# Business cycles fluctuations in three-sector intertemporal equilibrium models

Kazuo Nishimura, Florian Pelgrin, Alain Venditti

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Kazuo Nishimura, Florian Pelgrin, Alain Venditti. Business cycles fluctuations in three-sector intertemporal equilibrium models. Journal of Economic Theory, 2025, 226, pp.106010. 10.1016/j.jet.2025.106010. hal-05085177

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Submitted on 26 May 2025

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ations. Notably, the instability of the steady state—combined with nonlinear interactions among aggregate variables — naturally gives rise to deterministic limit cycles.<fnref n="1" /> More recent contributions by Beaudry et al. (2015, 2017, 2020) have extended this analysis to stochastic settings, identifying endogenous medium-term cycles that align with economic periodicities observed in US trendless macroeconomic data.<fnref n="2" />

Building on these insights, this paper introduces a frictionless, deterministic two-consumption good, three-sector intertemporal equilibrium model, unveiling a novel driver of endogenous cycle fluctuations. While conventional wisdom attributes these fluctuations primarily to sectoral capital intensity differences, our model uncovers a richer, preference-driven mechanism that redefines intratemporal decisions and intertemporal consumption trade-offs. This fundamentally reshapes the model's dynamics, shedding new light on the origins of self-sustaining economic cycles.

The paper makes three primary contributions to the literature on endogenous fluctuations. First, we propose a *minimal* three-sector model with a pure consumption good, a *mixed* good, and a pure investment good. The mixed good serves as both a final consumption good and a capital-accumulating investment good, whereas the pure investment good is used solely for capital accumulation. We consider general preferences that value a Cobb-Douglas aggregator of the pure consumption good and the mixed consumption-investment good, alongside an additive-separable endogenous labor supply. As highlighted in the literature, the consideration of at least three sectors and a four-dimensional dynamical system is a *necessary* condition for the occurrence of deterministic endogenous fluctuations driven by complex characteristic roots. In other words, the three-sector specification, combined with the dual-purpose good, produces a four-dimensional dynamical system capable of generating limit cycles through a Hopf bifurcation.<fnref n="3" /> This structure captures the essential trade-offs between consumption, investment, and labor decisions, making it the *simplest* macroeconomic setting that generates self-sustained periodic cycles under a plausible set of structural parameters' values, even in the absence of market frictions.<fnref n="4" />

In so doing, we introduce a novel preference-based mechanism behind endogenous business cycles based on the introduction of a *mixed consumption-investment good*. Unlike standard models, which feature only distinct consumption and investment goods, the introduction of a third, hybrid good modifies both intratemporal decisions and the intertemporal consumption trade-off. Consequently, this new preference-based mechanism complements the well-known Rybczinski and Stolper-Samuelson effects, which stem from sectoral capital intensity differences.

On the one hand, the mixed good smooths consumption adjustments over time, making the economy's dynamics more sensitive to small capital intensity differences. This generates more persistent cycles, even when differences in sectoral capital intensity are small, with the preference mechanism becoming the key driver of business cycles. On the other hand, the dual role of the mixed good introduces a second layer of trade-offs, linking consumption and investment decisions more tightly. This amplifies the oscillatory dynamics caused by technological conditions. Endogenous fluctuations then naturally emerge from sectoral capital intensity differences, intertemporal consumption trade-offs between the two consumed goods, or the interaction of both mechanisms.

A simple comparative static exercise illustrates that endogenous fluctuations can arise under more plausible discount rates, intertemporal substitution elasticities, and consumption shares, regardless of whether the preference-based or technology-based mechanism is the primary driver of business cycles. In this context, the simplifying assumption of a mixed consumption-investment good reduces the excessively high elasticity of intertemporal substitution typically required by one-consumption-good multi-sector models.

Second, the model highlights the role of *endogenous labor supply* as a key amplifier or dampener of cyclical dynamics. When labor supply is highly elastic, the consumption-leisure trade-off smooths fluctuations, stabilizing the economy dynamics. Conversely, lower labor supply elasticity strengthens the feedback loop between consumption and labor, intensifying oscillations. This dual effect enriches the cyclical behavior of the model and makes labor elasticity a key determinant of the emergence of endogenous fluctuations.

Third, the paper also offers a methodological contribution. We fully characterize the bifurcation conditions and the periodicity of cycles in a four-dimensional dynamical system. While the literature provides sufficient conditions for the existence of real roots (see Magill and Scheinkman, 1979), deriving clear-cut, sufficient conditions for the emergence of complex characteristic roots in dynamical systems of dimension higher than two has remained an open challenge (see Magill, 1977, 1979a, 1979b, for necessary conditions). Moreover, existing results often rely on the restrictive assumption of a linear utility function.<fnref n="5" /> In this context, we establish explicit conditions for the existence of complex characteristic roots and Hopf bifurcations, even under strictly concave utility functions.<fnref n="6" />

<footnote n="1">¹ See also Becker and Foias (1994) and Becker and Tsyganov (2002) for the existence of endogenous cycles in intertemporal equilibrium models with heterogeneous agents. Boldrin and Montrucchio (1986) show that complex chaotic fluctuations can also arise but at the cost of a high degree of impatience, or an implausibly large elasticity of intertemporal substitution, or a high degree of complementarity between inputs in the production functions (see e.g. Benhabib and Rustichini, 1990; Sorger, 1992; Nishimura and Yano, 1995).</footnote>

<footnote n="2">² Other strands of the literature discussing the emergence of limit cycles include contributions on innovation cycles and growth (Matsuyama, 1999; Growiec et al., 2018), on endogenous credit and productivity cycles (Gorton and Ordoñez, 2020), on endogenous credit cycles in OLG models (Azariadis and Smith, 1998; Myerson, 2012; Gu et al., 2013), on endogenous learning- and bounded rationality-based business fluctuations (Hommes, 2013).</footnote>

<footnote n="3">³ A four-dimensional dynamical system can also be derived from a *two-sector* model that includes a mixed consumption-investment good and a pure capital good. However, if a fixed factor such as labor is included, the model introduces three inputs but only two goods, resulting in a non-square input coefficient matrix, which is thus non-invertible. Consequently, it becomes impossible to express outputs, wages, and interest rates as functions of capital stocks and prices (see Brock, 1973; Brock and Burmeister, 1976).</footnote>

<footnote n="4">⁴ Note that the *frictionless* assumption simplifies the analysis and isolates the *minimal* conditions required to generate endogenous fluctuations. The inclusion of market imperfections would amplify the likelihood of endogenous cycles (see Benhabib et al., 2008; Nishimura and Shigoka, 2006, 2019).</footnote>

<footnote n="5">⁵ Cartigny and Venditti (1994) and Venditti (1996) provide some results for general multi-sector models with a non-linear utility function but do not provide conditions for complex roots.</footnote>

<footnote n="6">⁶ See also Dufour et al. (2015) where the Hopf bifurcation is shown to be relevant from an empirical perspective in two-sector models with productive externalities and sunspot fluctuations.</footnote>
The periodicity of cycles is another key aspect of our analysis. For linear homogeneous preferences, we derive closed-form solutions that link cycle duration to technological parameters and consumption shares. Under strictly concave preferences, we rely on numerical approximations to show that endogenous cycles remain highly sensitive to investment, labor-market and consumption dynamics, even in more realistic settings. This deeper characterization of periodicity provides new insights into the relationship between technological and preference parameters and the duration of endogenous economic cycles. It also highlights the need for future research to assess the theoretical predictions of endogenous fluctuation models against the observed periodicity of business cycles.

