1 - Introduction

1 G. L. S. Shackle argues in his well-known book The Years of High Theory that “there began in the mid-1920s an immense creative spasm, lasting for fourteen years until the Second World War, and yielding six or seven major innovations of theory, which together have completely altered the orientation and character of economics” [Shackle, 1967: 5]. Two important developments of this period, Keynes's General Theory and the theory of imperfect competition developed by Joan Robinson [1933] and Edward Chamberlin [1933] led to a profound renewal of the analysis of income distribution and of the trade cycle.

2 Rejecting the model of free competition, several authors, principally under the lead of Harrod [1936] and Kalecki [1936], developed important arguments for building a bridge between macroeconomics and imperfect competition [see d'Aspremont C., Dos Santos Ferreira, R. and Gérard-Varet L.-A., 2007]. In particular, the invalidation of the wage-employment relationship conjectured by Keynes in the General Theory raised a debate through which Kalecki tried, by resorting to imperfect competition, to renew the analysis of income distribution [2]. In 1938, referring to Chamberlin's theory [1933], Kalecki explicitly linked the determinants of the relative shares in income distribution to the degree of competition. His argument went as follows: Marginal real costs are supposed to remain constant in the presence of excess capacity. Entrepreneurs are assumed to maximize profits by setting the volume of output at a level for which the marginal revenue equals the marginal cost. Then, at a short period equilibrium, we reach the conclusion that income distributive shares are determined by the degree of monopoly equal to the inverse of demand elasticity for the product of the firm. With constant elasticity and constant cost, Kalecki showed there exists a constant relationship between real wages and output. Later Dunlop [1938], Tarshis [1939], Harrod [1939] and Kalecki [1939] referred explicitly to the works on oligopolistic competition, including mainly price rigidities and collusion, and delivered a richer analysis of income distribution.

3 Within a two-class model, the analysis of income distribution based on imperfect competition has strong implications on the theory of the multiplier. For when agents have different propensities to consume, the social propensity to consume is related to income distributive shares. Kalecki [1939] and Kaldor [1940] perceived this point clearly. Convinced by the stability of income distributive shares, Kalecki conceded that there existed a constancy in the propensity to save and the stability of the multiplier. On the contrary, Kaldor was ready to admit that income distribution and the value of the multiplier could vary during the business cycles. As regards formal dynamics, this leads to important differences. In Kalecki's model, endogenous fluctuations results only from the properties of the investment function, assumed to be nonlinear and taking an S-shaped form. By contrast, in Kaldor's analysis endogenous fluctuations can result from the properties of the saving function, which, owing to income distribution effects, can be nonlinear. In consequence, while in Kalecki's model income distribution effects have no impact on the dynamics, these effects play a central role in Kaldor's model.

4 The present paper is an attempt to examine the determinants of income distribution shares in relation to imperfect competition, and to further explore the role played by income distribution effects in two formal trade cycle analyses developed in the late thirties. As will be shown in the light of recent theoretical developments along the lines of nonlinear macro dynamics, these theories can be formulated in terms of nonlinear differential equations whose essential features can clearly be demonstrated.

5 The distribution of income between wages and profits was studied successively under monopolistic competition and oligopolistic competition. The first section centers on the determinants of income distributive shares under monopolistic competition, showing how income distributive shares are likely to change during the business cycle. Other imperfections in the product markets are only mentioned in passing. Kalecki's 1936 review of the General Theory under monopolistic competition will be our point of departure. The second section review then how product market imperfections were integrated to the analysis of endogenous cycle, focusing successively on Kalecki's 1939 and Kaldor's 1940 models.

2 - The Determinants of Distributive Shares under Monopolistic Competition

6 The main works on imperfect competition were in development at the eve of the Second World War. In his review article of Keynes's General Theory (published in 1936 in the main Polish review) the essential part of which was integrated one year later in his 1937 article on business cycle theory, Kalecki adopted the marginalist framework developed by E. Chamberlin [1933]  [3].

7 This work shows how the microfoundations of the General Theory might be presented in terms of imperfect competition. In a way, Kalecki tried to clarify the problem of the microfoundations of the General Theory, in which Keynes considered the degree of competition as one of the 'givens' for the determination of the level of employment and output [4].

