## Monthly Archives: September 2014

## Malthusian regime: Conclusion

However, if population growth is positive in che Modern Growth regime, then education and technological progress will continue to rise, and, similarly, if population growth is negative they will fall. In fact, the model makes no firm prediction about what the growth rate of population will be in the Modern Growth regime, other than that population growth will fall once the economy exits from the Malthusian region. It may be the case that population growth will be zero, in which case the Modern growth regime would constitute a global steady state, in which e and g were constant. Alternatively, population growth could be either positive or negative in the Modern Growth regime, with e and g behaving accordingly add comment.

Concluding Remarks

This paper develops a unified endogenous growth model in which the evolution of population, technology, and output growth is largely consistent with the process of development in the last millennia. The model generates an endogenous take-off from a Malthusian Regime, through a Post-Malthusian Regime, to a demographic transition and a Modern Growth Regime. In early stages of development – the Malthusian Regime – the economy remains in the proximity of a Malthusian trap, where output per capita is nearly stationary and episodes of technological change bring about proportional increases in output and population.

## Malthusian regime: The Dynamical System 6

Over time, the slow growth in population that takes place in the Malthusian regime will raise the rate of technological progress and shift the p(et+i) locus in Figure 4a upward so that it has the configuration shown in Panel B. At this point, the dynamical system of education and technology will be characterized by multiple, history-dependent steady states. One of these steady states will be Malthusian, characterized by constant resources per capita, slow technological progress, and no education electronic-loan.com.

The other will be characterized by a high level of education, rapid technological progress, growing income per capita, and moderate population growth. For the story that we want to tell in this paper, however, the existence of multiple steady states turns out not to be relevant. Since the economy starts out in the Malthusian steady state, it will remain there. If we were to allow for stochastic shocks to education or technological progress, it would be possible for an economy in the Malthusian steady state of panel В to jump to the Modern Growth steady state, but we do not pursue this possibility.

## Malthusian regime: The Dynamical System 5

Panel b. In later stages of development as population size increases sufficiently, the joint evolution of education and technology is characterized by multiple locally stable temporary steady-state equilibria, as depicted in Figure 4b. The corresponding EE Locus, depicted in the space (et, xt) in Figure 5b, consists of 3 vertical lines corresponding the three steady-state equilibria for the value of et. That is, e = 0, e = eu, and e = eh. The vertical lines e = and e = eh shift rightward as population size increases. Furthermore, the global dynamics of et in this configuration are given by:

## Malthusian regime: The Dynamical System 4

Hence, the XX Locus, as depicted in Figure 5 in the space (et) xt), is a vertical line above the Conditional Malthusian Frontier at a level e.

Lemma 3 holds as long as consumption is above subsistence. Click Here In the case where the subsistence constraint is binding, the evolution of xti as determined by equation (26), is based upon the rate of technological change, gu the effective resources per-worker, xt as well as the quality of the labor force, et.

## Malthusian regime: The Dynamical System 3

Global Dynamics

This section analyzes the evolution of the economy from the Malthusian Regime, through the Post-Malthusian Regime, to the demographic transition and the Modern Growth regime.

Phase Diagrams

The global analysis is based a sequence of phase diagrams that describes the evolution of the system within each regime and in the transition between the different regimes. The phase diagrams, as depicted in Figure 5, are based on three central elements:

The Malthusian Frontier

As was established in (27) and (28) the economy exits from the subsistence consumption regime when potential income, z^ exceeds the critical level z = c/(l — 7)(l+r). This switch of regime changes qualitatively the nature of the dynamical system from a two to a three dimensional system.

## Malthusian regime: The Dynamical System 2

In the second regime the subsistence consumption constraint is not binding and the evolution of the economy is governed by a two dimensional non-linear first-order autonomous system:

In both regimes, however, the analysis of the dynamical system is greatly simplified by the fact that, as follows from Lemma 1, (21), and (A4), the joint evolution of et and gt is determined independently of the xt. Furthermore, the evolution of et and gt is independent of whether the subsistence constraint is binding, and is therefore independent of the regime in which the economy is located- The education level of workers in period t +1 depends only on the level of technological progress expected between period t and period t-f 1, while technological progress between periods t and t -f 1 depends only on the level of education of workers in period t. Thus we can analyze the dynamics of technology and education independently of the evolution resources per capita.

The Evolution of Technology and Education

The evolution of technology and education, given (A4), is characterized by the sequence {gtl that satisfies in every period t the equations gt+1 = g(et), and ei+\ —

e(gt+1). This dynamical sub-system consists in fact of two independent one dimensional, non-linear first-order difference equations that can be written as,

where the rate of technological change from period 0 to period 1, #i, is determined uniquely by eo; gi = g{eo)- Hence, the optimal sequence can be derived directly from (29) and the sequence can be generated via (30), or via the static relationship = g(ei+1).

Although the evolution of the sequences {e/.}^0, and {#m}Sch are fundamentally disjoint, and hence can be analyzed in either the plain (et+i, e*), or the plain Q^+i, gt), the structure of this sub-system becomes more apparent in the context of the two dimensional system depicted in the plain (eugt).

In light of the properties of the functions and g(et) given in Lemma 1, (A3)-(A4), and (21)-(22), it follows that in any time period, if population size does play a role in technological progress, this dynamical sub-system is characterized by three qualitatively different configurations, which are depicted in Figure 4. The economy shifts endogenously from one configuration to another as population increases and the curve g(et) shifts upward to account for the effect of an increase in population.

In Figure 4a, for a range of small population sizes, the dynamical system is characterized by globally stable steady-state equilibria. For a given population size in this range, the steady-state equilibrium is (e,g) = (0,gl). As implied by (21), the rate of technological change in a temporary steady state increases monotonically with the size of population, while the level of education remains unchanged.

In Figure 4b, for a range of moderate population sizes, the dynamical system is characterized by three steady-state equilibria. For a given population size in this range, there exist two locally stable steady-state equilibria: (e,g) — (0,gl) and (e,~g) = (ehygh), and an interior unstable steady-state (ё,7/) = (еи,ди). {eh,gh) and gl increase monotonically with the size of population.

Finally, in Figure 4c, for a range of large population sizes, the dynamical system is characterized by globally stable steady-state equilibria. For a given population size in this range, there exists a unique globally stable steady-state equilibrium: (e,g) — (eH,gH). These temporary steady-state levels increase monotonically with the size of population.