Synchronization of endogenous business cycles

Consider $N$ agents, indexed by $i\in\mathcal{I}=\{1,\ldots,N\}$. Denote by $x_{i}$ an accumulation variable representing a stock of agent $i$, and by $y_{i}$ a decision variable representing a flow. We can think of $y$ as production and of $x$ as inventories; we can also think of $y$ as investment or consumption of durable goods, and of $x$ as capital or net worth, respectively.

The time evolution of the accumulation variable $x_{i}$ is simple. At each time step $t$, it depreciates by a factor $\delta$, and increases by the decision variable $y_{i,t}$. In formula, $x_{i,t+1}=(1-\delta)x_{i,t}+y_{i,t}$.

The dynamics of the decision variable $y_{i}$ is more involved and captures the effect of the interactions with the other agents. The planned level of $y$ for agent $i$, $y_{i,t+1}$, depends most importantly on a term $\overline{y}_{i,t}$, capturing the interactions among the decision variables $y$ of the agents. The interaction term $\overline{y}_{i,t}$ is defined as $\overline{y}_{i,t}=\sum_{j=1}^{N}\epsilon_{ij}y_{j,t}$, where $\epsilon_{ij}\in[0,1]$, such that $\sum_{j}\epsilon_{ij}=1$. Each term $\epsilon_{ij}$, which can be thought of as an interaction coefficient, is the weight that the decision variable of agent $j$ has in determining the value of $\overline{y}_{i,t}$ (self-interactions $\epsilon_{ii}$ are also important). The values of $\epsilon_{ij}$, for all $i$ and $j$, define a weighted, directed, interaction network.

The effect of $\overline{y}_{i,t}$ is mediated by a non-linear function $F\left(\cdot\right)$ that determines the effect of interactions. Suppose that the decision variables of the agents with whom agent $i$ interacts increase, so that $\overline{y}_{i,t}$ becomes larger. If, as a consequence, agent $i$’s marginal payoff goes up, one says that there are strategic complementarities between agent $i$ and the agents with whom she interacts (Cooper and John, 1988).⁵⁵5Letting $V_{i,t}$ be the payoff function for agent $i$ at time $t$, strategic complementarities correspond to the condition $\frac{\partial^{2}V_{i,t}}{\partial y_{i,t}\partial\overline{y}_{i,t}}>0$. Intuitively, in case of strategic complementarities, agent $i$ decides to increase her decision variable if the agents with whom she interacts do the same, i.e. $\partial y_{i,t+1}/\partial\overline{y}_{i,t}>0$.

We now complete the description of the model. We first assume that $y_{i,t+1}$ also linearly depends on one-step lagged values of $x$ and $y$, with coefficients $\alpha_{1}$ and $\alpha_{2}$ (further to an intercept $\alpha_{0}$). We take $\alpha_{1}$ to be negative, and $\alpha_{2}$ to be positive and bounded between zero and one (Beaudry et al., 2015, 2020). These assumptions reflect, respectively, decreasing returns to accumulation, i.e. willingness to avoid excessive stocks of inventories, capital or net worth, and sluggishness in the adjustment of the decision variable, i.e. difficulty to quickly modify the level of production, investment, or consumption of durable goods. The assumption on $\alpha_{1}$ is instrumental to avoid explosive dynamics, as it helps keeping the dynamics bounded when it wanders away from the steady state. At the same time, the assumption about $\alpha_{2}$ introduces realistic smoothness in the dynamics.⁶⁶6A problem of many endogenous business cycle models is that they generate a “sawtooth” dynamics that has no persistence, in stark contrast to empirical time series.

We finally add idiosyncratic shock terms $u_{i,t}$ to the evolution of the decision variables. We parameterize this shock process as an AR(1), i.e. $u_{i,t+1}=\rho u_{i,t}+\iota_{i,t}$, where $\rho\in[0,1]$ is a persistence parameter and $\iota_{i,t}$ is white noise, normally distributed with mean zero and standard deviation $\sigma$.

