1.Pasinett'i s theorem and the duality theorem

1.1 Pasinetti's theorem vs.the duality theorem

Pasinetti（1962）presented a proposition that asserts:On the full employment steady state growth path, the rate of profit does not depend on the rate of saving of the working class.This proposition was derived by the Kaldorian effective demand principle and the necessary conditions for the steady state growth path.The model employed for the proof was, however, not complete to determine production and investment at each time point.The effective demand principle works only in the equality of investment and saving，and the independence of the investment from the saving is neglected in his proo.f Thus, the effective demand principle is not crucial for Pasinetti's theorem.

Indeed, Pasinetti's theorem can be derived simply from the necessary condition for the steady state growth path.The amount of capital owned by the capitalist class Kc, brings about the profit income rKc, where r is the rate of profit.Since the capitalist class saves sc of income, capital increases by scrKc.Hence, their capital is accumulated by the rate of s cr.In the steady state growth path, all real variables grow in the same rate, so that s cr is equal to the rate of labour growth n.Then, one obtains .This is Pasinett'i s theorem.Since Pasinetti's theorem is derived from the Kaldorian distribution basis, in what follows we call it the Kaldor-Pasinetti model, in short K-P, or K-P model.

Pasinetti's theorem left a point to be discussed, however.Mead（1966）and Samuelson-Modigliani（1966）, abbreviated as S-M in the following, argued as follows.

Let K denote the total amount of capital, and the dot differentiation by the time.Di-f ferentiating with respect to time, one obtains Since proportions of real variables never change on the steady state growth path, the differentiation should be 0.is the rate of accumulation of total capital, and it should be equal to n for full employment.Hence, it follows that the necessary condition for the steady state growth path is reduced to

The set of solutions of this equation includes , in addition to s cr=n.Thus, there are 2 types of the steady state growth path:a path that justifies the Pasinetti Theorem and a path with The former path is denoted as P-path and the latter as D-path throughout this paper.S-M called their conclusion the duality theorem. In appearance, the duality theorem is more comprehensive.Is it really so?

1.2 Frameworks of the analyses

In order to compare Pasinetti's theorem with the duality theorem, one must consider their background.

To prove the duality theorem, S-M employed a neoclassical growth model, which incurs a problem.The neoclassical growth model employs the marginal productivity theory in determining the rate of profit, but this mechanism is completely different from Kaldor's theory of distribution（Kaldor,1956）that underlies Pasinetti's theorem.The question is whether the marginal productivity theory which replaces Kaldor's distribution theory makes the duality theorem valid, or Pasinetti's original model is not free from the possibility of the stable D-path.In other words, the problem is to what an extent the duality theorem depends on the framework of the neoclassical growth model.

Needless to say, the necessary condition（1.1）cannot avoid the solution Hence, the problem is reduced to whether the solution, D-path, is asymptotically stable or not.S-M have shown that D-path is asymptotically stable under some values of parameters and a class of production functions. Hence, our task is to check the stability in Pasinetti's original framework.

One of the common features of Keynesian models is the independence of investment functions.The effective demand principle asserts that the investment is determined independent of the determination of the savings.Therefore, in order to examine how much the duality theorem is intrinsic to the neoclassical growth model, we investigate the asymptotic stability of D-path within a growth model equipped with an independent investment function.This is the main objective of this paper.

1.3 Stability of the growth path and the adj ustment process

In the Kaldorian theory of distribution, distribution changes so as to raise savings which is equivalent to the investment.This variable rate of savings characterizes Kaldor's model, that dealt with the instability in Harrod's model.In order for savings to change in accordance with changes in distribution, there should exist multiple classes of agents who have different propensities to save:the model needs the capitalist.

If there is no class of agents with higher saving rate, i.e., then Kaldor's distribution mechanism can never work, so that the rate of savings will be fixed.Hence, to keep the stability D-path needs a distribution mechanism which replaces the Kaldor's mechanism of distribution and adjusts the savings rate.Without such a mechanism the duality theorem does not hold in Kaldor's world.Conversely, if an appropriate mechanism of distribution is set up and a short run equilibrium is guaranteed to exist, then there is a possibility that the D-path is asymptotically stable for some combination of parameters and a class of the production function.

The S-M model is a neoclassical model with variable capital coefficient and Say's law which ensure the long-run stability.Irrespective of the existence of capitalists, in other words, even if the rate of savings is not variable, the economy is flexible enough to secure the stability of D-path.

Darity（1981）has introduced the dynamic distribution model with an independent investment function.The investment function had three variables:the marginal productivity of capital, the rate of interest, and the level of employmen.t Although he analyzed the stability of the steady state growth path, his model is not appropriate to examine the duality because he introduced the monetary authorities that function as an adjusting mechanism.The stability is ensured by the assumption even if no capitalists exist. His model will be examined later in this paper.

As is shown in the following analysis, the stability of the growth path, the conclusion, depends on the setting of the short-run economic framework.This is inevitable because what matters here is the stability of D-path in the economy which embodies the Kaldorian distribution theory.

The plan of the paper is as follows:Section 2 will be dedicated to the setting of our model.Section 3 will focus on the equilibrium points of the model and their asymptotic stability, and presents three propositions with their proof.