1 – Introduction
1In post-war macroeconomic analysis, economic dynamics generally refers to two main phenomena: economic growth and business cycles. This view implicitly implied the existence of a strict dichotomy between short run and long-run dynamics. The convergence between some general dynamic equilibrium business cycles models and endogenous growth theory in the nineties has changed the relation between growth and cycles in modern macroeconomics. As Aghion and Howitt summed it up briefly, “one of the general implications of endogenous growth theory is that growth and cycles are related phenomena, with causation going in both directions.” [Aghion and Howitt, 1998:4] Two main approaches then emerged. The first corrects the short-comings of early RBC models, offering a more sophisticated analysis of technological innovations and of their economic effects. The second approach refers to creative destruction and establishes, in line with Schumpeter, “causal links from cycles to growth or the reverse causality.” [Aghion and Howitt, 1998: 233] This new conception contrasts with those defended by the first modern business cycle model-builders and growth theorists. Thus, in 1960, Pasinetti characterized the various models combining multiplier effects and some form of the acceleration principle noting that “the situation is that, on the one hand, the macro-economic models which provide a cyclical interpretation of the economic activity cannot give any explanation of economic growth and, on the other hand, those theories which define, or rely on, the conditions for a dynamic equilibrium to be reached and maintained cannot give an explanation of business cycle.” [Pasinetti, 1960, p.69] The origin of this “situation” is to be found in the treatment of investment and its consequences on business cycles. Business cycles had to be interpreted as a pure short period phenomenon disconnected to long run accumulation. Furthermore, this approach neglected the reverse influence of economic growth on business cycles [Arena and Raybaut 2003]. 
3On the contrary, the postwar extension of Keynesian concepts to problems of economic growth has been progressively accompanied since the mid-fifties by a shift from the investigation of the requirements for pure cyclical behavior towards the study of the interrelation of cycle and trend. The period under consideration in this paper is characterized by the emergence of various approaches of the problem, so much that Matthews  could write that “there is less agreement among economists about the relation between trend and cycle than about most other topic in the theory of the cycle” [Matthews R.C.O,1959: 253].
4A first difference refers to the conception of the growth process.
5On the one hand, the approach developed by Duesenberry  in Business Cycles and Economic Growth which combines the traditional ingredients of business cycle theory to yield steady growth instead of fluctuations. Fluctuations are due to causes exogenous to the process of growth and are separately explained. In particular, if speculation and financial disturbances are avoided, steady growth is possible without cycles.  On the other hand, the majority of the contributors adopt Harrod’s view, trying to describe in terms of endogenous factors what happens when the economy is not on the steady growth path. In this perspective, the question is mainly to determine by what mechanism, if any, the rate of growth of effective demand and the rate of growth of productive capacity are brought into alignment.
6A second difference refers to the very relation between cycles and growth.
7To begin with, a trend can be introduced into a model of the trade cycle or a trade-cycle theory can be used to “explain” the trend as well. This is in particular the case of Felner  or of Tinbergen and his successors, notably Klein for the US economy. To some extend Kalecki is also in the same position. The cycle refers equally to a static or a growing economy. Certain variables, which were held constant for business-cycle analysis, are subject to long-run changes to produce a trend. Tough, “in his latest work, Kalecki has attempted to produce a separate theory of trend. But he has still not shown how the process of growth itself can generate fluctuations.” [Smithies, 1957:4]
8Second, the trend can be conceived as an essential ingredient of the explanation of the cycles. Hicks’1950 trade cycle theory is representative of the this view. As we know, the trend is an essential element of his theory of economic fluctuations. Accordingly, Hicks insists on the necessity of dealing of the business cycle as a problem of an expanding economy. But the way he incorporated as a deus ex machina a linear steady state growth trend in an unstable multiplier/accelerator business cycle framework without changing its basic characteristics remains “quite unconvincing in spite of its brilliancy.” [Ando and Modigliani, 1959: 502] This influential approach has given birth to several variants in the fifties. Among others, let mention the model of Minsky . Floors and ceilings in the linear multiplier accelerator framework are interpreted as imposing new initial conditions reflecting different effective supply constraints or institutional and financial environment. Another interesting reinterpretation of Hicks’ model is due to Higgins . For him, “the relationship between cycles and growth is a two-way one.” [Higgins, 1955: 595] One can conceive of economic growth without fluctuations and of economic fluctuation without economic growth. But, “At the same time, actual fluctuations will be markedly influenced by the growth factors, and the actual trend will be very much affected by economic fluctuations.” [Ibid.] The models first considers briefly as to whether autonomous investment can conceivably produce smooth growth without significant fluctuations. Then, he analyzes more fully the effects of the autonomous trend on the amplitude and duration of economic fluctuations. The analysis relies on the role of expectations effects and Marrama’s “rationality factor”, showing notably that after the floor is reached, it may take a longer period for sustained increase in output to start induced investment and launch an upswing.
