What really happens in the Solow model: technological progress versus population growth?

Quinn, Kevin ; Hoag, John

This paper has two objectives. First, it presents the Solow model in a way that is intuitive for students. Second, it applies the presentation to show that more can be learned by looking at the model this way.

In the standard model, the basic expository device is the Solow diagram where the levels of output, saving and break-even investment per efficiency unit are plotted as function of capital per efficiency unit (k and y hereafter). This exposition identifies steady state levels of K and Y per efficiency unit. Comparative static exercises then look at how the steady state levels change when parameters such as the savings rate, the rate of population growth (n) or the rate of growth of efficiency (g) are changed. Often the results are coupled with a picture of the implied time-path of k or y from the initial to the new steady state.

There are two problems with this approach. First, it is difficult for the student to make the connection between the diagram involving levels and the time path of the transition from one steady state to another. In addition, because the diagram concentrates on capital and output per efficiency unit, it is difficult to see the implication for the growth rates and time-paths of capital and output per capita, which are the variables which matter for welfare.

Second, and more importantly, focus on the per efficiency unit variables obscures the huge differences for welfare between an increase in n, the population growth rate, and an increase in g, the growth rate of efficiency. Both of these changes reduce the steady-state value of output and capital per efficiency unit, but the one decreases while the other increases the level of capital and output per capita at every date in the future compared to what they would have been with no change.

Our solution to both problems is to do the analysis of the steady state not with the Solow diagram, but with growth rates, not levels, of K/L or Y/L (not K/LE or Y/LE) expressed as functions of capital per efficiency unit. It is much easier to see the connection with the time-path for the levels of each when we look at the model expressed in terms of growth rates. We work explicitly with growth rates of the per capita variables, so different implications of changes in the same direction of n and g are obvious.

I. The growth rate of K/L and why it matters

As a first step, we will find the relationship between the percentage growth of k and the percentage growth of (K/L), which we denote as K. By definition, k = (K/L)/E = [kappa]/E. Therefore, two important relationships hold.

ln k = ln([kappa]) - ln E, so ln [kappa] = ln k + ln E (1)

Before proceeding to the implications of this equation for [kappa], we relate K to what really matters: per-capita output. In general the growth in Y/L, per-capita output, is directly related to k and, therefore to growth in k and E. In the case of Cobb-Douglas production with constant returns to scale, Y = [K.sup.[alpha]][(LE).sup.1-[alpha]] so that Y/L = [(K/L).sup.[alpha]][E.sup.1-[alpha]]. It follows that [DELTA](Y/L)/(Y/L) = [alpha] [DELTA]([kappa])/([kappa]) + (1 - [alpha]) [DELTA]E/E. The growth rate of Y/L is a weighted average of the growth rates of [kappa] and E. This holds for all constant returns production functions.

Now we proceed with our treatment of K/L. The derivative of (1), provides this expression:

[DELTA][kappa]/[kappa] = [DELTA]k/k + g, (2)

where g = [DELTA]E/E. In order to gain further insight, we use the accumulation equation, as developed by Mankiw (2009: 223). Also, see Jones (2002: 39-43).

[DELTA]k = sf(k) - ([delta] + n + g)k (3)

Divide both sides by k:

[DELTA]k/k - sf(k)/k - ([delta] + n + g). (4)

Substitute this expression into (2) to obtain the following.

[DELTA][kappa]/[kappa] = sf(k)/k - ([delta] + n), (5)

where s is the saving rate, f(k) is the production function, [delta] is the depreciation rate, and n is the population growth rate.

Because of diminishing returns sf(k)/k is a decreasing function of k. That makes [DELTA][kappa]/[kappa] also a decreasing function of k. Figure 1 shows the components of Equation (2).

Steady state occurs where the two lines cross, at k*. We can easily see that the equilibrium is stable. Suppose that k > k*. In that case, g > [DELTA][kappa]/[kappa] so by (2), [DELTA]k/k < 0, and k falls. The same type argument holds for k < k*.