Overall, our model offers a parsimonious yet insightful framework for understanding the intrinsic drivers of business cycles. By blending preference-based and technology-driven mechanisms, it captures a richer set of cyclical dynamics and fluctuation sources, even in a frictionless setting. Furthermore, it provides new theoretical results into the periodicity of business cycles.

The rest of the paper is organized as follows. Section 2 introduces the model and characterizes the intertemporal equilibrium. Section 3 examines the dynamics under a linear homogeneous utility specification, while Section 4 extends the analysis to strictly concave preferences with endogenous labor. Section 5 concludes. The main proofs are contained in a final Appendix and additional details are provided in an online Appendix.

## 2. The model

This section first describes the production structure and the preferences of our 3-sector model. Then we discuss the intertemporal equilibrium and the steady state. Finally, we characterize the Jacobian eigenvalues associated with linearization around the steady state.

### 2.1. The 3-sector economy

We consider an economy composed of three sectors, each producing a distinct type of good: a pure consumption good $y_0$, a mixed good $y_1$, and a pure investment good $y_2$ (Assumption A.1). The mixed good serves both as a consumption and investment good, whereas the pure investment good is used solely for capital accumulation.<fnref n="7" />

**Production technology** Each good is produced using two types of capital, $k_1^j$ and $k_2^j$, and labor $l^j$, with sector-specific Cobb-Douglas production functions (Assumption A.2):

$$y_j = A_j(k_1^j)^{\theta_1} (k_2^j)^{\theta_2} (l^j)^{1-\theta_1-\theta_2}, \quad j = 0, 1, 2, \quad (1)$$

where $A_j$ denotes the total factor productivity in sector $j$, and the parameters ($\theta_1, \theta_2$) capture the capital intensity in each sector, with ($\alpha_1, \alpha_2$), ($\beta_1, \beta_2$), and ($\gamma_1, \gamma_2$) corresponding to the technological parameters in sectors 0, 1, and 2, respectively. The total endowments of labor and capital satisfy the resource constraints:

$$\ell = l^0 + l^1 + l^2, \quad k_1 = k_1^0 + k_1^1 + k_1^2, \quad k_2 = k_2^0 + k_2^1 + k_2^2. \quad (2)$$

The capital good $k_2$ represents traditional investment goods such as machinery and equipment, while $k_1$ refers to capital goods that can also be consumed, such as vehicles or computers. This structure resembles recent models exploring the network origins of fluctuations, where input-output linkages propagate shocks across sectors (e.g., Acemoglu et al., 2012).

**Endowments** The economy is populated by a large number of identical infinitely-lived agents. Without loss of generality, we assume that the total population is constant and normalized to one. At each period, the representative household is endowed with $\bar{\ell}$ units of time and amounts of capital in each of the two sectors $k_1(t)$ and $k_2(t)$.

**Consumption, investment and capital accumulation** Given the output prices $p_0(t)$, $p_1(t)$, and $p_2(t)$, the rental rates of capital $r_1(t)$ and $r_2(t)$, and the wage rate $w(t)$, the household decides how to allocate its available time between hours worked $\ell(t)$ across the three sectors, and how to share its consumption between the pure consumption good $c_0(t)$ and the mixed good $c_1(t)$. Meanwhile, the representative household also owns the representative firm in each sector, and the economy features two types of capital accumulation. The mixed good $y_1$ is partially allocated to investment $i_1$ and partially to consumption $c_1$, while the pure investment good $y_2$ is entirely devoted to capital formation. The capital accumulation dynamics are then governed by:

$$y_1(t) = i_1(t) + c_1(t) = \dot{k}_1(t) + gk_1(t) + c_1(t), \quad y_2(t) = i_2(t) = \dot{k}_2(t) + gk_2(t) \quad (3)$$

where $g > 0$ is the depreciation rate, assumed to be identical across the two capital types for simplicity. This formulation highlights the dual role of the mixed good as both a consumption and investment good, creating a trade-off between current consumption and future productive capacity.

<footnote n="7">⁷ This production structure departs from standard multi-sector models by introducing a mixed consumption-investment good. Indeed, traditional multi-sector models generally consider either (i) only one pure consumption good and n sectors producing mixed goods which are consumed and used as investment intermediate goods in the production of all sectors (Benhabib and Nishimura, 1979a, 1979b, 1981; Huffman and Wynne, 1999) or (ii) n + 1 sectors producing mixed goods used as both consumption and investment goods in all sectors (Acemoglu et al., 2012; Long and Plosser, 1983).</footnote>
**Equilibrium production and pricing** For given $z \equiv (k_1, k_2, y_1, y_2, \ell)$, choosing the pure consumption good as the *numéraire* (i.e., $p_0(t) = 1$), profit maximization in each sector determines the demand functions for labor and capital, which depend on the capital stocks $k_1$ and $k_2$, the production levels of the investment goods $y_1$ and $y_2$, and total labor $\ell$. The optimal production of the pure consumption good is given by:

$$c_0 = y_0 = T(z) \equiv A_0 (k_1^0(z))^{\alpha_1} (k_2^0(z))^{\alpha_2} (\ell^0(z))^{1-\alpha_1-\alpha_2} \quad (4)$$

where $T(z)$ defines the value function determining the optimal output level of the pure consumption good. Using the envelope theorem, the equilibrium prices and rental rates are given by the partial derivatives of $T(z)$:

$$r_1 = T_{k_1}(z), \quad r_2 = T_{k_2}(z), \quad p_1 = -T_{y_1}(z), \quad p_2 = -T_{y_2}(z), \quad w = T_{\ell}(z).$$

## 2.2. Preferences

The representative household's instantaneous utility is specified by the following function, based on a standard constant relative risk aversion (CRRA) formulation (Assumption A.3):

$$u(c_0, c_1, \ell) = \frac{(c_0^\theta c_1^{1-\theta})^{1-\eta} - 1}{1-\eta} - \frac{\ell^{1+\chi}}{1+\chi},$$

where $c_0$ denotes consumption of the pure consumption good and $c_1$ denotes consumption of the mixed consumption good, with $\eta \ge 0$, $\theta \in (0, 1]$ and $\chi \ge 0$. The parameter $\theta$ measures the share of the pure consumption good $c_0$ within total utility, while $\chi$ drives the elasticity of labor supply $\varepsilon_l = 1/\chi$. Specifically, when $\theta = 1$, one retrieves the standard formulation proposed by MaCurdy (1981).

Four points are worthy of note. First, the mixed and pure consumption goods can be interpreted in distinct ways. To some extent, the pure consumption good can be viewed as a nondurable item, while the mixed consumption-investment good can be classified as a durable good. Our specification, however, assumes full depreciation of “new” machines allocated to consumption. A more realistic formulation could assume that “consumer durables” depreciate gradually over time, providing a flow of services throughout their useful life. Meanwhile, this extended formulation, though more complex in terms of notation, would not affect our main conclusions regarding the existence of endogenous cycles. Indeed, a fundamental characteristic of a durable good is that it introduces a service flow into the utility function proportional to the stock of the durable good. Consequently, any change—whether an increase or decrease—in durable consumption decisions at one period generates a propagation effect on subsequent periods' decisions through the service flow, which can amplify the initial fluctuations (see Utaka, 2011).

