"One can make the same point in terms of the rates of saving, population growth, and so on that determine capital per worker. For example, the elasticity of y* with respect to s is /{1 - aic) (see [1.22]). Thus accounting for a difference of a factor of ten in output per worker on the basis of differences in s would require a difference of a factor of a hundred in s it = and a difference of a factor of ten if . Variations in actual saving rates are much smaller than this.

"" can try to avoid this conclusion by considering production functions where capitals marginal product falls less rapidly as rises than it does in the Cobb-Douglas case. This

ratios are roughly constant over time. Thus the U.S. capital stock per worker is roughly ten times larger than it was a hundred years ago, not a hundred or a thousand times larger. Similarly, although capital-output ratios vary somewhat across countries, the variation is not great. For example, the capital-output ratio appears to be two to three times larger in the United States than in India; thus capital per worker is "only" about twenty to thirty times larger in the United States. In sum, differences in capital per worker are far smaller than those u?eded to account for the differences in output per worker that we are trying to understand.

The second difficulty is that attributing differences tn output to differences in capital without differences in the effectiveness of labor implies immense variation in the rate of return on capital (Lucas, 1990a). If markets are competitive, the rate of return on capital equals its marginal product, f(k), minus depreciation, 8. Suppose that the production function is Cobb-Douglas (see equation [1.5]), which in intensive form is f(k) = k". With this production function, the elasticity of output with respect to capital is simply a. The marginal product of capital is

f(k) = ak"-

(1.27)

Equation (1.27) impUes that the elasticity of the marginal product of capital with respect to output is -(1 - a)/a. If a = , a tenfold difference in output per worker arising from differences in capital per worker thus implies a hundredfold difference in the marginal product of capital. And since the return to capital is f(k) - 8, the difference in rates of return is even larger.

Again, there is no evidence of such differences in rates of return. Direct measurement of returns on financial assets, for example, suggests only moderate variation over time and across countries. More tellingly, we can learn much about cross-country differences simply by examining where the holders of capital want to invest. If rates of return were larger by a factor of ten or a hundred in poor countries than in rich countries, there would be immense incentives to invest in poor countries. Such differences in rates of return would swamp such considerations as capital-market imperfections, government tax policies, fear of expropriation, and so on, and we would observe immense flows of capital from rich to poor countries. We do not see such flows.

approach would encounter two major difficulties. First, since the marginal product of capital would be similar in rich and poor countries, capitals share would be much larger in rich countries. Second, and similarly, real wages would be only slightly larger in rich than in poor countries. These implications appear grossly inconsistent with the facts.

Thus differences in physical capital per worker cannot account for the differences in output per worker that we observe, at least if capitals contribution to output is roughly reflected by its private returns.

The other potential source of variation in output per worker in the Solow model is the effectiveness of labor. Attributing differences in standards of living to differences in the effectiveness of labor does not require huge differences in capital or in rates of return. Along a balanced growth path, for example, capital is growing at the same rate as output; and the marginal product of capital, f(k), is constant.

The Solow models treatment of the effectiveness of labor is highly incomplete, however. Most obviously, the growth of the effectiveness of labor is exogenous: the model takes as given the behavior of the variable that it identifies as the driving force of growth. Thus it is only a small exaggeration to say that we have been modeling growth by assuming it.

More fundamentally, the model does not identify what the "effectiveness of labor" is; it is just a catchall for factors other than labor and capital that affect output. To proceed, we must take a stand concerning what we mean by the effectiveness of labor and what causes it to vary. One natural possi-biUty is that the effectiveness of labor corresponds to abstract knowledge. To understand worldwide growth it would then be necessary to analyze the determinants of the stock of knowledge over time. To understand crosscountry differences in real incomes, one would have to explain why firms in some countries have access to more knowledge than firms in other countries, and why that greater knowledge is not rapidly transferred to poorer countries.

There are other possible interpretations of a: the education and skills of the labor force, the strength of property rights, the quality of infrastructure, cultural attitudes toward entrepreneurship and work, and so on. Or a may reflect a combination of forces. For any proposed view of what a represents, one would again have to address the questions of how it affects output, how it evolves over time, and why it differs across parts of the world.

The other possible way to proceed is to consider the possibility that capital is more important than the Solow model impUes. If capital encompasses more than just physical capital, or if physical capital has positive externalities, then the private return on physical capital is not an accurate guide to capitals importance in production. In this case, the calculations we have done may be misleading, and it may be possible to resuscitate the view that differences in capital are central to differences in incomes.

These possibiUties for addressing the fundamental questions of growth theory are the subject of Chapter 3.

nt) Kit) dY{t) K(t) LWdYWat) A(t)r(t)A(f) Y(t) Y(t) dKit) Kit) Yit) dLit) Ut) Y(t) dAit) Ait)

(1.29)

Here aiit) is the elasticity of output with respect to labor at time f, ajf(f) is again its elasticity with respect to capital, and Rit) = [Ait)IYit)][dYit)ldAit)\{Ait)IAit)l Subtracting i(f)/I(t) from both sides and using the fact that aiit) + anit) = 1 (see Problem 1.7, at the end of this chapter) gives us an expression for the growth rate of output per worker:

Yit) Ut)

1 - = - )

kit) Lit)

Kit) Lit)

+ Rit). (1.30)

The growth rates of Y,K, and L are straightforward to measure. And we know that if capital earns its marginal product, can be measured using data on the share of income that goes to capital. Kit) can then be measured as the residual in (1.30). Thus (1.30) provides a way of decomposing the growth of output per worker into the contribution of growth of capital per worker and a remaining term, the Solow residual. The Solow residual is sometimes interpreted as a measure of the contribution of technological progress. As the derivation shows, however, it reflects all sources of growth other than the contribution of capital accumulation via its private return.

This basic framework can be extended in many ways (see, for example, Denison, 1967). The most common extensions are to consider different types of capital and labor and to adjust for changes in the quality of

1.7 Empirical Applications

Growth Accounting

In the Solow model, long-run growth of output per worker depends only on technological progress. But short-run growth can result from either technological progress or capital accumulation. Thus the model implies that determining the sources of short-run growth is an empirical issue. Growth accounting, which was pioneered by Abramovitz (1956) and Solow (1957), provides a way of tackling this subject.

To see how growth accounting works, consider again the production function Y(t) = F{K(t),A{t)Ut)). This implies

dV/dL and / denote ldY/d{AL)]A and [SY/d{AL)]L, respectively. Dividing both sides by Y{t) and rewriting the terms on the right-hand side yields