Paul Samuelson and the dual Pasinetti theory.

Ramrattan, Lall ; Szenberg, Michael

Introduction

In this paper, we examine Paul Samuelson's contribution to a very specific brand of post-Keynesian economics. One major aspect of post-Keynesian economics is to delineate mechanisms whereby monetary and fiscal policies affect the economy. In particular, post-Keynesians are concerned with showing how changes in thriftiness, propensity to consume, investment, government expenditures, and taxes channel to changes in GNP, money value of output, prices, and production even if the money supply remains constant.

Samuelson follows his dictum that states, "Post Keynes, ergo different from neoclassical macroeconomics." (1, 2) He classifies himself as a "post-Keynesian" (3) with Modigliani and others, and sometimes as just Keynesians, writing that "The Keynesians I admire--like Franco Modigliani, James Tobin, Robert Solow--are not the same in this decade as they were in the last, and the next decade they will be something else again." (4)

Others have taken a different view. Geoffrey Harcourt thinks that Nicholas Kaldor, Joan Robinson, Pierro Sraffa, Michael Kalecki, and Luigi Pasinetti are the "really seminal people among the post-Keynesians." (5) Edward Nell made the distinction that sets post-Keynesians apart from Modigliani and Samuelson: "Rather than adapt Keynes to neo-Classical microfoundations, post-Keynesians have sought to defend and develop Keynesian thinking." (6) According to Davidson, this line of reasoning would separate Samuelson from the post-Keynesian school, because "Samuelson (1947) asserts that the foundation of economics requires several classical axioms that were rejected in Keynes' General Theory." (7) Davidson also separates Pierro Sraffa from the post-Keynesian school because "Sraffians, reject Keynes's notion of the importance of uncertainty (i.e., nonergodicity) in determining the effective demand equilibrium solution." (8)

The approach we take in this chapter is to elaborate the thoughts of Samuelson on the dual Pasinetti theorem and raise them to a more general level without the weight of the post-Keynesian distinction. The generalization focuses on whether the propensity to save by both workers and capitalists is significant for the determination of the profits in income and on capital.

Pasinetti originally proposed his theorem, or paradox, in relation to Kaldor's theory on the distribution of income between capitalists and workers. In response, Samuelson and Modigliani pointed out that the result is more general, applying to any golden-age growth theory. While Pasinetti thought that the dual theory is an apology for neoclassical economics, Samuelson and Modigliani perceived the subject matter as a general theorem. They wrote that, "As is the case for duality relations, there is a complete symmetry between the Primal and Dual equilibria. Neither is more general than the other. This symmetry Dr. Pasinetti once more denies. He continues to regard his golden-age equilibrium as a more general one, being in some special sense relevant independently of marginal productivity assumption." (9)

Pasinetti's Theorem and Its Background

The Pasinetti theorem or paradox "gives a neat and modern content to the deep-rooted old Classical idea of a certain connection between distribution of income and capital accumulation." (10) The paradox is important because it is "regarded as a system of necessary relations to achieve full employment." (11) Its methodology is based on classical rather than on neoclassical economics. Its practitioners claim that it is more in line with pure Keynesian thought than with those thoughts that are layered with a classical overview. The theorem that Pasinetti advanced and proved is as follows:

 The equilibrium rate of profit is determined by
 the natural rate of growth divided by the capitalists'
 propensity to save; independently of
 anything else in the model. (12)

Pasinetti's theorem can be traced to concerns with the distribution of income between wages and profits. Once capital and labor produce goods and services jointly, the problem of how each is rewarded for their efforts must be solved. The most popular solution is to treat profits as a residual, meaning that the capitalists take what is left over, if any, after rewarding labor. According to C. Ferguson, "the first of the modern 'alternative' theories of distribution is Kalecki's." (13) Here, the word alternative is used in the sense of a theory based on demand and grounded in Keynesian thought. Ignoring foreign trade, government expenditures, taxes, and workers' savings, Kalecki wrote that "Gross profits = Gross investment + Capitalists' consumption." (14) To show how this equation is derived, Kalecki borrowed from the Keynesian demand side concepts, which allows National income = Gross profit (P) + Wages (W). National income is also equal to Total consumption (C) + Gross investment (I). (15) Total consumption, = Capitalists' Consumption ([C.sub.c]) + Workers' Consumption ([C.sub.w]). Kalecki assumed that workers consume all their income or wages, [C.sub.w] = W. Collecting the terms, we have: P + W = I + [C.sub.c] + [C.sub.w] = I + [C.sub.c] + W. Solving for P yields the desired expression for gross profits that Kalecki noted.

