II. FRISCH’S PROPAGATION MODEL

The main difficulty of Frisch’s model lies in the fact that it mixes both continuous and discrete time. Among the three equations of his model, some of them rely on delays, the dependence of current values of a variable on its previous values, while others rely on the rate of change of a current variable. This makes Frisch’s system one of mixed difference-differential equations with a number of properties that are not found in a simpler model.

One of these properties is that the model’s solution can take the form of an infinite sum of exponential functions with real and complex roots. A homogeneous dynamic system usually has a number of independent solutions; this number is related to the number of rates of change in the case of a differential equation and to the number of lags in a difference equation, and is infinite in the case of a mixed difference-differential equation. Each of these independent solutions corresponds to a root, which is one of the solutions of a characteristic equation, a function of the model’s parameters. A general solution is a linear combination of all these independent solutions. This solution is general because all the independent solutions can be obtained from it by an appropriate choice of constants by which the solutions are linearly combined (I note that this is what Tenenbaum and Pollard [Reference Tenenbaum and Pollard1985, p. 208–201] explained in the passage quoted by Ginoux and Jovanovic (p. 6); Tenenbaum and Pollard do not talk about the superposition principle).

This solution is also general in the sense that the constants by which independent solutions are combined are arbitrary only to a certain extent: they are in fact determined by initial conditions. If a numerical solver on a computer is used to simulate a dynamical system, it will ask for initial conditions, and the simulated solution will be the same as the linear combination of the general system, with the coefficient of the combination being determined by the initial conditions chosen. These two points will be important in the following.

Frisch started his solution by producing a characteristic equation, in fact a system of two equations (Frisch Reference Frisch1933, p. 184), and proceeded to find the first four roots of this system with the help of his assistants (Frisch Reference Frisch1933, pp. 186–187). In computing those roots, Frisch realized that one of them was real and three were complex. What this means is that there would be one purely exponential solution, the “trend” (Frisch Reference Frisch1933, p. 188), and at least three oscillating solutions. Frisch presented the general form of those solutions in equations (16) and (18) of his paper (Frisch Reference Frisch1933, pp. 188–190). But he then proceeded to give different initial conditions to the trend and the cycles, by considering them as if they were differential equations isolated from each other (Frisch Reference Frisch1933, pp. 188, 190, 192). In fact, as these components are the product of the same mathematical system of equations, they depend on the same initial conditions.

Because initial conditions determine the amplitude of the cycles and the trend in the general solution, this led Frisch to give a higher amplitude to his first cycle than for the trend. This prevented him from seeing that the superposition of the cycles and the trend would not produce an apparent oscillation for the parameters he chose originally. This explains why Stefano Zambelli (Reference Zambelli and Velupillai1992, Reference Zambelli2007), when he simulated the model with Frisch’s parameters, found a monotone return to equilibrium. This result is a consequence of the fact that, when started from the same initial conditions, the cycles have a very small amplitude and are dominated by the real exponential solution.

To find a solution to Frisch’s model, I started reading the subsequent literature on the models built by Frisch, Jan Tinbergen, and Michał Kalecki. I realized that Richard Bellman, a mathematician who is known to economists primarily for the development of dynamic programming, contributed during the 1950s to the theory of difference-differential equations, eventually publishing a textbook with Kenneth Cooke (Bellman and Cooke Reference Bellman and Cooke1963; see also Bellman and Danskin Reference Bellman and Danskin1954). What was particularly interesting is that these researchers referred to the papers by Frisch and Harold Holme (Reference Frisch and Holme1935) and by Kalecki (Reference Kalecki1935), and one of the contributions of my article published in the Journal of the History of Economic Thought is to trace out the way in which this literature developed (Carret Reference Carret2022a, sec. IV).

Bellman and Cooke made extensive use of the Laplace transform and of contour integration to study difference-differential equations. Ginoux and Jovanovic argue that:

Reading Deakin’s paper, one would learn in fact that:

I take “front-line” here as referring to the handful of mathematicians who were working on the modern Laplace transform; Deakin’s point, made again later, is that engineers used different tools at that time but that they converted very quickly between the end of the 1930s and the 1950s (Deakin Reference Deakin1992, p. 272). Perhaps Frisch had heard about the transform and about contour integration, but he did not use it in the 1930s; and the first textbook expositions were published after he wrote, both for the Laplace transform and for the theory of differential-difference equations.

