The nonlinear economy: How resource constraints lead to business cycles Open Access

Business cycles bear similarities to self-sustained oscillations in nonlinear dynamics. The periodic occurrence of boom, recession, depression, and recovery phases in economic systems is an empirical fact. However, the reasons for business cycles are still debated. Are they induced by exogenous shocks, or do they result from the endogenous nonlinear coupling of economic dynamics? We support the endogenous explanation by providing a model that generates business cycles when considering a depleting resource. This depletion is reflected in a production function for economic output dependent on the input of capital and energy. Using this production function, we derive a nonlinear dynamics that allows for the coexistence of limit cycles and stationary solutions of high productivity.

For Jason Gallas, nonlinear dynamics was not just a research domain, but a lifelong passion. His impressive list of publications covers various application areas, ranging from geophysics to cancer. Economics, however, was only touched indirectly, when studying bifurcations in competition models.¹ This is understandable. Compared to the complex dynamics of, e.g., the Belousov–Zhabotinsky reaction, one of Jason’s favorite topics,² nonlinear economic models are rather simple, and studying macroeconomic models with the arsenal of nonlinear dynamics was fashionable from the 1980s to 2000,^3–5 but not so much today.

We take the opportunity to change this perspective a bit with the hope to renew the attention of physicists and applied mathematicians for these kinds of models. Our aim is not to present a completely new approach. After all, economists would not (yet?) see a need for this. In this paper, we first remind on how these macroeconomic models have been established, to then propose some extensions, and conclude by linking our approach to current research on active matter in physics. The latter becomes possible because, from an abstract thermodynamic perspective, economies are non-equilibrium open systems. They depend on the influx of resources that are transformed into valuable output, similar to the alchemistic idea of transforming lead into gold. That means, like other living systems, economies are not conservative but dissipative systems. The transformation of resources is inherently tied to the production and export of entropy.

The relation between the thermodynamics of irreversible processes and economic production has been discussed already since the 1980s, starting from the pioneering work of Georgescu–Roegen.⁶ It still plays a role in evolutionary economics and environmental economics, the former focusing on dynamic aspects and self-organization,^7,8 the latter on resource constrains and sustainability.^9,10 With our approach, we somehow combine these two perspectives, investigating the impact of a depleting resource, “energy,” on the dynamics of production.

The issue of resource constraints has been broadly discussed in the literature, for instance, as “resource dependence theory” in management science,¹¹ but not as a model that could be formally explored. In economics, exhaustible resources play a role since Ricardo’s time.^12,13 Notably, Hotelling made important contributions to formalize the discussion.^14–16 The main approach, however, is different from ours in that it primarily deals with how the timing of resource extraction affects its value and availability over time, balancing the diminishing stock with factors such as commodity prices, wages, profit rates, and demand.

Our starting point is the neoclassical growth model where production depends on the input of capital and labor. However, these are not modeled as resources that deplete during production. Instead, they continuously grow: Labor force because of population growth and capital stock because of investments. Therefore, we propose to consider a resource that is consumed during production. We use the term “energy” for it but interpret it very broadly as a natural resource. As a consequence, output is constrained by the availability and the renewal of this resource. Growing production implies decreasing energy. This denotes an important difference to capital stock, which is assumed to increase with growing production.

Our second contribution is the formal derivation of a production dynamics, starting from our production function. In economics, this dynamics is often studied in the so-called “multiplier-accelerator” models.^17,18 The multiplier describes the impact of an input, e.g., capital, on the expansion of production. The accelerator describes the feedback of the growing output on the input variable, e.g., the growth of capital stock through the investment of a fraction of the output. The dynamics assumes lags in this feedback process and can produce different types of steady-state solutions, including fix points, limit cycles, damped oscillations, but also unstable solutions, i.e., growing oscillations or even chaos.