We turn now to how Kalecki determined the value of output, and the share of output that goes to labor and capital. To value output, he used a price mark-up on prime costs, k > 1, which he applied to wage, W, and material costs, M. Gross profits then yields: P = k(W + M) - (W + M) = (k - 1)(W + M) Similarly, National Income = P + W = (k - 1)(W + M) + W. We can now express workers' share of national income as: w = W/[(k - 1)W + M) + W]. Kalecki (16) further refined this equation by dividing the numerator and the denominator of the expression by W, and substituting j = M/W, yielding a refined expression for workers' share of output:

w = 1 / 1 + (k - 1)(j + 1).

One interpretation of the expression for the workers' share of output is to consider that k, the mark-up, is a measure of the degree of monopoly power, and that j, the ratio between the cost of materials and labor is the terms of trade between them. As the degree of monopoly power and the terms of trade increase, the workers' share of output will fall.

To determine profits Kalecki postulated that "capitalists may decide to consume and to invest more in a given period than in the preceding one, but they cannot decide to earn more." Kalecki stated that his concept of the determination of profits can be viewed from the works of Karl Marx's three departments as well. In Department I, investment goods are produced; in Department II, consumption goods for capitalists or luxury goods are produced, and in Department III, consumption goods for workers or wage goods are produced. (17) After the capitalists have produced consumption goods in Department III, they will sell an amount to the workers in Departments I and II, and keep the remainder as profits. In Departments I and II, the capitalists have been producing investment and consumption goods for themselves. In modern textbooks, the determination of profits from all three departments is the sum of their profits on both constant and variable capital. (18)

In an early version, Kalecki used the expression: P = A + C to indicate that profits equal investments plus capitalists' consumption. (19) He then disaggregated the consumption of capitalists into two components. One component is made up of a fraction of gross profits, [lambda]P. The other component is a stable component, [B.sub.0]. Profits can then be expressed as: P = A + [B.sub.0] + [lambda]P = (A + [B.sub.0)/(1 - [lambda]). John Eatwell has remarked that this represents "a theory of aggregate profits, determined by the volume of autonomous investment and the propensity to save out of profits (the propensity to save out of wages being set equal to zero)." (20) According to Robert Solow, "... we have here already the nucleus of the Widow's Cruse model of profits, which Kaldor and Joan Robinson adapt for their neo-Keynesian macroeconomic theory of distribution." (21) The Widow's Cruse model plays on the earnings and spending terms: capitalists earn what they spend, and workers do the reverse, spend what they earn.

For our elaboration of the Modigliani dual Pasinetti theorems, it is important to note the source of Kalecki's thinking. According to George Feiwel, Kalecki's theory is based on the degree of monopoly and is independent of the neo-classical tradition. (22) Kalecki emphasized that the ratio of the wage bill to profit, k, is a constant. By applying this constant to profits and adding the wage bill, we obtain Kalecki's theory of income determination: Y = W + P = (1 + k)P = (1 + k)([B.sub.0] + A)/(1 - [lambda]). (23) Since Pasinetti has built his theorem of distribution on Kaldor's model, we will next review the latter's views.