Our model for endogenous business cycles is thus fully specified by the following equations, for all agents $i$:

In the case in which all interaction coefficients $\epsilon_{ij}$ are identical and equal to $1/N$ (complete and homogenous interaction network), and all agents behave alike, $x_{i,t}=x_{j,t}$ and $y_{i,t}=y_{j,t}$, for all agents $i$ and $j$ and for all times $t$ (and shocks are identical across all agents), Eqs. (1) recover the model in Beaudry et al. (2015).⁷⁷7 A minor difference with respect to Beaudry et al. (2015) is that $y_{i,t+1}$ depends only on variables at $t$, while in Beaudry et al. (2015) it depends on both variables at $t$ and (contemporaneous) variables at $t+1$. As there did not seem to be much difference in the implied dynamics, we chose this form as it was computationally simpler. Economically, it corresponds to assuming that the decision variables of the agents with whom agent $i$ interacts have a lagged effect on her decision.

We now interpret the agents in the abstract model as countries. Following the discussion in the literature review section, we interpret the interaction coefficients as trade linkages. As we stressed already, we do not aim at building a particular microfounded trade model, but rather show how much comovement a general model with connections parameterized as trade linkages can produce under exogenous vs. endogenous business cycles.

A practical difference with respect to the abstract model described in the previous section is that countries have diverse sizes. This would imply that the accumulation and decision variables would be on different scales, making it necessary to consider a different set of parameters for each country. To avoid this impractical situation, we work with “oscillation” variables $x_{i,t}$ and $y_{i,t}$ that define the oscillations independently of the scale, while moving all the effects of the scale to the interaction coefficients.

To do so, we let total investment (or production, or durable consumption) be given by $Y_{i,t}=\widetilde{Y}_{i}y_{i,t}$ and total capital (inventories, stock of durables) be $X_{i,t}=\widetilde{Y}_{i}x_{i,t}$. In these expressions, $\widetilde{Y}_{i}$ denotes the steady state value of $Y_{i,t}$, because with our parameter restrictions $y_{i,t}$ oscillates with mean one. Then

In this application, we take the interaction term $\overline{y}_{i,t}$ to represent the oscillation of the total demand received by country $i$. Letting total demand be denoted by $D_{i,t}$, we assume that its steady state $\widetilde{D}_{i}$ is equal to the steady state of the flow variable $Y_{i,t}$, that is $\widetilde{D}_{i}=\widetilde{Y}_{i}$. This corresponds to assuming market clearing in the steady state: depending on the interpretation for the decision variable, it could be that total demand of intermediate and final goods equals production, or that total demand of capital goods equals investment, or that total demand of durable goods equals realized consumption of such goods. Under this assumption, we can write $D_{i,t}=\widetilde{D}_{i}d_{i,t}=\widetilde{Y}_{i}d_{i,t}$, where again $d_{i,t}$ is the oscillation of demand around its steady-state value. So, in this application $\overline{y}_{i,t}:=d_{i,t}$.

The key behavioral assumption to compute total demand is that countries demand fixed proportions of imports from other countries. Letting $\widetilde{Y}_{ij}$ be the steady-state trade flow from country $i$ to country $j$, and being $a_{ij}=\widetilde{Y}_{ij}/\widetilde{Y}_{j}$ the fraction of imports from country $i$ over country $j$’s output, we write the total demand that country $i$ receives as $D_{i,t}=\sum_{j}a_{ij}Y_{j,t}$. This means that, whether country $j$ produces more or less than the steady-state value $\widetilde{Y}_{j}$, it still demands the fixed proportion $a_{ij}$ of imports from $i$. Equating the two formulations for $D_{i,t}$, $D_{i,t}=\widetilde{Y}_{i}d_{i,t}$ and $D_{i,t}=\sum_{j}a_{ij}Y_{j,t}$, we can express the oscillations of demand $d_{i,t}$ in terms of the oscillations of the decision variables $y_{j,t}$, as:

where the coefficients $\epsilon_{ij}=\widetilde{Y}_{ij}/\widetilde{Y}_{i}$ are the fraction of the output of country $i$ that is exported to $j$. Importantly, this includes the fraction of output that remains in country $i$, $\epsilon_{ii}$, representing domestic demand. These coefficients satisfy the conditions laid out in Eq. 1: $\epsilon_{ij}\in[0,1]$ and $\sum_{j}\epsilon_{ij}=1$.

In sum, the abstract model introduced in Section 3.1 can be interpreted as a model of business cycles of countries coupled by an international trade network. Under strategic complementarities, an increase in domestic and international demand may prompt firms in several countries to increase their decision variables, further increasing domestic and international demand. This could make the steady state unstable, and, as we discuss next, the model could produce endogenous business cycles.

