Jekyll2021-01-26T21:25:40+01:00https://hetmodels.com/feed.xmlHET ModelsVincent CarretKaldor’s model of the trade cycle: dynamics and a numerical example2021-01-15T00:00:00+01:002021-01-15T00:00:00+01:00https://hetmodels.com/articles/Kaldor-nonlinear-cycles<p>In 1940, Nicholas Kaldor published in the <em>Economic Journal</em> a famous article presenting a new model of the trade cycle.<sup id="fnref:kaldor"><a href="#fn:kaldor" class="footnote">1</a></sup> Its main mechanism relied on the discrepancies between ex-ante saving and investment to explain the changes in the level of activity, and it introduced two important variations to this idea: (i), saving and investments were themselves functions of the level of activity, and (ii) the stock of capital was slowly varying. This make of his model a medium-run model of economic fluctuations, where oscillations arise endogeneously because of the non-linear relations between the two curves of investment and savings.</p>
<p>While Kaldor did not construct a fully worked out model, he suggested a graphical interpretation that was very clear and ensured his ideas had a great influence. Several economists sought to translate them into a formal model, and one of the most successful of such interpretations was given by Chang and Smyth in an article published in the <em>Review of Economic Studies</em> in 1971.<sup id="fnref:chang_smyth"><a href="#fn:chang_smyth" class="footnote">2</a></sup> In this post, I go back on the original mechanism of Kaldor’s model and compare it with the model formulated by Chang and Smyth. I then build an interactive version of the model that tries to encapsulate their ideas, with an estimation of the nonlinearity in the savings function.</p>
<h2 id="the-mechanism-of-fluctuations-in-kaldors-original-model">The mechanism of fluctuations in Kaldor’s original model</h2>
<p>While there had already been many influential models of the trade cycle, Kaldor thought that they lacked some explanatory powers, in particular because they postulated either too much or too little stability, in particular because they were based on mostly linear relationships:<sup id="fnref:but_tinbergen"><a href="#fn:but_tinbergen" class="footnote">3</a></sup></p>
<blockquote>
<p>“Since thus neither of these two assumptions can be justified, we are left with the conclusion that the I(x) and S(x) functions cannot both be linear, at any rate over the entire range. And, in fact, on closer examination, there are good reasons for supposing that neither of them is linear.” (Kaldor, 1940: 81)</p>
</blockquote>
<p>The curved form of the investment and saving functions were based on first principles, for instance the fact that high incomes meant not only that a higher amount was saved, but a higher <em>proportion</em> of total income was saved. At the opposite end, under a certain level of income, nothing was saved. Inversely, for investment, the nonlinearities can be thought of as maximum and minimum levels of investments determined by general technical and institutional conditions (such as the current stock of capital). The form of the curves ensured that they met at three points corresponding to three different levels of income, where the highest and lowest equilibrium are stable, and the middle one unstable.<sup id="fnref:stability"><a href="#fn:stability" class="footnote">4</a></sup> The economy can thus settle either at a high level of activity (<script type="math/tex">X</script>) or a low level, which can be seen in the “Stage I” picture of the following figure, taken from Kaldor’s article.</p>
<p><img src="/assets/kaldor_1940/trade_cycle.png" /></p>
<p>In Kaldor’s model, fluctuations actually happen between the high and the low equilibria, because these equilibria are only stable in the short run: “as activity continues at either one of these levels, forces gradually accumulate which sooner or later will render that particular position unstable” (<em>ibid</em>.: 83).</p>
<p>To explain why the equilibria become unstable, it is necessary to introduce the medium-run evolution of both investment and saving curves when the economy is at the high or at the low equilibria. At the high equilibrium, the stock of capital is gradually increasing, and so is the total amount of consumption goods produced. The level of savings is increasing and the S curve shifts upward, while investment is depressed and shifted down by the tightening of profit opportunities, due to the accumulation of capital. At some point the economy will reach stage three where there are only two equilibria left, and the high equilibrium is unstable downward: savings become higher than investment and the level of activity will decrease until they are equal again - at the low equilibrium A.</p>
<p>Once the economy reaches the low level equilibrium, the inverse movement will happen: the level of savings decrease and S shifts downward, while investment opportunities arise from the decumulation of capital. At some point there is two equilibria again, but this time the low equilibrium is upward unstable and the high equilibrium becomes a stable attractor, and the cycle between the two level of activity will endlessly repeat - unless something is done to stop it, that is, to prevent the high equilibrium from becoming unstable.</p>
<p>Kaldor thus ends his exploration of this idea with some considerations on the amplitude and duration of those cycles, and the implications for economic policy. The most important consequence is that it is necessary to ensure the level of investment is high enough to avoid that the high equilibrium becomes unstable; there is obviously a strong role the government can play to keep investment opportunities high, and to intervene directly on this schedule when the private sector is letting it shift down.</p>
<h2 id="chang-and-smyths-model-capital-and-income-dynamics">Chang and Smyth’s model: capital and income dynamics</h2>
<p>Chang and Smyth proposed a dynamic formalisation of Kaldor’s model, that clarified important points that were missed by Kaldor, in particular the fact that the initial conditions and the parameter values were crucial for a limit cycle to arise and to avoid a monotonous convergence to a stationary-state equilibrium.</p>
<p>The model is based on a system of nonlinear differential equations with two variables, capital and income. Investment and savings depend on the level of income (positively) and the stock of capital (negatively). The dynamic of income depends on the difference between investment and savings, as in Kaldor’s model, and on a speed of adjustment <script type="math/tex">\alpha</script>:</p>