The literature attributes to Kaldor, a "Classical Savings Function," and to Keynes, a psychological law of savings. In a letter to Keynes, Kaldor explained that the classical point of view implies a stable equilibrium in the sense that "at a given level of money wages there is only one level of employment which secures equilibrium." (24)

Kaldor made the point that, "It really is the assumption that savings vary with real income, which constitutes the main difference between the classical economics and the Keynesians." (25) He emphasized the rate of change in savings with respect to the interest rate, dS/dr, in equilibrium. The classics held a "partial differential quotient" view on how changes in the rate of interest affect savings. Holding income constant, the classics argued that a fall in interest rate would reduce savings, and vice versa, i.e., dS/dr > 0. The Keynesians hold a "general quotient" view, to the effect that "a fall in the interest rate will increase savings, if the effect on investment and income is taken into account." (26) Keynes argued that "a reduction in the rate of interest, whether or not it increases the propensity to consume out of a given income, increases the amount of savings owing to its effect on the amount of income through the stimulus of investment. This ceases to be true when a state of full employment is reached, when dS/dr may become zero. But a state of affairs in which dS/dr is positive would, on my argument, be extremely unusual and paradoxical." (27)

In order to explain how Pasinetti built his model on Kaldor's ideas, we tabulate their equations and assumptions side-by-side in Table 1. Kaldor introduced a new take on the theory of distribution. (28) As indicated by equation 1.1, Kaldor viewed income as distributed between wages and profits. He explained that "the wage-category comprises not only manual labour but salaries as well, and profits income of property owners generally, and not only of entrepreneurs; the important difference between them being the marginal propensities to consume (or save), wage-earners' marginal savings being small in relation to those of capitalists." (29) Equations 1.2 to 1.4 show the steps involved in deriving the profit-income ratio.

[TABLE 1 OMITTED]

Equation 1.5 in the Kaldor column shows the profit-income ratio as a linear function of the investment-income ratio. The corresponding equation 1.5a in the Pasinetti column is different. Pasinetti observed a slip in the specification of Kaldor's model, which he amended, as described in the next section.

Pasinetti's "Correction" of the Kaldor's Model

Starting with the national income identity in the first row of Table 1.1, Pasinetti split the profit term of equation 1.1a into two components--profits to workers, and profits to capitalists. The split is shown as two additional equations, 1. lb and 1. lc, in the Pasinetti column. The split is necessary because Pasinetti argued that Kaldor's model didn't consider that workers also own capital. He noted that "... when any individual saves a part of his income, he must also be allowed to own it, otherwise he would not save at all. This means that the stock of capital which exists in the system is owned by those people (capitalists or workers) who in the past made the corresponding savings." (30) The entries in the Pasinetti column of the table from the first row down to the profit-income ratio equation show step-by-step how the corrections are implemented to the corresponding equations in the Kaldor's column.

Thus, with the correction made, Pasinetti ended up with equations 1.5a and 1.6a in Table 1, which are different from Kaldor's equation. Equation 1.5a reflects a distribution between capitalists and workers. To reflect on the distribution between wages and profits, we need to add the share of workers' profits to it. Similarly, we also need to do the same for equation 1.6a in order to get a relationship for total profits to capital and not just the profits of the capitalists to capital. Thus, equations 1.5a and 1.6a can be modified to account for the amount of capital the workers own, [K.sub.w] and the return they get from loaning it out, r. When the adjustment is made, we still need "a theory of the rate of interest," (31) and to equate it with the rate of profits. The result is that the profit-capital and profit-income ratios can be expressed without the workers propensities to save. The startling implication of this is that "workers' propensity to save, though influencing the distribution of income between capitalists and workers ... does not influence the distribution of income between profits and wages ... Nor does it have any influence whatsoever on the rate of profit-equation." (32)

In the absence of growth the model took the investment-income ratio as fixed. In a growth environment, as discussed below, the investment-income ratio can be variable.

Growth Elements of Pasinetti's Theorem

The growth theory part of the Pasinetti theorem starts with the Harrod-Domar model that makes savings a driver of growth in output and income. The explanation we give is more in line with Domar's derivation. From the Keynesian multiplier theory we have Y = I/s, or I = sY, where Y is national output or income, I is investment, and s is the marginal propensity to save. From the naive accelerator theory we have I = v[DELTA]Y, where v is the capital-output ratio. Equating the two equations for investment provides a growth path for income that is driven mainly by savings, as the capital-output ratio is constant, i.e., Y = A[e.sup.(s/v)t], where t is time, and s/v is the warranted-growth rate.

