I. CARRET’S HYPOTHESES HAVE NO ECONOMIC JUSTIFICATION

In his criticism of our work, Vincent Carret (Reference Carret2022a) states that we made a mistaken interpretation of Ragnar Frisch’s paragraph in which he expressed a condition that must be respected in order to obtain the general solution. Frisch called it a “proviso.” According to the Cambridge Dictionary, a proviso is “a statement in an agreement, saying that a particular thing must happen before another can.” In modern terms, Frisch’s proviso is a closure relation, according to which the sum of the coefficients k_j, which are the weight of each cycle, must be equal to unity. With this closure relation, Frisch normalized the weight of each cycle.

We recently demonstrated that this closure relation must be considered in Frisch’s work; a point that Lionello Punzo (Reference Punzo2022) confirmed in a symposium published in History of Economic Ideas following our demonstration that Stefano Zambelli’s assertion is based on a mathematical error, and consequently does not hold (Ginoux and Jovanovic Reference Ginoux and Jovanovic2022a). Punzo (Reference Punzo2022) also validated our result by claiming that Zambelli failed to prove that “the rocking horse does not rock.” Consequently, Zambelli’s assertion is pure speculation, without any mathematical or numerical demonstration. Taking Zambelli’s work as a starting point, Carret reproduced the same errors. Moreover, since the value of our coefficients respect Frisch’s closure relation, they necessarily respect Frisch’s initial conditions and cannot be arbitrary (Ginoux and Jovanovic Reference Ginoux and Jovanovic2022b, pp. 180–181), a fact that Carret fails to understand, as shown by his criticism in his JHET article.

We would like to address here another fundamental problem that Carret avoids answering in his criticism of our work: Why did he ignore this closing relationship in his publications? Indeed, he has never explained his choice to ignore this condition that must be respected by definition. This question is crucial since Frisch used this condition for some economic reasons, as we will clarify here.

In his model, Carret does not normalize the weight of each cycle, as Frisch did; on the contrary, Carret assumes that all coefficients k_j have the same value, namely 1, and then that $ {k}_1={k}_2={k}_3=\dots =1 $ . Does Carret’s hypothesis make sense from an economic viewpoint? With his model, Frisch reproduced two cycles observed in the literature at his time, i.e., the primary cycle of 8.57 years and the secondary cycle of 3.50 years (the tertiary cycle of 2.20 years is a prediction made by Frisch). 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. Carret, however, states that these different cycles impact the economic activity with the same amplitude. Unfortunately, there is no economic justification for Carret’s hypothesis, as explained, for instance, by Joseph Schumpeter (Reference Schumpeter1939) and more recently by Muriel Dal Pont Legrand and Harald Hagemann (Reference Dal Pont Legrand and Hagemann2007, p. 12).

This is not the only fundamental problem that Carret avoids answering. What business cycle did Carret try to replicate with his model? Because Carret (Reference Carret2022a) ignored Frisch’s closure relation, which ensures that Frisch’s system oscillates, he was obliged to change the value of the parameters to obtain oscillations (Ginoux and Jovanovic Reference Ginoux and Jovanovic2022b). The problem here is that Carret dismisses the consequences of his choice too easily. Indeed, Carret (Reference Carret2022a, p. 632) obtained “a primary cycle with a period of about 6.5 years, a secondary cycle with a period of about 3.2 years … all values rather close to those in Frisch’s article.” 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).

Of course this is a problem. Contrary to what Carret claims, the period of his primary cycle (6.5 years) is not “rather close to” the cycle observed in the literature (an 8.5-year cycle) and which was reproduced by Frisch. An 8.5-year cycle is very different from a 6.5-year cycle: over twenty years, we will have two cycles, and therefore two economic recessions in one case, and three in the other. That is very different. Moreover, a 6.5-year cycle does not correspond to any known cycle in history of economics (Juglar cycles are eight to ten years).

Then, Carret (Reference Carret2022a, p. 633) admits that “[t]here is, however, one caveat, compared with Frisch’s original article: in order to obtain apparent cycles at the aggregate level, we had to decrease the damping of the system. In fact, the return to equilibrium is much longer than in Frisch’s original article.”

Indeed, Carret’s “damping exponent” for the first cycle is 8.5 times lower than Frisch’s.Footnote ¹ This means that Carret’s general solution takes 8.5 times more time to damp, as seen in Carret’s figures 1 and 2 (see the horizontal axis, which extends for a century). What economic significance should be made of an econometric model that takes 100 years to return to the stationary level? Did we observe such behavior in Frisch’s time? Not at all.

To make his general solution oscillate for a while, Carret tuned the values of Frisch’s parameters, and left out elements of Frisch’s demonstration when it did not suit him. For instance, his parameter λ is six times higher than Frisch’s in order to increase the importance of the transient regime until reaching the stationary level (horizontal asymptote at 0.18 in Carret’s figures 1 and 2), which obviously does not oscillate around it. In fact, Carret worked on a solution of Frisch’s model that is different from the original one. As we can see in Table 1, all of Carret’s parameters are different from Frisch’s, except ε and c.