The main advantage of the Laplace transform was that it allowed me to obtain the same sum of exponential functions that Frisch had worked with but with an analytic solution showing the dependence of the coefficients combining the independent solutions on the same initial conditions (a continuous function over an entire interval in the case of mixed difference-differential equations). The Laplace transform is useful because it transforms linear equations involving derivatives and differences into linear equations that do not involve them (Bellman and Cooke Reference Bellman and Cooke1963, p. 1). We can solve these transformed equations, and then use the inverse transform to obtain the solution of our initial problem. The inversion algorithm I used followed the contour integration described by Bellman and Cooke and applied by them to other differential-difference equations (Reference Bellman and Cooke1963, pp. 9ff. and ch. 3). In addition, to find the location of the roots on the complex plane, I used the Lambert W function (Corless et al. Reference Corless, Gonnet, Hare, Jeffrey and Knuth1996). This function has the added bonus that it gives an intuition of why there can be one real solution and an infinity of cycles for certain parameters, as is described in Assous and Carret (Reference Assous and Carret2022, app. to chs. 2, 3, and 5).Footnote ²

Another advantage of the Laplace transform is that the initial conditions are “built in,” as Roy Allen pointed out (Allen Reference Allen1959, p. 159), although Ginoux and Jovanovic misrepresent what Allen said. They write (p. 508):

Allen’s quote is: “when the solution is obtained [with the Laplace transform], it has the initial conditions ‘built in’; there is none of the bother about a solution with n arbitrary constants, to be ‘fitted’ or evaluated with great labour from the initial conditions” (Allen Reference Allen1959, p. 159). What Allen argued is that the Laplace transform allows us to obtain a solution where the relationship between initial conditions and amplitude or phase are directly obtained from the transform and its inverse. On the other hand, when we solve a simple differential equation without using the Laplace transform, a general solution of the form $ x(t)={c}_1{e}^{\lambda_1t}+{c}_2{e}^{\lambda_2t} $ has two arbitrary constants $ {c}_1 $ and $ {c}_2 $ that have to be evaluated with initial conditions, for instance at $ x(0) $ and $ \dot{x}(0) $ . Allen pointed out that the Laplace transform would give a solution in terms of $ x $ and its first derivatives evaluated at $ t=0 $ , “a particular, though very common, case” (Allen Reference Allen1959, p. 159). This was actually the case considered by Frisch, who gave initial conditions on each cycle so that “it shall be zero at origin and with velocity = $ \frac{1}{2} $ ” (Frisch Reference Frisch1933, p. 192). In the case of a differential-difference equation, it also makes it clear that the initial condition we need is actually a continuous function over an initial interval.

Yet another useful consequence of the Laplace transform is that it directly gives a particular solution for the model when the system is not homogeneous, as was the case in Frisch’s model. Frisch used throughout his paper a constant c equal to 0.165 (Frisch Reference Frisch1933, p. 188), the same I used throughout my work. I did not state explicitly that I used the same numerical value for this constant, or the initial conditions that I used for the published figures, although they were written in my code; the initial development was to maintain the model at 20% of its equilibrium level and then releasing it.

III. FRISCH’S “PROVISO”

Ginoux and Jovanovic argue that Frisch (Reference Frisch1933, p. 191) gave a “proviso” that “the sum of the coefficients $ {k}_j $ , which are the weight of each cycle, must be equal to unity” (p. 2). My interpretation of Frisch’s paragraph, which I detailed in my contribution to a symposium in History of Economic Ideas (Carret Reference Carret2022c), is that Frisch was merely pointing out that a particular solution of an non-homogeneous system should be counted only once in the general solution. A homogeneous system is one where the dynamic equation is equal to zero. A non-homogeneous system is one where the dynamic system is equal to a constant or a forcing function. The homogeneous system is solved by each of the individual solutions in (18) given by Frisch, and also by any linear combination of them. But Frisch said, rightly, that in the case of the non-homogeneous system,

The terms $ {a}_{\ast } $ , $ {b}_{\ast } $ and $ {c}_{\ast } $ are the particular solution of his system. Frisch’s point, particularly apparent in the last sentence, is that when we add two functions that have both one solution of the homogeneous system and one particular solution, if we want to count the particular solution only once, the coefficients by which we add the two functions must sum to 1. I have not “ignored a well-known theorem” (p. 5) but tried to make sense of what Frisch was saying with a simple example (Carret Reference Carret2022c). This is in any event a hypothetical case that does not give license to add cycles together with ad hoc “weights”; the “proviso” is just an illustration of a well-known fact, and any solution having only one particular solution will satisfy Frisch’s proviso.