9Finally for a third approach, cycles and trend are so intertwined as to be indistinguishable. Schumpeter was obviously the leading exponent of this view, but in the fifties his position is reasserted by younger scholars who try to reformulate the problem within the post-war macroeconomic framework. In the light of the recent Neo-Schumpeterian advances on cycles and growth mentioned above, we will focus in this paper on this view. Two main contributions are particularly representative of this approach. The first one is the attempt by A. Smithies  to link more closely fluctuations and growth using ratchet effects. The second one is the nonlinear approach of cyclical growth initiated by Goodwin . 
2 – The ingenious contribution of Smithies
10This interesting attempt which deserved a lot of attention in the period was developed by A. Smithies in a long article published in Econometrica [1957, 25(1): 1–52]. According to Smithies, the existing bridges between business cycles and growth theories of the post-war are rather “infrequent and by no means satisfactory.” [Smithies, 1957: 1] For instance, Tinbergen and his successors made a clear separation between cycles and trends in the data, and the cycle is entirely analyzed as deviations from trend, “Thus the possibility of interaction between long-run growth and cyclical fluctuations is removed before the analysis even commences.” [Ibid.] The position of Kalecki seems to him rather similar, since “his cycle works equally well in a static or in a progressive economy.” [Ibid.] An attempt to produce a separate theory of trend exists in Kalecki’s work, “But he has still not shown how the process of growth itself can generate fluctuations.” [Ibid.] Concerning the growth issue, the suitable point of departure is obviously the contribution of Harrod. Indeed, “Harrod himself would agree that displacements from steady growth are more likely to generate further displacements than a return to the original position… One of my major purposes is to produce a theory that takes Harrod’s steady growth as a path of reference, but describes what happens when the economy is not on that path.” [Ibid.: 4] From this standpoint, Hicks’s Trade Cycle is an important, although unsuccessful, attempt to link the explanation of growth and fluctuations. But, for Smithies there is no evidence at all that a full-employment ceiling is responsible for cyclical downturns, in addition, there is also no empirical evidence for the value of the accelerator during the upswing or the downswing. Finally, the trend is a deus ex machina. Indeed, “what motivates autonomous investment is entirely unexplained, and it has no influence in increasing capacity output. And since it is the factor that explains economic growth, it urgently needs explanation. The Hicks cycle may well be a trivial addition to the main phenomenon of investment which the theory does not touch.” [Smithies, 1957: 5] It is thus necessary to take another direction which tries to overcome these difficulties.
11In this perspective, according to Smithies the main reference in the field, which interrelates growth and cycles, is the theory advanced by Schumpeter in the Theory of Economic Development and in Business Cycles.  If the Schumpeterian concept of innovation has clearly won its place in economic thinking, for Smithies it is not completely the case for Schumpeter’s conception of equilibrium in his theory of economic dynamics. Smithies refers here in particular to the idea that the phases of expansion are characterized by disruption of an initial equilibrium position and the depressions by convergence towards a new equilibrium. Consequently, a process of cyclical expansion is generated “whereby the economy moves through successively ‘higher’ equilibrium positions. [Smithies, 1957: 6] Smithies understands clearly that the crucial point is precisely the new potential equilibrium established in every cycle “holds the economy up, and prevents a return during the depression to the initial (equilibrium) starting point. In this manner Schumpeter explains growth and cycles as part of the same dynamic process.” [Ibid.]
12As we know in Business Cycles Schumpeter argues that equilibrium may never be actually achieved, but that it exists a tendency towards it. This tendency refers notably to the concept of a normal business situation “present in the minds of businessmen-as evidenced by directors’ reports which usually describe the affairs of the business or the economy by reference to a norm.” [Smithies, 1957: 6] On this basis, Smithies contends that business men generally consider that “neither booms nor depressions will last forever but will make their investment and production plans with reference to a normal situation and that the normal itself will change as development continues.” [Ibid.] However for him, Schumpeter’s argument calls for additional considerations on the demand side since “Businesses will not increase production simply because they believe business is subnormal.” [Ibid.] For Smithies, the suitable mechanism refers here to an extended use of the ratchet effects recently introduced by Modigliani and Duesenberry to explain the relation between consumption and income. To put the argument in a nutshell, consumption depends not only on the current level of income but also on the highest levels previously achieved. Consequently, if current income is lower than some previous level, that level will exert an influence on current consumption, and “in this way the theory recognizes that consumption standards and patterns once achieved are not lightly abandoned and the retention of those standards provides the demand needed to spark the equilibrium tendency.” [Smithies, 1957: 7] Moreover, these ratchet effects may involve different aspects than the mere relation of consumption to income. In particular, Smithies believe these effects play an important role in investment decisions.