<script type="math/tex; mode=display">\frac{dY}{dt} = \alpha[I(Y,K) - S(Y,K)]</script>
<p>For the dynamics of capital, several interpretations have been proposed in the litterature, as underlined by Chang and Smyth (1971: 39, n.2). They retain the simplest possible evolution, where the accumulation of capital is only a function of net investment:</p>
<script type="math/tex; mode=display">\frac{dK}{dt} = I(Y,K)</script>
<p>The authors then proceed to show that Kaldor’s conditions for the presence of limit cycles are not necessary nor sufficient in the general case by using the Poincaré Bendixson theorem. It is however regrettable that they do not give a numerical example for their model, and limit themselves to a hand-drawn figure of the locuses of points where investment equals savings and where net investment is null (that is, the set of points in the Y-K plane where income and capital are respectively stationary). Their diagram is reproduced below.</p>
<p><img src="/assets/kaldor_1940/chang_smyth.png" /></p>
<p>It is apparent from their model that there is one point of stationary equilibrium, <script type="math/tex">e</script>, where both capital and income are stationary. The last section of their paper proves that Kaldor’s model will usually converge to this point, although there exists a set of points who give rise to the type of limit cycle represented in the picture. There can be several limit cycles, in which case the initial conditions will determined the ultimate cyclical behaviour of the economy.</p>
<p>Contrary to Kaldor, they do not seem to assume that the dynamics of the capital stock depends on its depreciation, although this was a crucial hypothesis in Kaldor’s article, who argued that “at the level of investment corresponding to A investment is not sufficient to cover replacement, so that net investment in industrial plant and equipment is negative” (Kaldor, 1940: 84). This “decumulation” of capital had the effect to increase investment opportunities and to decrease savings “in so far as it causes real income per unit of activity to fall” (<em>ibid</em>.: 85).</p>
<p>This makes the solution of their system quite strange in fact because if one looks at the two equations above (which are directly taken from their theorem p.41), we can readily see that income and capital will be stationary only when investment and savings are both zero. I suppose one could argue that this corresponds to a situation where we only take into account net investment and savings but I do not see how to reconciliate this with the picture drawn by Kaldor where investment and savings are always positive. Thus we slightly change their equation for the evolution of the capital stock to show the effect of capital depreciation, which allows us to obtain stationary points for positive values of investment and savings:</p>
<script type="math/tex; mode=display">\frac{dK}{dt} = I(Y,K) - \delta K</script>
<p>Where <script type="math/tex">\delta</script> is a measure of the rate of depreciation of the stock of capital. To understand this behavior better, and because this blog is more oriented toward numerical computation, I skip their proof and present in the last section of this post a specification of the forms of I and S that give rise to a limit cycle for different values of parameters.</p>
<h2 id="numerical-specification-of-chang-and-smyths-model">Numerical specification of Chang and Smyth’s model</h2>
<p>In this last part, I search for a model of the investment and saving functions that make sense economically, and that exhibit the required properties giving rise to limit cycles. To simplify the task, I use data from the St. Louis Federal Reserve<sup id="fnref:data_source"><a href="#fn:data_source" class="footnote">5</a></sup> and use simple regression methods to find statistically significant relations between income, capital, investment and savings.<sup id="fnref:not_ideal"><a href="#fn:not_ideal" class="footnote">6</a></sup></p>
<p>While both Kaldor (1940: 82) and Chang and Smyth (1971: 38) underline that it is not necessary for both the investment and saving curves to be nonlinear, I found that it was easiest to find the right locus in the K-Y plane with nonlinear forms for both. The model retained for the investment function is:</p>
<script type="math/tex; mode=display">I(Y,K) = I_y \cdot Y + I_k \cdot K + I_{ky} \cdot Y \cdot K + \bar{I}</script>
<p>Where <script type="math/tex">I_y</script>, <script type="math/tex">I_k</script>, <script type="math/tex">I_{ky}</script> and <script type="math/tex">\bar{I}</script> are all parameters estimated with a simple least square regression. The coefficients are summarised in the following table:</p>
<p><img class="coef_table" src="/assets/kaldor_1940/inv_coefs.png" /></p>
<p>A nonlinear form of the savings curve can be found in an article by Klein and Preston,<sup id="fnref:klein_preston"><a href="#fn:klein_preston" class="footnote">7</a></sup> who estimate a logistic relation between consumption and income based on UK data, to construct a nonlinear limit cycle of Kaldor’s type. At first I just used a cubic model that showed an undeniable improvement over simple linear or quadratic relations while a quartic model did not bring anything to the fitting. The cubic model also exhibits the required properties (see the figure below).</p>
<p><img id="plot" src="/assets/kaldor_1940/savings.png" /></p>
<p>The final model also includes a cross term between income and capital that gave rise to a better behaved locus of stationarity in the phase plane:</p>
<script type="math/tex; mode=display">S(Y,K) = S_{y} \cdot Y + S_{y2} \cdot Y^2 + S_{y3} \cdot Y^3 + S_{yk} \cdot Y \cdot K + \bar{S}</script>
<p>We obtain the parameters summarised in the following table for this model.</p>
<p><img class="coef_table" src="/assets/kaldor_1940/sav_coefs.png" /></p>
<p>The fact that we have a negative intercept is in line with the idea that under a certain level of income, savings are equal to zero. Inserting those two equations into our dynamic system, we get a nonlinear model where limit cycles easily arise. In the following application, the initial conditions for income and capital are set to their 2013 values, the depreciation of capital is set to <script type="math/tex">\delta = 0.02</script> and the speed of adjustment is set to <script type="math/tex">\alpha = 0.1</script>. The parameters of the investment and saving functions are set to the estimated parameters above but they can be changed as well. A description of the different panels and possible movements follows the application.</p>
<iframe src="/assets/kaldor_1940" width="100%" height="700px"></iframe>
