regression now produces an estimate of b of -0.566, with a standard error of 0.144. Thus accounting for the selection bias in Baumols procedure eliminates about half of the convergence that he finds.

The second problem that De Long identifies is measurement error. Estimates of real income per capita in 1870 are imprecise. Measurement error again creates bias toward finding convergence. When 1870 income is overstated, growth over the period 1870-1979 is understated by an equal amount; when 1870 income is understated, the reverse occurs. Thus measured growth tends to be lower in countries with higher measured initial income even if there is no relation between actual growth and actual initial income.

De Long therefore considers the following model:

1 [( / ) 979] - 1 [( / ) 87 ]* = + / + (1.33)

1 [( / ), 87 ] = ln[(y/N)u87o]* + Ul, (1.34)

where 1 1( /N)i87o]* is the true value of log income per capita in 1870 and ln[(y/N)i87o] is the measured value, a and are assumed to be uncorrelated with each other and with 1 [( / )187 ]*-

Unfortunately, it is not possible to estimate this model using only data on ln[(y/N)i87o] and 1 [( / )1979]. The problem is that there are different hypotheses that make identical predictions about the data. For example, suppose we find that measured growth is negatively related to measured initial income. This is exactly what one would expect either if measurement error is unimportant and there is true convergence or if measurement error is important and there is no true convergence. Technically, the model is not identified.

De Long argues, however, that we have at least a rough idea of how good the 1870 data are, and thus have a sense of what is a reasonable value for the standard deviation of the measurement error, o- = 0.01, for example, implies that we have measured initial income to within an average of 1 percent; this is implausibly low. Similarly, ou = 0.50-an average error of 50 percent-seems implausibly high. De Long shows that if we fix a value of au, we can estimate the remaining parameters.

Even moderate measurement error has a substantial impact on the results. For the unbiased sample, the estimate of b reaches 0 (no tendency toward convergence) for an =0.15, and is 1 (tremendous divergence) for Pu =: 0.20. Thus plausible amounts of measurement error eliminate most or all of the remainder of Baumols estimate of convergence.

It is also possible to investigate convergence for different samples of countries and different time periods. Figure 1.9 is a convergence scatterplot analogous to Figures 1.7 and 1.8 for virtually the entire non-Communist world for the period 1960-1985. As the figure shows, there is little evidence of convergence. We return to the issue of convergence at the end of Chapter 3.

2.0

1.5 -

S 1.0-

0.5 -

-0.5 -

-1.0,

"+

+ -H-+ +

+ ++ +

6 7 8 9 1

Lofi income per capita in 1960 (1985 intemational prices)

FIGURE 1.9 Initial income and subsequent growth in the postwar period (data from Summers and Heston, 1991)

Saving and Investment

Consider a world where every country is described by the Solow model and uhere all countries have the same amount of capital per unit of effective labor. Now suppose that the saving rate in one coimtry rises. If all of the additional saving were invested domestically, the marginal product of capital in that country would fall. There would therefore be incentives for residents of the country to invest abroad. Indeed, in the absence of any impediments to capital flows, the investment resulting from the increased saving would be spread uniformly over the whole world; the fact that the rise in saving occurred in one country would have no special effect on investment there. Thus there would be no reason to expect countries with high saving to also have high investment.

Feldstein and Horioka (1980) examine the association between saving and investment rates. They find that, contrary to this simple view, saving and investment rates are strongly correlated. Specihcally, Feldstein and Horioka run a cross-country regression for 21 industrialized countries of the average share of investment in GDP during the period 1960-1974 on a constant and the average share of saving in GDP over the same period. The results are

(I/Y), = 0.035 + 0.887(S/y),, (0.018) (0.074)

= 0.91,

(1.35)

Investment, Population Growth, and Output

According to the Solow model, saving and population growth affect output per worker through their impact on capital per worker. A country that saves more of its output has more capital per worker, and hence more output per worker; a country with higher population growth devotes more of its saving to maintaining its capital-labor ratio, and so has less capital and output per worker.

The model makes not just qualitative but quantitative predictions about the impact of saving and population growth on output. We saw in Section 1.5 that the elasticity of output on the balanced growth path with respect to s is a/(l - a), where a is capitals share. Similarly, one can show that its elasticity with respect to n + g + is -a/(l - a) (see Problem 1.5). Thus,!

"One can also derive (1.36) by assuming that the production function is Cobb-Douglas; in this case, no approximations are needed (see Problem 1.2).

where again the numbers in parentheses are standard errors. Thus, rather than there being no relation between saving and investment, there is an almost one-to-one relation.

There are various possible explanations for Feldstein and Horiokas finding (see Obstfeld, 1986, for a discussion). One possibility, suggested by Feldstein and Horioka, is that significant barriers to capital mobility exist. In this case, differences in saving and investment across countries would be associated with rate of return differences.

Another possibility is that there are underlying variables that affect both saving and investment. For example, high tax rates can reduce both saving and investment (Barro, Mankiw, and Sala-i-Martin, 1995). Similarly, countries whose citizens have low discount rates, and thus high saving rates, may provide favorable investment climates in ways other than the high saving; for example, they may limit workers ability to form strong unions.

Finally, the strong association between saving and investment can arise from government policies that offset forces that would otherwise make saving and investment differ. Governments may be averse to large gaps between saving and investment-after all, a large gap must be associated with a large trade deficit (if investment exceeds saving) or a large trade surplus (if saving exceeds investment). If economic forces would otherwise give rise to a large imbalance between saving and investment, the goverrmient may choose to adjust its own saving behavior or its tax treatment of saving or investment to bring them into rough balance.

In sum, the strong relationship between saving and investment differs dramatically from the predictions of a natural baseline model. Whether this difference reflects major departures from the baseline (such as large barriers to capital mobility) or something less fundamental (such as underlying forces affecting both saving and investment) is not known.