13On this basis Smithies develops its own explanation of the interaction between fluctuations and growth “through the operation of endogenous forces.” [Smithies, 1957: 1] The model consists of two interrelated parts. The first one relates to the dynamics of aggregate demand explained by induced investment and the multiplier. The second part is dedicated to the dynamics of full-capacity output.
14The level of gross profits is the principal determinant of induced investment. The direct influence of profits is twofold. They provide a basis for expectations concerning the returns to be obtained from investment, and in line with Kalecki they influence the conditions on which firms can finance their investment. But additional factors are taken into account. First, investment also depends on the full-capacity output YF. Excess capacity will tend to reduce investment, while a general shortage of capital will encourage it. In addition, there exists a ratchet effect on profits captured by Ȳ, the highest level of GNP. Finally, “exogenous” trends may be introduced, in particular the rate of population growth. Smithies assumes that gross profits are proportional to GNP. Thus, the variable Y is used in the he investment function instead of profits. The investment function is linear with one period lags and writes:
16The first term captures the direct dependence on profits, the second one a ratchet level effect on profits and the third the capacity effect. The final term refers to the exogenous influence of trends, such as population growth.
17Consumption C depends on disposal income and its ratchet value captured by GNP, Y, and ratchet GNP, Ȳ. All the relations are linear and the the consumption function is
19Consequently, the long-run marginal propensity to consume when Y = Ȳ, is 1 – α1 + α2 and the short-run one in a fluctuating economy is 1 – α1.
20Then, investment and consumers’ expenditures determine in a standard way the level of Y at any particular time with Y = C + 1 or Y – C = I = S and “in that sense, the system is in short-run equilibrium” [Smithies, 1957: 18].
21The model is completed by a relation depicting the effect of investment on full capacity. output YF. That is:
23where D1 = δ1YF–1 and D2 = δ2(YF–1 – Y–1) refer respectively to physical depreciation and extraordinary obsolescence. A trend factor lt is introduced in order to take into account changes in capacity independent of the rate of investment.
24On the basis of these simple set of linear relations, the core of Smithies’ explanation of the interaction between fluctuations and growth lies in the distinction between two regimes, state 1 and state 2. In the first one the ratchet effect is discarded and Ȳ = Y–1 in relation (1) and Ȳ = Y in (2). This distinction results in two sets of equations, those that hold when the ratchets are operating and those that are in effect when the economy expands beyond the ratchet level of output. Consequently, the dynamics of the model consists of two simultaneous linear difference equations, one with actual output Y and the other with full-capacity output YF as the dependent variable. We have for state 1:
26When the ratchet effect is operative the system for state 2 is:
28It is thus easy to solve these two linear systems in Y and YF. As well known, monotonous or oscillatory solutions may be obtained with real or complex roots of the corresponding characteristic equations. Indeed, if state 1 is a monotonous process of growth with Y continually increasing, state 2 never becomes relevant. On the contrary, if Y oscillates it may sometimes fall below Ȳ, and the economy enters into state 2 until Y again exceeds Ȳ. Smithies considers also the possibility that the economy may enter in state 2 and remain in this state without the capacity to return to to State 1 “a situation which did not seem an absurd idea in the 1930’s.” [Smithies, 1957: 34] Consequently, the first step is to examine closely the solution of state 1. To begin with, let us notice as Allen  put it that Smithies’s state 1 bears a close family relationship to Duesenberry’s simplified model. It can be shown to be its formal equivalent if Smithies’s YF is replaced by σK where σ is Domar’s coefficient.