<p>There are four panels in the application. The first one is a 3d plot with our data points, fitted values, and the trajectory of investment and savings determined by the model. For the initial parameters chosen, we see that investment, savings, capital and income follow somewhat closely the real values, and converge to a point of equilibrium rather close to the real point for 2019, the last point in our data set (does this mean that, without the current crisis, we would have reached an equilibrium? At least that’s what our peculiar model seems to tell us).</p>
<p>The second panel pictures the investment and savings curve estimated above, and parameterized with the initial level of capital. The phase diagram (third panel), where the locus of stationarity for income (blue) and capital (green) are represented, makes clear that we have attained an upper equilibrium in a model with three equilibria. Had we started from a different initial position, as can be easily seen by changing <script type="math/tex">Y(0)</script> and <script type="math/tex">K(0)</script>, our model economy could have actually collapsed to the lower equilibrium. The middle equilibrium is of course unstable. Finally, we can see the evolution of the stock of capital and of income in the fourth panel (‘Trajectories’).</p>
<p>To visualize the mechanism of Kaldor’s model, we need to find a limit cycle. The reader that has gone this far will have no trouble to find one by changing just about any parameter in this (kind of wacky) model. Let’s just point out that, as remarked by Chang and Smyth, changes in the speed of adjustment can have dramatic results on the behavior of the model, without changing the phase diagram and the number of equilibria. Increasing the value of <script type="math/tex">\alpha</script> to 0.3 thus will give rise to a limit cycle of gigantic proportions (unfortunately, most limit cycles for those parameters bears those proportions that lead to a complete collapse of income below the zero level - but now that we have come so far, let’s not allow this to stop us from interpreting the cycle).</p>
<p>While the limit cycle is perhaps most clearly seen in the ‘Phase diagram’ and ‘Trajectories’ panels (increasing the “Final time” also helps), the 3d plot in the first panel gives us the most interesting way to illustrate Kaldor’s mechanism. After an initial increase much higher than with the lower speed of adjustment, both investment and saving show a sharp reversal, with savings consistently above investment for the whole way down to extremely low levels of income, capital, investment and savings. This corresponds to stage III of Kaldor’s model, when the savings curve is above the investment curve and the economy slides to the lower equilibrium (in the (gdp, savings / investment) plane of Kaldor’s figure 6).</p>
<p><img src="/assets/kaldor_1940/way_down.png" /></p>
<p>At some point during this catastrophic fall, investment starts to go up again and eventually crosses above the level of savings at a very low level of income and capital. The economy can start its long path toward an even higher equilibrium than before.</p>
<p><img src="/assets/kaldor_1940/way_up.png" /></p>
<p>While savings are initially way below investment (or in other words, people are borrowing heavily and investment soars), it starts to soar when income has sufficiently increased and the rapid accumulation of capital slows down the increase of investment, until savings catches up and after overtaking investment, triggers a new collapse. After that, the cycle repeats in a rather pleasant looking loop, although not a very accurate one from a purely economic point of view. But the movement corresponds rather well to the mechanism described in 1940 by Kaldor.</p>
<p><img src="/assets/kaldor_1940/limit_cycle.png" /></p>
<h2 id="notes">Notes</h2>
<div class="footnotes">
<ol>
<li id="fn:kaldor">
<p>Kaldor, Nicholas. 1940. “A Model of the Trade Cycle.” <em>The Economic Journal</em> 50(197):78–92. doi: 10.2307/2225740. url: <a href="http://www.jstor.org/stable/2225740">http://www.jstor.org/stable/2225740</a> <a href="#fnref:kaldor" class="reversefootnote">↩</a></p>
</li>
<li id="fn:chang_smyth">
<p>Chang, W. W., and D. J. Smyth. 1971. “The Existence and Persistence of Cycles in a Non-Linear Model: Kaldor’s 1940 Model Re-Examined.” <em>The Review of Economic Studies</em> 38(1):37–44. doi: 10.2307/2296620. url: <a href="http://www.jstor.org/stable/2296620">http://www.jstor.org/stable/2296620</a>. Another mathematical formalisation of Kaldor’s model can be found in Black, J. 1956. “A Note on Mr. Kaldor’s Trade Cycle Model.” <em>Oxford Economic Papers</em> 8(2):151–63. doi: 10.1093/oxfordjournals.oep.a042259. url: <a href="https://academic.oup.com/oep/article/8/2/151/2362202">https://academic.oup.com/oep/article/8/2/151/2362202</a> <a href="#fnref:chang_smyth" class="reversefootnote">↩</a></p>
</li>
<li id="fn:but_tinbergen">
<p>Although my current work with Michaël Assous on the early macrodynamic models has shown that this was not always the case; in particular, Jan Tinbergen built several models that could explain both local, limited fluctuations maintained around an equilibrium point and the possibility of global instability and collapse either to a low equilibrium level or to a zero level in the most extreme cases. See in particular our WP “Jan Tinbergen’s early contribution to macrodynamics (1932-1936): multiple equilibria, complete collapse and the Great Depression” <a href="https://halshs.archives-ouvertes.fr/halshs-03087375">https://halshs.archives-ouvertes.fr/halshs-03087375</a> and a forthcoming book on the subject. <a href="#fnref:but_tinbergen" class="reversefootnote">↩</a></p>
</li>
<li id="fn:stability">
<p>The stability of those equilibria can easily be seen graphically: whenever savings is above investment, the activity will have a tendency to decrease to reestablish the equilibrium between them. On the contrary if investment is above saving, the activity will increase until savings are at the level of investment. Thus we see that going above and on the right of point C, the level of activity will have a tendency to increase to point B, etc. <a href="#fnref:stability" class="reversefootnote">↩</a></p>
</li>
<li id="fn:data_source">
<p>The data used to estimate the relationships between investment, savings, GDP and the stock of capital are taken from the Federal Reserve Bank of Saint Louis website. The series used are <a href="https://fred.stlouisfed.org/series/PSAVERT">Personal Saving Rate</a>, <a href="https://fred.stlouisfed.org/series/GDPC1">Real Gross Domestic Product</a>, <a href="https://fred.stlouisfed.org/series/W171RC1Q027SBEA">Net domestic investment</a> and <a href="https://fred.stlouisfed.org/series/RKNANPUSA666NRUG">Capital Stock</a>. <a href="#fnref:data_source" class="reversefootnote">↩</a></p>
