Geography, Health, and Demo-Economic Development

If an equilibrium exists, it may be unstable. Stability requires that a “Malthusian mecha-
nism” operates according to which ∂gL∕∂y > 0 implying that there exists at most one stable
equilibrium. In order to prove this claim we exclude the degenerate case where gL is exactly
at max(gL). Then, the hump-shaped curvature of the gL(y) curve ensures that either none or
two equilibria exist, and - if two equilibria exist - that we observe ∂gL∕∂y > 0 at the first one
(labelled y*) and ∂gL∕∂y < 0 at the second one (labelled y), see Figure 3.

The elements of the Jacobian matrix J of system (14) evaluated at an equilibrium (15) are

Local stability requires that both eigenvalues are smaller than one in absolute terms which
translates into the condition |1 +J1| < (1 +J1) - J2J3. Because J2 > 0, this necessarily requires
J₃ < 0 i.e. ∂gL∕∂y > 0. Thus, the equilibrium at y is never stable.

Only the equilibrium at y* - where the Malthusian mechanism operates - is a potential
candidate for a stable poverty trap. Substituting (14a) evaluated at the equilibrium into (16)
we observe that the condition ∂gL∕∂y > 0 becomes sufficient together with

I .,,^dhy ¹Λ∣ ,.√^{d (1+} gL) У                          hft

^{1 + α}∂yh> 2^{(1 - α}) -ιyr- τ+gL.                (¹7

Condition (17) requires that the income elasticity of population growth is not too large. If it
is too large, equilibrium (τ5) becomes unstable and low incomes economies converge towards
subsistence level. It can be verified numerically that (τ7) is not restrictive and y^* is - if it
exists - stable for any reasonable parameterization of the model. Note, however, that y ^* is only
locally stable. Any big push or series of small positive shocks that drives y in regions where
∂gL∕∂y < 0 enables an escape towards successful demo-economic development. The following
theorem summarizes the results.

16

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