29First, Smithies considers the conditions of existence of a continuous growth path in which . This equilibrium path refers to Harrod warranted rate and “represents an orderly and simple state of the economy and is a useful path of reference for considering other situations.” [Smithies, 1957: 23] The other noncyclical case that is economically interesting is the case where Y grows over time and “is or tends” to be greater than YF. This case is called “persistent exhilaration” by Smithies. The other noncyclical solutions, persistent excess capacity or continual decline in both, are dismissed since they “reveal unstable states of the model rather than significant economic possibilities.” [Op. cit.:36] Second, Smithies examines the possibility of cyclical solutions in state 1. The model in this initial state can exhibit cyclical solution if the characteristic equation of system (6)(7) admits complex roots. That is if . The interpretation given by Smithies is that cyclical fluctuations arise from the influence exercised by the difference between Y and YF on investment such that, “if this influence is great enough (but not too great) in the model it will produce fluctuations rather than continued growth or decline.” [Ibid.] However, the economy may fluctuate and remain in State 1 without never interring in State 2, then “we have a model of the conventional Tinbergen-Kalecki type where the cyclical process does not engender growth, and growth depends entirely on trends.” [Op. cit.: 29] Consequently, the most interesting configuration for our issue is clearly that State 1 admits complex roots of modulus greater than the unity. This condition discussed in length in the mathematical section is simply ad – bc > 1. In addition, it is shown that the roots of state 1 are complex, those of State 2 are necessarily complex. Moreover, if the solution of State 1 are explosive, that of state 2 may be explosive or damped and if solution of State 1 is damped, state 2 is also damped. Finally, the period of the oscillations in state 2 are greater than in state 1, since , where θ′ and θ are respectively composed of the coefficients of the model in state 2 and state 1. Thus, “Provided state 1 is explosive, the ratchet effects can be such as to meet the required conditions; and a process of endogenous fluctuating growth is possible.” [Ibid.] In this case, the economy may alternate between states 1 and 2 in a process of fluctuating endogenous growth in which “each depression and each boom respectively occurs at a higher level of output than the preceding one.” [Op. cit.:42]
30The crucial element is the difference between actual and full capacity output. The mechanism can be summarized as follows. A boom ends and a depression begins when YF has sufficiently increased relatively to Y such that that total investment is reduced, and this in turn reduces consumption. The depression continues until depreciation and obsolescence reduce sufficiently YF in relation to Y to produce a revival of investment, and consequently of consumption. In addition, the mere fact that during the prosperity YF increases faster than Y sets the stage for another depression. Smithies insists on the fact that if State 1 were the only state of the economy, such mechanism would simply describes a process of explosive or damped deviations from equilibrium growth, “it is therefore the existence of State 2 that opens up other possibilities.” [Ibid.]
31He adds that this process is obtained without the introduction of any exogenous trend. Finally, Smithies discusses the potential influence of the introduction of exogenous trends in the model. These trends are likely to play the role of exogenous shocks and may modify the endogenous dynamics described above. For instance, a downward or an upward trend in the investment function can offset the process of endogenous expansion, keeping the economy indefinitely in state 2 or conversely can push the economy outside this state even with damped solutions.
32The different possibilities with fluctuations are illustrated on the following figure [Smithies, 1957: 30] reproduced below.
33The solid lines refers to actual output Y and the dotted lines to YF. B represents the case where the ratchet effect is not strong enough to produce growth, such that the model exhibits explosive fluctuations without growth. In C, the endogenous forces for expansion are so weak that the economy remains perpetually in state 2. In D trend influences are superimposed on C, and are strong enough to bring the model into State 1 during prosperity periods. Finally, only A refers to a configuration in which the economy alternates between state 1 and state 2 leading to endogenous growth cycles in which each peak and each depression occur at a higher level of Y than the preceding one.
34Smithies concludes: “Reasonable values of the coefficients of the model are consistent with continued economic growth or with fluctuations. Neither possibility can be excluded without specific knowledge of the economy under consideration.” [Smithies, 1957: 23] In addition if we consider a specific country, these conditions may change from time to time, “so that in one decade a continued growth model may be appropriate while in another a fluctuating model is needed, depending on conditions outside the present frame of reference.” [Ibid.] Smithies contends from the beginning of the paper that his framework is empirical in the sense that “any of the possibilities it suggests can be excluded by the available evidence” [op. cit.: 1], but this framework “is not numerical.” [Ibid.] More generally, Smithies has doubt about the usefulness of time series to test a simple model of long run dynamics, since the structure of the model is likely to change within the period. This is precisely refers to one of the difficulties of providing a theory of both cycles and growth raised by Pasinetti . The outcomes of the models are generally obtained for a limited range of the parameters, which raises the question of their stability over time. These parameters are particularly sensitive to variations in expectations. Now, “if the parameters expressing the behavior of entrepreneurs in their decisions to invest are bound to assume different values in time, the situation certainly cannot be solved as trade cycle and growth theorists have practically done by choosing arbitrarily particular values of such parameters and then carrying on one’s analysis on the assumption that such values remain constant.” [Pasinetti, 1960] 
