APPENDIX: The CEIS/IEIS Model
This appendix presents both the CEIS and the IEIS models. It initially presents the IEIS model, which is more general. Then it solves the CEIS model as a special case in which c0 = 0 and z1 = 1.
The general problem can be written using a Hamiltonian:
The first order conditions are:
From the budget constraint we get:
From the first FOC:
Substituting the second FOC:
Substitute and get:
On the other hand, from the production function we get:
Substitute and get:
Now, we have two differential equations in three variables, c/k, y/k and z1:
The solution for y/k and c/k in the steady state is:
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This paper is a revised version of the first chapter of my Ph.D. thesis, presented to Harvard University in July 1994. I would like to thank all the participants in the Macro-Growth Seminar at Harvard, and especially Professor Robert Barro, for helpful comments and discussions.
The assumption about the nature of the technological progress is irrelevant in the case of Cobb-Douglas technology.
In the growth literature, it is usually defined as the “Solow Residual” or the “Total Factor Productivity Growth”.
An alternative way is to assume that human and physical capital are perfect complements in the production function. In this case the two types of capital will be accumulated in fixed proportions and again in our analysis we can concentrate on the accumulation of physical capital.
In the case of less developed countries, this progress in knowledge can be interpreted as the rate of assimilation of the knowledge that flows from more developed countries.
In every other respect, except the population size, all economies are assumed to be identical.
We strongly believe that the methodology we use to analyze the system (using output/capital and consumption/capital ratios) is superior to the traditional methodology (that uses a normalization of k defined as “capital per effective worker”). The reasons are: (1) many variables in the model are linear in y/k, both during transition and in the steady state; (2) the y/k and c/k ratios have a strong intuitive meaning, as opposed to the “capital per effective worker” variable. However, all the results in our analysis can easily be replicated using the traditional methodology.
Special circumstances may include the aftermath of a major war that devastates the capital stock but not the technological know-how.
For example, analyzing the original data on output and capital stocks in the United States used by Solow (1956), one can easily check that the output/capital ratio actually increased over the sample period (1909-1949).
If most capital is in the form of land and this land produces food, then an increase in the production of food means that the output/capital ratio must also increase, at least in the beginning of the development process.
Under our assumptions regarding the parameter values, the output/capital ratio that corresponds to a rate of technological progress of 1% is 0.2125, while the one that corresponds to a zero rate of technological progress is only 0.1000. In general, the output/capital ratio in the steady state is always a positive function of the rate of technological progress. See the Appendix for proof.
This is the explanation for the present differences in income per person across countries: they are all identical, but some started to grow earlier than others.
This result of course makes the Stone-Geary preferences an unattractive alternative to the standard model. Simulating the IEIS model, we will prove that this is not a general result of using these preferences.
In general, as noted by Rebelo (1992), the saving rate will be extremely low for low-income economies that are in the first stages of development.
Introducing leisure in the utility function and work in the production function and taking first order conditions with respect to consumption and leisure, we obtain the following expression:
leisure = β z1 (1 - s)/[β z1 (1 - s) + (1 - α)]
where s is the saving rate, β is a positive parameter and z1 = 1 - c0 / c. In the CEIS model z1 = 1 and this implies a negative correlation between leisure and savings. In the IEIS model, on the other hand, z1 is between 0 and 1 and increasing. In this case, a positive correlation between leisure and savings is possible.
A good example of this kind of study is Barro and Sala-i-Martin (1992). Figures 1, 2 and 3 in their paper show convergence between rich countries and regions, while Figure 4 shows no evidence of convergence between poorer countries. Figure 5 is especially interesting, showing no convergence between the poorest countries but strong convergence among the richest countries. The emphasis in their paper, however, is on the conditional convergence findings and therefore they do not discuss this feature of the absolute convergence findings.
Anecdotal evidence for this pattern of growth is most evident in South-East Asia, first in Japan, and then in South-Korea, Taiwan, Singapore and Hong Kong. The experience of each individual country in this group confirms the hump-shape pattern of growth rates. Furthermore, each one of these countries achieved the peak in growth rates at a comparable income level. Recently, poorer countries such as China and Malaysia are achieving similar growth rates.