Second, the utility function specification is flexible enough to encompass various interesting cases studied in the multi-sector model literature, including linear utility functions (Shea, 2002), Cobb-Douglas preferences (Acemoglu et al., 2012), and log utility functions (Shea, 2002), contingent on the values of the parameters $\theta$ and $\eta$. Third, the model assumes the consumption of only two goods, which we consider a *minimal assumption*. It can be shown that introducing a third consumed good, through a second mixed consumption-investment good, neither leads to a new mechanism nor yields new results compared to our benchmark model. Fourth, the preference specification implies that the (pure) elasticities of intertemporal substitution in pure and mixed consumption goods $c_0$ and $c_1$, $\varepsilon_{00}$ and $\varepsilon_{11}$, and the (cross-) elasticities of intertemporal substitution between the two goods, $\varepsilon_{01}$ and $\varepsilon_{10}$, are given by:

$$\begin{align}
\varepsilon_{00} &= -\frac{u_1}{u_{11}c_0} = \frac{1}{1-\theta(1-\eta)}, \quad \varepsilon_{01} = -\frac{u_1}{u_{12}c_1} = -\frac{1}{(1-\theta)(1-\eta)}, \tag{5} \\
\varepsilon_{10} &= -\frac{u_2}{u_{21}c_0} = -\frac{1}{\theta(1-\eta)}, \quad \varepsilon_{11} = -\frac{u_2}{u_{22}c_1} = \frac{1}{1-(1-\theta)(1-\eta)}. \nonumber
\end{align}$$

Importantly, $\varepsilon_{00}$ and $\varepsilon_{11}$ remain finite as long as $\theta < 1$, even in the case of a non-strictly concave utility with $\eta = 0$. Of course, if $\theta = 1$, we obtain the standard case of a unique pure consumption good, and the elasticity of intertemporal substitution in consumption is given by $\varepsilon_{00} = 1/\eta$, resulting in infinity when the utility is linear.

The intertemporal optimization problem of the representative agent is then given by:

$$\max_{\{c_1(t), k_1(t), k_2(t), y_1(t), y_2(t), \ell(t)\}} \int_0^{+\infty} \left( \frac{[c_0(t)^\theta c_1(t)^{1-\theta}]^{1-\eta} - 1}{1-\eta} - \frac{\ell(t)^{1+\chi}}{1+\chi} \right) e^{-\delta t} dt \quad (6)$$

s.t. (3), (4) and $k_1(0), k_2(0)$ given

where $\delta \ge 0$ is the discount rate.

## 2.3. Decentralized intertemporal equilibrium and steady state

We are now in a position to characterize a decentralized equilibrium path and to prove the existence of a unique steady state. From this point onward, we define $\mathbf{p} \equiv (p_1, p_2)$ and $\mathbf{k} \equiv (k_1, k_2)$.

**Proposition 1 (Intertemporal equilibrium).** At equilibrium, wage rate and rental rates are $C^1$ functions of the output prices only, $w = \omega(\mathbf{p})$, $r_1 = r_1(\mathbf{p})$ and $r_2 = r_2(\mathbf{p})$, while outputs are $C^1$ functions of the capital stocks and the output prices, $y_j = y_j(\mathbf{k}, \mathbf{p})$, $j = 0, 1, 2$, with $c_0 = c_0(\mathbf{k}, \mathbf{p})$ and $c_1 = c_1(\mathbf{k}, \mathbf{p}) = c_0(1-\theta)/(\theta p_1)$, and the optimal labor supply is given by:
$$ \ell = \left[ w \theta \left( \frac{1-\theta}{\theta p_1} \right)^{(1-\theta)(1-\eta)} c_0^{-\eta} \right]^{1/\chi} \qquad (7) $$

A decentralized competitive equilibrium path is then a solution {$k(t), p(t)$}$_{t \ge 0}$ of (i) the following motion equations,

$$ \begin{gathered} \dot{k}_1 = y_1(\mathbf{k}, \mathbf{p}) - gk_1 - c_1(\mathbf{k}, \mathbf{p}), \quad \dot{k}_2 = y_2(\mathbf{k}, \mathbf{p}) - gk_2 \\ \dot{p}_1 = \frac{(\delta+g)p_1 - r_1(\mathbf{p}) + F_1(\mathbf{k}, \mathbf{p})}{E_{\eta, \chi}^{\theta}}, \quad \dot{p}_2 = \frac{(\delta+g)p_2 - r_2(\mathbf{p}) + F_2(\mathbf{k}, \mathbf{p})}{E_{\eta, \chi}^{\theta}} \end{gathered} \qquad (8) $$

where $F_1(\mathbf{k}, \mathbf{p})$, $F_2(\mathbf{k}, \mathbf{p})$, and $E_{\eta, \chi}^{\theta}$ are given in Appendix 6.1, and of (ii) the transversality conditions:

$$ \lim_{t \to +\infty} e^{-\delta t} p_1(t) k_1(t) = 0 \quad \text{and} \quad \lim_{t \to +\infty} e^{-\delta t} p_2(t) k_2(t) = 0. $$

In this respect, Proposition 2 states the existence of a unique steady state of the dynamical system (8).

**Proposition 2 (Steady state).** There exists a unique steady state ($c_1^*, \mathbf{k}^*, \mathbf{p}^*$) > 0 of the dynamical system (8) with $c_0^* = c_0(\mathbf{k}^*, \mathbf{p}^*)$ and $c_1^* = c_0(\mathbf{k}^*, \mathbf{p}^*) \frac{1-\theta}{\theta p_1^*}$. Moreover, the labor supply at the steady is given by (7) with $w = w(\mathbf{p}^*)$.

As expected, the absence of market failures implies that the decentralized competitive intertemporal equilibrium defined in Proposition 1 is fully equivalent to the solution of the planner's problem.

**Remark 1 (Externality and inefficiency).** Our model can be extended to include externalities in the utility function, such as those associated with public expenditures, which disrupts the equivalence between the decentralized solution and the optimal social planner solution, thereby introducing inefficiencies. Indeed, consider an economy where the government provides public services, denoted by $G(t)$, which directly influence the utility function as follows:

$$ u(c_0, c_1, \ell, G) = \frac{(c_0^\theta c_1^{1-\theta})^{1-\eta-1}}{1-\eta} - \frac{\ell^{1+\chi}}{1+\chi} + \frac{G^\phi}{\phi} $$

where $\phi \in [0, 1)$. Following Giannitsarou (2007) and Nishimura et al. (2013), these public services are financed through a balanced-budget constraint:

$$ G(t) = \tau c_0(t) + \tau p_1(t) c_1(t) = G_0(t) + p_1(t) G_1(t) $$

where $\tau \ge 0$ represents a constant consumption tax rate. It can then be demonstrated that the decentralized competitive intertemporal equilibrium corresponds to the equilibrium characterized in Proposition 1, with the exception of the optimal labor supply, which is adjusted by a scale factor dependent on the constant tax rate, $1/(1+\tau)$:

$$ \ell = \left[ w \frac{\theta}{1+\tau} \left( \frac{1-\theta}{\theta p_1} \right)^{(1-\theta)(1-\eta)} c_0^{-\eta} \right]^{1/\chi} $$

Consequently, the steady state is slightly altered, but the stability properties of the decentralized equilibrium path remain unchanged.<fnref n="8" />

## 2.4. Dynamics characterisation

Characterizing the occurrence of complex dynamics requires analyzing the local stability conditions at the steady state (Proposition 2). Linearizing the dynamical system around the steady state ($c_1^*, \mathbf{k}^*, \mathbf{p}^*$) yields a 4 × 4 Jacobian matrix and the corresponding characteristic polynomial, both detailed in Appendix 6.3. Proposition 3 establishes key properties of the Jacobian eigenvalues under either a linear homogeneous or a strictly concave utility function, underscoring the role of labor supply elasticity as a driver of endogenous fluctuations.

