progress or without delays. In particular, varying the time delay, the system is able to produce stability switches and Hopf bifurcations.
Economics > Econometrics
Since problems concerning economic growth and knowledge accumulation are usually studied on BGP balanced growth path or when the steady state of the model is stable, we believe that our dynamical analysis may be useful to understand the short run fluctuations of the economic dynamics in a theoretical model.
Some possible extensions of the present analysis should also be mentioned. First, a more general structure of time delays may be introduced different delays for technological and physical productive factors. Second, the allocative process may be endogenized. The authors declare that there is no conflict of interests regarding the publication of this paper. National Center for Biotechnology Information , U.
Journal List ScientificWorldJournal v. Published online Mar Author information Article notes Copyright and License information Disclaimer. Received Aug 30; Accepted Feb Guerrini and M. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We introduce a time-to-build technology in a Solow model with bounded technological progress.
Stability and Existence of Hopf Bifurcation In this section, we mainly study stability and existence of limit cycle of system 3. Numerical Simulations In this section, we study how the long-run dynamics of the dynamical system 3 change when the time delay parameter varies. Open in a separate window. Figure 1.
Figure 2. Conclusions We have analysed the dynamical properties of a Solow model with bounded technological progress and time-to-build technology. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. References 1. Jones CI. Journal of Political Economy. Romer P. Endogenous technological change. Time series tests of endogenous growth models. Quarterly Journal of Economics. Too much of a good thing?
Journal of Economic Growth. Solow RM. A contribution to the theory of economic growth. Zak PJ.