</li>
<li id="fn:not_ideal">
<p>This econometric approach is obviously not very rigorous and it should in no way be taken at face value but rather as an easy way to specify our functions’ shapes to explore numerically the perception of cycles contained in Kaldor’s article and Chang and Smyth’s article. The whole point of this blog is really to develop new computational tools and visualization to understand the history of economic thought and models and I hope the reader will excuse the casual treatment of econometric relationships. <a href="#fnref:not_ideal" class="reversefootnote">↩</a></p>
</li>
<li id="fn:klein_preston">
<p>Klein, L. R., and R. S. Preston. 1969. “Stochastic Nonlinear Models.” <em>Econometrica</em> 37(1):95–106. doi: 10.2307/1909208. url: <a href="http://www.jstor.org/stable/1909208">http://www.jstor.org/stable/1909208</a> <a href="#fnref:klein_preston" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>Vincent CarretIn 1940, Nicholas Kaldor published in the Economic Journal a famous article presenting a new model of the trade cycle.1 Its main mechanism relied on the discrepancies between ex-ante saving and investment to explain the changes in the level of activity, and it introduced two important variations to this idea: (i), saving and investments were themselves functions of the level of activity, and (ii) the stock of capital was slowly varying. This make of his model a medium-run model of economic fluctuations, where oscillations arise endogeneously because of the non-linear relations between the two curves of investment and savings. Kaldor, Nicholas. 1940. “A Model of the Trade Cycle.” The Economic Journal 50(197):78–92. doi: 10.2307/2225740. url: http://www.jstor.org/stable/2225740 ↩Simulation and decomposition of Ragnar Frisch’s rocking horse model (1933)2020-12-05T00:00:00+01:002020-12-05T00:00:00+01:00https://hetmodels.com/articles/Frisch-1933<p><em>This post is based on a working paper accessible <a href="https://halshs.archives-ouvertes.fr/halshs-02969773v3">here</a>.</em></p>
<!-- référence du papier -->
<p>In 1933, Ragnar Frisch built one of the first determinate models of the business cycle, and introduced it under the new term of “macrodynamic”.<sup id="fnref:ppip"><a href="#fn:ppip" class="footnote">1</a></sup> This article has been celebrated ever since its publication, in large part because of the two mechanisms Frisch introduced to explain the business cycle: a propagation mechanism, explaining the return to equilibrium, and an impulse mechanism, explaining the perturbation of the system that prevented it from reaching its equilibrium level. This distinction is still central to modern models in macroeconomics.</p>
<p>Although celebrated, this model has also been the object of controversy: in 1992 and 2007, Stefano Zambelli published two papers<sup id="fnref:zambelli"><a href="#fn:zambelli" class="footnote">2</a></sup> where he explained that Frisch had failed to construct a propagation mechanism generating fluctuations, arguing that it would be so “for all possible combinations of different economically significant sets of parameters” (Zambelli, 2007: 153).</p>
<p>This is a striking result, and it would indeed be puzzling that neither Frisch nor any one else that worked on this model found out about it earlier. But does it really hold? Although we believe that Zambelli is right when he says that the original parameters chosen by Frisch do not yield an oscillation, in fact a complete simulation of the model shows that only a slight change in parameters will make this oscillation apparent.</p>
<p>To clarify matters, one needs to go back to Frisch’s original methodology. In the early 1930s, Frisch was heavily influenced by the mostly empirical work that had been done on the business cycle during the preceding decade. In particular, he had worked himself on a new method to decompose time series, aimed to overcome the shortcomings of harmonic analysis and the periodogram.<sup id="fnref:1931-program"><a href="#fn:1931-program" class="footnote">3</a></sup> In 1933, his model took a different view of the same problem: instead of decomposing the observed trajectory of a process, Frisch tried to explain how this process was generated. To obtain this explanation, he used a system of equations that he knew would produce an infinite sum of periodic cycles (sinusoidal curves), where each cycle’s frequency could be compared to the frequencies already identified with a decomposition of an observed time series.</p>
<p>Frisch knew that simple ordinary differential or difference equations would not be able to generate such an infinite sum of cyclical components. However, he had already been introduced to the peculiar characteristics of mixed difference and differential equations (today known as delay differential equations, DDE) by Jan Tinbergen, who had build a model based on those equation, and lectured on their uses at the first European meeting of the Econometric Society in 1931.<sup id="fnref:tinbergen"><a href="#fn:tinbergen" class="footnote">4</a></sup> This explains, at least partly, why Frisch’s model has more often been praised than solved: the theory on DDEs had not even begun in the early 1930s, and it is still an ongoing area of research. However, we have today at our disposal the necessary tools to simulate the model, and more importantly to compute its component cycles very efficiently. In turn, this allows us to obtain new insights on the behaviour of Frisch’s model and the ideas that the Norwegian economist tried to encapsulate in it.</p>
<p>In the rest of this post, we will present our methodology to compute the components, and illustrate the different possible trajectories with an application reproducing the model, its solutions and their decomposition for a wide choice of parameters.</p>
<p>There are three equations in Frisch’s model:</p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{align}
\dot x_t &= c - \lambda(r \cdot x_t + s \cdot z_t)\\
y_t &= m \cdot x_t + \mu \cdot \dot x_t \\
z_t &= \frac{1}{\epsilon} \int_{t-\epsilon}^{t} y(\tau) d\tau
\end{align} %]]></script>