**Proposition 3 (Jacobian eigenvalues and labor supply elasticity).** For any $\eta \ge 0$, $\theta \in (0, 1]$ and $\chi \ge 0$, the Jacobian eigenvalues associated with the degree-4 characteristic polynomial are necessarily either:

(i) four real and distinct roots;

(ii) two pairs of non-real complex conjugate roots;

(iii) two real double roots.

Notably, if $\lambda$ is an eigenvalue of the Jacobian matrix, then $\bar{\lambda}$, $\delta - \lambda$ and $\delta - \bar{\lambda}$ are also eigenvalues. Furthermore, endogenous business cycles fluctuations can emerge under the following conditions:

<footnote n="8">⁸ A similar property can be obtained under labor income taxes (see Abad and Venditti, 2021). More details are provided in an online Appendix.</footnote>
- In the presence of a linear homogeneous utility function ($\eta = 0$), irrespective of the (inverse of) labor supply elasticity $\varepsilon_l = 1/\chi$;

- In the presence of strictly concave utility function ($\eta > 0$) under either an inelastic or an elastic labor supply.

The results concerning the complex-to-real characteristic roots and their multiplicities are consistent with the findings of Kurz (1968) and Levhari and Liviatan (1972). A noteworthy aspect of Proposition 3 is that, under a linear homogeneous utility function ($\eta = 0$), the labor supply elasticity, governed by $\chi$, has no influence on the local stability properties of the steady state. In other words, the occurrence of endogenous fluctuations in the context of a linear homogeneous utility function is entirely independent from (the value of) the labor supply elasticity.

The rationale behind this new result goes as follows. According to Proposition 1, the mixed-to-pure consumption good ratio along the equilibrium path is defined by $c_1(t)/c_0(t) = (1 - \theta)/(\theta p_1(t))$. Consequently, the two components of consumption are positively correlated and, for a given price of the mixed consumption good $p_1(t)$, vary linearly. Using equation (7), it follows that the labor supply becomes independent of consumption when $\eta = 0$, varying solely across time through changes in the wage rate and the price of the mixed consumption good. As a result, consumption fluctuations are not directly impacted by the labor supply elasticity. In contrast, this separation result between the labor supply elasticity and the Jacobian eigenvalues, and thus the characterization of fluctuation dynamics, breaks down in the presence of strictly concave preferences.

In the sequel, building upon Proposition 3, we first consider the case of a linear homogeneous utility function ($\eta = 0$). Initially focusing on a single consumption good, i.e. $\theta = 1$, we outline the existence of a technological trade-off, as shown by Benhabib and Nishimura (1979a, 1985). This trade-off hinges on specific conditions related to capital intensity differences across sectors, providing a rationale for fluctuations in output, capital stocks, interest rates, and prices. We further extend these conditions to ensure the existence of complex roots at the core of the Hopf bifurcation and endogenous cycles. Subsequently, we explore the formulation with a mixed consumption-investment good, $\theta \in (0, 1)$, where an intertemporal consumption trade-off between the two consumption goods emerges as a new driver of endogenous fluctuations. Finally, we consider the complete model with a strictly concave utility function and endogenous labor, and underscore the role of the labor supply elasticity.

### 3. Endogenous cycles under linear homogeneous preferences

In this section, we investigate the emergence of endogenous fluctuations under linear homogeneous preferences. Rather than providing an exhaustive stability analysis, we build on the results of Benhabib and Nishimura (1979a) and focus on the case with complex characteristic roots—the essential condition for generating persistent cyclical dynamics through a Hopf bifurcation.

#### 3.1. Capital intensity and technological trade-offs

We start by introducing the following core assumptions:

**Assumption 1 (Sectoral capital intensity ranking).**

* Sector 2 is the most capital-intensive in capital good 1, followed by sector 1, then sector 0:

$$ \frac{k_2^2}{l^2} > \frac{k_1^1}{l^1} > \frac{k_0^0}{l^0}. $$

* Conversely, sector 0 is the most capital-intensive in capital good 2, followed by sector 2, then sector 1:

$$ \frac{k_2^0}{l^0} > \frac{k_2^2}{l^2} > \frac{k_2^1}{l^1}. $$

**Assumption 2 (Large capital intensity difference).** The gap between sectors 0 and 1 in terms of capital intensity is sufficiently large:

$$ \frac{k_1^1}{l^1} - \frac{k_0^0}{l^0} \text{ is large enough.} $$

**Assumption 3 (Small capital intensity difference).** The gap between sectors 0 and 1 is relatively small:

$$ \frac{k_1^1}{l^1} - \frac{k_0^0}{l^0} \text{ is small enough.} $$

Assumption 1 establishes the sectoral capital intensity hierarchy, while Assumptions 2 and 3 are mutually exclusive and introduce two contrasting, fundamental scenarios based on the magnitude of the capital intensity difference. These assumptions directly influence the economy's ability to generate endogenous cycles: larger capital intensity gaps drive stronger technological trade-offs, whereas smaller gaps require the reinforcement of preference-based mechanisms to sustain fluctuations.

Taken together, Assumptions 1 and 2 closely align with the model analyzed by Benhabib and Nishimura (1979a), which assumes a utility function with a single pure consumption good (corresponding to $\theta = 1$ in our framework). Note that similar but more restrictive conditions have been identified by Nishimura and Takahashi (1992). Theorem 1, however, extends this analysis by considering the case of two consumption goods ($\theta \in (0, 1)$).
<table><caption>**Table 1**
Conditions for a Hopf bifurcation</caption><thead><tr><th></th><th>Hopf Bifurcation parameter</th><th>Conditions</th><th>Assumptions</th></tr></thead><tbody><tr><td>Case A-1</td><td>δ* > 0</td><td>δ > δ*</td><td>1 and 2, θ = 1</td></tr><tr><td>Case A-2</td><td>θ* ∈ (0, 1) and ̂θ ∈ (θ*, 1)</td><td>δ < δ* and θ ∈ (θ*, ̂θ)</td><td>1 and 2, θ ∈ (0, 1)</td></tr><tr><td>Case B</td><td>θ* ∈ (0, 1) and ̂θ ∈ (θ*, 1)</td><td>θ ∈ (θ*, ̂θ)</td><td>1 and 3, θ ∈ (0, 1)</td></tr></tbody></table>

## 3.2. Characterizing endogenous fluctuations

Under these assumptions, the following Theorem formalizes the conditions under which endogenous cycles arise, regardless of the elasticity of labor supply.