Theoretical economists, physicists, and applied mathematicians have particularly studied a variant of such time-delayed dynamics, the Kalecki–Kaldor model.^19–24 One of the reasons for this interest is the complex dynamics of the model. Such a complexity is seen not as a drawback, but as an advantage, because it offers ample possibilities to generate a wealth of dynamic patterns. These are considered a precondition to explain a complex real world phenomenon: business cycles,^25–27 originally dubbed trade cycles. The question whether such delays are really needed to generate the dynamics of business cycles was negatively answered already by Baron Kaldor himself, who wrote in 1940: “Previous attempts at constructing models of the Trade Cycle—such as Mr. Kalecki’s or Professor Tinbergen’s—have thus mostly been based on the assumption of statically stable situations, where equilibrium would persist if once reached; the existence of the cycle was explained as a result of the operation of certain time lags, which prevented the new equilibrium from being reached, once the old equilibrium, for some external cause, had been disturbed. In this sense, all these theories may be regarded as being derived from the ‘cobweb theorem.’ The drawback of such explanations is that the existence of an undamped cycle can be shown only as a result of a happy coincidence, of a particular constellation of the various time-lags and parameters assumed. The introduction of the assumption of unstable positions of equilibrium at and around the replacement level provides, however, [ $…$] an explanation for a cycle of constant amplitude irrespective of the particular values of the time-lags and parameters involved. The time-lags are only important here in determining the period of the cycle, they have no significance in explaining its existence. Moreover, with the theories of the Tinbergen-Kalecki type, the amplitude of the cycle depends on the size of the initial shock. Here the amplitude is determined by endogenous factors and the assumption of ‘initial shocks’ is itself unnecessary.”²⁸ 

This long quotation sets a nice stage for our own investigations. In line with the cited research, we attempt to explain business cycles as endogenously created by coupled nonlinear dynamics. This contrasts other explanations of business cycles as the result of exogenous perturbations of an otherwise stable dynamics. But differently from the cited research, we will not utilize delayed differential equations to generate cycles, nor propose ad hoc nonlinear functions. Instead, we will derive the non-linear dynamics from the production function, using suitable assumptions for the dynamics of capital and energy. This will shed new light on the formal preconditions for obtaining limit cycle dynamics.

The test case for our approach is a reformulation of Kaldor’s model of business cycles. Kaldor argued about the importance of nonlinear functions, but himself never proposed an analytic expression for the dynamics. Therefore, it is challenging to see how these nonlinear functions can be obtained from our framework. As we will show, deciding between different possibilities and solutions eventually requires us to resort on the economic arguments provided by Kaldor.

The variables $X 1(t), X 2(t),…$ denote different inputs, e.g., capital, labor, and resources, and the function $F[⋅]$ describes how the input is transformed into a valuable output. The normalization $Y 0$ can be seen as an equilibrium state with baseline economic activity.

For the output $Y[K,L]$⁠, an instantaneous adjustment is assumed. Instead of $dY/dt$⁠, the economic model only considers $dK/dt$ and $dL/dt$ and postulates that $Y$ takes its new level immediately after $K$ and $L$ change. In physics, this is known as a separation of time scales. Compared to the slow change of $K$ and $L$⁠, $Y$ changes fast; therefore, it can be assumed in quasi-stationary equilibrium. This reduces the discussion to a comparison of the different values of $Y$ before and after the changes of $K$ and $L$⁠.

