2.The Model

2.1 Variables, parameters and basic relations

The variables and constants that are employed in the following will be defined as follows：

Let Y, Kc, Kw, K, I and L denote, respectively, the real level of production, the amount of capital owned by the capitalist class, the amount of capital owned by the working class, the total amount of capital, the level of investment, and the amount of labour.Let P be the nominal price of products, W be the money wage rate, and r be the rate of profit.All those are variables.

As for constants:n, sw, sc, α and βdenote respectively the growth rate of labour, the saving rate of the working class, the saving rate of the capitalist class, the adjustment speed of prices in the products market, and the adjustment speed of money wages in the labour market.

Variables are transformed for the sake of brevity into simpler forms as follows：

In addition, one has

It is assumed that the economy is equipped with a well-behaved production function Y=F（K, L）, that is homogeneous with degree 1：

This defines the function f（·）.One obtains that

We assume the production function well behaved：

K-P took up the steady state growth path with full employment.The assumption of full employment affects much to the setting of the model, so that some preliminary considerations are necessary before considering the price adjustment process that is related to investment and saving.Full employment considered by K-P indicates that the rate of investment exceeds the rate of labour growth along the steady state growth path, so that labour demand is always greater than labour supply.It is, so to speak, “restricted golden age”（Robinson,1962）.The following analysis is confined to full employment that is brought about by the bottleneck due to the natural growth rate.K-P assumed that in such a situation price adjustment is faster than changes in demand, and analyzed the adjustment mechanism of distribution.If the assumption of full employment is removed, price adjustment will not be relatively fast enough, so that one should introduce the quantity adjustment mechanism with unemployment into the model.Then, distribution is adjusted by a different mechanism. Darity's model assumes that the monetary authority takes appropriate measures so as to realize full employment.

In the vicinity of the steady state growth path with full employment, the condition that labour demand exceeds labour supply is equivalent to the condition that planned investment exceeds savings, as will be shown in the next section.Therefore, if full employment in the above meaning is assumed, then disequilibrium that ex ante investment is always greater than ex post savings arises.In this paper, in order to cope with this disequilibrium, it is assumed that part of investment exceeding savings will never be realized.In other words, the part that matches ex ante savings is realized.This assumption is highly neoclassical, but it does not influence the conclusion.This is because, irrespective of adjustment of short-run disequilibrium, the price adjustment mechanism determines the property of the model. Even if one assumes, for example, that a certain part of investment and/or savings is realized, the qualitative nature of the conclusion will not be altered.Remark that under this assumption, consumption of the working class is realized，so that it is consistent with full employment.

In the next place, take savings and capital accumulation.Labourers save sw of wage income wL and profit income rK w, whilst capitalists do s c of profit income rK c.From the assumption mentioned above, savings are all realized, so that capital accumulation is expressed by

The total of savings in the economy is given by sw（wL+rKw）+scKc.

If ex ante investment exceeds savings, the product price rises in proportion to the difference between investment and savings.Letαdenote the adjustment speed, and

With this price adjustment mechanism, changes in prices take place swiftly to changes in demand, especially investment, so that the rate of profit varies.Thus, this price adjustment mechanism plays a central role in the K-P distribution mechanism.This mechanism is called the Keynes-Wicksell adjustment by Stein（1966,1970）, that discussed the adjustment of disequilibrium between savings and investment in a monetary growth model.

In the third place, consider the adjustment of money wages in the labour market.Labour supply increases with a fixed rate（natural rate of growth n）, that is

Full employment is defined as a state with labour demand exceeding labour supply. Let Xd stand for labour demand.Since the production function is well-behaved, one has w=FL（K, Xd）.Let w*=FL（K, L）, and, in the vicinity of full employment, one obtains

Xd-L～w*-w～FL（K, L）-w

Therefore, the full employment assumption is represented by

Remark that the above（2.10）does not identify any basic parameters of the model, but it restricts the equilibrium point to confine the analysis within full employment situation.