However, Ginoux and Jovanovic take this paragraph as a license to add cycles with arbitrary numbers in front of them. With their latest set of “weights” (Ginoux and Jovanovic Reference Ginoux and Jovanovic2022), they multiply the amplitude of the first cycle by 1, the amplitude of the second by 30, and the amplitude of the third by -30. But an elementary fact of dynamic systems is that the amplitude (and the phase in the case of cycles) is given by initial conditions of the system, as Frisch himself knew (Frisch Reference Frisch1933, p. 183). When they multiply Frisch’s cyclical components by a factor of 30, Ginoux and Jovanovic are multiplying the amplitudes of these cycles without relating them to any initial conditions. There are many sets of three numbers that sum to 1; the key is not to produce those numbers out of thin air but to relate them to initial conditions, which govern the amplitude of the cycles. Frisch’s “proviso” was not a license to multiply the amplitude of individual cycles but the mere recognition that one should not write the particular solution more than once in a general solution of the dynamic system.

I also note that there is no “closure relation,” and Ginoux and Jovanovic transform what Lionello Punzo said when they refer to his comments published in History of Economic Ideas (Punzo Reference Punzo2022). Ginoux and Jovanovic cite his comment as supporting their demonstration (pp. 2, 8), but Punzo is talking about the parameters of the system and not “weights” when he speaks of a “closure set”:

What Punzo argued is that Frisch chose parameters on a special area in the parameter space, a choice that made his model not fluctuate. Reading Punzo, I find in fact that he agrees with Zambelli that Frisch’s model does not fluctuate for his original parameters but argues against Zambelli that it was wrong to say that this was a general case. What Zambelli has not proven is that the model could never fluctuate, and in fact, with my solution using the Laplace transform, I was able to find other parameters for which Frisch’s propagation model oscillates.

The “solution” given by Ginoux and Jovanovic rests on one paragraph of Frisch that led them to introduce new constants whose origin is unknown, and unrelated to any initial conditions. This interpretation, I argued here, is not correct. I pointed out this problem to the authors in my comments published in History of Economic Ideas (Carret Reference Carret2022c), including the fact that the sum of these constants was not even equal to 1 as they claimed (Ginoux and Jovanovic Reference Ginoux and Jovanovic2023, p. 19). The authors recently published a correction of one of their papers with new constants without acknowledging our debate and my comments (Ginoux and Jovanovic Reference Ginoux and Jovanovic2022).

IV. AN ECONOMIC ARGUMENT?

Ginoux and Jovanovic try to elaborate an economic justification for the existence of “weights” in front of the cycles at the beginning of their comments. They start by arguing that the “proviso” is necessary for economic reasons: “The closure relation implies that Frisch considered that these different cycles (such as Kitchin and Juglar cycles), which do not have the same origin according to the economic theories, do not impact the economic activity with the same amplitude” (p. 504). But in fact all of Frisch’s cycles, as well as the trend, come from the same economic mechanism. Ginoux and Jovanovic go on to say that “Carret, however, states that these different cycles impact the economic activity with the same amplitude” (p. 504). I did not claim that, but in fact the opposite: “it [the primary cycle] has a relatively high magnitude (compared with the trend and other cyclical components)” (Carret Reference Carret2022a, p. 632). Figure 2 of my paper shows clearly that the first cycle has a bigger amplitude than the second. And, again, the amplitude of each cycle is determined by the initial conditions and not by “weights” that would be arbitrarily chosen. When Ginoux and Jovanovic chose “weights” of 1, 30, and -30, they did not change the amplitude of the 8.5-year cycle but multiplied the amplitude of the 3.5-year cycle by 30 and of the smaller, 2.2-year cycle by -30. They provide no economic explanation for this.

The second “fundamental problem” is: “What business cycle did Carret try to replicate with his model?” (p. 3). I was not trying to reproduce a business cycle but to show how we could obtain apparent fluctuations along a trend line in Frisch’s model, and I changed Frisch’s parameters because his model does not fluctuate along the trend for his original parameters. Ginoux and Jovanovic also misrepresent what I argued (p. 3). I indeed obtained a primary cycle that was shorter than Frisch’s, but when Ginoux and Jovanovic say, “In Carret’s view, such differences do not represent an issue: ‘do[es] not think that it necessarily is [a problem]’ (Reference Carret2022a, p. 633)” (p. 3), they are mixing different paragraphs of my paper. In the complete paragraph I explain the “caveat” that the model’s damping is much slower with my parameters:

Let me point out again that I was interested in showing an example of the superposition of trend and cycles, and not in reproducing a business cycle. I also invite the reader to try other parameters by visiting the interactive application I built.Footnote ³

Was my work without merit? Beyond my historical contributions, I was able to point to several properties of Frisch’s model through my solution using the Laplace transform. I checked my results by solving the model using two more additional approaches, by discretizing the model, and by using a differential equation solver in R. The discretized solution can be found in my working paper (Carret Reference Carret2020, pp. 25–26), in the book with Assous (Assous and Carret Reference Assous and Carret2022, pp. 83–84), and in the code I wrote in R.Footnote ⁴