8 Macroequilibrium is shown in a diagram presenting microcurves, which are summed up to bring about the macroaggregates. Every imperfectly competitive firm produces so as to equate marginal “value added” and marginal cost. The marginal “value added” curve is the marginal revenue curve of any firm, given (as we would now say) the conjecture that other firms’ prices are constant. The intersection of the curve of marginal revenue and marginal cost is thus a representation of profit maximization. Once the costs of materials inputs are deduced from both curves, we obtain Kalecki's diagram:

Figure 1 - Kalecki's 1936 diagram

Figure 1 - Kalecki's 1936 diagram

9 In the labor market, workers are price takers at a given nominal wage. Looking at the first unit of output along the horizontal axis, the marginal revenue curve shows the revenue produced by that unit of output. The wage component is the marginal cost where MPL represents the marginal product of labor. The remainder, the difference between marginal revenue and marginal cost is the amount of profit associated with the first unit of output. Similarly, each subsequent unit of output is associated with an amount of profit. Total profit at the equilibrium is equal to the sum of the profits associated with each unit of output up to equilibrium and is hence equal to the area under the marginal revenue curve and above the marginal cost.

10 In scaling from micro to macro the diagram reconciles microanalysis and macroaggregates. The area OABC is national income, the area above the marginal cost is global profits, and the residual area, aggregate wages. The distributive shares deduce directly as the ratio of both areas.

11 It is clear from this diagram that any attempt to explain the observed cyclical behavior of distributive shares implies questioning the determinants of the shape of the cost functions as well as the evolution of the degree of competition. Following Kalecki's 1939 presentation [5], under monopolistic competition, the micro demand curve are a family of equations where y_i is micro quantity, p_i the individual demand price, P, the general price level, ε, elasticity of demand, Y aggregate demand, and m, the number of firms present in the economy [see Kalecki, 1939]. Obviously, any change in aggregate demand entailing iso-elastic shifts in marginal revenue curve along increasing marginal real cost provokes a change in income distribution in favor of profits. Taking into account a given degree of competition and admitting the assumption of increasing marginal real cost allows us to establish the procyclicality of the profit share.

12 Looking at the problem from the point of view of costs, monopolistic competition allows us to do without the classical assumption of increasing marginal real cost in a short time period. In the case of excess capacity, it is indeed possible to refer to the idea that marginal real cost can be taken as approximately constant, an idea first presented by Harrod [1936] and developed by Kalecki in 1938 in a paper quoted by both Dunlop [1939] [6] and Keynes [1939]. This is the assumption that the marginal cost and average cost coincide up to a point where the normal utilization of capacity is reached, and begins to rise afterwards.

13 As Dunlop noted it, Kalecki's reverse L-shaped cost curve undermined the stability properties of the economic system:

14

“Economists have a strong bias in favor of steeply sloped cost curves, since curves approaching the horizontal over large range of output make small changes in demand result in large ranges of output; the system becomes highly unstable.”[Dunlop, 1939: 528]

15 Dealing with the stability of the system thus implied a mechanism relating to the location of the microdemand curves and the aggregate demand curve. This was accomplished by Kalecki by assuming that shifts in aggregate demand was materialized as income effect entailing either rightward or leftward shifts in the microdemand curve and producing successively higher or lower microequilibria. It is worth emphasizing here that at a macrolevel, any disequilibrium between aggregate demand and aggregate supply will cause a change in output, which will itself act as an equilibrating force. That, if aggregate demand increases, then the resulting increase in output, and hence income, will stimulate supply until it has brought the economy to a new macroequilibrium. At a microlevel, this means that the importance of the shifts in individual demand curve will decline till it reaches a given position. Correspondingly, a crucial assumption of Kalecki's analysis implies that, for a given level of investment, the social propensity to consume is lower than one. For, if the marginal propensity to consume is equal to unity, no equilibrating mechanism would be activated by the changes in output. Specifically, as income (output) varies, spending would vary by exactly the same amount, so that any initial difference between aggregate demand and supply would remain unchanged. The system would hence be unstable.

16 Kalecki's initial approach [1933, 1934], based on free competition, was not identical to Keynes's one. Keynes assumed a simple linear relationship between consumption and income, partly because he made no strong distinction between workers’ and capitalists’ consumption. Whereas by contrast, in Kalecki's formulation, the social propensity to consume varies with output changes. More precisely, with procyclyclical profit share, when workers are assumed to consume their entire wages, the social propensity to consume decreases when output increases. These properties do not, however, make the system more unstable insofar as the social propensity to consume never exceeds one, since workers will never receive the entire national income.

