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.K. Foley Economics Letters 68 2000 309 –317
Fig. 1. The theoretical income–fertility relation has both an upward sloping segment, corresponding to Malthus’ assumption of rising fertility with income, and a downward sloping segment, corresponding to the demographic transition in which fertility falls with income.
There are two equilibrium levels of per-capita output, at which total fertility equals 2, the Malthusian equilibrium x , and the Smithian
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equilibrium, x .
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4. The demographic transition and Smithian demographic equilibrium
Malthus’ postulate on the relation between income and fertility was incorrect, or at least incomplete. Fertility eventually falls with rising incomes, so that there are two potential demographic
equilibria, as Fig. 1 illustrates, the low per-capita output Malthusian equilibrium and another high per-capita output equilibrium, the Smithian equilibrium, x .
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The Smithian equilibrium will be stable if an increase in the population raises productivity and
Fig. 2. When the population–per-capita output relation has both a rising portion, representing the effects of the division of labor, and a falling portion, representing the effects of diminishing returns, there are potentially two Malthusian and two Smithian equilibria. The
low-population Malthusian equilibrium and the high-population Smithian equilibrium are unstable, while the low-population Smithian equilibrium and the high population Malthusian equilibrium are stable.
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incomes. If incomes rise above the Smithian equilibrium, fertility will fall below its replacement level and the population will start to decline. If incomes are an increasing function of population size, as
Smith argued in his discussion of the division of labor, incomes will decline with the fall in population, and fertility will increase, tending to restore the equilibrium.
The stability of Smithian demographic equilibrium requires that the economy lie on a rising portion of the population–per-capita output relation. Smith’s theory of the division of labor suggests that this
curve has a rising portion, in which the division of labor effect is dominant, and then perhaps a falling portion in which diminishing returns due to land limitations take over, as in Fig. 2.
If the effects of the division of labor are sufficiently strong so that per-capita output can reach the levels required for the Smithian demographic equilibrium, there are actually four equilibrium levels of
population, two Malthusian, and two Smithian. The Smithian equilibrium on the rising part of the population–per-capita output curve and the Malthusian equilibrium on the falling part are stable,
while the other two are unstable. There is another stable equilibrium at the origin. The two unstable equilibria mark the boundary between the basins of attraction of the stable equilibria.
The stable Smithian equilibrium occurs at a relatively high average per-capita output and is stabilized by the fall in fertility occasioned by increases in per-capita output and hence household
income. The stable Smithian demographic equilibrium has features which are counter-intuitive from the
point of view of diminishing returns. An autonomous rise of the population–per-capita output relation at the stable Smithian equilibrium would lead to a lower equilibrium population, with an unchanged
level of equilibrium output per capita. An autonomous fall in in the income–fertility relationship at the stable Smithian equilibrium would lead to a lower equilibrium level of per-capita output and, for a
given population–per-capita output relation, a lower population.
In order for the economy to fall into the Malthusian-Ricardian equilibrium of generalized misery, the population would have to overshoot not just the stable Smithian equilibrium, but also the unstable
Smithian equilibrium, entering the realm of strongly diminishing returns, at a low enough level of per-capita output that fertility is above replacement level.
5. How close is the world to demographic equilibrium?