35Nevertheless, some attempts have been made to test empirically Smithies’s model. Let us mention in particular the studies by Choudhry and Mohabbat in the early sixties. The authors notice that even if we overcome the initial reluctance of Smithies, “the manner in which Smithies puts together his equations and the variables he includes in them create some formidable methodological-conceptual problems in the construction of time-series as well as in the estimation of equations. Our effort to cope with them may be deserving of interest.” [Choudhry and Mohabbat, 1965:90] Among these problems, two need special mention, the first one refers to the choice of the best method of estimation in view of the equations of the model. The second one is the problem involved by the construction of time-series on full capacity output. The authors are aware that a model as simple as this one can hardly capture the characteristics of the dynamics of a complex industrial economy, but the study confines to the Smithies’s simplest case due in particular to limitations of data. The test consists first in estimating the parameters a, b, c, d, g in relation and and second in examining the time path of GNP induced by these values. A clear negative result concerning the fluctuating growth approach is obtained. The two roots of the difference equation are real and less than unity in absolute value, thus the system converges monotonically. This result does not really come as a surprise for the authors. Accordingly, “by now the knowledge that simple systems like these could evince the cyclical-growth characteristics of a modern industrial economy only under very special assumptions about the parameters therein is indeed hoary. And yet, to the extent that our findings lend support to the thesis that the economy is an essentially stable process relying chiefly on exogenous impulses and random displacements to account for its cyclical dynamics, they may deserve to be noticed.” [Op. cit.:96]. This conclusion directly refers to Duesenberry’s view which, on this aspect, prefigures the approach of the eighties. In the same perspective during the seventies the study of Kromphardt and Döfner  examines closely Smithies’s attempt. The authors first recall that Evans had also shown numerically in 1965 that the range of parameters required to generate both cycles and growth in the model could hardly fit with their empirical relevant values. Then, they show analytically and numerically that it is actually impossible to found a set of relevant parameters compatible with the existence of cyclical solutions and endogenous growth with positive ratchet effects. Thus, they conclude that contrary to Smithies’s claim, the mere introduction of ratchet effects on consumption or saving is not sufficient to explain endogenously both cycles and growth, an exogenous trend remains necessary.
36These diriment criticisms confirm the logical difficulties to combined endogenously cycles and growth, even by using the ingenious solution developed by Smithies, in a linear framework. From this standpoint, let us now examine how the approach developed at the same period by Goodwin in a nonlinear perspective, may address these difficulties.
3 – Interrelations of growth and cycles in a nonlinear perspective: the first models of Goodwin
37Goodwin’s idea to build a model which would fuse indissolubly the cycle and the trend was initially developed in the article “The problem of trend and cycle” [Goodwin, 1953], and began to take shape in 1955. According to Goodwin, the issue has been stated clearly by Harrod who “holds that the same theory must explain both growth and cycles, but it is, I think fair to say that few have been convinced that he has provided such a theory.” [Goodwin, 1955: 123] Goodwin was initially convinced that the main difficulty was that Harrod did not develop a fully-fledged theory of the cycle.  Let us recall that Goodwin was very impressed by Tinbergen’s formal judgment on Harrod. His theory could not emerge onto a cycle theory because of the impossibility to obtain self-sustained oscillations from linear first order differential equations. However, as Goodwin himself will clarify later [Goodwin 1982], he was convinced from the beginning that Harrod and not Tinbergen had the sounder view of economic dynamics. Indeed, as Harcourt put it, with the development of nonlinear dynamics Goodwin “had the answer as to why Harrod’s intuition had been correct, that Harrod did have a cycle in his 1930s book.” [Harcourt, 1985: 417] In the fifties, as regards the recent attempt made by Hicks in his trade cycle theory, the trouble for Goodwin is that Hicks “losses the theory of the trend in all but a trivial sense… his trend is unrelated to required capital or indeed to anything except the necessity to get a trend.” [Ibid.] On the other hand, the decomposition of time series into cycles, trend and residual or the pioneer works on pure business cycles, including his own business cycles models, is “definitively not permissible.” [Goodwin, 1955: 122] In a linear world, thanks to the superposition theorem it is always possible to decompose and add the different elements to obtain an adequate representation of the true dynamics. In a nonlinear world, this theorem does not hold and decomposition is invalid. It would remain of course necessary to prove empirically that important nonlinearities exist, nevertheless “for the time being, we must, …, regard with grave misgiving the removal of trend, since we might remove much of the real problem.” [Ibid.] As a consequence, the formal solution is to adopt a nonlinear framework, as Goodwin did previously in his business cycles models.
