Adjustment Pocesses:
Solow vs. Harrod

a dark satanic mill


The argument of the Solow-Swan growth model, as we have presented it, seems straightforward enough. But it gives the impression that the reason k adjusts to steady-state k* comes out merely from the "technical" aspects of the model -- from the properties of a constant returns to scale production function and national income accounts identities. If there is an "economic theory" in there, it seems to be well-disguised.

But there is a lot of economics in it -- we just have to look at the details. From the outset, the first thing we need to fix in our minds is that the production function ¦ (k) is not merely a "technical" thing; it contains within it an intricate and completely Neoclassical economic theory.

To see the underlying argument, it is useful to compare the Solow-Swan model with the Harrod-Domar model. Let us recall that the Harrod-Domar problem exhibited two knife-edges: the balance between the actual and warranted rates of growth ("macroeconomic stability") and the balance between warranted and natural rates of growth ("employment stability"). The Solow-Swan model does not address macroeconomic stability but only employment stability.

The question of macroeconomic stability is ignored in Solow-Swan by the assumption that planned investment equals planned savings at all times. No consideration whatsoever is paid to the underlying "macroeconomic" adjustment process that makes this true. But the "knife-edge" Harrod and Domar focused on was precisely that one. As Frank Hahn notes:

"It will be noted straightaway that [Solow's] argument has no bearing on Harrod's knife-edge claim. Harrod had not proposed that warranted paths diverge from the steady state but that actual paths did. The latter are neither characterized by a continual equality of ex ante investment and savings nor by continual equilibrium in the market for labour. Thus although Solow thought he was controverting the knife-edge argument he had only succeeded in establishing the convergence of warranted paths to the steady-state." (F.H. Hahn, 1987)

The intricacies of macroeconomic adjustment, explicit theories of interest and expectations, which were the main concerns of Harrod and Domar, are completely missing in Solow-Swan. And the cost of ignoring them is high. As Frank Hahn (1960) has himself demonstrated, when even the slightest attention is paid to the underlying macroeconomic adjustment of a Solow-Swan model in a proper manner, the stability of its steady-state can be cast into serious doubt.

Still, let us turn to the second Harrodian knife-edge, "employment instability". We can translate the Solow-Swan model into Harrod-Domar terms as follows. Recall that 1/v is the slope of a ray from the origin to the intensive production function. There is a different 1/v ray for every k. Let v* be the capital-output ratio associated with the steady-state k*. Now, at steady state, we know that s¦ (k*) = nk*, or:

¦ (k*)/k* = n/s

But notice that ¦ (k*)/k* = 1/v*, thus at k*:

s/v* = n

which is the Harrod-Domar equilibrium for the second knife-edge. Now, let v1 be the capital-output ratio associated with the disequilibrium capital-labor ratio k1 (in Figure 1). Obviously, ¦ (k1)/k1 = 1/v1. As n and s are constant and 1/v1 > 1/v* (slope of the ray associated with k1 is steeper than that associated with k*), then we know 1/v1 > n/s, or simply:

s/v1 > n.

which is a Harrod-Domar employment disequilibrium. However, we know k* is stable, so k1 approaches k*, implying that v1 falls to v*, so that the ratio s/v1 declines to meet n. Thus, it is through adjustments in the capital-output ratio, v, that the Solow-Swan model "solves" Harrod-Domar's second instability problem.

Fig. 1 - Adjustment to Steady-State

What is the exact economic mechanism that permits this? Nothing less than assuming that the factor markets clear at all times and that the marginal productivity theory of distribution holds. Underlying the Solowian production function is a detailed factor market adjustment process. To see this in action, examine Figure 2. For a constant returns to scale intensive production function, at any k, the slope of a tangent ray is ¦ k = Fk, the marginal product of capital while the intercept of the tangent ray on the vertical axis is merely y - ¦ kk = FL, the marginal product of labor.

Let r denote the real rate of return on capital and let w denote the real wage. By the Neoclassical theory of distribution, the marginal productivity of a factor will constitute the demand for that factor. In equilibrium, factor demand equals factor supply, and thus at the market-clearing factor prices, Fk = r and FL = w. Thus, in Figure 2, we denote the slope of the tangent ray as r and the intercept on the vertical axis as w. One more thing can be deciphered from the diagram: namely, the point where the tangent curve intersects the horizontal axis in the left quadrant yields the factor price ratio, w = w/r.

Fig. 2 - Factor Price Adjustment

So now let us turn to the Solow-Swan adjustment process. Suppose we begin at k1, as shown in Figure 2. The underlying market-clearing factor prices are r1 and w1, or in ratio form, w 1. However, we know that k1 is not a steady state, so, eventually, k1 will rise to k2. In this movement, the supply of capital increases more than the supply of labor, implying that if factor markets are to continue to clear, then the rate of profit must fall (r1 declines to r2) and the wage must rise (w1 increases to w2). In the movement from k1 to k2, the factor-price ratio rises from w 1 to w 2. This is exactly what we see when comparing the intercepts of the tangent lines in Figure 2.

To see the factor market adjustment process more clearly, it is useful to translate Figure 2 into isoquant form, which we do in Figure 3. Beginning with the given amounts of capital, K1 and L1, we have factor market clearing, as the profit-maximizing firm will produce Y1 and the wage-profit ratio, w 1 is precisely what is needed for this. Thus, point 1 = (L1, K1) in Figure 3 corresponds to position k1 in Figure 2.