<p>The first equation gives the changes of consumption <script type="math/tex">x</script>. <script type="math/tex">c</script> is a constant level of increase, while <script type="math/tex">-\lambda (r \cdot x_t + s \cdot z_t)</script> represents the increasing tensions on consumption of the <em>encaisse désirée</em>, that is, the demand for cash coming from consumption and production <script type="math/tex">z</script>. <script type="math/tex">\lambda</script>, <script type="math/tex">r</script> and <script type="math/tex">s</script> are coefficients of sensibility. The second equation, <script type="math/tex">y</script>, is an accelerator relationship explaining the level of investment as a function of current consumption and changes in consumption. Finally, <script type="math/tex">z</script> is the production currently being done, and is a function of the investment decisions decided over the previous period from <script type="math/tex">t-\epsilon</script> to the present.</p>
<p>By substitution of the second into the third, and the third into the first equation, we obtain the following integro-differential equation with a delay in the state variable:</p>
<script type="math/tex; mode=display">\begin{equation}
\dot x(t) + \lambda (r +\frac{s\mu}{\epsilon}) \cdot x(t) - \frac{\lambda s \mu}{\epsilon} \cdot x(t-\epsilon) + \frac{\lambda s m}{\epsilon} \int_{t-\epsilon}^{t} x(\tau) d\tau = c \label{eq_gen}
\end{equation}</script>
<p>To solve this equation in the form of an infinite series of components, as did Frisch in 1933 with the limited tools at his disposal, we use the Laplace transform and its inverse. The methodology of using this transform on DDEs has been described in particular by Bellman and Cooke in a 1963 monograph, which capped two decades of explorations on the subject of DDEs and opened up new areas of research on those equations.<sup id="fnref:bellman-cooke"><a href="#fn:bellman-cooke" class="footnote">5</a></sup></p>
<p>Applying term by term the substitution will give an expression of the form <script type="math/tex">X(s) = \frac{F(s)P(s)}{H(s)}</script>, where <script type="math/tex">F(s)</script> is the transform of the non-homogenous term, <script type="math/tex">P(s)</script> comes from the initial conditions of the system between <script type="math/tex">t-\epsilon</script> and <script type="math/tex">t</script>, and <script type="math/tex">H(s)</script> is a characteristic polynomial. We refer the reader to the appendix of our <a href="https://halshs.archives-ouvertes.fr/halshs-02969773v3">working paper</a> for the whole detailed computation giving us the transformed expression of <script type="math/tex">x(t)</script>:</p>
<script type="math/tex; mode=display">\begin{equation}
X(s) = \frac{e^{- s \epsilon} \left(c+s \cdot x(\epsilon) - (s \cdot b-d) \int_0^\epsilon x(t) e^{-s t} dt - d \int_0^\epsilon x(\tau) d\tau \right)}{s^2 + a s + b s e^{-s \epsilon} +d - d e^{- s \epsilon}}
\end{equation}</script>
<p>To get back from the s-domain, we apply the <a href="https://en.wikipedia.org/wiki/Inverse_Laplace_transform">Mellin inversion formula</a>, also called a Bromwich integral: <script type="math/tex">f(t) = \frac{1}{2 \pi i} lim_{T \to \infty} \int_{\gamma - T i}^{\gamma + T i} F(s) e^{st} ds</script>. Thanks to the residue theorem the integral will become a sum determined by the zeros of <script type="math/tex">H(s)</script> (our characteristic polynomial) that are singularities of <script type="math/tex">X(s)</script>, and the <script type="math/tex">\frac{1}{2 \pi}</script> will cancel with the <script type="math/tex">2 \pi</script> of the sum of residues. Because the poles are simple, we can obtain them with the classic formula <script type="math/tex">Res(F,c) = lim_{z \to c}(z-c)F(z) = \frac{g(c)}{h'(c)}</script>. Replacing <script type="math/tex">g(.)</script> and <script type="math/tex">h(.)</script> by their expressions above, we obtain the following solution for <script type="math/tex">x</script>:</p>
<script type="math/tex; mode=display">\begin{equation}
x(t-\epsilon) = \sum_{i=0}^\infty \frac{c+r_i \cdot x(\epsilon) + (d-r_i \cdot b) \int_0^\epsilon x(t) e^{-r_i t} dt - d \int_0^\epsilon x(\tau) d\tau }{2r_i+a+be^{-r_i \epsilon}-b\epsilon r_i e^{-r_i \epsilon} + d \epsilon e^{- r_i \epsilon}} e^{r_i (t- \epsilon)} = \sum_{i=0}^\infty k_i e^{r_i (t- \epsilon)} \label{solution_x}
\end{equation}</script>
<p>Where the <script type="math/tex">r_i</script> are the zero of <script type="math/tex">h(.)</script>. The conjugate of each complex root will also be a solution, and its coefficient will be <script type="math/tex">\bar{k_i}</script>.</p>
<p>We can see right away that <script type="math/tex">0</script> will be a solution of <script type="math/tex">h(s)=0</script>. If we assume that the economy was at equilibrium and is suddenly displaced away from it, we can replace this into our solution above, and we will obtain an equilibrium level for our trajectory. To find the other zeros of <script type="math/tex">h(s)</script>, we can remark that as <script type="math/tex">s</script> tends to <script type="math/tex">\infty</script>, the polynomial will approach the expression
<script type="math/tex">s(s + a + b e^{-\epsilon s}) = 0</script>. To find the zeros of the expression in parenthesis, we can use the Lambert <script type="math/tex">W</script> function (this function gives the solutions of <script type="math/tex">we^w=z</script> as a multibranch function <script type="math/tex">w = W_k(z)</script> for some integer <script type="math/tex">k</script> giving a particular branch of the function). After some changes of variables, we obtain the expression <script type="math/tex">s = \frac{W_k(-\epsilon b e^{a\epsilon})}{\epsilon} - a</script> for the zeros of our polynomial, aside from the trivial root. Because the expression inside <script type="math/tex">W_k(.)</script> is always positive for economically relevant parameters, this expression will always give a nontrivial real root generating a monotonous trajectory and an infinity of complex roots generating cyclical solutions. The roots are then improved using a simple Newton algorithm which works well for the complex roots although special care must be taken when applying it to the nontrivial real root.</p>
<p>This allows us to obtain a general solution with three different parts: an equilibrium level, a monotonous trajectory that Frisch called a secular trend, and an infinite sum of cyclical solutions.</p>
<script type="math/tex; mode=display">\begin{equation}
x(t-\epsilon) = \frac{c}{\lambda (r + sm)} + k_1 e^{r_1 (t-\epsilon)} + \sum_{i=2}^\infty A_i e^{\alpha_i (t-\epsilon)} cos(\beta_i (t-\epsilon) + \phi_i)
\end{equation}</script>
<p>The damping and frequency of the sinusoidal functions is given by <script type="math/tex">r_i = \alpha_i + j\beta_i</script>, <script type="math/tex">j^2 = -1</script>, while their amplitude and phase if given by <script type="math/tex">A_i = 2 Mod(k_i)</script> and <script type="math/tex">\phi_i = Arg(2k_i)</script> (the factor of two comes from the complex conjugate). Because the <script type="math/tex">k_i</script> will quickly go to zero in the infinite sum, in practice a limited number of cyclical components is needed to have a good approximation of the general solution.</p>