Before we continue with the description of the model laid out in Table 1.1, it is worth mentioning a significant methodological difference between the views of growth theories that underlie the growth model developed here. If we take the limit of the growth path equation of the Harrod-Domar model above, as time approaches infinity, the system will diverge from its equilibrium growth path. Such a behavior has been called "knifed-edge" instability in the sense that as the economy is displaced from its equilibrium growth path, it will not return to that path. This divergence has been troubling to neoclassical economists who emphasized a "Steady State" growth rate. To put this matter in perspective, we compare the Harrod-Domar growth path with that of the classical and neoclassical economists' growth paths. The classical economists' growth model tended to a "Stationary State" in which net saving and net investment are zero. The neoclassical model tended to "The Steady State in which its constant growth rate, admitted positive saving ... quantities of inputs and outputs did not remain unchanged over time, their ratios did. In ratio terms, the Steady State was still quite stationary." (33) The neoclassic economists were able to prevent unstable (knife-edge) problems by comparing s/v in their "Steady State" model with the growth rate of population, assuming depreciation. They were able to attain a stable equilibrium growth path where all ratios grow at the same rate. Kaldor and Pasinetti, however, sided with an unstable system.

Equation 1.6 indicates how the Harrod-Domar warranted growth rate, s/v, can be derived for the Kaldor case. (34) But the warranted growth rate, [G.sup.w], defined as "a rate at which producers will be content with what they are doing" (35) may not equal the natural growth rate, [G.sup.n]. The distinction between the two growth rates is that the natural growth rate is the sum of the population, n, and output per labor growth rates, [lambda]. Apparently, Domar did not take the latter into account. It was an addition of Harrod, according to Pasinetti. (36)

Following equation 1.7, the warranted growth rate is attained. An adjustment process exists whereby the natural and warranted rates tend to equality. Using Harrod-Domar notations, if [G.sup.w] < [G.sup.n], then investment will take place, and by equation 1.5 profits will increase to a point that will restore equality between the warranted and the natural rate of growth. The argument works in the opposite direction as well, if [G.sup.w] > [G.sup.n]. In Samuelson's notation, Kaldor viewed savings, investments, and the equilibrium as adjusting through two processes: S(Y, P) = I(t), and dP/dt = K(Y - [Y.sup.*]), where P is the profit to income ratio, Y is income and employment, and asterisks denote full employment. (37)

Equation 1.7 is one way to derive the Harrod-Domar growth condition from I/Y. By formulating it as I/Y = (I/K)(K/Y), we bring out the accumulation of the capital concept and the accelerator concept, or what Kaldor called "the rate of growth of output capacity (G), and the capital/output ratio, v," (38) respectively. Equations 1.8 to 1.10 show how the profit in the income ratio relates to the natural growth rate and the capitalists' propensity to save. From equation 1.10, we can have P/K = r, the rate of profit. Generally, the rate of interest is used when the notion of capital is a fund in the financial sense, and the rate of profit assumes a technical production relationship. (39) Now, we obtain the bottom line relationship that the rate of profit is dependent on the capitalists' propensity to save, i.e., equation 1.11, where r = G/s. Equation l.la, through equation 1.1 la under the Pasinetti column of Table 1 show how the same conclusion that the rate of profit is dependent on capitalists' saving is achieved for Pasinetti's correction of Kaldor's model. The next section, demonstrates how Modigliani and Samuelson developed a dual theory reaching this conclusion.

