The issues, which Uzawa (1961, 1963) examined, emerged as an extension of the seminal Solow-Swan neoclassical growth model (Solow, 1956; Swan, 1956). The neoclassical model gave a set of equilibrium growth paths for the economy. The paths are determined by the value of a single independent variable (such a variable in the model is the capital-labor ratio k=K/L – the ratio of capital K to labor L). Immediately the question arose of finding the optimal path to maximize consumption. Soon, the condition for such an optimum was obtained for the simple model, r=g (the equality of the profit rate and the growth rate of aggregate output and income). It is often called as the Phelps (1961) golden rule. However, it seems to me, the meaning of this rule was not clearly understood. After all, it declares the equality of profit and investment necessary for balanced growth of the economy, rK=gK. The last equality determines the distribution of total income between social classes: the capitalists completely invested their profits, and, consequently, the workers consumed their wages. This pattern of income distribution has been used as an assumption for a long time; Uzawa called it the “classical hypothesis”.

Let us follow the course of Uzawa’s thought. He considers a two-sector model that takes into account two industrial sectors producing investment and consumer goods. It is important that he uses the classical hypothesis as an assumption, and thereby declares the equality r=g. And then Uzawa considers different equilibrium paths, depending on the only independent variable, for which he uses the ratio of wages to the profit rate, ω=w/r. Sraffa (1960) meticulously showed that if production technologies (or production functions) are given, then there is only one independent variable that determines the distribution of total income in the equilibrium state. As such a variable, one can use either the capital-labor ratio k (as in the Solow-Swan model), or ω=w/r (as in Uzawa), or simply the profit rate r (as in Yashin, 2017).

main results

When the independent variable is chosen, all other quantities are uniquely determined by its value. It is important that by changing of the selected independent variable, we inevitably shift the distribution of total income.

However, Uzawa already accepted r=g as the initial classical hypothesis. The output growth rate in the model is equal to the population growth rate, g=n, that is, g is exogenously given. Thus, Uzawa initially had fixed his independent variable, because ω and r are uniquely linked. Then his further attempts to change the “independent variable” ω (and hence r) contradict the original classical hypothesis r=g. Thus, unfortunately, Uzava’s study, as well as mine (Yashin, 2017), has an internal contradiction and leads to a dead end.

conclusion

In conclusion, we focus on the linear relationship between r and g, assumed by the classical hypothesis, r=g. Such a linear relationship is specified not only by this hypothesis and by the golden rule, but also by the Ramsey formula, which is also derived to maximize consumption. The Kaldor-Pasinetti equation, r=g/s_c is also the proportionality between r and g. This equation is obtained for balanced growth in a model where workers are allowed to receive capital income and capitalists consume a part of their income (s_c is the propensity to save of the latter). As we noted above, the relationship between r and g determines the distribution of total income. Thus, the three similar formulas mentioned in this paragraph, describe the optimal income distributions for the three models. In all three cases, a linear relationship between r and g is implied, which is equivalent to the proportionality between profit and investment. All three formulas were obtained independently; the slight differences between them are due to differences between the three models.

References

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