c(0) = {7-g)

K(0) (1 - a)b K(0)

y-m (3.33)

= ia-a)b + r-g]

where the last line uses the fact that r = ab.

We can now verify that this consumption behavior causes the capital stock to grow at rate J. Since and are both growing at rate g and since there are L households, (3.33) implies that total consumption, c(t)L, is given by

C(f) = (b g)K(t). (3.34)

Substituting (3.34) and the production function, Y = bK, into the equation of motion for K, = Y C, yields

k(t) = bK{t)-{b-g)K{t)

(3.35)

= gK(t).

Thus consumption, capital, and output all grow at a constant rate.

This analysis implies that if the economy is subjected to some kind of shock (a change in p, for example), the ratio of consumption to the capital stock jumps immediately to its new balanced-growth-path value, and consumption, capital, and output all immediately begin growing at a constant rate. Thus there are no transitional dynamics to reach the balanced growth path. Intuitively, the fact that production is linear means that there is nothing special about any particular level of the capital stock or of the

"in addition, one can show that this Is the only equilibriimi. To see this, suppose C(0) exceeds {b - 7J)K{0). Then consumption must be higher at every point in time than under (i.M) (since must grow at rate g), and capital must therefore be lower. This implies that the present value of lifetime consumption is higher than before and that the present value of lifetime labor income is lower. Bui this means that households are violating their budget constraint, and thus that this path is not possible. A similar argument shows that if C(0) is less than {b - g)K{0), the present value of lifetime consumption is less than lifetime wealth.

If the capital stock grows at rate the wage at time r is (1 - a)bK(0)eS/L (see [3.31]). Since the interest rate is constant at P, this implies that the representative households initial wealth plus the present value of its lifetime labor income is K{0)Jl. + (1 «)M(0)/f(T - )L\. And since consumption grows at rate g, the present value of Ufetime consumption is c(0)/(r - g). Equating the present value of lifetime consumption with lifetime wealth and solving for c(0) yields

capital-labor ratio. For example, if a war suddenly halves the capital stock, households respond simply by halving their consumption at every date.

The growth rate of the economy, g, is (ab-p)la. Since k{t)/K{t) = sit)b, where s(f) is the fraction of output that is saved, the fact that k(t)/K(t) is constant and equal to (ab - p)la implies that s(t) is constant and equal to (ab - p)lab. Thus, for example, a lower value of households discount rate, p, raises the saving rate and thereby increases long-run growth. A higher value of Ol also increases saving and growth: the higher the private marginal product of capital (ab) is relative to the social marginal product (b), the more households save, and thus the higher growth is. One implication is that unless a equals 1, the growth rate produced by the decentralized equilibrium is less than the socially optimal growth rate: a social planner would account for the full marginal product of capital rather than just the private marginal product, and would thus choose a saving rate oi (b - p)/ab, and hence a growth rate of (b - p)/cr.

3.6 Models of Knowledge Accumulation and the Central Questions of Growth Theory

Our analysis of economic growth is motivated by two issues: the growth over time in standards of living, and their disparities across different parts of the world. It is therefore natural to ask what the models of R&D and knowledge accumulation have to say about these issues.

With regard to worldwide growth, it seems plausible that the forces that the models focus on are important. At an informal level, the growth of knowledge appears to be the central reason that output and standards of liv ing are so much higher today than in previous centuries. And as described in Chapter 1, formal growth-accounting studies attribute large portions of the increases in output per worker over extended periods to the unexplained residual component, which may reflect technological progress.

It would of course be desirable to refine the ideas we have been considering by improving our understanding of what types of knowledge are most important for growth, their quantitative importance, and the forces determining how knowledge is accumulated. But it seems likely that the kinds of forces we have been considering are important. Thus, the general directions of research suggested by these models seem promising for understanding worldwide growth.

With regard to cross-country differences in real incomes, the relevance of the models is less clear. There are two difficulties. The first is quantitative. As Problem 3.11 asks you to demonstrate, if one believes that economies are described by something like the Solow model but do not all have access to the same technology, the lags in the diffusion of knowledge from rich to

poor countries that are needed to account for observed differences in incomes are extremely long-on the order of a century or more. It is hard to beheve that the reason that some countries are so poor is that they do not have access to the improvements in technology that have occurred over the past century.

The second difficulty is conceptual. As emphasized in Section 3.5, technology is nonrival: its use by one firm does not prevent its use by others. This naturally raises the qu°stion of why poor countries do not have access to the same technology as rich countries. If the relevant knowledge is publicly available, poor countries can become rich by having their workers or managers read the appropriate literature. And if the relevant knowledge is proprietary knowledge produced by private R&D, poor countries can become rich by instituting a credible program for respecting foreign firms property rights. With such a program, the firms in developed countries with proprietary knowledge would open factories in poor countries, hire their inexpensive labor, and produce output using the proprietary technology. The result would be that the marginal product of labor in poor countries, and hence wages, would rapidly rise to the level of developed countries.

Although lack of confidence on the part of foreign firms in the security of their property rights is surely an important problem in many poor countries, it is difficult to believe that this alone is the cause of the countries poverty. There are numerous examples of poor regions or countries, ranging from European colonies over the past few centuries to many countries today, where foreign investors can establish plants and use their know-how with a high degree of confidence that the political environment will be relatively stable, their plants will not be nationalized, and their profits wUl not be taxed at exorbitant rates. Yet we do not see incomes in those areas jumping to the levels of industrialized countries.

One may reasonably object to this argument on the grounds that the difficulty that such countries face is not lack of access to advanced technologies, but lack of ability to use that technology. But this objection implies that the main source of differences in standards of living is not different levels of knowledge or technology, but differences in whatever factors allow richer countries to take better advantage of advanced technology. Understanding differences in incomes therefore requires understanding the reasons for the differences in these factors. This task is taken up in Part of the chapter.

3.7 Empirical Application: Population Growth and Technological Change since 1 Million B.C.

The discussion in the previous section may seem to imply that models of endogenous knowledge accumulation are almost untestable. The models pre-