17 Under the assumption of a constant degree of competition, the articulation between the micro-and macrolevel becomes even clearer. In that case, indeed, one can move directly to a relation between pricing processes via the markup and the constancy of distributive shares as shown by the diagram drawn by Kalecki in 1938:

Figure 2: Kalecki's 1938 diagram

Figure 2: Kalecki's 1938 diagram

18 The area OABC still represents the total value revenue obtained by the firm producing 0C while the sums of the area below the marginal cost represent the cost composed of wages and raw materials. The area above the average cost curve represents profits. The ratio of this area to area OABC is equal to the profit share. As Kalecki says:

19

“with a given degree of monopoly the relation of price to marginal cost is a constant 1/(1 - µ) [where µ denotes the inverse of demand elasticity] Thus, if output remains below OC, the price corresponding to it is represented by the curve [AB] […]. The ratio of the shaded area, representing profits, interest, depreciation, and salaries, to the unshaded area, representing wages and the cost of raw materials, is equal to 1/(1 - µ)” [Kalecki, 1939: 244].” [Kalecki, 1939: 244].

20 With constant distributive shares, the social propensity to consume is now constant and one can obtain a model isomorphic to the Keynesian model, with a linear aggregate demand curve, which can be reproduced by the familiar 45° diagram which has served to express the core of the General Theory.

21 Besides this discussion about the shape of the marginal real cost curve, another related line of reasoning for explaining the evolution of the income distributive share was concerned with the variability of the degree of market imperfection within the cycle. Having in mind Chamberlinian monopolistic competition, Harrod [1936] addressed this issue explicitly in his 1936 article “Imperfect Competition and the Trade Cycle” [see d'Aspremont C., R. Dos Santos Ferreira and L.-A. Gérard-Varet, 2007]. According to the first order condition of profit maximization, the Lerner [1934) index of each firm degree of monopoly is equal to the reverse of the individual demand elasticity. Harrod thought that when income increases, the elasticity of demand would decrease, so that the degree of monopoly increases. This is expressed in his “Law of Diminishing Elasticity of Demand” [Harrod 1936b], resulting from the fact that the expected value for a consumer of searching for better opportunities among close substitutes is a decreasing function of his income. If one takes for granted the procyclicality of the degree of monopoly, the profit share will now be either increasing or decreasing, depending of the shape of the cost curve. If marginal cost curves are flat, the profit share will be necessarily procyclical [7]. Inversely, if marginal cost curves are increasing, and the degree of monopoly is more sensitive than marginal cost to a change in income, the profit share will be countercyclical.

22 Harrod's conclusion was, however, not generally accepted. J.^_Robinson [1936]–in her review of Harrod's article–and Kalecki [1938] indicated two countervailing factors which may affect the degree of monopoly; namely, the varying degree of concentration along the cycle and the change in the regime of competition resulting from the variability of the aptitude of firms to collude.

23 Robinson disagreed with Harrod’s assertion that the variability of the number of active firms along the cycle could affect the degree of monopoly:

24

“The degree of monopoly does not depend only on the imperfection of the market for a commodity, but also on the number of separate units of control engaged in selling it.”[Robinson, 1936: 59]

25 Kalecki's suggest that collusion between firms could also change:

26

“Mr. Harrod was rightly criticized in that there exist other factors which influence the degree of monopoly in the opposite direction. For instance in the slump, cartels are created to save profits and this of course increases the degree of monopoly, while they are afterwards dissolved in the boom because of improving prospects of independent activity and the emergence of outsiders.”[Kalecki 1938: 311]

27 In other words, taking into account the creation and destruction of firms over the business cycle, as well as the intensity of collusion between firms, is likely to countervail the influence of Harrod's Law and to establish the countercyclicality of markups.

28 The evolution of the relative share in national income over the cycle would be the result of mutual change in the degree of monopoly–increasing on account of the greater concentration –and changes in marginal costs. For Kalecki, the long term stability of the profit share would hence be due to the fact the changes in the degree of market imperfection is exactly offset by changes in marginal cost (including raw material costs), increasing in the boom and decreasing in the slump [8].

3 - Income Distribution Effect and Non-Linear Theories of Economic Cycles

29 As regards dynamics analysis, what is actually necessary for endogenous cycles is to assume that either the investment or the saving function is nonlinear. In a two-class model, the marginal propensity to save depends on the determinants of income distributive shares. Convinced by the stability of income distributive shares and the constancy of the multiplier, Kalecki resorted exclusively on the nonlinear properties of the investment function. On the contrary, following Harrod [1936], Kaldor thought that change in income distributive share and the propensity to save was likely to explain the intrinsic instability properties of the capitalist economies.