The Dual Pasinetti Theorem--Samuelson and Modigliani's Version

Samuelson and Modigliani proposed a dual to Pasinetti's original theorem. (40) While Pasinetti's theorem emphasizes the capitalists' propensity to save, the dual theorem emphasizes the workers' propensity to save. The primal theorem relates to the profit-capital ratio, while the dual theorem relates to the output-capital ratio, also referred to as the inverse of the naive accelerator, or the average product of capital. This point needs some emphasis. While the Harrod-Domar growth model perceived the capital-output ratio, v, as a constant in the warranted growth expression, s/v, the dual theory considers it as a variable. For Kaldor and Pasinetti, the focus on capital-output ratio was important only for a first state of growth when the labor supply is not fully absorbed. But when capital has sufficiently accumulated to absorb the existing labor force, they gave attention to the saving propensity, and by assuming that workers do not save, then the rate of profit becomes dependent only on the capitalists' propensity to save. Kaldor's and Pasinetti's rate of profit results rest on the condition that the investment to full output ratio, I/Y, must lie between the two savings propensities: [S.sub.w] < I/Y < [S.sub.c]. One of the purposes of the dual Pasinetti theorem is to describe what occurs outside of this range, which results in the conclusion that the full employment output to capital ratio will depend on the workers' saving propensity.

Samuelson and Modigliani suggested three purposes in their original paper: to establish the dual theorem, to disassociate it from Kaldor's theory of distribution, and to establish an "asymptotic" stability relation for their model, which is Pasinetti's "instantaneous" stability consideration. (41) The following analysis, which follows Henry Wan, expounds the dual theory. (42)

Samuelson and Modigliani started with the neoclassical approach in which the shares of the factors of production are equal to their marginal products. The proof that the factors are rewarded their marginal product has given rise to the controversy over the theory of capital. Briefly, if we start with Y = F(K, L), the question arises as to whether we measure capital, K, and labor, L, in physical or value terms. If, from a physical point of view, we can measure labor in hours or days of work, then the reward to labor can be measured in wage per hour, or wage per day. The idea of measuring capital in physical terms, however, has met with great opposition. It appears that, in physical terms, we can measure capital in terms of the number of machines or we can make an index to represent the physical quantity of capital. When capital is measured in physical terms, however, we have a problem making the rate of profit (reward to capital) equal to (or commensurate with) its value (quantity of capital time price), because prices depend on the rate of profits. The Wicksell effect refers to the situation where the marginal product of capital is not equal to the rate of profit, or the rate of interest. The reason is that as capital changes, its price changes as well. In the case of a single good economy, the price of capital will be the same as the price of the single good; but with multiple goods, this equality is not sustained. The controversy developed over how to aggregate heterogeneous capital goods in order to form the production function. While the Pasinetti's camp holds that this question must be answered definitively, the dual camp holds that progressive research can be carried out with an aggregate production function.

To establish the dual theory, we will express it on a per capita basis. Given the neoclassical production function, Y = F(K, L), the per capita form is attained by dividing all variables by labor to obtain: y = Y/L = (F(K/L, L/L) = F(k, 1) = f(k). With these notations, the critical dual dependent variable, the average product of capital, can be represented as Y/K =f(k)/k = A(k). The return to capital given capital per head, can be written as or(k) = [alpha]k/f(k).

Following standard textbook derivation, we can now obtain the rate of growth of capital per head as: k = sf(k) - nk. The solution to this differential equation is a path that leads to an equilibrium level of capital per head ratio, [k.sup.[infinity]]. The saving function sf(k) = nk yields the expression y = f(k) = (n/s)k. It explains that saving equals investment at a point necessary to equip the additional workers with capital that would maintain the capital per head ratio. It also invites the conclusion that the marginal product of capital is y' = f'(k) = r = n/s. We bifurcate then the solutions k = sf(k) - nk into classes representing capitalists, and workers.

To demonstrate the dual theorem, we make the following additional derivations.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12)

Equation 1.12 tells us that the growth rate of capital that the capitalists own, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] minus the growth rate of labor, n, is the growth rate of capital per head. To get a similar expression for the workers, we subtract from total income, Lf(k) the capitalists' income, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We then apply the workers' saving propensity to get the rate at which they save and calculate their growth rate by dividing by the capital they own. Finally, we express the growth rate of the capital that the workers own on a per capita basis by subtracting the growth rate of labor.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.13)

Equations 1.12, and 1.13 lead into the dual Pasinetti theorem. By specifying the equilibrium growth conditions the two equations will be equal to zero; i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.13a)

Pasinetti's Case

From 1.12a, we get the equilibrium growth rate:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.14)

This equation provides the Pasinetti case if capitalists do not provide labor services. It means that the steady state condition, [k.sup.[infinity].sub.c] > 0.