This setup has become the canonical model for the exogenous explanation of business cycles.³⁸ Subsequent works have introduced additional assumptions to endogenize the causes of business cycles.^39–42 

“The greater the prestige, the greater the opposition” also applies to the neoclassical growth model. Instead of reviewing the many criticisms and the various economic debates that followed, we will concentrate on some nonlinear dynamic aspects. To introduce these, we replace labor force as the relevant input variable. There were certainly periods in history where economic growth was predominantly driven by an exponential increase in the population. However, nowadays, this growth does not easily translate into the exponential growth of labor force. A refined dynamics of $L(t)$⁠, Eq. (13) should also reflect labor related issues such as unemployment, unskilled labor and working poor, and lack of specialized workforce, which are not further discussed here.⁴¹ 

Instead, we consider, in addition to capital, $K(t)$⁠, a different input variable, energy, $E(t)$⁠. It is a general form of “energy” to also reflect other material resources needed for production. Considering resources that are depleted denotes a conceptual change. The production function $Y[K,L]$ uses capital and labor as essential inputs, but the process of production does not reduce any of the inputs. Thus, $K$ and $L$ act as catalysts for the production, very similar to catalysts in chemical reactions. They are needed for the “reaction,” but are not consumed during the production. The input variable $E$⁠, however, is consumed, i.e., the initial resource is diminished.

The driven dynamics $Q[K,E]$ reflects changes in production resulting from the dynamic input of capital and energy, which are the driving variables. As already mentioned, from a thermodynamic perspective, the economy is a pumped system. Hence, instead of a classical conservative system, we model a dissipative system.

Before we solve this dynamics numerically, let us summarize the relation to the previous macroeconomic models. Regarding the input variables, we use the established dynamic equation for capital, comprised of the two terms for investment into capital stock and capital depreciation. The second input variable, energy, uses the same dissipative core dynamics [Eq. (21)], but the decay term additionally reflects that energy is consumed during the production. So, different from, e.g., the neoclassical growth model, we explicitly consider the depletion of a resource.

The nonlinear dynamics for production has no counter party in the neoclassical reference model because an instantaneous relaxation of production is assumed there. Instead, we consider that production changes at a time scale $ε Ydt$⁠. The nonlinear terms directly result from the dynamics of the input variables, as Eq. (20) makes clear. Specifically, the quadratic and cubic terms reflect the increase in production because of the use of energy. The terms related to capital refer to capital depreciation, and the remaining terms to the eigendynamics of production. To demonstrate the value of our explicit dynamics for production, we will map it to the investment and savings functions used in Kaldor’s model of business cycles in Sec. IV C. These functions were not formally specified, so there is no comparison possible. However, we show below that economic arguments will help us to distinguish between different solutions.

As illustrated in Fig. 1, from numerically solving the three coupled dynamics, we find two significantly different outcomes: (i) a stationary production and (ii) sustained oscillations. Unfortunately, the stationary solution is $Y(t)→ Y 0$⁠, i.e., after some intermediate oscillations, we are back at the baseline scenario, while hoping for a case with $Y(t)≫ Y 0$⁠. This is obtained in the second scenario where we can verify that $⟨ Y ( t ) ⟩> Y 0$⁠. Thus, the average production is, indeed, above the baseline. More important, during certain time periods, $Y(t)$ is much larger that $Y 0$⁠, i.e., we see a boom phase of the economy. However, this does not last long, and is followed by a steep decline of production that can even reach negative values, very similar to the business cycles. These comprise four phases of different durations: (a) a short boom phase, (b) a long recession phase, (c) a short depression phase, and (d) a long recovery phase, after which a new cycle starts. This is, indeed, captured with our dynamics of production.

The two-dimensional dynamics defined in Eq. (42) allows to calculate a bifurcation diagram from $Y ˙=0$ and $K ˙=0$⁠. Because of the cubic term in the production dynamics, we can expect two stationary solutions, denoted by $Y ⋆$ and $K ⋆$⁠. We need to know how these solutions change if we vary the control parameters of the dynamics. We have chosen $ε K$ because the consideration of different time scales for the dynamics of production, energy, and capital is a main feature of our model. After eliminating $E$ from the equations, $ε E$ plays no role. However, energy still implicitly impacts the dynamics of production via the parameters $d 1$ and $d 2=−ζ$⁠, which define $Y s=− d 1/ d 2$ [Eq. (27)]. The influence of energy as a depleting resource on production is a main contribution of our paper; therefore, we have chosen $d 1$ and $ζ$ as additional control parameters. Figure 2 shows the bifurcation diagrams for the three different control parameters.