**Theorem 1 (Endogenous fluctuations for any labor supply elasticity).** Let Assumption 1 hold and $η = 0$. Then, for any nonnegative value of the inverse of the labor supply elasticity $χ ≥ 0$:

A - For large capital intensity differences (Assumption 2):

1. When only the pure consumption good is consumed ($θ = 1$) and the discount factor is sufficiently high, endogenous cycles emerge via a technological mechanism driven by the Rybczinski and Stolper-Samuelson effects.

2. When both pure and mixed consumption goods are valued ($θ ∈ (0, 1)$) and the discount factor is low enough, endogenous cycles arise through a combined mechanism where technology-driven dynamics are amplified by intertemporal substitution effects.

B - For small capital intensity differences (Assumption 3), endogenous cycles emerge predominantly through a preference-based mechanism, which reinforces the weaker technological trade-off. This occurs independently of the discount factor, provided that the share of the pure consumption good is sufficiently large.

Endogenous fluctuations in this framework correspond to closed orbits around the steady state, triggered by a Hopf bifurcation. The bifurcation parameters and conditions vary across cases, as summarized in Table 1.<fnref n="9" />

Two points are worth commenting on. First, as discussed in Section 3.4, the linear homogeneous specification of the utility function ensures that labor supply is independent of consumption decisions. Consequently, as established in Theorem 1, the existence of endogenous fluctuations is unaffected by the elasticity of labor supply, denoted $1/χ$. Notably, when $η = 0$, the elasticities of intertemporal substitution in pure and mixed consumption goods—$ε_{00}$ and $ε_{11}$, defined in (5)—reach their maximum values. Specifically, with $θ = 1$, where only the pure consumption good is consumed, $ε_{00}$ becomes infinite. In this case, if the representative consumer sufficiently substitutes consumption over time to offset fluctuations—such as periods of reduced utility—her labor decisions, derived from the consumption-leisure trade-off, do not influence the intertemporal consumption trade-off.

Second, the critical values $δ^*$ (case A-1), $θ^*$ and $̂θ$ (cases A-2 an B) are *Hopf bifurcation values* that typically give rise to closed orbits around the steady state. These steady states, and the closed orbits, may be either saddle-point stable or totally unstable, depending on whether the bifurcation occurs to the right or left of $δ^*$, $θ^*$, or $̂θ$.<fnref n="10" /> The Hopf bifurcation theorem does not provide a straightforward condition to distinguish between supercritical and subcritical bifurcations. The distinction depends on the sign of a coefficient from the third-order Taylor approximation of the dynamical system (8) on the two-dimensional center manifold corresponding to the bifurcating eigenvalues. However, Benhabib and Nishimura (1979a) show that in the supercritical case, the periodic orbit is stable within a two-dimensional submanifold due to the Hamiltonian structure of the model. This stability explains the term “saddle-point stable”.

## 3.3. Endogenous cycles drivers

We now turn to the two primary and complementary drivers of endogenous cycles:

1. A **technological trade-off** driven by capital reallocations across sectors, amplified by the Rybczinski and Stolper-Samuelson effects.

2. An **intertemporal consumption trade-off**, arising from the interactions between pure and mixed consumption goods. This mechanism can either amplify the technological effect or initially trigger endogenous fluctuations, even in the absence of significant sectoral capital intensity differences.

**The technological trade-off** In Case A, where only the pure consumption good is consumed ($θ = 1$), the sectoral capital intensity ranking (Assumption 1) and the large capital intensity gap (Assumption 2) trigger cyclical reallocations. These properties, combined

<footnote n="9">⁹ In the technical Appendix, we also show that endogenous fluctuations may arise in case A-2 when $δ > δ^*$.</footnote>

<footnote n="10">¹⁰ From a technical standpoint, case A-1 steady state is saddle-point stable with oscillating convergence for $δ ∈ (0, δ^*)$ and totally unstable for $δ > δ^*$. For cases A-2 and B, the steady state is saddle-point stable with oscillating convergence if $θ < θ^*$, totally unstable if $θ ∈ (θ^*, ̂θ)$ and again saddle-point stable with oscillating convergence if $θ > ̂θ$.</footnote>
with the Rybczinski and Stolper-Samuelson effects, explain the emergence of endogenous fluctuations (Benhabib and Nishimura, 1985). For instance, suppose there is an instantaneous increase in the first type of capital stock $k_1(t)$. This generates two opposing mechanisms:

1. **An initial trade-off:** The capital intensity ranking favors the production of capital good 2 over pure and mixed consumption goods. As a result, the Rybczinski effect raises sector 2 output $y_2(t)$ while reducing output in sectors 0 and 1, $y_0(t)$ and $y_1(t)$.

2. **A subsequent adjustment:** This leads to higher investment in $k_2(t)$ and lower investment in $k_1(t)$ in the next period, reversing the initial effects through the Rybczinski effect. Consequently, outputs in sectors 0 and 1 increase, while sector 2 output decreases, provided the capital intensity difference $\frac{k_1^1}{l_1^1} - \frac{k_0^0}{l_0^0}$ remains large. As a result, $k_1(t)$ rises while $k_2(t)$ declines.

Of course, the Stolper-Samuelson effect generates corresponding fluctuations in interest rates and relative prices of both capital stocks.

Therefore, two conditions are necessary for endogenous fluctuations to occur in case A-1: (1) a sufficiently large capital intensity difference ($\frac{k_1^1}{l_1^1} - \frac{k_0^0}{l_0^0}$) and (2) a discount rate $\delta$ high enough to prevent intertemporal arbitrage opportunities that would smooth fluctuations.

**_The intertemporal consumption trade-off_** This trade-off complements (Case A-2) or strengthens (Case B) the technological trade-off, generating or amplifying endogenous fluctuations.

- **Case A-2: Periodic fluctuations with a large capital intensity difference.** When the economy features a mixed consumption good and a sufficiently large capital intensity difference (Assumption 2), periodic fluctuations arise even with a low discount rate. Theorem 1 shows that a Hopf bifurcation occurs whenever the share of the pure consumption good in utility is sufficiently large ($\theta > \theta^*$).

Under Cobb-Douglas preferences, both the pure good $c_0(t)$ and the mixed good $c_1(t)$ exhibit procyclical behavior, as the income effect dominates the substitution effect. Since the size of the substitution effect increases with $\theta$, a lower $\theta$ leads to smaller consumption adjustments.

Following an increase in the capital stock $k_1(t)$, both $c_0(t)$ and $c_1(t)$ decrease. However, due to the positive share $1 - \theta$ of the mixed good in consumption, the substitution effect remains small relative to the potentially large income effect. This causes $c_1(t)$ to fall sharply after the initial drop in $c_0(t)$. In the following periods, as $k_1(t)$ declines, both consumption levels recover, with $c_1(t)$ exhibiting a stronger rebound. This asymmetry amplifies capital stock fluctuations through the differentiated responses of pure and mixed goods, even when the discount rate is low.

- **Case B: Persistent cycles with small capital intensity differences.** When capital intensity differences are small (Assumption 3), the traditional technological trade-off alone is insufficient to generate endogenous fluctuations. However, the introduction of the mixed consumption good ($\theta \in (0, 1)$) triggers an intertemporal trade-off that sustains periodic fluctuations.