<p>The application below is initially set with Frisch’s original parameters. The reader can modify any parameter of the system to see the impact on the general trajectory of consumption, as well as the different components of this trajectory. An increase in <script type="math/tex">\lambda</script> is sufficient to obtain an apparent cycle in the general solution. In 1933, Frisch did not see that all the components would depend on the same initial conditions, and he gave a different initial condition for the “trend” component and for the cyclical components. Zambelli was the first one to remark that the parameters he used generated a total solution that was monotonous but we cannot agree with him that this is a general case: it is only an unfortunate coincidence as can be seen by changing other parameters. Thus the rocking horse is rocking again!</p>
<iframe src="/assets/frisch_1933" width="100%" height="700px"></iframe>
<p><u>Note on the app:</u> For computational reasons, we use a discrete-time version of the model to approximate the general solution of the model. The equation behind the discretized version for an arbitrary step is also presented in our <a href="https://halshs.archives-ouvertes.fr/halshs-02969773v3">working paper</a> (see in particular Appendix II) and we do not reproduce it here. The individual components are computed with the algorithm described above. Another version of this app, using directly the sum of components, can be found <a href="https://cbheem.shinyapps.io/Frisch/">here</a>.</p>
<div class="footnotes">
<ol>
<li id="fn:ppip">
<p>Frisch, Ragnar. Propagation Problems and Impulse Problems in Dynamic Economics. Oslo: Reprinted from <em>Economic Essays in Honour of Gustav Cassel</em>. Universitetets Okonomiske Institutt, 1933. <a href="#fnref:ppip" class="reversefootnote">↩</a></p>
</li>
<li id="fn:zambelli">
<p>Zambelli, Stefano “A Rocking Horse That Never Rocked: Frisch’s ‘Propagation Problems and Impulse Problems.’” <em>History of Political Economy</em> 39, no. 1 (March 1, 2007): 145–66.</p>
<p>Zambelli, Stefano. “The Wooden Horse That Wouldn’t Rock: Reconsidering Frisch.” In <em>Nonlinearities, Disequilibria and Simulation: Proceedings of the Arne Ryde Symposium on Quantitative Methods in the Stabilization of Macrodynamic Systems Essays in Honour of Björn Thalberg</em>, edited by Kumaraswamy Velupillai, 27–56. London: Palgrave Macmillan UK, 1992. <a href="#fnref:zambelli" class="reversefootnote">↩</a></p>
</li>
<li id="fn:1931-program">
<p>See in particular the program laid out in Frisch, Ragnar. “A Method of Decomposing an Empirical Series into Its Cyclical and Progressive Components.” <em>Journal of the American Statistical Association</em> 26, no. 173S (1931): 73–78. <a href="#fnref:1931-program" class="reversefootnote">↩</a></p>
</li>
<li id="fn:tinbergen">
<p>Tinbergen, Jan “L’utilisation Des Équations Fonctionnelles et Des Nombres Complexes Dans Les Recherches Économiques.” <em>Econometrica</em> 1, no. 1 (1933): 36–51.</p>
<p>Tinbergen, Jan. “A Shipbuilding Cycle.” In <em>Jan Tinbergen: Selected Papers</em>, edited by Leo H. Klaassen, L. M. Koyck, and H. J. Witteveen, 1–14. North-Holland Publishing Company, 1959 [1931]. <a href="#fnref:tinbergen" class="reversefootnote">↩</a></p>
</li>
<li id="fn:bellman-cooke">
<p>Bellman, Richard Ernest, and Kenneth L. Cooke. <em>Differential-Difference Equations</em>. Santa Monica: Rand Corporation, 1963. <a href="#fnref:bellman-cooke" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>Vincent CarretThis post is based on a working paper accessible here.ISLM and the Aggregate Demand curve2020-11-29T00:00:00+01:002020-11-29T00:00:00+01:00https://hetmodels.com/articles/ISLM-AD<p>In a <a href="/articles/ISLM">previous post</a> we constructed the IS-LM model from a demand for goods, a demand for money, and two equilibrium conditions. By adding a very simple liquidity trap, this model was already able to tell us some things about the Keynes effect and the Fisher effect in different situations.</p>
<p>In this model, we will make consumption depend on real balances, just as we did for investment before. This represents the Pigou effect <a href="https://doi.org/10.2307/2226394">(Pigou, 1943)</a>, which is assumed to run inverse to the evolution of prices: a fall in prices will increase real balances and consumption, shifting the IS curve to the right.</p>
<p>The Keynes effect also represents a negative relationship between prices and income, via the interest rate: a fall in prices increases the real money supply and shifts LM to the right. On the other hand, the Fisher effect <a href="https://doi.org/10.2307/1907327">(Fisher, 1933)</a> impacts the demand for goods negatively through a rise in real debt when prices fall to a lower level. Thus, all other things equal, when prices fall to a lower level, investment decreases and the IS curve is shifted to the left.</p>
<p>When the Pigou effect and the Keynes effect dominate the Fisher effect, the relation between prices and income will be negative. On the contrary, when the Fisher effect dominates, this relation becomes positive and the aggregate demand curve is upward sloping! This situation will arise most likely when the economy is stuck in the liquidity trap: the Keynes effect will be inefficient, a shift of LM to the right after a fall in prices will not increase income.</p>
<p>This is largely in reaction to this idea that Pigou argued in the 1940s that Keynes had not taken into account the effect on real balances that bears his name. The real balance effect was much debated, but for our purposes here we can note that it can still be dominated by the Fisher effect when prices fall too low. This means that with a liquidity trap, our aggregate demand function can have a very odd shape in the (y,p) plane, decreasing for high enough prices, but increasing for a low level of prices.</p>
<p>To see this effect, we start from our previous model, adding in the IS equation a term making consumption depend on real balances (we will use the same <script type="math/tex">\frac{D}{p}</script> as in the investment function for the Fisher effect). We obtain the following equation for IS:</p>
<script type="math/tex; mode=display">\begin{equation}
r = \frac{(1-\alpha - I_y) \cdot y + \alpha t - C_{\pi} \frac{D}{p} - I_D \frac{D}{p} - \bar{I} -g}{I_r}
\tag{IS}
\end{equation}</script>