For our analysis, we have chosen a set of parameters that yields three fixed points. For all three control parameters, one of these fixed points is at $Y 0$ (equal to 1.25). Additionally, we find different regimes where either three fixed points exist or only one. The bifurcation diagram $Y ⋆( ε k)$ in Fig. 2(a) clearly shows this. It is the simplest because $ε K$ only affects the stability and not the values of the fixed points. For small $ε K$ (below 0.048), the lower and upper fixed points are unstable and the central one is a saddle point. In this case, all trajectories converge to a limit cycle, similar to the one shown in Fig. 1(c) for the three-dimensional system.

Increasing the value of $ε K$ above 0.048, we observe for the upper fixed point, a transition from unstable to stable. That means the former limit cycle now coexists with the stable fixed point (for $ε K<0.071$⁠). Trajectories originating close to the fixed point will converge to the fixed point, while other trajectories will converge to the limit cycle. Increasing $ε K$ further makes the limit cycle to disappear and all trajectories to converge to the upper fixed point. For large values of $ε K$ (above 0.235), the lower fixed point (at $Y 0$⁠) also undergoes transition from unstable to stable. Depending on the initial conditions, now trajectories will converge either to the lower or the upper fixed point. The middle fixed point always remains a saddle. This means that trajectories can cross it.

The two bifurcation diagrams for the energy related control parameters $d 1$ and $ζ$ are similar in that they show an inverted- $s$ curve for $Y ⋆$⁠. To better understand this, we can treat $K$ as a parameter. Doing so, $Y ⋆$ shows three branches: the upper and lower ones are stable while the middle one is unstable. This means that all trajectories will be attracted toward these two stable branches. We verify that both control parameters have a strong impact on the position of the upper branch that grows with $d 1$ and $1/ζ$⁠.

These three branches yield fixed points only when, in addition to $Y ˙=0$⁠, also, $K ˙=0$⁠. That means $K$ is, in fact, not a parameter, but a linear curve $K ⋆=(s/κ) Y ⋆$ [Eq. (23)] that can intersect with $Y ⋆$ up to three points, depending on the parameters. There are two stable points possible if $K ⋆( Y ⋆)$ crosses the upper or lower branches of $Y ⋆$⁠, and a saddle point where $K ⋆( Y ⋆)$ crosses the unstable branch of $Y ⋆$ only once.

We take a closer look at Fig. 2(b), starting on the right and decreasing $ζ$⁠. For large values of $ζ$⁠, only one stable fixed point at $Y 0$ exists and all trajectories will converge to it. Decreasing $ζ$ below 0.047, this fixed point changes from stable to unstable and the limit cycle appears. This transition resembles a van der Pol oscillator. At $ζ=0.218$⁠, a second fixed point appears, which subsequently bifurcates into an upper unstable point and a middle saddle point. Further decreasing $ζ$ to values below 0.02, the upper fixed point undergoes a transition from stable to unstable. The limit cycle briefly coexists with the stable fixed point, i.e., trajectories reach the limit cycle or converge to the fixed point depending on the initial conditions. Slightly decreasing $ζ$⁠, the limit cycle disappears and all trajectories converge to the upper fixed point whose value becomes larger with smaller $ζ$⁠. Interestingly, because $Y 0$ remains unstable, all trajectories originating below the middle fixed point first need to go below $Y 0$ before they can overcome the unstable point and converge to the stable one.