38From an analytical point of view, the most fruitful source of inspiration in order to deal simultaneously with cycles and growth is obviously Schumpeter.  This theory is of course difficult to formulate in simple mathematical terms. Therefore according to Goodwin, him, “such use I shall made of it may seem rather a caricature,” [op. cit.: 124] as regards Schumpeter’s own theory and rich empirical illustration in Business Cycles. In addition, Goodwin contends that he cannot follow Schumpeter in his rejection of the Keynesian framework. From a theoretical point of view, there is no objection for him to combine Schumpeter’s insights with effective demand analysis. On the contrary, Keynesian tools can be used to strengthen Schumpeter’s system. Accordingly, “by incorporating effective demand it is possible to lessen the excessive strain that he put on his special theory on the innovational process. Essentially, he places the explanation of the cycle on the internal dynamics of successful innovation… It may sometimes be so, but that it has always been so over more than 150 years is difficult to accept.” [Ibid.] In addition, the combination of the innovational mechanism and the role of effective demand makes more plausible Schumpeter’s idea of clustering of innovations [see e.g. Goodwin, 1946].
39Keeping these arguments in mind, Goodwin develops two simple models. The first one is a purely exercise which “is not designed to do justice either to [Schumpeter] or to economic reality.” [Goodwin, 1955: 125] To begin with, Goodwin considers a curve p(k) of present value of all capital projects as a function of the quantity of capital k. This curve shifts to the right at a regular rate due to progress. The simple purpose of the model is to show that the mere introduction of a nonlinearity (a hook) in the function of the supply price of new capital as a function of the rate of net output of new capital, opens the room to growth cycles. This process is analyzed graphically by Goodwin on the following figure.
40Due to “all the reasons which Schumpeter has widely describe” [op. cit.:125], the range of profitable opportunities increases, but nothing is done until they are potentially so high, B – A, that an innovator may succeed. Then he is followed by many imitators with a burst of investment at a rate F declining to E. At this point net investment ceases, and again the economy slowly accumulate investment possibilities as p raises to B. At this point, the system jumps to C and so on and so forth. If progress is such as to require a rate of accumulation between O and E this process is self-sustained. Consequently, in this simple framework, “the growth is the explanation of the cycle, and the latter would not exist without the former.” [Op. cit.: 126]
41The second model is based on more standard macroeconomic tools of the period.
42To begin with, concerning the investment function, Goodwin considers a flexible nonlinear accelerator principle. From this perspective, investment will be undertaken as long as desired capital, ξ is greater than actual capital. Desired capital is determined by:
44where υ is the accelerator coefficient, y is output and β(t) is a parameter associated with technical change. Consequently, the accelerator means that more capital is desired with increased output, while “innovation means that more capital is desired with a given output.” [Op. cit.: 129] In addition, the pressure to expand is implicity reasons for proportional to the difference ξ – k. That is:
46where g is an increasing linear piecewise function. In addition, Goodwin assumes that full employment of the capital goods coincides with “full employment generally”. A related question discussed by Goodwin is whether or not innovations should raise the level of desired capital and the acceleration coefficient. Goodwin recalls that his “own inclination has always been to assume that the coefficient is increased” [op. cit.: 131], and that is due to the persuasive arguments of Harrod and J. Robinson that he changed his mind. The argument is that new methods of production require additional capital, but also destroy capital by obsolescence or by reducing replacement costs, therefore, “we may continually adding to our capital autonomously, but find that, after many years a routine addition to output requires the same addition to capital as before.” [Op. cit.: 131]. It is thus simplest to assume that ν remains constant.
47Effective demand is introduced via the standard relation
49where γ(t) refers to public spending and the slope of f is the multiplier. Goodwin follows on saving the analysis developed by Duesenberry. Thus, concerning the slope of f, he assumes that “when going down, we follow the steeper slope, and only when we venture again into new high ranges of income does saving again grow only slowly.” [Op. cit.: 132] It is interesting to note that Goodwin gives also another possible explanation of behavior referring to Marshall: The long period can be much longer in the downward than in the upward direction and “outlays expand easily but resist contraction to an extraordinary degree” [ibid].
50Combining these equations and rearranging the terms, we obtain the following differential equation
52Goodwin considers that in this rough approximation of the connection between cycle and growth, it is convenient to assume that all growth of the full employment ceiling occurs only during booms. Using the notation , an increase in β(t) implies a shift of the curve φ to the right. As shown on the figure below, this means that the flat top to slowly rises, “which raises the required capital and prolongs the boom.” [Op. cit.: 135] Thus, the following diagram is obtained:
53The dynamics follows a horizontal line which is steadily rising, “so that in fact we follows a slowly rising curve” [ibid.]. When a sufficient level of capital is accumulated, investment will slow down, which ends the boom. As a consequence, the model describes a process of cyclical growth in which “the economy surges forward and plunges down, but does not go back to its old low level” [Goodwin, 1955: 136], which captures one of the essential features of the Schumpeterian view on the connection between cycles and growth.