<p>Where <script type="math/tex">C_\pi</script>, previously an autonomous level of consumption, is now a coefficient of sensibility of consumption to real balances, assumed positive, and the other parameters and variables are the same as in the <a href="/articles/ISLM">previous post</a>.</p>
<p>We change our LM equation a bit more to obtain a good liquidity trap from which we will derive an aggregate demand curve with the properties explained above. Our equation takes the following form:</p>
<script type="math/tex; mode=display">\begin{equation}
r = r_{min} + \frac{r_{max}}{1+e^{\frac{L_y}{L_r}(y-\frac{M^s}{p})}}
\tag{LM}
\end{equation}</script>
<p>We have introduced a maximum and a minimum level for the interest rate r and the other parameters remain unchanged. An increase in the real stock of money will shift LM to the right, a change in the coefficient of sensibility of the demand for money to the interest rate (<script type="math/tex">L_r</script>) or to income (<script type="math/tex">L_y</script>) will change the slope of the LM curve in its upward sloping part (this equation is a simple logistic curve).</p>
<p>With this equation for LM, it is not so easy to determine the equation for aggregate demand in the (y,p) plane, but this is where graphical techniques are really helpful, and we can easily visualize the form of aggregate demand in the following application, where the parameters will yield different forms for AD.</p>
<p>The strength of the Pigou effect can be adjusted with the value of <script type="math/tex">C_\pi</script>: a higher value will increase its strength and we can see that it does not take much to have a Pigou effect always dominating the Fisher effect. The latter is controlled by <script type="math/tex">I_D</script>: a lower value of this parameter will increase its strength.</p>
<iframe src="/assets/islm-ad" width="100%" height="750px"></iframe>Vincent CarretIn a previous post we constructed the IS-LM model from a demand for goods, a demand for money, and two equilibrium conditions. By adding a very simple liquidity trap, this model was already able to tell us some things about the Keynes effect and the Fisher effect in different situations.The ISLM Model2020-11-24T00:00:00+01:002020-11-24T00:00:00+01:00https://hetmodels.com/articles/ISLM<p>We present a simple IS LM model with two markets: the goods market and the “money” market. The goods market is describe in a <a href="/articles/IS_KC">previous post</a> We linearise the model so that we have a very simple solution, graphically straightforward.</p>
<p>The equations are the following</p>
<script type="math/tex; mode=display">Y^d = \alpha \cdot (y-t) + C_{\pi} + I_y \cdot y + I_{D} \cdot \frac{D}{p} + I_r \cdot r + \bar{I} + g</script>
<p>With <script type="math/tex">Y^d</script> global demand, <script type="math/tex">\alpha</script> the share of income going to workers, <script type="math/tex">y</script> the income / production, <script type="math/tex">t</script> taxes, <script type="math/tex">C_{\pi}</script> an autonomous level of consumption (assumed to come from “capitalists” in our <a href="/articles/IS_KC">previous post</a>), <script type="math/tex">I_y</script>, <script type="math/tex">I_D</script> and <script type="math/tex">I_r</script> the sensibility of investment to income, real debts and the interest rate, <script type="math/tex">\bar{I}</script> the level of autonomous investment and <script type="math/tex">g</script> public expenditures. <script type="math/tex">\frac{D}{p}</script> is the level of real debts, nominal debts are fixed here to <script type="math/tex">2000</script>. The introduction of real debts in our investment function allows us to capture at least some part of the argument made by Irving Fisher in 1933, that is, the adverse effect of deflation on investment (see <a href="https://doi.org/10.2307/1907327">Fisher, 1933</a>).</p>
<p>The equilibrium condition is simply that</p>
<script type="math/tex; mode=display">Y^d = y</script>
<p>The demand for money is:</p>
<script type="math/tex; mode=display">M^d = p(L_y \cdot y + L_r \cdot r)</script>
<p>Where <script type="math/tex">M^d</script> is the demand for money,<script type="math/tex">p</script> is the price level, <script type="math/tex">L_y</script> and <script type="math/tex">L_r</script> are the sensibilities of the demand for money to income and interest rate. The money supply is <script type="math/tex">M^s</script> and real money supply is denoted <script type="math/tex">\bar{M}</script>, our equilibrium condition is that <script type="math/tex">\frac{M^d}{p} = \bar{M}</script>.</p>
<p>From these two equations and the equilibrium conditions we can deduce the equations of IS and LM.</p>
<script type="math/tex; mode=display">\begin{equation}
r = \frac{(1-\alpha - I_y) \cdot y + \alpha t - C_{\pi} - I_D \frac{D}{p} - \bar{I} -g}{I_r}
\tag{IS}
\end{equation}</script>
<script type="math/tex; mode=display">\begin{equation}
r = \frac{1}{L_r}(\frac{\bar{M}}{p} - L_y y)
\tag{LM}
\end{equation}</script>
<p>Because <script type="math/tex">I_r</script> is assumed negative (investment evolves inversely to the interest rate), the condition for IS to be downward sloping in the <script type="math/tex">(y,r)</script> plane is that <script type="math/tex">1-\alpha - I_y > 0</script>, that is <script type="math/tex">1- \alpha > I_y</script>, the saving function has a higher slope than the investment function. For LM, <script type="math/tex">L_y</script> is assumed positive, a higher level of income increases the demand for money, while <script type="math/tex">L_r</script> is negative, because an increase in the interest rate will mean that bonds will have a greater yield and the opportunity cost of holding money goes up. This means that LM will be upward sloping, except that the interest rate cannot go below some value <script type="math/tex">r_{min}</script> (liquidity trap).</p>
<p>To find the equilibrium solution we can reorganize these two equations so that the two unknowns are on the left:</p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{align}
(1-\alpha-I_y) \cdot y - I_r \cdot r &= -\alpha t + C_{\pi} + I_{D} \cdot \frac{D}{p} + \bar{I} + g\\\\
L_y \cdot y + L_r \cdot r &= \frac{\bar{M}}{p}
\end{align} %]]></script>
<p>The solutions for <script type="math/tex">r</script> and <script type="math/tex">y</script> will be:</p>
<script type="math/tex; mode=display">% <![CDATA[
\begin{align}
y* &= \frac{(-\alpha t + C_{\pi} + I_{D} \cdot \frac{D}{p} + \bar{I} + g)L_r+I_r \frac{\bar{M}}{p}}{L_r(1-\alpha-I_y) + I_r L_y}
\\\\
r* &= \frac{\frac{\bar{M}}{p}(1-\alpha-I_y) - L_y(-\alpha t + C_{\pi} + I_{D} \cdot \frac{D}{p} + \bar{I} + g)}{L_r(1-\alpha-I_y) + I_r L_y}
\end{align} %]]></script>