We formally show under which conditions the steady state loses stability in Appendix A.1, where we use the strength of strategic complementarities at the steady state as a bifurcation parameter.⁸⁸8A possible criticism of this paper is that we are using strategic complementarities both as a way to get nonlinear dynamics and as a way of generating comovement, thus making it obvious that nonlinear dynamics corresponds to high comovement. This criticism is misplaced because using other bifurcation parameters while keeping the strength of strategic complementarities fixed leads to the same results (Appendix D.2.5). We also present numerical simulations representing this dynamics, with and without shocks, in Appendix A.2. Here, we just provide the intuition for endogenous dynamics in the abstract framework, and then discuss its economic interpretation.

Consider a country whose dynamics is purely driven by internal demand by identical agents (and thus drop the subscript $i$). Starting from a situation below the steady state, the regime of strategic complementarity makes the agents increase their decision variables $y$ when the other agents are doing the same, in a way that they overshoot the steady state. However, as the agents start to over-accumulate $x$ because the value of $y$ more than offsets the depreciation $(1-\delta)x$, the rise of $y$ slows down. This is both because the agents dislike accumulation and because, as $y$ increases, the function $F(\cdot)$ stops increasing or even decreases (Figure 6B in Appendix A.1). In this situation of large $y$, strategic complementarities are very weak, or we may even be in a regime of strategic substitutability, where an increase of $y$ by one agent makes the other agents decrease their decision variables. Soon, $y$ reverts and starts to decrease, leading to a prolonged recession. At some point, the decrease in $y$ slows down as agents liquidated excessive stock, and we are back to a regime of strategic complementarity. The process restarts.

Depending on the interpretation for the accumulation variable $x$, the decision variable $y$, and the function $F$, the narrative above could correspond to several economic forces causing endogenous business cycles. Beaudry et al. (2020), for example, consider a microfounded model in which $x$ is net worth, $y$ is consumption of durable goods, and $F$ models banks’ willingness to give loans. In good times, lending is perceived safe, and so agents can borrow to consume more durable goods. This further strengthens the economic boom, making lending to be perceived even safer (strategic complementarity). When the economy slows down because of overaccumulation, lending instead starts to be perceived less safe, making banks cut back on credit (strategic substitutability). This behavior of banks can easily trigger a recession, which lasts until agents have liquidated assets in excess, at which point the cycle starts again. This narrative (Beaudry et al., 2020) is similar to many of the previously proposed narratives for finance-based endogenous business cycles (Nikolaidi and Stockhammer, 2017). However, the strategic complementarity-substitutability framework easily lends itself to other narratives, such as ones based on overinvestment in the Keynesian tradition (Kaldor, 1940; Hicks, 1950; Goodwin, 1951), and ones based on temporal clustering of technological innovations (Judd, 1985; Shleifer, 1986; Matsuyama, 1999).

We build on the master stability approach originally proposed by Pecora and Carroll (1990, 1998). Appendix B.1 derives all the equations, while here we give some intuition into how the approach works.

Master stability analysis starts by assuming that all agents are in a synchronized state $\boldsymbol{s}_{t}=(x^{s}_{t},y^{s}_{t})$, in which they all behave alike. We now express the dynamics of individual agents by $x_{i,t}=x^{s}_{t}+x^{\xi}_{i,t}$ and $y_{i,t}=y^{s}_{t}+y^{\xi}_{i,t}$, where $\boldsymbol{\xi}_{i,t}=(x^{\xi}_{i,t},y^{\xi}_{i,t})$ denotes a small deviation from the synchronized state, for example due to idiosyncratic shocks hitting agent $i$. By performing a Taylor expansion of the dynamical equations around the synchronized state, it is possible to derive the evolution of $\boldsymbol{\xi}_{i,t}$, for all agents $i$. As we show in Eq. (11) in Appendix B.1, $\boldsymbol{\xi}_{i,t+1}$ depends on all the deviations of the other agents $j$, $\boldsymbol{\xi}_{j,t}$. So the dynamics of each agent are coupled to those of all other agents. The coupling is mediated by the normalized Laplacian, a matrix derived from the interaction network whose eigenvalues and eigenvectors give key information about the structure and dynamics of the network.