In this case, the mixed good creates stronger intertemporal linkages between consumption and investment decisions. As a result, even small capital intensity differences induce pronounced consumption adjustments, amplifying oscillations in outputs and capital stocks. This mechanism generates persistent cycles, albeit with smaller amplitude compared to Case A-2.

Overall, the specification of a mixed consumption good in the utility function introduces a novel mechanism that amplifies fluctuations in consumption and capital stocks, offering a new perspective on the dynamics of persistent cycles. It turns out that the intratemporal substitution effect, combined with the intertemporal consumption trade-off, can sustain endogenous fluctuations even when the technological trade-off alone is insufficient. Importantly, fluctuations in interest rates—arising from the Stolper-Samuelson effect tied to capital stock variations—further amplify these dynamics, underscoring the critical role of the mixed consumption good in driving persistent and pronounced cyclical behaviors.

### 3.4. Cycle periodicity

Following Theorem 1, an approximation of the cyclical periodicity can be derived.

**Corollary 1 (Periodicity of endogenous cycles).** Let $\eta = 0$. Suppose that $\delta$ and $\theta$ are near their respective Hopf bifurcation values $\delta^*$, $\theta^*$ and $\theta$. Then, the periodicity of endogenous cycles $T$ decreases with capital intensity differences between sectors, and increases with the share of the pure consumption good ($\theta$).

Corollary 1 goes beyond the theoretical insights from Benhabib and Nishimura (1979a) regarding cycle periodicity. Indeed, we provide an analytical approximation of cycle periodicity in terms of key economic fundamentals and show that the cycle period is inversely related to sectoral capital intensity differences but increases with the share of the pure consumption good, $\theta$.<fnref n="11" />

<footnote n="11">¹¹ The periodicity of endogenous cycles depends on the size of the imaginary part of the (oscillatory) eigenvalues, denoted by $\sqrt{-\Delta\theta}$, evaluated at $\theta = 1$ in the presence of a pure consumption good (case A-1 of Theorem 1), and $\theta = \theta^*$ or $\theta = \bar{\theta}$ (cases A-2 and B of Theorem 1) in the presence of both pure and mixed consumption goods.</footnote>
In the case of a pure consumption good ($\theta = 1$), larger capital intensity differences between sectors amplify capital stock adjustments, leading to shorter cycles. For example, when sector 2 is significantly more capital-intensive than sector 1, and sector 0 more capital-intensive than sector 2, these imbalances drive dynamic adjustments that accelerate cyclical behavior. Conversely, when mixed consumption goods are present ($\theta \in (0, 1)$), the cycle length is influenced by both technological parameters and $\theta$. A smaller $\theta$ reduces the substitution effect relative to the income effect, leading to sharper fluctuations in the consumption of mixed goods and, consequently, shorter cycles.

This result once again highlights the pivotal role of mixed consumption goods combined with technological trade-offs in shaping cyclical periodicity. Our framework uniquely quantifies the relationship between sectoral capital intensities, the share of pure consumption goods, and the cycle period.

**Remark 2 (About the percentage deviation from the steady state).** As demonstrated by Hopf (1942), in the supercritical case of a stable periodic orbit, the radius of the orbit grows proportionally to $\sqrt{|x - x^*|}$, where $x$ refers either $\delta$ or $\theta$, and $x^*$ represents the bifurcation values $\delta^*$, $\theta^*$, or $\hat{\theta}$. Consequently, the radius of the periodic orbit is influenced by the stability parameter, which is derived from a third-degree Taylor expansion of the dynamical system (8) on the two-dimensional center manifold corresponding to the bifurcating eigenvalues (See Guckenheimer and Holmes, 1983). To approximate the dynamics, we can interpret the periodic orbit as a circle with periodicity $T$ corresponding to the circumference. Thus, the radius $R$ of the circle can be computed as $R = \frac{T}{2\pi}$, which represents the maximum amplitude of deviation of the variables on the periodic orbit from their steady-state values. As a result, an approximate maximum standard deviation of capital stocks and prices can be calculated as $\sigma = \frac{R}{\sqrt{2}}$.

# 4. Endogenous cycles under strictly concave preferences

This section extends our analysis by considering a strictly concave utility function ($\eta > 0$). In line with Proposition 3, we explicitly account for the influence of labor supply elasticity on the emergence of business cycle fluctuations.

We demonstrate that, even in the presence of endogenous labor, the inclusion of a mixed consumption good preserves the preference-based mechanism as one key driver of endogenous fluctuations. In particular, we establish that a sufficiently low labor supply elasticity ($\chi$ large) strengthens the occurrence of periodic cycles through the same technology-based and preference-based mechanisms identified in the previous sections.

## 4.1. Hopf bifurcation with labor supply elasticity

In the spirit of Section 3, Theorem 2 establishes the conditions under which endogenous fluctuations emerge when labor supply is elastic.

**Theorem 2 (Endogenous fluctuations with labor supply elasticity effects).** Let Assumption 1 hold, and assume the labor supply is elastic, $\chi \in (0, +\infty)$. If Assumption 2 or 3 also holds, there exist some thresholds of the relative aversion parameter $\eta_\theta^* > 0$ and the labor supply parameter $\chi_{\theta,\eta}^* > 0$ such that, for $\eta \in (0, \eta_\theta^*)$, endogenous fluctuations arise through the same technology-based and preference-based mechanisms, provided the labor supply elasticity is sufficiently low, $\varepsilon_l < \varepsilon_l^* = 1/\chi_{\theta,\eta}^*$.

Theorem 2 presents the same conditions for the existence of a Hopf bifurcation as those summarized in Table 1, while also requiring $\eta \in (0, \eta_\theta^*)$. This condition holds regardless of the labor supply elasticity, $\varepsilon_l = 1/\chi$,<fnref n="12" /> and is consistent with Rockafellar (1976), who demonstrates that the steady state exhibits the saddle-point property when the degree of concavity of the utility function is sufficiently high. The existence of an upper bound for $\eta$, along with the possibility of small values $\eta \in (0, \eta_\theta^*)$, ensures that the representative agent would be willing to substitute consumption over time. This, in turn, enables the preference-based mechanism to drive endogenous fluctuations.

Moreover, for any given $\eta \in (0, \eta_\theta^*)$, there exists a critical threshold for labor supply elasticity, $\varepsilon_l^* > 0$, that governs the stability of the steady state. Specifically, when $\varepsilon_l \in [0, \varepsilon_l^*)$, the steady state is fully unstable, while for $\varepsilon_l > \varepsilon_l^*$, it becomes saddle-point stable with damped oscillations. The threshold $\varepsilon_l^*$ represents the Hopf bifurcation value, leading to the emergence of non-constant, stable or unstable closed orbits around the steady state. This bifurcation mechanism arises because, when $\eta > 0$, labor supply is directly tied to the consumption of the pure good $c_0$, breaking the consumption/leisure separation result described in Proposition 3. In this framework, the intertemporal consumption trade-off persists even in the absence of intertemporal arbitrage when labor supply elasticity is low (i.e., $\chi$ is large). Conversely, a highly elastic labor supply dampens consumption fluctuations, acting as a stabilizing buffer in the frictionless three-sector model.