Table 1. Comparison between the Parameters Used

^** : Carret did not provide the value he used for c in his demonstrations. However, we were able to calculate it based on the value he used for $ {a}_{\ast } $ . By using Frisch’s characteristic equations (1933, p. 184, eq. 12 and 13) and Carret’s parameters, we found $ {a}_{\ast }=0.1833 $ . This value is consistent with what we observe on Carret’s fig. 1 (Reference Carret2022a, p. 632). Then, since Frisch (Reference Frisch1933, p. 188, eq. 17) stated $ c=\lambda {a}_{\ast}\left(r+ sm\right) $ , we have $ c=0.3\ast 0.1833\ast \left(1+2\ast 1\right)=0.165 $ .

In his work, Carret does not make it possible to reproduce the observed cycles that Frisch sought to reproduce. Therefore, he ignores the problem that Frisch was faced with. Moreover, because his model does not have the same economic behavior as Frisch’s, his results cannot be directly compared with those of Frisch. Carret has not explained why he ignored Frisch’s closure condition (or closure relation). By ignoring its role, Carret’s results have no merit in either mathematics or economics.

II. CARRET IGNORED A WELL-KNOWN THEOREM

In footnote 19, to criticize our work, Carret refers to another article (Reference Carret2022b) in which he claimed to have demonstrated our mistake. Specifically, Carret (Reference Carret2022b, p. 168) states that Frisch’s paragraph on page 191 refers to the resolution of a system with a homogenous part and a non-homogeneous part. He clarifies by stating that “[a] combination of a homogeneous solution [i.e., Frisch’s eq. 18] and a particular solution [i.e., $ {a}_{\ast } $ , $ {b}_{\ast } $ and $ {c}_{\ast } $ ] will thus solve the system,” and “the functions in [Frisch’s eq.] (16) solved [the non-homogeneous system] because they were composed of the sum of the particular solutions a _*, b _* and c _* and the trends which solved respectively the nonhomogeneous and the homogeneous parts of the system.”

Then, to explain Frisch’s proviso, Carret claims that

$$ {r}_0{x}_0={r}_1{x}_1={a}_{\ast}\left({r}_0+{r}_1\right)+{r}_0{a}_0{e}^{\rho_0t}+{r}_1{A}_1{e}^{-{\beta}_1t}\sin \left({\phi}_1+{\alpha}_1t\right). $$

Obviously, Carret is referring indirectly to the Superposition Theorem.

Unfortunately, Carret has misunderstood this theorem and applied it poorly. Indeed, this theorem is “an existence and a uniqueness theorem” (Tenenbaum and Pollard Reference Tenenbaum and Pollard1985, p. 208). It states that we must look:

• – first, for a particular solution to the non-homogeneous equation for $ c\ne 0 $ ;

• – second, for a general solution to the homogeneous equation for $ c=0 $ , which is given by a linear combination of linearly independent solutions;

• – and third, for the general solution of the non-homogeneous equation, which will be equal to the sum of the particular and general solutions (Tenenbaum and Pollard Reference Tenenbaum and Pollard1985, p. 208; Warusfel Reference Warusfel1966, p. 138).

Given that $ {a}_{\ast } $ is a particular solution, and that $ {a}_0{e}^{\rho_0t} $ and $ {A}_1{e}^{-{\beta}_1t}\sin \left({\phi}_1+{\alpha}_1t\right) $ are general solutions, Carret clearly did not respect the theorem. Indeed, he made a linear combination of a mixture of both particular and general solutions that led him to count the particular solution twice, which we never did. The superposition theorem requires precisely that the particular solution be counted only once.

In fact, Carret (Reference Carret2022a, p. 631, eq. 5) should have written the general solution of Frisch’s system as:

(1) $$ x(t)=\underset{(P)}{\underbrace{a_{\ast }}}+\underset{(H)}{\underbrace{x_0^H(t)+{x}_C^H(t)}}=\underset{(T)}{\underbrace{a_{\ast }+{a}_0{e}^{-{\beta}_0t}}}+\underset{(C)}{\underbrace{\sum \limits_{j=1}^{\infty }{k}_j{A}_j{e}^{-{\beta}_jt}\sin \left({\phi}_j+{\alpha}_jt\right)}} $$

Where (P) represents the particular solution, (H) the homogeneous solution, (T) the trend, and (C) the cyclic components;Footnote ² $ {a}_{\ast } $ is the particular solution while $ {x}_0^H(t)={a}_0{e}^{\rho_0t}={a}_0{e}^{-{\beta}_0t} $ and $ {x}_C^H(t)=\sum \limits_{j=1}^{\infty }{k}_j{A}_j{e}^{-{\beta}_jt}\sin \left({\phi}_j+{\alpha}_jt\right) $ are the general solutions of the homogeneous equation. This latter is a linear combination of Frisch’s cyclical components.

