The steady state in the Solow model

The steady state in the Solow model.

This does not mean that an increase in the rate of population growth has no effect at all in the Solow model. It lowers the steady-state level of the per capita capital stock, expressed in units of capital per effective unit of labor, and in this way affects the level of per capita income, expressed again in units of effective labor. The easiest way to see this is to recall Figure 3.4, which we reproduce here as Figure 9.4.

Recall that the steady state k*, expressed in terms of effective units of labor, is found as the intersection of two graphs. These are, respectively, the left- and right-hand sides of the equation that describes the evolution of capital stocks in the Solow model with technical progress:

It’s now easy to see that if n goes up, this “swivels” the left-hand side of (9.3) upward and brings down the steady-state level of the capital stock, expressed as a ratio of effective labor. This means that although the long-run rate of growth is unaffected by a change in the rate of population growth, the entire trajectory of growth is shifted downward. See Figure 9.5 for a depiction of this scenario.

Figure 9.5. Growth rates are unaffected, but the levels shift down.

Thus increased population growth has negative level effects in the standard growth models of Chapter 3. These effects are perfectly intuitive, although as we have seen, they may manifest themselves differently in different models. Population growth means that a given level of output must be divided among an increasing number of people, so that an increase in population growth rates brings down the size of the per capita cake. In the Harrod- Domar model, the effect is resoundingly negative, because population growth has no redeeming features, such as increasing the productivity of capital when more labor is available. In the Solow model, this redeeming feature is partially present. An increase in the population growth rate both increases the demands on the national cake and expands the ability of capital to produce the national cake. The net effect on long-run per capita growth rates is zero. Nevertheless, the level of per capita income at any given point in time is lowered.13 This comes from the assumption in the Solow model that there are diminishing returns to every input, so that an increase in the labor

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intensity of production (necessitated by increased population growth) reduces the long-run per capita level of output relative to efficiency units of labor.

The growth models teach us that once the Malthusian assumption of unrestrained procreation is dropped, population growth certainly does not condemn society to everlasting subsistence. Growth in per capita income is still possible. At the same time, increased population growth does adversely affect this growth rate (as in the Harrod-Domar model), and even if it keeps the long-run growth rate unchanged, it affects the level of the trajectory (as in the Solow model).

Population and savings There is yet another negative effect of population growth that is not considered in the growth models already presented, but is easy enough to incorporate. Faster population growth lowers the aggregate rate of savings. This happens simply because population growth eats into aggregate income. If it is true that the rich save a higher fraction of their income, savings rates may be adversely affected. More importantly, faster population growth shifts the age structure of the population toward the very young and in so doing increases the dependency ratio in families. Because children consume more than they produce, this tends to lower savings rates as well. This is one of the effects emphasized by demographers Coale and Hoover [1958] in their classic work on the subject.

The savings effect works in very much the same way as the direct population growth rate effects. In the Harrod- Domar model, it exacerbates the reduction in growth rates [allow s to fall as well in equation (9.2)]. In the Solow model, there continue to be no growth effects, but the long-run time trajectory of per capita income is shifted down.

Population, inequality, and poverty A high rate of population growth will exacerbate the poverty problem, if the arguments in the previous section are valid. It will also worsen inequality if population growth among the poor is disproportionately larger.

Do the poor have more children? From the discussion in the previous sections of this chapter, that would appear to be the case, although the connections are not unambiguous by any means. It is more likely that the poor need children for old-age support. It is more likely that infant mortality rates are higher for the poor, so having a larger number of children to compensate is more likely to occur for the poor. We already know that this will translate into a higher expected number of surviving children (because risk-averse couples generally overcompensate for these risks).

It is somewhat harder to compare the relative costs of child bearing. Poor families are likely to have a higher degree of labor force participation by females, simply because additional income is of greater importance. This raises the opportunity costs of having children. However, it is also true that growth in income creates a quantity– quality trade-off in children. Richer households may want to invest proportionately greater sums in the education of their children. Consequently, the costs of an additional child (given the quality considerations) are proportionately much higher, which brings down the total number of children desired.

These considerations suggest that the poor may have higher fertility rates than the rich. To the extent that this is true, a high overall rate of population growth will have a disproportionately heavy impact on those who are already poor, or on the threshold of poverty.