<footnote>This periodicity is then approximately equal to $T^{\delta*} = \frac{2\pi}{|\sqrt{-\Delta^1}|}$, when the time preference $\delta$ is in a neighborhood of the Hopf bifurcation value $\delta^*$, and $T^{\theta*} = \frac{2\pi}{\sqrt{-\Delta^{\theta*}}}$ and $T^{\hat{\theta}} = \frac{2\pi}{\sqrt{-\Delta^{\hat{\theta}}}}$ when the pure consumption good share $\theta$ is close to the Hopf bifurcation values $\theta^*$ and $\hat{\theta}$, respectively.</footnote>

<footnote n="12">¹² Comparable results are obtained by Garnier et al. (2007) concerning the existence of sunspot fluctuations in a two-sector continuous-time model with sector-specific externalities.</footnote>
<table><caption>**Table 2**
Technological parameters</caption><thead><tr><th></th><th colspan="2">Sector 0<br>Consumption</th><th colspan="2">Sector 1<br>Mixed</th><th colspan="2">Sector 2<br>Investment</th><th>Capital Intensity<br>Difference</th><th>Assumptions</th></tr></thead><tbody><tr><th>Θ</th><td>α<sub>1</sub></td><td>α<sub>2</sub></td><td>β̂<sub>1</sub></td><td>β̂<sub>2</sub></td><td>γ<sub>1</sub></td><td>γ<sub>2</sub></td><td>(k<sub>1</sub>/l) - (k<sub>2</sub>/l)</td><td></td></tr><tr><th>Θ<sub>1</sub></th><td>0.09</td><td>0.60</td><td>0.20</td><td>0.30</td><td>0.20</td><td>0.46</td><td>0.11</td><td>1 and 3</td></tr><tr><th>Θ<sub>2</sub></th><td>0.06</td><td>0.60</td><td>0.20</td><td>0.30</td><td>0.20</td><td>0.49</td><td>0.246</td><td>1 and 2</td></tr></tbody></table>

Finally, introducing an (additively separable) endogenous labor supply reinforces the core mechanisms of our three-sector model. Sectoral capital intensity disparities and intertemporal consumption trade-offs between pure and mixed goods are further amplified by the low substitutability of labor supply, echoing the role of the discount rate $\delta$ discussed in Section 3.

## 4.2. Cycle periodicity when $\eta > 0$

In the presence of a strictly concave utility function, the absence of closed-form solutions for the eigenvalues prevents explicit computation of the Hopf cycle periodicity, as is possible with a linear utility function. However, the cycle length can still be evaluated numerically. Similar to *Corollary 1*, the periodicity depends on the imaginary part of the bifurcating eigenvalues, which is primarily influenced by the technological parameters and the share $\theta$ of the pure consumption good in the utility function, rather than the intertemporal substitution parameter $\eta$.

The results from Sections 3 and 4 regarding cycle periodicity can be generalized to both per capita and per GDP variables. On the one hand, Theorems 1 and 2 establish the conditions for periodic cycles in the capital stocks $k(t)$ and the associated prices $p(t)$, while *Corollary 1* provides a way to approximate their periodicity. Additionally, Proposition 1 shows that sectoral outputs $y(t) = (y_1(t), y_2(t))$ and consumption levels $c(t) = (c_0(t), c_1(t))$ also follow periodic orbits, provided the capital stocks and prices lie on the periodic orbit's closed curve. Since the total population is normalized to one, $y(t)$, $k(t)$, and $c(t)$ naturally represent per capita variables.

On the other hand, the gross domestic product, defined as

$$Y(t) = c_0(t) + p_1(t)y_1(t) + p_2(t)y_2(t),$$

allows us to define per-GDP variables as

$$\tilde{x}(t) = \frac{x(t)}{Y(t)} \quad \text{for } x(t) \in \{k(t), p(t), y(t), c(t)\}.$$

In this context, Proposition 4 summarizes the periodicity properties of these variables.

**Proposition 4 (Periodicity of per capita and per GDP variables).** Let $T$ denote the periodicity of the capital stocks $k(t)$ and prices $p(t)$ along the periodic orbit generated by the Hopf bifurcation. Then, along the periodic orbit, per capita sectoral outputs $y(t)$, consumption levels $c(t)$, gross domestic product $Y(t)$, and per GDP variables $\tilde{x}(t)$ all share the same periodicity $T$.

In the absence of other sources of endogenous fluctuations (e.g., externalities, informational asymmetries, or market imperfections), the periodicity of the capital stocks aligns with that of the relative prices, as both sets of variables are assumed to follow the same periodic orbit. Proposition 4 further highlights that, along this periodic orbit, per capita sectoral outputs, consumption levels, GDP, and per GDP variables all exhibit identical periodicity, reflecting the synchronized cyclical behavior induced by the Hopf bifurcation.

## 4.3. Numerical illustration

As a preliminary and purely theoretical exercise, we contrast the results of Theorems 1 and 2 using a numerical illustration. This involves calibrating the depreciation rate, the discount factor, and the Cobb-Douglas technological parameters (Assumption A.3). We assume one period represents one year, with the discount factor ($\delta$) set at 0.01 and the annual depreciation rate at 10%, corresponding to a standard quarterly depreciation rate of 2.5%. Two sets of technological parameters, $\Theta_1$ and $\Theta_2$, are considered to capture cases of small and large capital intensity differences, corresponding respectively to Assumptions 2 and 3.

However, calibrating the mixed consumption-investment good sector lacks solid empirical or national accounting foundations. The assumption of a mixed investment-consumption good, while useful for theoretical exploration, is a simplification aimed at overcoming the curse of dimensionality inherent in more sophisticated preference specifications (e.g., introducing good varieties or a basket of durable and nondurable goods). In this context, the technology parameters for the consumption and investment sectors are chosen consistently with the three-sector literature on endogenous cycles (e.g., Benhabib et al., 2006). By contrast, the calibration of the mixed sector relies on an *inverse engineering approach*, where the values of $\beta_1$ and $\beta_2$ are selected to ensure consistency with the emergence of endogenous fluctuations. See Table 2. This approach highlights the potential theoretical mechanisms at play but does not aim to provide a robust empirical representation of the mixed good industry, nor an empirical validation of the mixed consumption-investment good.
<table><caption>**Table 3**
Results with a non-strictly concave utility</caption><thead><tr><th></th><th>δ<sup>*</sup></th><th>θ<sup>*</sup></th><th>ε<sup>*</sup><sub>00</sub></th><th>ε<sup>*</sup><sub>11</sub></th><th>T<sup>θ</sup></th><th>θ̂</th><th>ε̂<sub>00</sub></th><th>ε̂<sub>11</sub></th><th>T<sup>θ̂</sup></th></tr></thead><tbody><tr><th>Θ<sub>1</sub></th><td>—</td><td>0.6166</td><td>2.608</td><td>1.6218</td><td>9.7558</td><td>0.6262</td><td>2.6754</td><td>1.5968</td><td>9.8331</td></tr><tr><th>Θ<sub>2</sub></th><td>0.186</td><td>0.8145</td><td>5.3931</td><td>1.2276</td><td>14.2529</td><td>0.8362</td><td>6.1069</td><td>1.1958</td><td>14.4451</td></tr></tbody></table>

<table><caption>**Table 4**
Results with a strictly concave utility and endogenous labor</caption><thead><tr><th></th><th>θ</th><th>ε<sup>*</sup><sub>l</sub></th><th>ε<sup>*</sup><sub>00</sub></th><th>ε<sup>*</sup><sub>11</sub></th><th>T<sup>θ̂</sup></th></tr></thead><tbody><tr><th>Θ<sub>1</sub></th><td>0.6256</td><td>2.5</td><td>2.671</td><td>1.5985</td><td>9.8281</td></tr><tr><th>Θ<sub>2</sub></th><td>0.8359</td><td>2.941</td><td>6.0938</td><td>1.1963</td><td>14.442</td></tr></tbody></table>

In accordance with cases A-2 and B of Theorem 1, Table 3 presents the threshold values for the discount rate ($\delta^*$), the preference parameter ($\theta^*$ and $\hat{\theta}$), the elasticities of intertemporal substitution for the two goods ($\epsilon_{00}$ and $\epsilon_{11}$), and the cycle periodicity ($T$).