The key idea of the master stability approach is to perform a diagonalization that uncouples the deviations from the synchronized state into orthogonal independent components $\boldsymbol{\zeta}_{i,t}=(x^{\zeta}_{i,t},y^{\zeta}_{i,t})$, called eigenmodes. These components correspond to the eigenvalues and eigenvectors of the normalized Laplacian. By numerically computing the Lyapunov exponents of the eigenmodes dynamics, for each possible eigenvalue, one derives the so-called master stability function. It is then possible to see that eigenmodes corresponding to smaller eigenvalues have less negative Lyapunov exponents, and so take longer to return to zero after a shock (Figure 8 in Appendix B.1). The eigenvectors corresponding to each eigenmode relate the eigenmodes to the dynamics of the agents.

To illustrate this, consider the simplest possible case of a network of two agents (Appendix B.2.1), and assume that each agent follows limit cycle dynamics. The smallest eigenvalue of the normalized Laplacian of this simple network is always null, and the corresponding eigenvector is $(+1,+1)$. The Lyapunov exponent corresponding to this eigenvalue is zero: therefore, any shock hitting the agents does not decay in this eigenmode. Moreover, the eigenvector shows that the shock affects both agents in the same way, so this eigenmode can be interpreted as a shift of the common dynamics of the two nodes. Conversely, the second eigenvalue is positive. It corresponds to negative Lyapunov exponent, so any shock reverts to zero after a few time steps. The eigenvector is $(+1,-1)$, so this eigenmode can be interpreted as the relative shift in the dynamics between the two agents due to the shock. In this case, the master stability analysis indicates that the synchronized dynamics is stable, in the sense that agents synchronize again after a shock.

In a more general network, the eigenmodes illustrate how long different components of the network take to synchronize after a shock. Consider a network composed by six nodes, with two cliques of three nodes each, connected by a link between two of the nodes of the cliques (Appendix B.2.2). In this case, the second smallest eigenvalue is small, and the corresponding eigenvector (Fiedler vector) is positive at the nodes in one clique, and negative at the nodes of the other clique. All other eigenvalues are much larger. Here, synchronization occurs first within cliques and last between cliques. The second eigenmode corresponds to between-clique synchronization.

In conclusion, master stability analysis is a principled method to understanding when synchronized dynamics are stable, and, if they are, how quickly the agents revert to the synchronized state after they are hit by an idiosyncratic shock. In the case of a complex network structure, the analysis quantifies how quickly different parts of the network synchronize. The approach is semi-analytical, in the sense that the relative strength of synchronizing and desynchronizing forces is obtained by numerically computing Lyapunov exponents, but once these are computed, the key insights are obtained analytically.

To conclude this section, we now discuss a real-world example. We first explain how we build the international trade network that specifies the coupling between countries, and then discuss the insights that can be gained by applying complete synchronization theory to these data.

Further to providing intuition into the mechanisms driving synchronization, the theory of complete synchronization helps explain how synchronization of endogenous business cycles could improve on exogenous business cycle models in terms of generating higher comovement between macroeconomic time series.

Consider two countries following two deterministic identical limit cycles in isolation. When coupled, absent shocks, after some time their dynamics perfectly align in the synchronized state $\boldsymbol{s}_{t}=(x^{s}_{t},y^{s}_{t})$. Suppose now that at time $\tau$, the decision variables $y_{1,\tau}$ and $y_{2,\tau}$ of the countries are hit by idiosyncratic shocks $y_{1,\tau}^{\xi}$ and $y_{2,\tau}^{\xi}$ respectively. The comovement between the time series of the decision variables, calculated for the time steps $t$ immediately following the shocks, is given by the Pearson correlation coefficient

In the above equation, cov denotes the covariance, std indicates the standard deviation, and we have used linearity of the covariance to decompose it in various terms. The term $\text{cov}\left(y^{s}_{t},y^{s}_{t}\right)$ indicates the perfect synchonization of the unperturbed deterministic dynamics; the terms $\text{cov}\left(y^{s}_{t},y_{2,t}^{\xi}\right)$ and $\text{cov}\left(y_{1,t}^{\xi},y^{s}_{t}\right)$ indicate the comovement between the deterministic dynamics and the deviations due to the shocks; and the term $\text{cov}\left(y_{1,t}^{\xi},y_{2,t}^{\xi}\right)$ indicates the correlation of the deviations (i.e., shock propagation). Compare this equation with the correlation between time series of two countries that are in the same stable steady state and are hit by shocks $(y_{1,t}^{\xi},y_{2,t}^{\xi})$:

If we assume that the variance of endogenous and exogenous fluctuations is the same, the correlation coefficient is only determined by the numerators of these expressions. Comparing (4) and (5), it is clear that the endogenous component of dynamics potentially increases correlation. In particular, when the variance of the shocks is small relative to the variance of $y^{s}_{t}$ (i.e., the typical amplitude of the endogenous cycles), the correlation coefficient tends to one when business cycles are purely endogenous.

To do so, we must measure empirical comovement in the first place. This is not a trivial task. Papers measuring business cycle comovement have been using all sorts of variables that measure economic activity and all sorts of filters to detrend the corresponding time series (De Haan et al., 2008).

Here, we consider annual data on employment and value added to measure economic activity. We obtain these data from the Penn World Tables, version 10.0 (Feenstra et al., 2015). More specifically, we use the number of persons engaged as a measure of employment, and real GDP at constant 2017 national prices to measure value added. For the countries considered in this paper, data are available from 1950 to 2019.

We cannot use these raw data to measure comovement, because they show clear common trends (confirmed by an augmented Dickey-Fuller test) that would bias upward the estimate of comovement at business cycle frequencies. To address this issue, we consider various modifications. First, we may divide employment or value added by total or working age population.⁹⁹9Total population is available in the Penn World Tables, while working age population is downloaded from the World Bank. Second, we filter the resulting data so as to remove long-run fluctuations from the sample. To do so, we use two filters. One is the well-known Hodrick-Prescott filter, with both the standard parameter $\lambda=100$ that is used for annual data, and the value $\lambda=6.25$ suggested by Ravn and Uhlig (2002). The other is the band-pass filter proposed by Christiano and Fitzgerald (2003), alternatively keeping fluctuations between 2-15 or 2-25 years.¹⁰¹⁰10We prefer the filter by Christiano and Fitzgerald (2003) over the Baxter and King (1999) filter because the latter loses some data points at the extremes of the series. For all these options for the filters, we either consider the cyclical component or the ratio between the cyclical and trend components. In conclusion, from all these combinations we get 27 alternative detrending procedures for both employment and value added.

We find that the mean pairwise Pearson correlation among countries’ employment is 0.23, with a standard deviation of 0.03 across detrending procedures. When considering value added, the correlation is 0.34, with a 0.05 standard deviation. These results are in line with estimates in the literature (Kose and Yi, 2006).

We now check whether endogenous business cycles are indeed necessary to reproduce the empirical level of comovement, or if instead shock propagation in an exogenous business cycle specification is sufficient. To do so, we simulate the model under three different parameterizations, and compare the comovement of the decision variables $y$ to empirical comovement as measured in the previous section. We perform this comparison at the aggregate, country and country-pair level.

We simulate the model under three baseline sets of parameters (many more parameter combinations are checked in Appendix D.2.3). These correspond to the deterministic skeleton of the model producing limit cycles, or converging back to the steady state with damped fluctuations (focus) or monotonically (node). We choose the parameters in Appendix A.2 as a baseline. Moreover, we parameterize the shock process as an AR(1) with persistence parameter $\rho=0.3$ and standard deviation $\sigma$ varying between 0.01 and 0.2. We simulate the model for 280 time steps after removing an initial transient. It makes sense to interpret time steps as quarters (Appendix A.2), so this leads to a 70-years period that is as long as the data we compare against (1950-2019). To calibrate the interaction network, we consider year 1990, which is in the middle of the available data (we will also consider time-varying interaction coefficients as a robustness test).

