3. Getting a Sense of Magnitudes

A Cobb-Douglas production function with equal shares of labor and capital implies that the capital accumulation equation can be written

(K_t+1/N- K_t/N)=s(K_t/N)^1/2- K_t/N.

In steady state,

s/δ=(K^*/N)^1/2=Y*/N.

In this case, a doubling of the saving rate leads to a doubling of output per worker in the long run. How fast does the capital stock increase? Suppose that s increases from 0.1 to 0.2, that the depreciation rate equals 0.1 initially, and that K/N=1. Using the dynamic equation (11.4), one can show that adjustment to the new steady state is only 63% complete after 20 years.

With the same Cobb-Douglas production function, consumption per worker can be written

C*/N=Y*/N-K*/N=(s/)-(s/)²=s(1-s)/,

which is maximized when s=1/2. Almost all countries have saving rates below this level, which suggests that most countries have capital stocks below their golden rule levels.

4. Physical versus Human Capital

The aggregate production function can be generalized to include human capital (H):

Y/N=f(K/N, H/N).

The conclusions derived previously can be interpreted as applying to the accumulation of physical capital for given levels of human capital, or to the accumulation of human capital for given levels of physical capital. Whether increasing both factors in the same proportion would result in permanently higher growth is a subject of current research, associated with the “endogenous growth” literature triggered by the work of Lucas and Romer. The text argues that the evidence thus far does not provide much support for this proposition. One might draw from this conclusion that an increase in the human capital saving rate, e.g., an increase in the proportion of output per worker spent on education, would not have permanent effects on the growth rate. On the other hand, it is possible that education might be related to the rate of technological progress. The next chapter looks at the sources and effects of technological progress.

V. Pedagogy

Students may be confused by the notion that an increase in the saving rate will increase steady-state consumption per worker, since IS-LM analysis suggests that an increase in the marginal propensity to save will reduce output. The differing results arise in different time frames, and both could be true. It may be worthwhile to clarify the short-run/long-run distinction in the context of this example. Moreover, the dynamic simulation in the text provides some idea of the length of the long run.

VI. Extensions

How could the introduction of human capital potentially allow for an increase in the saving rate to generate a permanently higher growth rate? The issue turns on whether the production function

Y/N=F(K/N, H/N)

exhibits constant returns to scale in its two arguments. If so, then equiproportional increases in human and physical capital per worker will generate proportional increases in output per worker forever. Thus, if saving can accumulate human as well as physical capital, then an increase in the saving rate will generate a permanently higher growth rate.