[FIGURE 1 OMITTED]

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From a welfare perspective, important issue is how [kappa] and k grow. In steady state, [kappa] and E must grow at the same rate for k to be constant at k*. Also, E grows at the rate g, so K must grow at the rate g as well. Because [DELTA][kappa]/[kappa] is the slope of In K, In [kappa] (and In E) are linear functions with slope g as shown in Figure 2.

Figure 2 presents the Solow model in terms of growth rates and time paths so that the student can more easily see what is going on in the model. It also facilitates seeing how changes in the parameters of the model differ, a difference that is not easy to see in the standard analysis.

II. The impact of a change in the population growth rate on [kappa].

An increase in n reduces [DELTA][kappa]/[kappa]; as (5) shows. Thus the graph would look as shown in Figure 3. The immediate impact at k* is that now [DELTA][kappa]/[kappa] < g, so k* falls to k** as described above.

[FIGURE 3 OMITTED]

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Figure 4 shows what happens to the growth path of [kappa]. Suppose that at [t.sub.o], n increases. The dashed line shows what the time path would have been if n had not increased. The solid line shows what actually happens to the path of ln([kappa]). The new steady state is reached at time [t.sub.l]. The smaller gap between ln([kappa]) and In E means that Ink has fallen. This clearly agrees with Figure 3.

III. The impact of a change in the growth of efficiency on [kappa].

Consider next an increase in the rate of growth of efficiency. Figure 5 shows that the g line shifts up to g'. The [DELTA][kappa]/[kappa] function is not affected. Clearly the equilibrium value of k* falls to k**. This is exactly the same impact on k as the increase in n. In this case, however, the growth rate of [kappa] never falls; rather, it rises steadily toward g' from g. By contrast, in the earlier scenario, the growth rate of K drops abruptly at the time of the change before steadily recovering to an unchanged g.

[FIGURE 5 OMITTED]

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The upper dashed line in Figure 6 shows the time path of ln([kappa]) if there had been no change. At [t.sub.o], g changes to g', and the ln E line changes slope abruptly. The path of ln([kappa]) now swings up and regains steady state at [t.sub.l]. Now ln([kappa]) grows at rate g' rather than g, but [k.sup.*] has fallen. After [t.sub.o] the path of ln([kappa]) is always above what it would have been if g had not changed.

IV. The impact of a change in n and a change in g on Y/L.

The growth rate of Y/L is a weighted average of the growth rates of E and [kappa]. Therefore, when [kappa] is growing faster than E, so is Y/L; when [kappa] is growing more slowly than E, so is Y/L; and when [kappa] grows at rate g, so does Y/L. Now we can put ln(Y/L) on the same graph with ln E and ln([kappa]), and ln(Y/L) will look qualitatively the same as ln([kappa]). We leave it to the reader to construct this graph.

Here is the important outcome of the exercise. When n increases, the growth rates of both Y/L and K/L immediately fall, the former by less than the latter. The faster growth of Y/L then raises Y/K, pushing both growth rates back toward g in the new steady state.

When g increases, the impact effect on the growth rate of K/L is zero, while the growth rate of Y/L will increase--but not to g'. Again, Y/L is growing faster than K/L raising Y/K and thus increasing both growth rates toward their new common steady state value, g'.

V. Summary

This paper develops a method to see more clearly the differing impacts of a change in population growth rate and a change in the rate of growth of labor efficiency in the Solow model. Both have the same impact on the efficiency-adjusted capital labor ratio, but they have very different impacts on capital and output per person. This difference is not so easy to see in the standard presentation. The approach provided here makes the difference apparent.

References

Jones, Charles. 2002. Introduction to Economic Growth. 2nd edition. London: Norton.

Mankiw, Gregory. 2009. Macroeconomics. 8th edition. Worth, New York: Worth.

Kevin Quinn, Associate Professor of Economics, Bowling Green State University, Bowling Green, Ohio 43043 Email: kquinn@bgsu.edu

John Hoag, Professor of Economics, Bowling Green State University, Bowling Green, Ohio 43403 Email: jhoag@bgsu.edu

The authors deeply thank two anonymous referees for who made substantial improvements.