54Accordingly, in this framework without lag it is assumed that “ordering new capital goods leads to immediate installation, and that the full effect of this on income also results immediately.” [Goodwin, 1955: 138] As a consequence, the economy jumps instantaneously from boom to deep depression. Accordingly, in a second approximation, Goodwin introduces an average lag θ, which captures the lag between the decision to invest and its outlays. At the same time, he makes two simplifying assumptions. First, instead of , he writes , where m is the constant multiplier. Second, he considers a linear investment function , where a is a constant, instead of . Then, he obtains the following linear second order differential equation: 
56This linear equation gives rise to explosive oscillations if amv > 1. Goodwin assumes that the accelerator ν and the multiplier m are both greater than unity and that a < 1. For example, with ν = m = 2, the solution will be unstable if . Thus, in order to get an endogenous cyclical processes Goodwin combines this explosive linear oscillation with the floor-ceiling framework described above. Consequently, he draws the following picture:
57This figure depicts an endogenous cycle “made up of two simple routines, the flat top and bottom, and the outward spiralling ends… [where] the harshness of transition from boom to depression and back again is entirely softened.” [Goodwin, 1955: 139]
58However, we have not yet addressed the question of the evolution of the factor β(t) capturing the Schumpeterian insights on innovative investment. Goodwin considers that in a progressive economy, β(t) increases through innovation. These may occur slowly at some times and rapidly at others. Accordingly, the factor β(t) is actually the product of many cumulated small events, but in a simple aggregated framework, it is likely to be continuous. However, Goodwin does not provide any explicit formalization of the dynamics of this factor in his 1955 model. In this perspective, it is interesting to refer to the relation introduced by Goodwin  in an early contribution which has directly benefited from Schumpeter’s critical and constructive remarks.  Accordingly, in this article dedicated to innovations and the irregularity of economic cycles, Goodwin develops one of the first formalizations of Schumpeter’s ideas on clustering, in which the waves of innovation are simply modeled as a continuous periodic function of time. We have:
60where, L defines the amplitude and ω the frequency of the motion.
61Let us now introduce this relation in a slightly modified version of Goodwin’s model with delays, in which the linear accelerator νy is replaced by a continuous sigmoid shaped function Φ()y). We obtain:
63The equation becomes nonlinear and can easily exhibit endogenous closed orbits. We will not enter into the formal details,  and simply give below a numerical example of this type of solution with the forcing effect on β(t) specified above.  The following results are obtained.
64The first figure displays the phase diagram of the model in the plane, showing the existence of a limit cycle. The second one displays the cyclical evolution over time of . The mere introduction in Goodwin’s framework of a forcing effect modeled by a periodic function of time makes possible to explain the irregular shape of the growth cycles. Indeed, as suggested in Goodwin , this example exhibits much shorter downswings than upswings.
65The last figure shows the dynamics of the , induced by a more complex forcing effect.  As noticed by Goodwin , β(t) is actually not sinusoidal and periodic as assumed hitherto. But, “the result of the simple case may be used however to study more complicated phenomena.” [Goodwin, 1946: 99] In particular, β(t) may be “approximately periodic” with an another shape. This makes possible, as shown on this example, to explain the irregular shape of the growth cycle.
66In this perspective, the trend and the cycle should be regarded as fused indissolubly and “there would be no meaning to drawing any smooth trend line through the time series of national income.” [Goodwin, 1955:136] Thus, this model extremely simple captures another Schumpeterian insight, the fact that steady accumulation is converted into bursts of development which eventually relax into depressions. Finally, as stressed by Goodwin , this first approximation may also explain the existence of a short as well as a longer cycle, fitting with Schumpeter’s formulation of the secondary wave.
67This contribution focussed on two attempts to model endogenously cycles and growth in a unified framework during the fifties. An important aspect of these approaches lies in the fact that they explicitly refer to Schumpeter’s view on cyclical growth combined with post-war macroeconomic advances. But they adopt two different perspectives. Smithies emphasizes Schumpeter’s conception of economic dynamics as a process of disruption of an initial stationary state or regular growth path and convergence towards another one into his two states linear model, while Goodwin draws a stylized picture of the role of technical change in a nonlinear world.