The adjustment in the labour market is made in such a way that the money wage rate increases in proportion to the difference between labour demand and supply Xd-L; letβ denote the adjustment speed, and one obtains

In the above, the marginal productivity notions as in（2.10）and（2.11）are introduced with a well-behaved production function.Remark, however, that except the case β→∞the setting alone does not determine the rate of profit.

Ex post investment is restricted by savings,（2.6）and（2.7）, and the above（2.10）and（2.11）are all deeply coloured by neoclassical tone, but these are employed here just to make model analysis clear.Even with these assumptions and settings, the rate of profit and distribution change in accordance with wage rise and increases in product prices. Therefore, the rate of profit is simultaneously determined with the equilibrium in the product market, where the effective demand principle governs.Distribution is not determined by marginal productivities alone.

As for the investment function, we employ the Robinson type animal spirits investment function.The higher the rate of profit, the greater the motivation of investment of capitalists.In the“restricted golden age”of Robinson（1962）, it is suggested that investment is controlled by the rate of interest when the investment motivation is not fully satisfied.Since our argument here is confined to the real terms, it is assumed, as suggested by Robinson, that investment is held down if employment is restricted, that is, if FL＞w or r＞fk. Since fk is a decreasing function of k, fk will deviate more from r, as k increases.It is also assumed that the effect of animal spirits is diminishing.Moreover, it is assumed that for investment to take place there is a positive ensured profit rate Yo.This minimum value may be interpreted as the smallest risk premium.

From the above, one obtains the following investment function：

with

where r0＞0.

This investment function is the simplest one with the characteristic nature common to Keynesian models.The capitalist class associated with this investment function is an active entreprenur, and it is faithful to K-P.

2.2 Comments on Darity

The above model differs from Darity's most remarkably in the adjustment mechanism.Since the working of the adjust-ment mechanism affects the choice of investment functions, some comments will be made on Darity's model.Darity introduced the investment function of the form

I=g（fk, r, e）K

with gf k＜0, gr＜0 and ge＞0, where r is a rental price of capital, so that it can be interpreted as the rate of interest, and e denotes the level of employment.

Monetary authorities adjust the rate of interest to achieve full employment：

The level of employment is determined by the equilibrium in the product market

I（fk, r, e）=S（k, e, t, r）

where S stands for savings of the economy as a whole.It is assumed that Ie＜Se.From this assumption, in the vicinity of full employment e=1，

e≷1⇔I（fk, r, e=1）≷S（k, e=1, t, r）

where k, r and t are historically given.Therefore, the adjustment process is rewritten as

As pointed out previously, whether the dual theorem holds or not depends on the existence of stable equilibrium in the economy without capitalists, that is, in the economy in which the rate of savings of the economy does not change in accordance with changes in distribution.In Darity's model, it is assumed that if there is no capitalist class（t=0）, savings do not depend on r that reflects distribution：

This is his adjustment rule on r.The process makes investment equal to savings and brings about a stable equilibrium, because Ir＜0.注1Hence, in the vicinity of t=0, Darity's model possesses similar characteristics with the S-M neoclassical growth model. Therefore, it is impossible to analyse the relationship between the K-P theorem and neoclassical growth model.Here, the stability of equilibria is ensured by the setting of the investment function, and not by something else.

注1More rigorously, one has to check the stability of solutions of the system of simultaneous equations, i.e.the monetary adjustment and capital accumulation.Capital accumulation is expressed by k=k{g（k, r, e）-n}It is easily seen that the stability condition for t=0 is gr＜0.

2.3 Summary of the model

In this paper, we employ a simple investment function, as mentioned before.Our investment function will not automati-cally ensure the stability of equilibrium.r remains to be the rate of profit.

The framework of our analysis is now complete. Variables that are controlled by（2.3）—（2.13）are K, Kw, Kc, P, W and L.on the given initial values of Y, r, g（r）and FL.

By transforming variables in view of（2.1）—（2.2）, one obtains