By claiming that we have misinterpreted Frisch’s paragraph, Carret attempts to draw readers into a false debate. Careful readers will have noted that in his criticism, Carret (Reference Carret2022b, p. 168) did not discuss the fact that “the sum of the coefficients [must be] equal to unity” in order to add the constant terms $ {a}_{\ast } $ , $ {b}_{\ast } $ , and $ {c}_{\ast } $ “to (18) in order to get a correct solution.” We suspect that Carret did not want to discuss the coefficients k_j, because this would lead him to question the superposition theorem and consequently the existence of a general solution in Frisch’s model; especially since the superposition theorem applies even when using the Laplace transform. Therefore, it is no coincidence that Carret criticizes our work in this paragraph: as soon as we respect Frisch’s closure relation, which follows from the superposition theorem, Carret’s whole demonstration collapses and his “results” on Frisch’s model would have to be abandoned.

III. SOLVING CARRET’S PUZZLE

Now let us solve Carret’s puzzle about the error he purports to have demonstrated in Frisch’s work. As Carret has rightly mentioned in all of his publications, Frisch gave different initial conditions for the trends and the cycles. Carret claims that this was a serious error: “Frisch erred, because he gave different initial conditions for the trend and for the cycles, even though they should depend on the same initial development” (Reference Carret2022b, p. 165).

This problem is crucial for Carret, since he has built his argumentation on this element and presents it as one of his major contributions: “With the hindsight of a more complete theory of mixed differential–difference equations, we can show analytically by using the Laplace transform that all components, whether cyclical or oscillatory, will depend on the same initial conditions” (Assous and Carret Reference Assous and Carret2022, p. 67).

In fact, there is no error in Frisch’s work. In the general case, Carret is correct: we should have the same initial condition for the trend and the cyclical components. But Frisch managed to find a particular case that works. Punzo (Reference Punzo2022, p. 174) also pointed this out, leading him to claim that it is “almost an honor” for Frisch to “find one such constellation,” i.e., the three cyclical components and their periods. Carret has missed this in all of his publications, including his JHET article (Carret Reference Carret2022a, pp. 634, 637), overlooking the originality of Frisch’s work. Indeed, this “honor” results from Frisch’s calibration with his econometric model (Ginoux and Jovanovic Reference Ginoux and Jovanovic2022b, p. 179).

This result demonstrates that Carret attempts to make the reader believe that he has shown something that is not in Frisch’s writings, but this is not true.

IV. CARRET’S DEMONSTRATION IS INCOMPLETE AND UNVERIFIABLE

Carret’s criticism hides a major methodological difference between our work and his: Carret does not give the information needed to reproduce his work; we do. Therefore, anyone can verify our conclusions, but no one can verify Carret’s work. For proof, in footnote 19 of his JHET article (2022a, p. 630), Carret criticizes our work by invoking his “solution based on the Laplace transform” and its inverse. However, he does not provide the demonstration of his analytical solution. He explains that “[the] full derivation of this solution is published in Assous and Carret (Reference Assous and Carret2022).”

Unfortunately, Michaël Assous and Carret (Reference Assous and Carret2022) did not provide all the calculations of the inverse Laplace transform they used. Carret’s work is therefore unverifiable. This is embarrassing, because the Laplace transform and its inverse underpin all Carret’s analysis and arguments. It is on this basis that he could claim that Frisch “erred,” or made “an error,” or that he “can give a more elegant answer than Frisch” (Carret Reference Carret2022a, p. 630). Moreover, according to Carret, the Laplace transform and its inverse are “modern mathematical tools that [Frisch] did not know” (Reference Carret2022a, p. 624). This is an astonishing claim to make, given that the Laplace transform was introduced in 1737, that the first use of its modern formulation dates back to 1910, and that in “the 1920s and 1930s it was seen as a topic of front-line research” (Deakin Reference Deakin1992, p. 265).

This is not our only issue with Carret’s “solution based on the Laplace transform.” As Roy Allen (Reference Allen1959, pp. 155–156) explained, the Laplace transform is a “trick” of mathematicians. One of the main problems with this trick is that when we use the Laplace transform and its inverse, we automatically introduce new constants (i.e., new initial conditions). Thus, “when the solution is obtained, it has the initial conditions ‘built in,’” and “n arbitrary constants, to be ‘fitted’ or evaluated with great labor from the initial conditions” (Allen Reference Allen1959, p. 159). In other words, to solve a system similar to Frisch’s with a Laplace transform and its inverse, we have to generate at least one or two new initial conditions that are arbitrary by construction.

It is surprising, even shocking, that Carret keeps silent about this problem in his work, while he constantly criticizes Frisch on the value of his initial conditions. More vexatious is the fact that in none of his publications does Carret provide the value for his initial condition(s), including the new ones he introduced with the Laplace transform. Specifically, since he did not provide the expression of x(t) for $ 0\le t\le \varepsilon $ , we cannot compute the values of his $ {A}_i $ and $ {k}_i $ used in his equation $ {A}_i=2\left|{k}_i\right| $ . It is legitimate to ask why he does not bother to provide the initial conditions he used in his work. By not providing his initial condition(s) and all of his calculations for his inverse Laplace transform, Carret ensures that no one can reproduce and thus verify his work, as is expected in a scientific work.