1 2* - Iny* = ~----(1 %2 - InSKl) +-----(lnSH2 " InSni)

l-a-p l-a-p

a + P „ , > , , (3.59)

[1 ( 2 +0)-ln(ni +0)]

l-a-p

= 1.4(ln2) + 1.6(ln2)-(31n0.8) 2.75.

Since 15.6, output per worJcer is almost 16 times larger in the second country. Thus differences in saving rates and population growth that are not extraordinary give rise to differences in incomes comparable to the vast gaps we are trying to understand.

The Solow model, as we saw in Chapter 1, cannot do this, tor the same parameter values (except with p set to zero), (3.58) imphes that the gap in log incomes is

1 2* - Inyf = ~(lnsK2 - ln%i) - -[1 { 2 + g) - 1 ( ] + g)] I ~ a 1 - a

(3.60)

= 4!(ln2-ln0.8)-0.49. 0.65

Since 6°* 1.6, the Solow model implies only a 60 percent difference in incomes.

Finally, because the model assumes diminishing marginal products of physical and human capital, it implies that rates of return are lower in rich than in poor countries. Thus the model does not answer the question of why capital does not flow to poor countries. At the same time, because the marginal products are only slowly diminishing, large differences in incomes are not associated with vast differences in rates of retum. One can show that the marginal products of physical and human capital on the balanced growth path are

MPK* =a(n+g)/sK. (3.61)

MPH* = pin +g)/s„ . (3.62)

(see Problem 3.16). Since the variation in population growth and saving rates needed to account for large income differences is limited, large income differences imply only moderate differences in rates of return.

In the example above, for instance, sk2 = 2ski, Sh2 = 2shi, and 2 + = 0.8(n, + g); thus (3.61) and (3.62) imply MPK- = OAMPK and MPH = 0.4MPH]*. Although these differences are substantial, it is not out of the question that tax policies, the possibility of expropriation, capital-market imperfections, and so on could cause capital not to flow to poorer countries in the face of such differentials. And if p is larger, the differences

The Case of Increasing Returns

AU of the analysis in this part of the chapter has assumed diminishing returns to physical and human capital together. But it is possible that there are constant or increasing returns to capital. There are several reasons that this might occur. First, as in the learning-by-doing model of Section 3.4, learning can occur as a by-product of capital accumulation. Second, there can be some other source of extemal economies ofscale. For example, the presence of other firms producing similar products can foster the development of a skilled labor force and of speciahzed support firms, and can therefore make production at a given firm more efficient. And third, there can be internal economies of scale: methods of production that are highly efficient at high levels of output may be impractical at low levels.

Relaxing the assumption of diminishing returns to physical and human capital together would have important implications for the analysis. As in the models of Part A of this chapter, growth rates would become endogenous and potentially ever-increasing. Changes in the resources devoted to capital accumulation could lead not just to large differences in the level of output per worker, but to permanent differences in growth rates. A simple example of such a model arises tn the model we have been considering in the special case of a + p = 1. This model, which is sirnUar to the linear growth models of Part A, is analyzed in Problem 3.17. More elaborate models of constant or increasing returns to capital include P. Romer (1986); Lucas (1988); Rebelo (1991); and Murphy, Shleifer, and Vishny (1989a, 1989b).

Increasing returns to capital provide a candidate explanation other than knowledge accumulation of worldwide economic growth. But, as with the endogenous growth models of Part A of this chapter, there is reason to be skeptical of the importance of increasing returns for cross-country income differences. The key issue is the area over which the increasing returns occur. Clearly they must occur at least at the scale of entire economies if they are to be the driving force of economy-wide growth. And there is no reason for them to be limited by political boundaries: surely firms in Luxembourg can take advantage of increasing returns as well as firms in Germany can.

"External economies of scale occur when a doubUng of inputs by a single firm only doubles its output, but a doubling of inputs by all firms together more than doubles their output. Internal economies of scale occur when a doubling of inputs by a single firm more than doubles its output.

in saving and population growth, and hence the differences in rates of return, needed to account for large income differences are even smaller. In the Solow model with a conventional value of capitals share, in contrast, we know from Chapter 1 that large income differences require vast differences in saving rates and rates of population growth, and thus that they imply vast differences in rates of return.

There is a large empirical literature on cross-country income differences. Among the factors that have been identifted as potentially important to income are political stability iBarro, 1991), equipment investment (De Long and Summers, 1991, 1992), the financial system (King and Levine, 1993a, 1993b; JappeUi and Pagano, 1994), microeconomic distortions lEasterly, 1993), corruption (Mauro, 1993), and inflation (Fischer, 1991, 1993).

When depreciation is included in the model, g + S appears in place of g in (3.57). The \alue of 0.05 is thus intended as an estimate ot g + S rather than of g.

If outputs are approximately linear in capital with little or no role for raw labor, then only by accident would total output (not output per worker) be higher in areas with larger populations. If there are constant or increasing returns to capital and a role for labor, there must be increasing returns to capital and labor together. But then, unless the increasing returns are worldwide, it is puzzling why output per imit of input is not much lower in such places as New Zealand and Hawaii, which are far removed from the rest of the world, than in such places as Western Europe and the Eastern United States. A final possibility is that the increasing returns are potentially worldwide but that economies differ in their ability to tap into those increasing retums. But then (just as with the argument that economies differ in their success in using worldwide knowledge) the source of cross-country mcome differences is not increasing retums, but whatever determines the differences in this success. Thus it appears to be difficult to use increasing returns to capital to account for income differences across parts of the world.

3.11 Empirical Application: Physical

and Human Capital Accumulation and Cross-Country Differences in Incomes

The previous section shows that, when we allow for human capital, variations in population growth and capital accumulation have the potential to account for large cross-country differences in incomes. Mankiw, Romer, and Weil (1992) address the question of whether those variations in fact account for the differences.!

As described in Section 1.7, Mankiw, Romer, and Weil find that the estimated impact of saving and population growth on income is far larger than predicted by the Solow model with a capital share in the vicinity of one-third. Since the model with human capital predicts much larger impacts of saving and population growth on output than the Solow model does, this finding is encouraging for the human-capital model.

Mankiw, Romer, and Weils strategy is to estimate equation (3.57), Iny* = [a/(l - a - pMnSK +[fi/a-a- /3)] lns„ - [(a + ;8)/(l -a~ fi)]ln(n + g). Their measures of y, sk, and n are discussed in Chapter 1. As before, g is set to 0.05 for all coimtries. They measure Sh as the average