Now, suppose capital supply increases to K2 and labor supply to L2. Obviously, as we see in Figure 3, capital increases a lot more than labor, so the new factor supply position will be at point 2 = (L2, K2), and thus the capital-labor ratio is higher (on ray k2). Now, suppose market prices do not adjust. In other words, suppose that the wage-profit ratio remains w1. In Figure 3, this is represented by the isocost line w1¢ , which has the same slope as the isocost curve w1, and thus represents the "old" factor prices, but passes through the new factor supply point 2. Notice then, that at point 2, with the old factor prices ruling, the marginal rate of technical substitution exceeds the factor-price ratio, ¦ L/¦ K > w/r.

Fig. 3 - Factor Market Adjustment

At these old factor prices, the profit-maximizing firms in our economy will attempt to produce output Y2¢. As shown in Figure 3, their demand for capital will consequently be K2¢ and their demand for labor is L2¢ . So, with old factor prices w1¢, new factor supplies are at 2 while new factor demands are at 2¢ . Notice immediately that K2¢¢ < K2 and L2¢ > L2, so there is excess supply of capital and excess demand for labor. We have factor market disequilibrium.

Consequently, assuming in a typical Neoclassical fashion that factor markets adjust automatically, then wages must rise and profits must fall, so the factor price ratio must increase from w1¢ to w2. Notice that at these new factor prices, w2, firms will produce Y2 and demand exactly what is supplied, K2 and L2. We have restored factor market equilibrium.

We can also depict this adjustment process in the intensive production function diagram in Figure 2. Specifically, if k1 increases to k2 but factor prices remain constant at w 1, then firms will be attempting to produce y2¢ in Figure 2, which lies above the intensive production function. At this (y2¢ , k2) position, we see that w/r = w 1 and ¦ L/¦ K = w2, thus ¦ L/¦ K > w/r again. This is equivalent to the factor market disequilibrium we depicted in Figure 3. If we permit the wage profit ratio to adjust freely so that w/r = w2, then ¦ L/¦ K = w/r is restored, and firms are now producing y2. This is equilibrium once again.

In sum, we see that a Neoclassical factor price adjustment process is exactly what is captured by the "curvature" of the intensive production function. Thus, when we posited the straight-line production function in our depiction of the modern Harrod-Domar model, the critical feature is not so much that we were assuming a single technology, but rather that we are not assuming that there was an underlying Neoclassical factor market clearing process.

This is the important point. Harrod-Domar did not believe that factor prices were driven by factor-market clearing, thus they did not incorporate a Solow-Swan type of CRS production function with flexible technology. Specifically, as Roy Harrod (1948, 1953, 1973) explains, following the Keynesian schema, the rate of interest, r, is governed by monetary phenomena; the real wage, well, by an assortment of other things, e.g. unions, etc. So, w = w/r is not determined by factor market clearing as Neoclassical theory (and Solow-Swan) assume. As Harrod did not know how (or why) the monetary authorities, labor unions, firms, etc., would adjust r and w so that the economy could be guided to the steady-state capital-labor ratio, he therefore assumed that the capital-output ratio v was a constant. [Note: in a revision of his model, Harrod (1960) does attempt to account for the influence of changing interest, importing, incidentally, the supply of saving from Ramsey (1928).]

Seen in this light, it is difficult to accept the common refrain that Solow-Swan "generalized" the Harrod-Domar model just because they allowed for flexible technology whereas Harrod-Domar did not. This is certainly what Solow insinuated, arguing that the "bulk of this paper is devoted to a model of long-run growth which accepts all the Harrod-Domar assumptions except that of fixed proportions." (Solow, 1956: p.66). But, as this discussion has hopefully made clear, it is not technology that is critically different. It is the adjustment process.

From the perspective of Ockham's razor, it may very well be that the Harrod-Domar model is "more general" than Solow-Swan. Harrod-Domar make fewer restrictive assumptions. Firstly, they do not assume an instantaneously stable macroeconomic equilibrium (as Solow-Swan do). Secondly, they do not assume any particular factor price adjustment mechanism (as Solow-Swan do).

The question of generality, then, can only be resolved by an empirical race between the two models. If Harrod-Domar can explain the same things as Solow-Swan, then, by Ockham's razor, Harrod-Domar is clearly "superior" because it does so with fewer assumptions. If Solow-Swan explains the data better than Harrod-Domar, then we would lean towards declaring Solow-Swan the "better" model.

However, as we shall see, it turns out that Solow-Swan performs poorly when confronted with empirical evidence. Substantial modifications have to be added (particularly regarding technical progress) to make it comply with the data. Interestingly, the kind of modifications to the Solow-Swan growth model that "endogenous growth theory" has proposed in recent years turn out to generate a reduced-form dynamical system that is virtually identical to the Harrod-Domar model. Taking an analogy from astronomy, economists have had to add epicycles upon epicycles upon epicycles to the Solowian growth model in order to have it explain what could be more simply explained by the Harrod-Domar model. The conclusion imposes itself.

Finally, we should remind ourselves why Solow-Swan is a "Neoclassical" and not a "Keynesian" growth model. From the outset, we have a very Neoclassical factor market equilibrium adjustment process. But, perhaps more strikingly, the macroeconomics are very different. A complete Keynesian growth model would have investment as a function of financial conditions and savings derived from investment via the multiplier. In the Solow-Swan model, not only are all Keynesian "financial" factors omitted, but the direction of causality between savings and investment is reversed. This is equivalent to re-imposing Say's Law. Thus, the Solow-Swan growth model is "Neoclassical" in every respect, and not an extension of "Keynesian" macroeconomics, as has occasionally been advertised. For attempts at introducing more "Keynesian" features into a growth model, see our review of Cambridge growth theory and Keynes-Wicksell monetary growth models.

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