The Dual Case

Here all owners of capital are workers, so that [k.sup.[infinity].sub.c] = 0. The result is that equation 1.14 does not apply because it becomes identically zero or in other words, it would vanish. From 1.13a we require a few operations:

First, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substituting the steady state conditions yields:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Substituting equation 1.13 yields:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.15)

By assuming that [k.sup.[infinity].sub.c] = 0, the expression becomes [k.sup.[infinity]] = [k.sub.w.sup.[infinity]] + [k.sub.c.sup.[infinity]] = [k.sub.w.sup.[infinity]] + 0 = [k.sub.w.sup.[infinity]]. Substituting the two equations above in equation 1.15 yields the dual theorem in which the workers' savings ratio dominates.

f([k.sup.[infinity]]) = n/[s.sub.w] [[k.sup.[infinity].sub.w] (1.16)

Equations 1.14 and 1.16 are results for the primal and dual Pasinetti theorem, respectively, from the production function perspective. The primal case indicates the equilibrium conditions for the profit-capital ratio, which we can plot on a horizontal axis. The dual case shows the equilibrium condition for the output-capital ratio, which we can plot on a vertical axis. With this view, a 45-degree line would represent the equality of these two growth rates. James Meade (43) has used such a diagram to summarize the implications of the literature on these two theorems. With this diagram, Meade was able to demonstrate that the production function was a useful device to indicate the long-run growth outcome, whether it will be the primal, dual, or no stated state outcome.

Conclusion

The debate on distribution theory has "mesmerized the economics profession for close to twenty years." (44) The Cambridge, U.K., view holds that long-run returns to capital will depend uniquely on capitalists' saving propensity, "even when workers save, provided that their propensity to save is less than the share of investment in income." (45) The algebra we have discussed on the Cambridge, U.K., model represents Kaldor's proof and Pasinetti's extended version that "additional saving flow generated by the laborer class ... does nothing to alter either the asymptotically attained balanced growth equilibrium rate of interest ... or the balanced growth capital/labor ratio ... written as a function of the rate of interest." (46)

The methodological areas of dispute between the two schools involve the neoclassical view on the MIT side, and the theories of alternative distribution on the Cambridge, U.K., side. Kaldor's "theory is a stark contrast to the neoclassical theory of distribution, based not on the relative scarcity of factors of production but on the dynamism of accumulation." (47) The trade mark of Kaldor is his "stylized" approach that is built around his appeal to realism, increasing returns, and complementarity. (48)

Pasinetti draws the implication that the irrelevance of the workers propensity to save is more general than is believed. The profit rate and income distribution between workers and capitalists would not change whether we have a disaggregated hypothesis of savings at the individual workers' level, or a highly aggregated hypothesis of savings at the macroeconomic level. A disaggregated view of workers' savings will affect savings behavior at the microeconomics level, but will not affect the total distribution of wages and profits, and the rate of profits. "Whatever the workers may do, they can only share in the amount of total profits which for them is predetermined: they have no power to influence it at all." (49)

The major condition for the Cambridge, U.K., implication to hold is that the capitalists' propensity to save must exceed the workers' propensity to save. This condition follows from solving the differential equation d/dt(P/Y) = f(I/Y - S/Y). (50) We see that the neoclassical position is anchored in the solution to another differential equation, k = sf(k) - nk discussed above.

Samuelson and Modigliani's view relates the output-capital ratio to workers' propensity to save, which requires a neoclassical production function for steady state growth of other variables. Their dual-Pasinetti conclusion holds that in growth equilibrium, the capitalists' propensity to save will decrease to an insignificant level. Kaldor explained this as follows:

"all savings get invested somehow, without disturbing full employment: because any excess of savings over its equilibrium level induces a corresponding excess of investment over its equilibrium level. It is a world in which excess savings in search of investment necessarily depress the rate of interest, r, to whatever level required to induce the necessary addition to investment, which means that, given a sufficient fall in r, a value of k/y can always be found (this is where 'well-behaved' production function come in) to make nk/y = [S.sub.s]." (51)

The latter is the Kalecki-Kaldor-Pasinetti result.