The bifurcation diagram for $d 1$ [Fig. 2(c)] has a similar interpretation. For low values of $d 1$⁠, only a stable fixed point at $Y 0$ exists. By increasing $d 1$⁠, we observe for this fixed point a transition from stable to unstable, when the limit cycle appears, similar to a van der Pol oscillator. For larger values of $d 1$⁠, a second fixed point appears, which then bifurcates into an unstable upper fixed point and a middle saddle point. When the upper unstable fixed point undergoes a transition to a stable point, the limit cycle first coexists with the stable point. When the limit cycle disappears, all trajectories converge to the upper fixed point. For even larger values of $d 1$⁠, the middle fixed point joins the lower point at $Y 0$ and then bifurcates again, yielding a saddle point at $Y 0$ and an unstable fixed point lower than $Y 0$⁠.

For $d 1>0.416$⁠, the lower fixed point changes to a stable fixed point. That means, depending on the initial conditions, the trajectories can converge either to a high value $Y≫ Y 0$ or a very low value $Y≪ Y 0$⁠. This is the regime we were interested to find a high and stable production. The fact that it coexists with a regime of low and stable production illustrates the risk for the economic system. For the same parameters, the initial conditions impact whether the system ends up in a fortunate or an unfortunate regime.

As the discussion shows, the relations between the two non-trivial manifolds resulting from $Y ˙=0$ and $K ˙=0$ generate a complex dynamics for our production model. A particularly interesting feature is the coexistence of stable fixed points and cycles for certain parameter ranges. Hence, an economic system would be stable while the macroscopic dynamics is completely different. This coexistence of different stable solutions was already explored in other macroeconomic dynamics. For example, for the Kaldor model, the coexistence of stable and oscillatory behaviors was obtained from an interplay between noise and periodic forcing.⁴⁸ In a series of publications,^49–51 Dieci et al. analyzed the bifurcation processes that lead to the coexistence of multiple attractors, including stable equilibria and limit cycles. They demonstrated how small changes in parameters or initial conditions can result in drastically different long-term outcomes. These findings have been further extended to time-discrete systems.⁵² 

A major motivation of our approach is to derive a formal model for the endogenous explanation of business cycles (see also Sec. V). While most models of the multiplier-accelerator type failed in this respect, the work of Kaldor provided an early solution, albeit based on arguments, not on explicit mathematical expressions. In the following, we will, therefore, try to close this gap.

First, Kaldor argues that both functions have to monotonously increase with production $Y$⁠. This rejects Eq. (46) [Fig. 3(a)] because these functions are non-monotonous in $Y$⁠. It also rejects Eq. (47) [Fig. 3(b)] because these functions are monotonously decreasing in $Y$⁠. The variant of Eq. (48) [Figs. 3(c), 3(d), and 3(e)], however, fulfills this requirement.

Second, Kaldor distinguishes the behavior of the two functions in the case of both high and low capital. If the capital is low, investments should increase with $K$ over time to make use of many good investment opportunities. Savings, however, should decrease with $K$ over time because of the rising prices. On the other hand, if the capital is high, investments should decrease, while savings should increase. This implies that both curves move against each other as indicated in Figs. 3(c), 3(d), and 3(e). If $I(Y,K)$ moves up, $S(Y,K)$ moves down (for low levels of $K$⁠) and the other way round (for high levels of $K$⁠). This requirement is met by the functions of Eqs. (46) and (48), but not by Eq. (47), because here $S(Y,K)$ does not depend on $K$ and, therefore, does not move. Nevertheless, Eq. (47) would generate oscillations like the other examples.

We conclude that our proposal for the two nonlinear functions given in Eq. (48) fulfills Kaldor’s requirements. It should be noted, however, that the concrete shape of our nonlinear functions depends on the chosen parameters; thus, statements about the slope and monotonous increase are restricted to this choice.

Kaldor²⁸ argues that two linear functions could still monotonously increase with $Y$ and correctly depend on $K$⁠, but they would only allow for one stationary solution. This solution would be unstable if the slope of $I(Y,K)$ is larger than the slope of $S(Y,K)$⁠. Hence, the economy would either only grow or only shrink. On the other hand, if the slope of $I(Y,K)$ is smaller than the slope of $S(Y,K)$⁠, the stationary solution would be stable and the economy would remain around this stable state. This, however, contradicts the broad observations of business cycles, which Kaldor wanted to explain with his investigations.

