Optimum Population

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In casual discourse it seems obvious that a population of some geographically delimited entity–a village, a city, a country, the world–can be too small or too large. People can be too few to sustain a productive economy or creative culture; or so many that congestion effects and environmental degradation detract from present and future well being. Hence, there must be some intermediate population (or population range) that is in some sense best or optimal. Unfortunately, translating this simple idea into a coherent and useful concept has proved elusive–to the extent that in the early twenty-first century the term is rarely used in writings on demography or population policy. Some of the difficulties encountered are discussed in this article.

Increasing and Diminishing Returns

The idea that there was some size of population under which, other things being equal, per capita economic wellbeing was maximized, was implicit in the writings of the classical economists in the nineteenth century, particularly John Stuart Mill (1806–1873). Economic (at the time meaning agrarian) output per head was seen as determined by the offsetting forces of diminishing returns to labor (as more labor was applied to the same amount of land) and technological progress. At a given technological level, as the number of workers increased, the output per worker might initially rise, for example, as a result of the division of labor. But as more workers were added, output would ultimately fall. The labor input at the point of peak average productivity corresponded to what came to be called the optimum population.

In more complex economies it is possible to make an analogous argument, although the diminished significance of natural resources and the options for trade and other factor flows make it much less cogent. The pervasiveness of technological change as a source of economic growth undermines the usefulness of the stipulation that other things are equal, which is needed to define an optimum. There was a period of enthusiasm for the concept of an optimum population in the early decades of the twentieth century, starting with English economist Edwin Cannan (1861–1935), which dissipated as its difficulties became apparent. French demographer Alfred Sauvy (1898–1990) made some use of the term, and defined a power optimum as distinct from theeconomic optimum.

The Social Welfare Function

The simplest economic growth models take income per capita as the measure of wellbeing. A straightforward generalization would add numerous other components to the welfare criterion aside from income. Some ethicists (including Peter Singer)–and a few economists (including the Nobelist James Meade)–have argued, following English philosopher Jeremy Bentham (1748–1832), that a more appropriate measure of social wellbeing is average wellbeing multiplied by the size of population. This is called the total welfare or Benthamite criterion. Under it, if everyone in a society enjoyed a given level of happiness, overall wellbeing would be improved by having more people to enjoy it. An optimum population under the Benthamite criterion would be much larger than under the per capita welfare criterion. Most people, however, strongly prefer the per-capita form. This issue is treated at length by the philosopher Derek Parfit. Related theoretical concerns are discussed in a literature in welfare economics on what may be called normative population theory–for example, by economists such as Partha Dasgupta and Charles Blackorby.

If the welfare criterion attaches a strong value to preservation of the natural environment the optimum population may be substantially diminished. This is the basis of the call by some environmentalists for a reduction of world population. However, global averages have little meaning for most such calculations in view of the great diversity of country and local situations. In more constricted regions, the disamenities that may be associated with continued population growth are clearer. For example, various modeling exercises have sought to calculate optimum city sizes under welfare criteria that take into account density and congestion costs. Purely qualitative assessments of diminution of the quality of urban environmental amenity as population increases are also frequently made, albeit confounding the effects of scale with those of public expenditure and aesthetic standards.

No reasonable welfare function should be timeless. Conditions change, and researchers know at least some things about the directions of change. Hence, the welfare function that is to be maximized is usually expressed formally as an integral over time:

where U (c, t) is the level of welfare at time t, c is a vector of components of wellbeing, and e-rt is a discounting factor to reflect a bias toward the near term by downweighting future welfare at a rate of r (a necessity if one is to ensure that the integral converges). A plausible welfare function might include the population growth rate or the age structure among its arguments as well as population size: People might reasonably prefer not to live in an excessively elderly population that would eventually be produced by very low fertility. Thus, there may be a conflict between a population size goal and an age-structure (hence population growth) goal, calling for specification of preferred trade-offs between them.

Optimal Population Trajectories

Population growth and economic growth are interrelated, and the combined system can be modeled. The models can be extremely simple, like the Solow neoclassical model, or extremely elaborate, like the large-scale economic-demographic planning models in vogue in the 1970s. The models could have been (although usually they were not) optimized over alternative feasible population trajectories–either analytically or numerically. Incorporating a cost of control is mathematically straightforward. Optimal steady-state solutions for simple classical and neoclassical growth models with age-structured populations are discussed in a 1977 article by W. Brian Arthur and Geoffrey McNicoll. Other formulations are examined by Paul A. Samuelson and Klaus F. Zimmermann.

A striking real-world application of optimal control theory to population lay behind the design of China's radical one-child policy that was introduced in 1979. In work done during the 1970s, Song Jian and his colleagues, systems engineers by training, calculated that the long-run sustainable population of the country was 700 million. They then formulated the population policy problem in control-theory terms: How should fertility evolve if the population is eventually to stabilize at 700 million, the peak population is not to exceed 1.2 billion, there are pre-set constraints on the acceptable lower bound of total fertility (a one-child average) and the upper bound of old-age dependency, and there is to be a smooth transition to the target population while minimizing the total person-years lived in excess of 700 million per year? The resulting optimized policy called for fertility to be quickly brought down to the allowable minimum, held there for 50 years (producing negative population growth), then allowed to rise back to replacement level. Notably not a part of the technical deliberations–or of the actual policy that was adopted–was consideration of the human costs that attainment of such a trajectory would entail.

Central planning has deservedly lost favor in the economic realm, a fortiori in the demographic. But the general issue of achieving balance between human population numbers and the natural world remains relevant, as does the issue of striking a balance between the benefits to a society of a desired demographic outcome and the costs to the society of achieving it. But before these matters can be usefully expressed in the language of optimization, there are important surrounding problems of values, levels of analysis, and delimitation of boundaries to be resolved–areas where most of the meat of the issue is likely to be found. Real-world population policy easily eludes formal characterization.

See also: Cannan, Edwin; Carrying Capacity; Limits to Growth; Mill, John Stuart; Population Policy.


Arthur, W. Brian, and Geoffrey McNicoll. 1977. "Optimal Growth with Age Dependence: A Theory of Population Policy." Review of Economic Studies 44(1): 111–123.

Blackorby, Charles, Walter Bossert, and David Donaldson. 1995. "Intertemporal Population Ethics: Critical-level Utilitarian Principles." Econometrica 63: 1,303–1,320.

Meade, James. 1955. Trade and Welfare. New York: Oxford University Press.

Parfit, Derek. 1984. Reasons and Persons. Oxford: Clarendon Press.

Robbins, Lionel. 1927. "The Optimum Theory of Population." In London Essays in Economics in Honour of Edwin Cannan, ed. T. E. Gregory and Hugh Dalton. London: Routledge.

Samuelson, Paul A. 1975. "The Optimum Growth Rate for Population." International Economic Review 16: 531–538.

Sauvy, Alfred. 1969. General Theory of Population, trans. Christophe Campos. New York: Basic Books.

Song, Jian, Chi-Hsien Tuan, and Jing-Yuan Yu. 1985. Population Control in China: Theory and Applications. New York: Praeger.

Zimmermann, Klaus F., ed. 1989. Economic Theory of Optimal Population. Berlin: Springer-Verlag.

Geoffrey McNicoll