The implication on the MIT side is that when the workers' savings propensity changes, "the composition of the total capital stock as between capitalists' capital and laborers' capital and the composition of saving as between capitalists' saving and laborer's saving" (52) will change. Finally, Samuelson and Modigliani's insight was to ask if there would be a point where the new composition of savings by workers would come to dominate the savings of capitalists.

References

Allen, R. G. D., Macro-Economic Theory, (London: St Martin's Press, 1968)

Burmeister, Edwin and A. Rodney Dorbell, Mathematical Theories of Economic Growth, (London: Macmillan Publishing Co., Inc, 1970)

Davidson, Paul, "Setting the record Straight on A History of Post Keynesain Economics," Journal of Post Keynesian Economics, Vol. 26, No. 2, (Winter 2003-4), 245-272

Eatwell, John, "Theories of Value, Output and Employment," in Keynes's Economics and the Theory of Value and Distribution, Edited, John Eatwell and Murray Milgate, (New York: Oxford University Press, 1983), 91-128.

Feiwel, George. "On Kalecki's Theory of Income Distribution." De Economist, Volume 2, No. 2, (July 1974), pp. 309-325.

Ferguson, C. E., The Neoclassical Theory of Production and Distribution, (Cambridge, UK: Cambridge University Press, 1969).

Harcourt, Goeff, "Interview," in J. E. King, Conversations with Post Keynesians, (London: St. Martin's Press, 1995), 168-186.

Harcourt, G. C., and N. F. Laing, Capital and Growth, Edited, (London: Penguin, 1971)

Hicks, John, The Economics of John Hicks, Edited by Dieter Helm, (Oxford, UK: Basil Blackwell, 1984)

Kaldor, Nicholas., "Letter," in John Maynard Keynes: The General Theory and After, Part II, Defense and Development, Edited by Donald Moggridge, (London: Royal Economic Society, 1973), 241-242.

Kaldor, Nicholas, "Alternative Theories of Distribution," The Review of Economic Studies, Volume 23, No. 2, (1955-1956), 83-100

Kalecki, Michael, Selected Essays on the Dynamics of the Capitalist Economy" (Cambridge, UK: Cambridge University Press, 1971).

Keynes, John Maynard, The General Theory and After, Part II, Defense and Development, Edited by Donald Moggridge, (London: Royal Economic Society, 1973), 241-242

Meade, J. E., "The Outcome of the Pasinetti-Process," Economic Journal, Vol. 76, 1966, pp. 161-165.

Nell, Edward J., The General Theory of Transformational Growth: Keynes after Sraffa, (Cambridge, UK: Cambridge University Press, 1998)

Pasinetti, L., and Scazzieri, R, "Capital Theory: Paradoxes," in J. Eatwell, M. Milgate, and P. Newman, Edited, The New Palgrave: A Dictionary of Economics, Vol. 1, (London: Macmillan, 1987), 363-368.

Pasinetti, Luigi L., Growth and Income Distribution: Essays in Economic Theory, (Cambridge, UK: Cambridge University Press, 1974).

Pasinetti, Luigi L., "New Results in an Old Framework," in The Review of Economic Studies, Vol. 33, No. 4, (October 1966), 303-306.

Pasinetti, Luigi L., "Rate of Profit and Income Distribution in Relation to the Rate of Economic Growth," The Review of Economic Studies, Vol. 29, No. 4 (October, 1962), 267-279.

Samuelson, Paul A., and Franco Modigliani, "The Pasinetti Paradox in Neoclassical and More General Models," in The Review of Economic Studies, Vol. 33, No. 4, (October, 1966), 269-301.

Samuelson, Paul A., and Franco Modigliani, "Marginal Productivity and the Macro-Economic Theories of Distribution: Reply to Pasinetti and Robinson," The Review of Economic Studies, Vol. 33, No. 4 (October 1966a), 321-330.