3.1 - Income Distribution Effects in Kalecki's Models [1939; 1943]

30 In his 1939 model, Kalecki proposed a quite sophisticated nonlinear model of business cycles which has recently been reformulated in the light of mathematical advances in the theory of nonlinear oscillations [Semmler, 1986] [9]. Kalecki relies on a geometric presentation of a business cycle model which depends on a nonlinear relation between income changes and capital stock changes which seems to generate self-sustained cycles without rigid specifications for the coefficients, time lags and initial shocks. More precisely, the geometric presentation of his model business cycle includes the possibility of limit cycles, i.e. asymptotically stable cycles regardless of the initial shocks and time lags. His ideas were formulated for a stationary economic system, and can be represented by a nonlinear differential equation in the following way [Chang and Smyth, 1971]:

31 Where α is a reaction coefficient, the rate of change of income, the rate of change of the capital stock, I = investment and S = saving. Saving is a function of the level of income alone and investment is a function of both income and the capital stock. According to the assumptions underlying the model, there is a unique singular point. The linear approximation of the system in the neighborhood of the equilibrium can be formulated by utilizing the Jacobian matrix

32 Where and . The determinant is – αS_YI_K which is positive because and . The element I_K is always negative. The Jacobian matrix represents at its core the investment-income dynamics according to which the change of income depends negatively on the level of the capital stock (αI_K) and the change of capital stock depends positively on the level of income (I_Y), but there is a negative feedback effect from the level of capital stock to the change of capital stock (I_K) and an ambiguous feedback effect from the level of income to the change of income α(I_Y –  S_Y). This will be explained subsequently. Analyzing the singular point one can conclude that the equilibrium is a focus or a node and that the equilibrium is stable or unstable accordingly as α(I_Y –  S_Y)+I_K is positive or negative. This singular point also allows for a limit cycle since the necessary condition for a limit cycle is that the dynamic system has an index of a closed orbit which is 1. This excludes a saddle point as a singular point. Moreover, the most interesting point in this dynamic system is the ambiguous element α(I_Y –  S_Y)+I_K. According to Kalecki's graphical presentation (reformulated by Kaldor and accepted by Kalecki in 1943) it is assumed that:

33 for a normal level of income
1) for abnormal high or abnormal low levels of income; and
2) the stationary state equilibrium has normal level of income.

34 This might be illustrated with Y the level of output which show that the normal level of Y is unstable and the extreme values of Y are stable. Mathematically this means that the trace α(I_Y –  S_Y)+I_K changes sign during industrial cycles, i.e. if the trace α(I_Y –  S_Y)+I_K does not change sign, limit cycles cannot exist. As proven by Chang and Smyth there indeed exists the possibility of limit cycles under the assumption proposed by Kaldor.

Figure 3: Kalecki's 1943 Business cycle model

Figure 3: Kalecki's 1943 Business cycle model

35 However, the three conditions as formulated above and originally formulated by Kaldor are not necessary for the existence of cycles. What is actually necessary for cycles is only that (i.e. that α(I_Y –  S_Y)+I_K switches sign) at some level of Y. Moreover, the singular point at the normal level of Y can be stable. In addition there is also the possibility that the system is globally stable. This is the case if 1) and 2) everywhere. The global asymptotic stability under these conditions follows from Olech's theorem [see Ito, 1978: 312].

36 In Kalecki's model, the coefficient S_Y is assumed constant during all the business cycle. The occurrence of endogenous cycles thus relies entirely on the change in the coefficient I_Y. One can hence see that income distribution effects play no role in Kalecki's model. Indeed, admitting that the propensity to save remains constant, the dynamics depends only on 1) the value taken by coefficient I_Y, assumed to depend on the change in expectation elasticity and 2) I_K related to the capacity effect.

37 A sharp way to see this point is to focus on the determinants of the saving function. In a two-class model, the propensity to save is directly linked to the determinants of the profit and wage shares. Indicating total real income of capitalists by π, capitalists' consumption is related to gross real profits by means of the linear function: where is a positive constant (fixed volume of capitalists' consumption) and λ is a positive constant, smaller than 1. Ignoring workers' saving, workers' consumption is equal to the total real wages received: C_w = wn (where n is the level of employment) and w the real wage. Finally, assuming that the relative share of wages in real national income is fixed, global consumption can be related in a linear form to global output. For since wn/Y=ω where ω is a fixed coefficient, we get:

38 The marginal propensity to consume of the community is equal to the product of the marginal propensity to save out of profit (1 – λ), and the coefficient of the share of profit in the national income (1 – ω). By this way, Kalecki arrived thus at defining a saving function isomorphic to the function defined by Keynes on psychological grounds.

39 As has already mentioned in section 2, under monopolistic competition the stability of relative share of wages during the cycle can be explained by either 1) the effect of constant elasticity and constant cost or 2) the effect of the opposite changes in the degree of monopoly and changes in the relation of the prices of raw materials to wages. Convinced by the stability of income distribution shares, Kalecki concluded the dynamics of the economy was not related to changes in income distribution. By hence resorting to imperfect competition and abandoning the free competition hypothesis, Kalecki disconnected the fields of income distribution analysis from the field of dynamics.