<p>The following application allows the user to change parameter values and observe the impact of this change on the equilibrium of the economy. A “shock” can be added on one of the parameters to keep the original parameters on the graph.</p>
<p>For instance, a shock on <script type="math/tex">g</script>, public expenditures, will increase income as well as the interest rate, creating an “eviction effect” of private investment. A second shock, an increase on <script type="math/tex">M^s</script> for instance, will mitigate the effect of the rise in the interest rate and a higher level of income will be obtained.</p>
<p>A shock on prices will impact both curves if <script type="math/tex">I_D</script> is different from 0 (if real debts have an effect on investment). At first, a decrease in prices will tend to move the equilibrium toward the right and bottom of the plane, but once the interest rate is at the level of the liquidity trap, income will rapidly decrease.</p>
<iframe src="/assets/islm" width="100%" height="700px"></iframe>
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<!-- <script src="/assets/islm/main.bdaea58b.chunk.js"></script>
-->Vincent CarretWe present a simple IS LM model with two markets: the goods market and the “money” market. The goods market is describe in a previous post We linearise the model so that we have a very simple solution, graphically straightforward.The Keynesian Cross and the IS curve2020-11-20T00:00:00+01:002020-11-20T00:00:00+01:00https://hetmodels.com/articles/IS_KC<p>In this post, we build the IS curve from the equilibrium positions on the goods market. The curve is later used in several models (see <a href="/articles/ISLM">this post</a> and <a href="/articles/ISLM-AD">that one</a>). The approach presented here is deliberately simplified, as a first step toward building the IS LM model and integrating different macroeconomic effects.</p>
<p>We start from a demand function made of consumption, investment and public expenditures:</p>
<script type="math/tex; mode=display">\begin{equation}
Y^d = c(y) + I(y,r) + g
\end{equation}</script>
<p>Keynes specified the consumption function by talking of a “fundamental psychological law” and the famous marginal propensity to consume. Instead of this approach, we would like rather to underline that consumption can be linked to the distribution of income. To understand this, we distinguish two categories of agents in our simplified model:</p>
<ul>
<li>the “workers”, who receive income on the form of a salary</li>
<li>the “capitalists”, who receive the returns from capital, the profits</li>
</ul>
<p>We make the further hypothesis that the workers will consume all their earnings, while capitalists only consume a fixed part of their profits (this last hypothesis is relaxed in <a href="/articles/ISLM-AD">this post</a> where capitalists’ consumption depends on their real wealth). This allows us to write the following consumption function:</p>
<script type="math/tex; mode=display">c(y) = \alpha y + C_{\pi}</script>
<p>Where <script type="math/tex">\alpha</script> is the share of income going to workers (<script type="math/tex">\alpha y = \frac{W N}{P}</script>), while <script type="math/tex">C_{\pi}</script> is the fixed consumption of capitalists. This model underlines that a large part of current consumption is dependent on current income, while part of this consumption will be untouched by variations in income. We see that <script type="math/tex">\alpha</script> will necessarily be inferior to one (the share of income accrued by workers is inferior to one), a good property to obtain a stable equilibrium. In this simplified model, income is equal to consumption plus savings, thus we have that:</p>
<script type="math/tex; mode=display">S = y - c(y) = (1-\alpha) y - C_\pi</script>
<p>Our investment function is dependent on income and the real interest rate. We suppose that the real interest rate will have a negative impact on investment by increasing the cost of borrowing, while current income will have a positive effect on investment through positive expectations. We assume for now a linear relationship between those variables, which include an autonomous term accounting for the investment that is independent of both income and the real interest rate:</p>
<script type="math/tex; mode=display">I(y,r) = I_y y + I_r r + \bar{I}</script>
<p>Where <script type="math/tex">I_y > 0</script>, <script type="math/tex">% <![CDATA[
I_r < 0 %]]></script>, that is, the coefficient of sensibility of investment to income is positive, and the coefficient of sensibility of investment to the real interest rate is negative.</p>
<p>Inserting into the first equation, we obtain</p>
<script type="math/tex; mode=display">Y^d = \alpha y + C_{\pi} + I_y y + I_r r + \bar{I} + g</script>
<p>From the equilibrium condition that income is equal to aggregate demand, we can derive the expression of the equilibrium income given a real interest rate (supposed here to be exogenous):</p>
<script type="math/tex; mode=display">y^* = \frac{C_\pi + I_r \bar{r} + \bar{I} + g}{1-\alpha-I_y}</script>
<p>Where <script type="math/tex">\bar{r}</script> is a fixed level of the real interest rate, supposed to be exogenous in this model of the goods market. The equation for IS can be derived similarly by isolating <script type="math/tex">r</script>:</p>
<script type="math/tex; mode=display">r = \frac{(1-\alpha-I_y)y-C_\pi-\bar{I}-g}{I_r}</script>
<p>The condition to obtain a downward sloping IS curve is that <script type="math/tex">1-\alpha-I_y>0</script> or alternatively <script type="math/tex">1-\alpha>I_y</script>, the propensity to save is higher than the propensity to invest, which can be easily visualized on the diagram representing income and investment - savings.</p>
<p>The IS curve can help us visualize the effect of a disequilibrium in the economy. On the right of this curve, we are in a situation of excess supply of goods, and the equilibrium will be reestablished by a fall in the real interest rate, which shifts the demand curve upward on the Keynesian Cross. Symmetrically, a situation of excess demand of goods on the left will be ended with an increase in the interest rate, shifting the demand curve downward because of the fall in investment.</p>
<iframe src="/assets/is-kc" width="100%" height="750px"></iframe>Vincent CarretIn this post, we build the IS curve from the equilibrium positions on the goods market. The curve is later used in several models (see this post and that one). The approach presented here is deliberately simplified, as a first step toward building the IS LM model and integrating different macroeconomic effects.