"This effect can be seen clearly in the models we have been considering in the case of constant returns to produced inputs and no population growth.

See Jones (1994) for tests of endogenous growth models that focus on recent history.

dictions concern worldwide growth; thus cross-country differences cannot be used to test them. In addition, since many factors, such as wars and business cycles, affect short-term growth substantially, short-run time-series data are also of little value. Thus we are left only with long-run data for the world as a whole, which one might expect to be insufficient to provide strong tests between alternative views.

Kremer (1993) demonstrates, however, that the hypothesis that growth arises from endogenous knowledge accumulation can be tested despite these difficulties. He first notes that essentially all models of the endogenous growth of knowledge predict that technological progress is an increasing function of population size. The reasoning is simple: the larger the population, the more people there are to make discoveries, and thus the more rapidly knowledge accumulates.

Kremer then argues that over almost all of human history, technological progress has led mainly to increases in population rather than increases m output per person. Population grew by several orders of magnitude between prehistoric times and the Industrial Revolution. But since incomes on the eve of the Industrial Revolution were not far above subsistence levels, it is not possible that output per person rose by anything close to the same amount. Only in the past few centuries, Kremer argues, has the impact of technological progress fallen to any substantial degree on output per person. Putting these observations together, Kremer concludes that models of endogenous technological progress predict that over most of human history, the rate of population growth should have been rising.

Kremers formal model is a straightforward variation on the models we have been considering. The simplest version consists of three equations. First, output depends on technology, labor, and land:

r(f) = R4A{t)L(t)]-", (3.36)

where R denotes the fixed stock of land. (Capital is neglected for simplicity, and land is included to keep population finite.) Second, the growth rate of knowledge is proportional to population:

= BUt). (3.37)

And third, population adjusts so that output per person equals the subsistence level, denoted y:

L{t) =

Aitf-R. (3.40)

This equation states that the population that can be supported is decreasing in the subsistence level of output, increasing in technology, and proportional to the amount of land.

Since and R are constant, (3.40) implies that the growth rate of I is (1 - a)/a times the growth rate of A. Expression (3.37) for the growth rate of A therefore imphes

= ). (3.41)

1(f) a

Thus, in this simple form, the model imphes not just that the growth rate of population is rising over time, but that it is proportional to the level of population.2

Kremer tests this prediction by using population estimates extending back to 1 million . . that have been constructed by archaeologists and anthropologists. Figure 3.11 shows the resulting scatterplot of population growth against population. Each observation shows the level of population at the beginning of some period and the average annual growth rate of population over that period. The length of the periods considered falls gradually from many thousand years early in the sample to 10 years at the end. Because the periods considered for the early part of the sample are so long, even substantia] errors in the early population estimates would have little impact on the estimated growth rates.

The figure shows a strongly positive, and approximately linear, relationship between population growth and the level of population. A regression of growth on a constant and population (in billions) yields

Uf = -0.0023 + 0.524 If, R = 0.92, D.W. = 1.10, (3.42) (0.0355) (0.026)

"For other recent growth models that treat population growth as endogenous, see Barro and Becker (1988, 1989) and Becker, Murphy, and Tamura (1990).

Kremer considers numerous variations on this model. Many of the simplifying assump tions prove not to be essential to the main results.

Aside from this Malthusian assumption about the determination of population, this model is similar to the model of Section 3.2 with = e = l."

To solve the model, begin by noting that (3.38) imphes Y{t) = yL(t). Substituting this into (3.36) yields

yl(f) = i?"fA(t)I(f)]i-", (3.39)

0.025 0.020

0.015

M 0.010 I

0.005

O.OOof -0.005

2 3 4 5

Population (billions)

FIGURE 3.11 The level and growth rate of population, 1 million . . to 1990 (from Kremer, 1993; used with permission)

where n is population growth and I is population, and where the numbers in parentheses are standard errors. Thus there is an overwhelmingly statistically significant association between the level of population and its growth rate.22

The argument that technological progress is a worldwide phenomenon fails if there are regions that are completely cut off from one another. Kremer uses this observation to propose a second test of theories of endogenous knowledge accumulation. From the disappearance of the intercontinental land bridges at the end of the last ice age to the voyages of the European explorers, Eurasia-Africa, the Americas, Australia, and Tasmania were almost completely isolated from one another. The model implies that at the time of the separation, the populations of each region had the same technology; thus the initial populations should have been approximately proportional to the land areas of the regions (see equation [3.40]). The model predicts that during the period that the regions were separated, technological progress was faster in the regions with larger populations. The theory thus predicts that, when contact between the regions was reestablished around 1500, population density was highest in the largest regions. Intuitively, inventions that would allow a given area to support more people, such as the domestication of animals and the development of agriculture, were much more likely in Eurasia-Africa, with its population of millions, than in Tasmania, with its population of a few thousand.

Ihe data confirm this prediction. The land areas of the four regions are 84 million square kilometers for Eurasia-Africa, 38 million for the Americas,

-The relationship appears to break down somewhat for the last two observations in the hgure, which correspond to the period after 1970. For this period it is plausible that the Malthusian model of population (equation [3.38]) is no longer a good hrst approximation.