40 Inversely, by admitting the saving function could be likely to vary during the business cycle, Kaldor placed income distribution effects at the center.

3.2 - Income Distribution Effects in Kaldor's Model

41 Following Chang and Smyth [1971], we can write the income-investment dynamics defined by Kaldor in the form of a system of nonlinear differential equations.

 

42 Where the Jacobian is

43 α is a reaction coefficient, α(I_K –  S_K) has a negative sign since and . The determinant is now α(S_KI_Y –  S_YI_K), which is positive because, for the existence of a unique singular point, it is assumed that . The element I_K is always negative. The Jacobian represents at its core the investment-income dynamics according to which the change of capital stock depends positively on the level of income (I_Y) and the change of capital stock depends positively on the level of the capital stock α(I_K –  S_K), but there is a negative feedback effect from the level of capital stock to the change of income α(I_Y –  S_Y). Analyzing the singular point, one can conclude, as for Kalecki's model, that the equilibrium is a focus or a node and that the equilibrium is stable or unstable accordingly as α(I_Y –  S_Y)+ I_K is positive or negative.

44 This singular point also allows for a limit cycle since the necessary condition for a limit cycle is that the dynamic system has an index of a closed orbit which is 1. This excludes a saddle point as a singular point. Moreover, the most interesting point in this dynamic system is still the ambiguous element α(I_Y –  S_Y)+ I_K. According to Kaldor's graphical presentation it is assumed as for Kalecki's model:

45

1. for a normal level of income^*
2. for abnormal high or abnormal low levels of income ; and^*
3. The stationary state equilibrium has normal level of income.

46 This might be illustrated with Y the level of output which show that the normal level of Y is unstable and the extreme values of Y are stable. Mathematically, this means that the trace α(I_Y –  S_Y)+ I_K changes sign during industrial cycles, i.e. if the trace α(I_Y –  S_Y)+ I_K does not change sign, limit cycles cannot exist. As proven by Chang and Smyth [1971] there indeed exists the possibility of limit cycles under the assumption proposed by Kaldor.

47 In Kaldor's model, both coefficient S_Y and I_Y are assumed to vary during the business cycle models. It is here worth stressing that now, income distribution effects alone are enough for generating self-sustained business cycle models. Indeed, even with given coefficient I_Y, one can obtain endogenous cycles. With the usual capacity effect, it is enough to assume a saving function of the following form:

Figure 4: Kaldor’s 1940 Business cycle model

Figure 4: Kaldor’s 1940 Business cycle model

48 Kaldor is the first, basing his idea on Harrod's insight, to envisage the implication of changing marginal propensity to consume in a two-class model. His innovation, from this point of view, is raising the possibility of an S-shaped saving curve.

49

“If income is high relative to the normal level, the marginal saving rates will rise, too. This is in accordance with the well-known two-class models in macro income theory. If income falls below its normal level, a point will be reached where absolute savings fall drastically. This sharp fall, which of course can lead to negative saving, implies that the marginal saving rate is higher than at the normal level.”[Kaldor, 1940]

50 In a two class-model, the saving rate depends on the determinants of income distributive shares, which depend themselves on market imperfections. In order to obtain a nonlinear saving function like the one defined by Kaldor, one must thus admit the markup set by firms-determined by both demand elasticity and the number of firms–is respectively low and high for normal and extreme values of output. Under monopolistic competition, this amounts to assuming that the effects of the rise in demand elasticity will be dominated by the effects of the rise in the number of firms for extreme values of output and not for normal value of output.

4 - Conclusion

51 The present study has shown how the trade cycle theories have made imperfectly competitive output markets a major theme of macroeconomics in the thirties, principally under the lead of Harrod and Kalecki. In the mid-thirties, Keynes and both Harrod and Kalecki were referring to a supposed feature of business cycles–namely the countercyclicality of real wages–which was, however, shortly about to be contested. Empirical evidence, as well as other more speculative considerations, induced an important flow of theoretical arguments developed by several authors during a very short period, on the eve of the second World War. Analysis under either monopolistic or oligopolistic competition reveals a satisfactory response to the challenge of explaining the evolution of distributive shares. Adopting the condition of imperfect competition especially allowed Kalecki to link, in a two-class model, the multiplier analysis to income distribution theory. Ironically, while Kalecki had gathered all the elements that allowed him demonstrate the nonlinearity of the saving function, it is Kaldor who showed first how income distribution effects could be, in given conditions, sufficient for developing an endogenous dynamics analysis.