CHAPTER 11. SAVING, CAPITAL ACCUMULATION,
AND OUTPUT
I.
MOTIVATING QUESTION
Does the Saving Rate Affect Growth?
In the long run, saving does not affect growth, but does affect the level of per capita output. An increase
in the saving rate can increase growth for some time, but not indefinitely, if the production function
exhibits decreasing returns to capital per worker.
II.
WHY THE ANSWER MATTERS
The comparatively low U.S. saving rate (relative to other OECD economies) becomes a policy issue from
time to time and is discussed fairly frequently in the press. Changes in the structure of the Social
Security system may also have implications for saving and capital accumulation. This chapter clarifies
the relationship between saving, per capita output, and growth, and discusses the likely effects of
increasing the saving rate.
III. KEY TOOLS, CONCEPTS, AND ASSUMPTIONS
1.
Tools and Concepts
i.
The chapter develops the Solow model of growth for the case of no technological change and no
population growth.
ii.
The golden rule level of capital per worker is the value of capital per worker that maximizes
steady- state consumption per worker.
iii.
The Cobb-Douglas production function is described in an appendix.
2.
Assumptions
This chapter assumes a closed economy, a fixed labor force, and a fixed level of technology.
IV. SUMMARY OF THE MATERIAL
1.
Interactions between Output and Capital
To save notation, write the aggregate production function of the previous chapter, Y/N=F(K/N,1), as
Y/N=f(K/N). This step is valid because the second argument of the original function is a constant.
Now assume:
i. No technological change.
ii. Population, the labor force participation rate, and the natural rate of unemployment are all
constant. Thus, N, interpreted as the natural level of employment, is also constant.
iii. The fraction of real GDP devoted to saving (the saving rate) is constant, i.e.,
S=sY.
(11.1)
This assumption captures the empirical regularities that the saving rate (s) does not appear
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to change systematically as a country increases its income and that savings rates in rich
countries do not appear to differ systematically from savings rates in poor countries.
iv. The economy is closed and the budget deficit is zero, so that goods market equilibrium is
equivalent to
I=S.
(11.2)
v. Capital depreciates at rate . Thus, the change in the capital stock over time is
Kt+1=(1-)Kt+It.
(11.3)
Investment creates new capital, but the existing capital stock depreciates.
2.
Implications of Alternative Saving Rates
The equations above together imply
(Kt+1/N- Kt/N)=sf(Kt/N)-δKt/N.
(11.4)
Capital per worker increases to the extent that total saving per worker exceeds depreciation of the
existing capital stock per worker. Figure 11.1 plots the separate components of equation (11.4).
Figure 11.1: Capital and Output Dynamics
To the left of point A, investment per worker (sf(K/N)) exceeds depreciation of the existing stock (K/N).
Thus, the capital stock per worker (K/N) is rising. As K/N increases, output increases less than
proportionately (given decreasing returns), and thus so does investment per worker. Depreciation, on the
other hand, increases proportionately with capital per worker, so eventually an equilibrium will be
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reached at point A. At this equilibrium, capital per worker is constant. From equation (11.4), a constant
value of K/N implies
sf(K/N)=  K/N.
(11.5)
Point A is called a steady state, since output per worker and capital per worker will remain constant at
this point. For this reason, time subscripts have been eliminated from equation (11.5). Point B
determines steady-state output per worker, which is given by
Y*/N=f(K*/N).
(11.6)
The dynamics of adjustment are indicated by the arrows on the horizontal axis of Figure 11.1. Note that
if capital per worker were to begin above its steady-state value, K/N would fall, since depreciation of
existing capital per worker would exceed total saving per worker.
Now consider an increase in the saving rate. In Figure 11.2, an increase in the saving rate from s to s'
shifts the sf(Kt/N) curve upward in proportion to the change in the saving rate. The new steady-state
equilibrium is given by point B. Notice that the steady-state growth rate (which is zero) is the same as
the original steady-state growth rate. Capital per worker, however, is higher at point B, so output per
worker is higher as well. These results imply that an increase in the saving rate will increase the growth
rate temporarily, since output per worker must increase to reach the new steady state, but not in the long
run.
Figure 11.2: Effects of Changes in the Saving Rate
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What is the optimal saving rate? A very low saving rate will result in very low steady-state output and
consumption per worker. A very high saving rate will waste resources on depreciation, since extra units
of capital per worker produce very little extra output per worker when the capital stock is high.
Somewhere in between is a saving rate that maximizes steady-state consumption per worker. This rate is
called the golden rule saving rate, which produces the golden rule capital stock.
Empirically, it appears that most countries have less than their golden rule levels of capital. Thus, it
appears that an increase in the saving rate could increase the consumption of future generations. On the
other hand, an increase in the saving rate would reduce the level of consumption for some time, until the
increased output generated by the higher capital stock compensated for the reduction in the proportion of
output consumed. A box in the text makes essentially the same point about a shift from a pay-as-you-go
to a fully funded Social Security system. A fully funded system might lead to a higher capital stock in
the long run (since Social Security contributions are invested, not simply redistributed as in a pay-as-yougo system). However, during the transition, some generations would have to contribute twice, once to
pay the benefits of current retirees and once to fund their own retirement. As a result, any shift to a fully
funded system would probably need to be gradual to spread the burden of adjustment across generations.
Even if the United States never shifts to a fully funded system, however, the text points out that some
changes in the Social Security system will be necessary to accommodate the projected increase in the
ratio of retirees to workers over the 21st century.
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
(Kt+1/N- Kt/N)=s(Kt/N)1/2- Kt/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
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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.
VII. OBSERVATIONS
Depreciation of the capital stock is necessary for the existence of a steady-state equilibrium in the growth
model presented in this chapter. An increase in the depreciation rate would reduce the steady-state
capital stock per worker.
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