Samuelson, Paul. A. The Collected Scientific Papers of Paul A. Samuelson, Vol. 2, Edited by Joseph Stiglitz, (Cambridge, MA: The MIT Press, 1966).

Samuelson, Paul A., The Collected Scientific Papers of Paul A. Samuelson, Vol. 5, Edited by Kate Crowley, (Cambridge, MA: The MIT Press, 1986).

Samuelson, Paul A., The Collected Scientific Papers of Paul A. Samuelson, Vol. 4, Edited by Hiroaki Nagatani and Kate Crowley, (Cambridge, MA: The MIT Press, 1977).

Samuelson, Paul A., "The Evaluation of 'Social Income": Capital formation and Wealth," in E A. Lutz and D. C. Hague, Edited, The Theory of Capital" Proceedings of a Conference held by the International Economic Association, (London: Macmillan, 1961), 32-72.

Targetti, F. and A.P. Thirlwell eds., The Essential Kaldor, (New York: Holmes and Meier, 1989).

Wan, Henry Y., Economic Growth, (New York: Harcourt Brace Jovanovich, Inc., 1971)

Notes

(1.) P.A. Samuelson, Vol. 4, 1977, p. 765.

(2.) P.A. Samuelson, Vol. 5, 1986, p. 263.

(3.) Ibid., p. 281.

(4.) Ibid., p. 277

(5.) G.C. Harcourt, 1995, p. 181.

(6.) E.J. Nell, 1998, p. 65.

(7.) P. Davidson, 2003, p. 246.

(8.) Ibid., p. 247.

(9.) P.A. Samuelson and F. Modigliani, 1966, p. 321.

(10.) L. Pasinetti, 1962, p. 267.

(11.) Ibid.

(12.) Ibid., p. 276.

(13.) C.E. Ferguson, 1969, p. 310.

(14.) M. Kalecki, 1971, p. 78.

(15.) Ibid., p. 36.

(16.) Ibid., p. 62.

(19.) M. Kalecki, 1971, p. 1.

(20.) J. Eatwell, 1983, p. 124.

(21.) R. Solow, 1975, p. 1333.

(22.) G. Feiwel, 1974, p. 325.

(23.) R. Solow, 1995, p. 1334.

(25.) Ibid., p. 242.

(26.) Ibid., p. 245.

(27.) J.M. Keynes, 1973, p. 243.

(28.) N. Kaldor, 1955-56, pp. 83-100.

(30.) L. Pasinetti, 1962, p. 270.

(31.) Ibid., p. 271.

(32.) Ibid., p. 272.

(33.) J. Hicks, 1984, p. 272.

(34.) C.E. Ferguson, op. cit., p. 316.

(35.) L. Pasinetti, 1974, p. 97.

(36.) Ibid., p. 96.

(38.) N. Kaldor, 1955-56, p. 96.

(37.) P. Samuelson, Vol. 2, 1966, p. 1547.

(39.) L. Pasinetti and R. Scazzieri, 1987, pp. 363-368.

(40.) P.A. Samuelson and E Modigliani, 1966, pp. 269-301.

(41.) Ibid., p. 270-271.

(42.) H. Wan, 1971, pp. 196-198.

(43.) J. Meade, 1966, pp. 161-165.

(44.) F. Targetti and A.P. Thirlwall, 1989, p. 10.

(45.) Ibid.

(43.) E. Burmeister and A.R. Dorbell, 1970, p. 46.

(47.) F. Targetti and A.P. Thirlwall, 1989, p. 8.

(48.) Ibid., pp. 13-15.

(49.) L. Pasinetti, 1974, p. 113.

(50.) Ibid., p. 116.

(51.) N. Kaldor in G.C. Harcourt and N. F. Laing, 1971, p. 300.

(52.) E. Burmieser and A.R. Dorbell, 1970, p. 46.

Lall Ramrattan, University of California, Berkeley Extension

Michael Szenberg, Lubin School of Business, Pace University