Population growth and the environment Recall the discussion on whether fertility is too high. In that discussion, one of the most important features is the underpricing of infrastructural resources. Government-provided education, health, and public transportation may all be subsidized. We also discussed why they are subsidized: it may be a second-best way to transfer resources to the poor. (Direct transfers may be infeasible because it may be impossible to credibly identify the poor.)

This observation has two corollaries. First, these resources must be consumed largely by the poor. Second, the inability of individuals to internalize the costs of these resources leads to higher fertility and consequent increased pressure on those very resources.

Under pricing arguments are not restricted to infrastructure alone. They apply to resources such as the commons (grazing land, fish stocks, ground-water) and the environment (forest cover, pollution, the ozone layer). Population growth places additional pressure on these scarce resources. Moreover, growth theory cannot be profitably applied to many of these resources: having more people around does not “produce” more forests, fish, water, or ozone. The

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effects are therefore stronger and more immediate.

9.4.2. Some positive effects In the previous section, we began with the naive argument that all that population growth does is eat into

available production. This is implicit in the Harrod-Domar model, for instance, but we know that population growth means a larger labor force, which contributes to additional production. Thus, at the very least, we have a tussle between the productive capabilities of a growing population and its consumption demands. The Solow model captured this well. Long-run growth of per capita income is unchanged in the Solow model because these two forces balance each other. We did note the existence of a level effect: there is more labor relative to capital on the long-run growth path. This brings down the level of income measured per unit of (effective) labor. This is an example of diminishing returns to labor at work. A higher ratio of labor to capital reduces its average product.

However, is that all labor is good for: production? In some broad sense, the answer is yes, but it is useful to return to a distinction between two notions of production: production using the same set of techniques, as embodied by the production function or technical know-how at any one point of time, and the production, invention, or application of new methods; in short, technical progress. Put another way, the pace of technical progress may be endogenous in the sense that it is affected by population size. Although we have discussed the endogeneity of technical progress before (see Chapter 4), the demographic effect on population growth merits additional attention.

The effect of population growth on technical progress can in turn be divided into two parts. First, population growth may spur technical progress out of the pressures created by high population density. This is the “demand- driven” view explored by Boserup [1981]. Second, population growth creates a larger pool of potential innovators and therefore a larger stock of ideas and innovations that can be put to economic use. This is the “supply-driven” view taken by Simon [1977] and Kuznets [I960].14

Population, necessity, and innovation Necessity is the mother of invention, and population pressure has historically created necessity. Nowhere is this more true than in agriculture, where increasing populations have historically placed tremendous pressure on the supply of food. It is certainly the case that such pressure was often relieved by the Malthusian weapons of famine and disease that wiped out large sections of the population. However, it is also true that scarcity drove man to innovate, to create, or to apply methods of production that accommodated the increased population by a quantum jump in food output.

Several indicators permit us to see evidence of this even in today’s world. Boserup [1981] classified countries into different grades by population density: very sparse, between 0 and 4 people per square kilometer; sparse, between 4 and 16 people per square kilometer; medium, between 16 and 64 persons per square kilometer; dense, between 64 and 256 persons per square kilometer; and very dense, 256 persons per square kilometer and upwards.15

Now consider an indicator such as irrigation. Which countries have more of it? Not surprisingly, the high- density countries do: in 1970, all the countries in Boserup’s sample (of fifty-six) with more than 40% of the arable land under irrigation were dense or very dense countries, in the sense defined in the previous paragraph. Alternatively, consider the use of chemical fertilizer: it increases systematically with population density. In addition, study multi-cropping: four out of five very dense countries (in a sample of twenty-four) had more than 50% of the land devoted to multiple cropping; no other country in the sample exhibited this sort of ratio. More generally, Boserup suggested the pairing of population densities and food supply systems shown in Table 9.4 as a summary of her overall observations.

The point to be made is simple, perhaps obvious. At least in agriculture, high population densities go hand in hand with technologically more intensive forms of farming. This by itself isn’t proof that such techniques were actually invented in high-density societies, although they almost surely were, but it does suggest that these methods, even if they were universally known, were applied more frequently in high-density societies.16


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