68These contributions lie at the cutting edge of the development of Post-War macrodynamics and constitute a first interesting Schumpeterian line of analysis of the issue before the modern neo-Schumpeterian advances. However they did not really succeed to overcome the major difficulties associated with the analysis of cycles and growth in a unified framework stressed by Pasinetti in 1960.
69On the one hand, the introduction of ratchet effects into consumption or investment functions does not provide, contrary to Smithies’s claim, an endogenous explanation of cycles and growth. The growth trend or the cycle can only be explained, as in other linear models of the period, by exogenous factors. On the other hand, Goodwin’s nonlinear approach does of course provide a promising perspective to overcome these difficulties. But, in the fifties his attempts to connect a Keynesian framework with Schumpeterian insights remain rather tentative. In particular the models capture in a very rough way Schumpeter’s conception of economic dynamics, reduced essentially to the mere introduction of an exogenous innovative investment function in a multiplier-accelerator model.  Goodwin’s reasoning was mainly based on a graphical apparatus, albeit with an implicit analytical and numerical background, as explicitly shown in different modern advances inspired by his approach, combining a Keynesian framework with Schumpeterian ideas in a nonlinear perspective.  But, we have to wait his 1967 seminal model for a fully-fledged growth cycle model in which Goodwin adopts a different perspective. 
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This dichotomy between growth and fluctuations prevailed until the end of the sixties. Lucas’s monetary equilibrium BC theory favored a completely new approach to fluctuations but maintained the idea that they were independent from economic growth. Even, the various generations of early RBC models did not improve this state of affairs.
As we may recall, early writers—such as Spiethoff or Tugan-Baranowsky—analyzed trade cycles stressing the essential role of investment and the accumulation of productive capacity in the genesis of both growth and fluctuations. They interpreted trade cycles as intrinsic components of the long-run process of growth [Arena and Raybaut, 2003].
The second characteristic of the model is its treatment of the relative importance of demand as the prime determinant of the rate of growth. Thus, “In normal circumstances factor limitations are held not to impose any significant constraint. Either there are plenty of resources or else the natural rate of growth adjusts itself to the requirements of demand.” [Matthews, 1959: 750]
Kaldor’s position regarding cycles and trends is somewhat ambiguous. In his the trade cycle model of 1940 cycles are obtained without growth. He begins his 1954 article by insisting that it is possible to conceive economic fluctuations without growth and to introduce an independent trend explained by technical progress. At the conclusion of his article, he insists on the complex connection between cycles and growth. In addition, in his final paragraph he refers to Schumpeter’s hero who is found to play “a key role, in the drama.” Thus Kaldor seems to be in the end on the side of those who contend that economic growth and the economic fluctuations have a common cause, without ever referring explicitly to a process of cyclical growth. However, all the ingredients for building such a process are present in Kaldor’s approach if we consider it in retrospect [Arena and Raybaut, 2003].
As we may recall, Smithies was personally in touch with Schumpeter during the last eighteen years of his life. He published several article on Schumpeter, notably on the relation between Schumpeter and Keynes and wrote in 1950 Schumpeter’s obituary in the American Economic Review.
As we know, on the basis of these remarks Pasinetti draws the conclusion that we have to abandon the aggregated macroeconomic approach in favor of a different perspective looking more deeply behind the factors explaining the dynamics of the parameters themselves.
For a modern restatement of Harrod’s dynamic theory, see Bruno and Dal-Pont .
As well known, Schumpeter was “one of Goodwin’s heroes and towards the end of Schumpeter’s life a close friend.” [Harcourt, 1985: 414]
By substitution into relation (11) and taking the two first terms of the Taylor series of .
See notably Goodwin  and Harcourt .
We obtain with a constant forcing effect β, a nonlinear second order differential equation of the type , studied in length in the fifties, notably by Goodwin himself in his nonlinear business cycle models [see Goodwin, 1982], by the “Japanese connection” [Velupillai, 2008] and in France by Rocard and Allais [Raybaut, 2014]. See e.g. Lorenz  and Sordi  for a complete analytical and numerical analysis of this type of Keynes-Goodwin-Schumpeter model of growth and cycles.
The parameters are selected in Goodwin  and Goodwin . We have, for example, m = 2, ω = .698 and a = 1.41 and the sigmoid function is specified by μ + tanh[x + ν], where μ and ν are two positive parameters greater than unity.
Accordingly, the amplitude and the period are no more constant, but vary over time. We have .
As we know, Schumpeter was very critical as regards the early developments of aggregated macrodynamic modeling.
For a complete treatment of the interaction of cycles and growth in this perspective, see e.g. Sordi .
As well known, in this model Goodwin uses a prey-predator framework in a neo-Marxian background.