Our model generates business cycles and provides functional expressions for the dynamics. Reproducing the asymmetric duration of the phases, however, is not the merit of our efforts; it shall be attributed to the underlying general dynamics of a van der Pol oscillator. This oscillator is a paragon of a non-linear system with nonlinear friction. To compensate this friction, oscillations require the input of energy. That means, we have a dissipative system and the oscillations can be classified as limit cycles. For a critical energy supply, we obtain a Hopf bifurcation.

There is an ongoing debate about the origin of business cycles. Are they exogenous, triggered by external shocks such as wars, natural catastrophes, and political collapse? Or, are they endogenous, that means, resulting from the internal dynamics of a country’s economy, changes in consumption, and inflation? The answer is probably that both endogenous and exogenous causes play a role. No surprise if big disruptions, like a major earthquake destroying production sites, induce a recession, even a depression of economic activities. From a modeling perspective, it is more interesting how small disruptions, e.g., the failure of single elements, can be amplified such that a whole system collapses.⁵⁴ This requires to understand the internal feedback mechanisms that generate the system dynamics endogenously. In this paper, we provide a minimal model that allows to study such endogenous effects on economic growth, in particular the appearance of business cycles, in a systematic manner. Unlike the previous models, which often relied on external shocks or specific nonlinear functions, our model demonstrates how these complex dynamics can arise endogenously from the interactions between production and resource constraints.

The starting point of our investigations was to propose a production function that, in addition to capital, depends on a generalized “energy”: a resource that is depleted during production. Thus, increasing production means decreasing energy. Our production dynamics results from this production function, together with assumptions about the dynamics of capital and energy. Here, we have chosen an ansatz that considers different time scales and a nonlinear dependence of production on energy. This way, we have derived a nonlinear dynamics of production that is able to generate business cycles endogenously, i.e., without assuming external shocks or time lags. Under the constraint of a specific coupling between capital and energy, we find additionally fixed points of the dynamics where the production is considerably higher than a baseline. While both cycles and fixed points are stable, our model is sensitive to parameter changes and initial conditions. This should be seen as an advantage because in complex systems, already small deviations can lead to instabilities, and economic systems are no exception.

The fact that energy is consumed during production drives the economy out of a thermodynamic equilibrium, which is reflected in our model. The take-up of energy, its transformation, and dissipation are features of the “active” matter.⁴⁵ Energy enables systems to evolve and self-organize.⁵⁵ However, in social and economic systems, this does not imply a desired outcome, as witnessed by business cycles in economic activities.

As a modeling consequence of such an out-of-equilibrium situation, we have to distinguish between driving and driven variables. In our case, energy is the driving variable. Its take-up and transformation increase production as the driven variable. This bears similarities to active motion, the directed movement of biological entities such as cells or animals .⁴⁷ In both cases, energy is consumed, that means decreased to increase production or speed. This makes our model different from the related models of business cycles, which do not reflect the consumption of energy. The constrained resource sets limits to the increase of production, resulting in a saturated growth dynamics. Our model considers, in addition to the driven dynamics, also the eigendynamics of the system. That means, it captures the production dynamics in the absence of additional input, a feature not addressed in simpler macroeconomic models. It is, in fact, the interplay between eigendynamics and driven dynamics that generates non-trivial stationary solutions, such as high stable production or limit cycles. The coexistence of stable fixed points and limit cycles, already observed in other models of business cycles, is particularly interesting. It allows to discuss intervention mechanisms that can not only stabilize economic dynamics, but also drive it to the preferred states,⁵⁶ offering a new perspective on business cycle dynamics.

The authors have no conflicts to disclose.

Frank Schweitzer: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Giona Casiraghi: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.