Figure 3 shows our key result. When the model produces limit cycles, the mean correlation coefficient across countries ranges from 0.99 in the case of idiosyncratic shocks with standard deviation $\sigma_{u}=0.01$ (or 1% of the steady state value of the decision variables) to 0.03 when the standard deviation of idiosyncratic shocks is $\sigma_{u}=0.20$ (20% of the steady state). By contrast, when the model produces fluctuations that converge to a stable steady state, the average correlation coefficient is never larger than 0.15, for any value of the standard deviation of idiosyncratic shocks. More specifically, when the steady state is a focus, the average correlation varies between 0.08 and 0.15, whereas when the steady state is a node it is close to zero. As empirical comovement is 0.23 when considering employment and 0.34 when considering value added, only in the limit cycle case the model can reproduce this level of comovement. This is confirmed by a t-test for the difference in means, with high statistical significance for all values of $\sigma_{u}$ ($t$-statistic $<-11$). In conclusion, under the assumptions in this section the model can only match the empirical level of comovement with endogenous business cycles, for standard deviation of idiosyncratic shocks between 5 and 10% of the amplitude of the limit cycles. This confirms the theoretical intuition in Section 4.3.

Next, Figure 4 zooms in at the level of countries. Here, we select a standard deviation of idiosyncratic shocks of 0.08, under which the limit cycle model produces a correlation that is close to the data (Figure 3). This figure conveys three results. First, for every country and not only for some, comovement is higher in the limit cycle case than in the focus and node regimes. Second, under limit cycles, empirical correlation coefficients are more similar to simulated ones (Pearson empirical-simulated coefficients: 0.69). This means that countries whose dynamics are more correlated to those of other countries in the data are also more correlated in the model. For instance, the mean correlation coefficient of the Netherlands is 0.46 in the data and 0.40 in the model, while the mean correlation coefficient of Brazil is 0.04 in the data and 0.12 in the model. This similarity between empirical and simulated correlation coefficients extends to the focus case, although it is slightly lower (Pearson 0.62). It is however completely absent when the steady state is a node (Pearson -0.15). The third result is that, in the limit cycle and focus cases, both empirical and simulated correlations are clearly linked to the position of the countries in the international trade network, as exemplified by the Fiedler eigenvector discussed in Section 4.2. Countries that are more correlated to the other countries in the sample tend to have intermediate values for their eigenvector components, whereas countries with extreme values such as Brazil tend to have lower average correlations.

These results are confirmed in Figure 5, which shows pairwise correlations. In the data, pairwise correlations are high within Europe, North America and Asia, but they are lower across continents. The simulations, both in the cycle and focus cases, generally reproduce these patterns, although they fail to yield the few negative correlations that can be seen in data. By contrast, when the model is parameterized as a node, pairwise correlations do not show a clear pattern (Figure 13 in Appendix D.1).

In conclusion, the results in this section support the main claim of the paper, namely that endogenous business cycles are needed to explain the level of international comovement that can be found in the data. They also show that the theory developed in Section 4 can be fruitfully employed to interpret synchronization.

In Appendix D.2 we check if the main result of the paper is robust to alternative specifications. We perform seven robustness tests. First, we investigate how higher persistence of the shock processes $\rho$ may influence the results. We find that, with high persistence, the model becomes more noisy for a given level of the standard deviation $\sigma$. So, in the limit cycle case, smaller values of $\sigma$ are needed for the model to match the empirical level of comovement. Still, the focus and node cases cannot match empirical comovement for any value of $\rho$ and $\sigma$. The second test is to relax the assumption that the interaction coefficients $\epsilon_{ij}$ have to be fixed. In particular, we consider time-varying interaction coefficients $\epsilon_{ij,t}$, with $t=1965,\ldots,2011$, representing varying shares of international trade over the data availability period. We find virtually identical results to the baseline. The third robustness test considers alternative parameters for the baseline specification of the interaction function $F$, which is a generic quartic function. It shows that in no case the model can match empirical comovement under focus or node parameterizations, whereas it can always do so with limit cycles, for an appropriate level of noise. The results are similar in the fourth robustness test, which considers an alternative logistic specification of the function $F$. The fifth robustness test considers other bifurcation parameters than the strength of strategic complementarities at the steady state. It still shows that node and focus parameterizations cannot produce a level of comovement that is as high as in the data, confirming that it is the type of dynamics and not the strength of strategic complementarities that substantially changes the level of comovement. The sixth robustness test shows that values of the $\alpha_{1}$ and $\alpha_{2}$ parameters that are heterogeneous across countries, although relatively close to the baseline, do not lead to almost any difference in the results. The seventh, and last, robustness test considers common shocks. We find that for sufficiently strong common shocks the focus parameterization does lead to sufficiently high comovement, but the node parameterization always fails to do so.