## Solow Growth Model

The growth model of Professor Solow* is purported to show, among other things, that the razor's-edge growth path of the Domar model is primarily a result of the

* Robert M. Solow, "A Contribution to the Theory of Economic Growth/' Quarterly Journal of Economics, February, 1956, pp. 65-94.

particular production-function assumption adopted therein and that, under alternative circumstances, the need for delicate balancing may not arise.

### The Framework

In the Domar model, output is explicitly stated as a function of capital alone: k = pK (the productive capacity, or potential output, is a constant multiple of the stock of capital). The absence of a labor input in the production function carries the implication that labor is always combined with capital in a fixed proportion, so that it is feasible to consider explicitly only one of these factors of production. Solow, in contrast, seeks to analyze the case where capital and labor can be combined in varying proportions. Thus his production function appears in the form

where Q is output (net of depreciation), K is capital, and L is labor—all being used in the macro sense. It is assumed that fK and fL are positive (positive marginal products), and fKK and flL are negative (diminishing returns to each input). Furthermore, the production function / is taken to be linearly homogeneous (constant returns to scale). Consequently, it is possible to write

In view of the assumed signs of fK and fKK, the newly introduced <p function (which, be it noted, has only a single argument, A:) must be characterized by a positive first derivative and a negative second derivative. To verify this claim, we first recall from (12.49) that hence fK > 0 automatically means §'(k) > 0. Then, since

'«-¿♦■w-^fï-♦-<*>I [see(,2 48)i the assumption fKK < 0 leads directly to the result *t>"(k) < 0. Thus the <j> function —which, according to (12.46), gives the APPL for every capital-labor ratio—is one that increases with k at a decreasing rate.

Given that Q depends on K and L, it is necessary now to stipulate how the latter two variables themselves are determined. Solow's assumptions are:

(14.26) = sQ [constant proportion of Q is invested]

(14.27) —j = X (À > 0) [labor force grows exponentially]

The symbol 5 represents a (constant) marginal propensity to save, and À, a (constant) rate of growth of labor. Note the dynamic nature of these assumptions; they specify not how the levels of K and L are determined, but how their rates of change are.

Equations (14.25), (14.26), and (14.27) constitute a complete model. To solve this model, we shall first condense it into a single equation in one variable. To begin with, substitute (14.25) into (14.26) to get

Since k = K/L, and K = kL, however, we can obtain another expression for K by differentiating the latter identity:

When (14.29) is equated to (14.28) and the common factor L eliminated, the result emerges that

This equation—a differential equation in the variable k, with two parameters s and A—is the fundamental equation of the Solow growth model.

### A Qualitative-Graphic Analysis

Because (14.30) is stated in a general-function form, no specific quantitative solution is available. Nevertheless, we can analyze it qualitatively. To this end, we should plot a phase line, with k on the vertical axis and k on the horizontal.

Since (14.30) contains two terms on the right, however, let us first plot these as two separate curves. The Ak term, a linear function of k, will obviously show up in Fig. 14.5a as a straight line, with a zero vertical intercept and a slope equal to A. The s<(>(k) term, on the other hand, plots as a curve that increases at a decreasing rate, like </>(&), since s<j>(k) is merely a constant fraction of the <f>(k) curve. If we consider K to be an indispensable factor of production, we must start the s<f>(k) curve from the point of origin; this is because if K = 0 and thus k = 0, Q must also be zero, as will be </>(/c) and s<i>(&). The way the curve is actually drawn also reflects the implicit assumption that there exists a set of k values for which s\$(k) exceeds Ak, so that the two curves intersect at some positive value of A\ namely k.

Based upon these two curves, the value of k for each value of k can be measured by the vertical distance between the two curves. Plotting the values of k against k, as in Fig. 14.56, will then yield the phase line we need. Note that, since the two curves in diagram a intersect when the capital-labor ratio is k% the phase line in diagram b must cross the horizontal axis at k. This marks k as the (intertemporal) equilibrium capital-labor ratio.

Inasmuch as the phase line has a negative slope at A7, the equilibrium is readily identified as a stable one; given any (positive) initial value of k, the dynamic movement of the model must lead us convergently to the equilibrium level k. The significant point is that once this equilibrium is attained—and thus the capital-labor ratio is (by definition) unvarying over time—capital must thereafter grow apace with labor, at the identical rate A. This will imply, in turn.