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:
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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,
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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.