For the parameter set $\Theta_1$, which corresponds to a small capital intensity difference, several points are noteworthy. First, endogenous cycles occur irrespective of the labor supply elasticity. Second, two Hopf bifurcation values, denoted by $\theta^*$ and $\hat{\theta}$ for the share of the pure consumption good, emerge in close proximity. Third, these bifurcation values yield plausible empirical estimates for the elasticities of intertemporal substitution, regardless of the technological configuration, and remain consistent with the literature (e.g., Gruber, 2013). Fourth, the periodicity of per capita and per GDP variables is close to 10 years, aligning with the findings of Beaudry et al. (2020).

For the larger capital intensity difference represented by the parameter set $\Theta_2$, the model still generates endogenous fluctuations. However, achieving this requires significantly higher—and potentially implausibly large—values for the elasticity of intertemporal substitution of the pure consumption good and for the cycle periodicity. This result is fully consistent with Corollary 1.

Considering now Theorem 2 with a strictly concave utility function and an endogenous labor supply, if the relative aversion parameter satisfies the necessary condition $\eta \in (0, \eta_{\theta}^*)$ and the share of the pure consumption good satisfies the conditions of Theorem 1 cases A-2 and B, $\theta \in (\theta^*, \hat{\theta})$, there exists $\chi_{\theta, \eta}^* > 0$ and a corresponding Hopf bifurcation value for the elasticity of labor supply, denoted by $\epsilon_l^* = 1/\chi_{\theta, \eta}^*$. See Table 4.

Compared with Table 3, the results for $\Theta_1$ exhibit minimal variation with endogenous labor, given that $\eta$ is close to zero ($\eta_{0.6256}^* = 3.1 \times 10^{-7}$). In this case, the presence of endogenous labor with a linear homogeneous utility function is nearly equivalent to the inelastic labor case. Moreover, the Hopf bifurcation value for the elasticity of labor supply aligns with recent macroeconomic estimates reported by Prescott and Wallenius (2011) and Rogerson and Wallenius (2009). However, as observed in the case of a non-strictly concave utility function, the elasticity of intertemporal substitution for the pure consumption good and the cycle periodicity become excessively large under Assumption 2 (Set $\Theta_2$).

Although primarily theoretical, this preliminary exercise suggests that the model's structure is consistent with certain structural parameter values commonly used in empirical studies.

## 5. Concluding comments

This paper considers a 3-sector intertemporal equilibrium model featuring a non-linear utility function and endogenous labor. Clear conditions for the existence of endogenous fluctuations through a Hopf bifurcation are provided, along with a characterization of the theoretical cycle periodicity. Notably, our results emphasize that the assumption of a single pure consumption good is insufficient to reproduce a reliable periodicity of endogenous fluctuations and leads to implausible discount rates and elasticities of intertemporal substitution. In contrast, considering both pure and mixed consumption goods (or a bundle of consumption goods) addresses these critical issues that arise within the standard one-consumption-good two-sector intertemporal equilibrium model. Simultaneously, endogenous fluctuations are likely to occur due to the relative sectoral capital intensity differences as well as the intertemporal consumption allocations of the two consumed goods.

Several avenues for future research emerge. On one hand, building on the works of Benhabib and Nishimura (1989) and Beaudry et al. (2020), the analysis could be extended to stochastic models, with particular attention to stochastic limit cycles, complemented by simulations using the methodology of Galizia (2021). The introduction of complementarity effects, frictions, and market failures could further deepen insights in this context. On the other hand, based on the simple evaluations of examples from Section 4, further empirical analysis of (stochastic) endogenous fluctuation models is needed to assess the relevance of structural preference and technology parameters, as well as the properties of these fluctuations (e.g., periodicity). Finally, a more comprehensive characterization of the local stability properties of limit cycles requires higher-order approximations, moving beyond the linear approximation of the dynamical system (Guckenheimer and Holmes, 1983).
# 6. Appendix 1

## 6.1. Proof of Proposition 1

The current value Hamiltonian associated to the optimization program (6) is given by<fnref n="13" />:

$$ \mathcal{H} = \frac{[T(\omega)^{\theta} c_1^{1-\theta}]^{-1-\eta}}{1-\eta} - \frac{\ell^{1+\chi}}{1+\chi} + q_1 (y_1 - gk_1 - c_1) + q_2 (y_2 - gk_2) $$

The first-order conditions are:

$$ q_1 = p_1 \theta c_0^{\theta(1-\eta)-1} c_1^{(1-\theta)(1-\eta)} \quad (9) $$

$$ q_2 = p_2 \theta c_0^{\theta(1-\eta)-1} c_1^{(1-\theta)(1-\eta)} \quad (10) $$

$$ q_1 = (1 - \theta) c_0^{\theta(1-\eta)} c_1^{(1-\theta)(1-\eta)-1} \quad (11) $$

$$ \ell^{\chi} = w \theta c_0^{\theta(1-\eta)-1} c_1^{(1-\theta)(1-\eta)} \quad (12) $$

$$ \dot{q}_1 = (\delta + g)q_1 - r_1 \theta c_0^{\theta(1-\eta)-1} c_1^{(1-\theta)(1-\eta)} \quad (13) $$

$$ \dot{q}_2 = (\delta + g)q_2 - r_2 \theta c_0^{\theta(1-\eta)-1} c_1^{(1-\theta)(1-\eta)}. \quad (14) $$

Using equation (9) into equations (11) yields the level of the mixed good in terms of the level of the pure consumption good:

$$ c_1 = \frac{1-\theta}{\partial p_1} c_0. \quad (15) $$

Using equation (12) with (15), we derive the optimal labor supply:

$$ \ell = \left[ w \theta \left( \frac{1-\theta}{\partial p_1} \right)^{(1-\theta)(1-\eta)} c_0^{-\eta} \right]^{1/\chi}. \quad (16) $$

Consider now the first-order conditions of the profit maximization program of the firms in the three sectors:

$$ \begin{gathered} w = \frac{(1-\alpha_1 - \alpha_2)y_0}{l^0} = p_1 \frac{(1-\beta_1 - \beta_2)y_1}{l^1} = p_2 \frac{(1-\gamma_1 - \gamma_2)y_2}{l^2}, \\ r_1 = \frac{\alpha_1 y_0}{k_0^1} = p_1 \frac{\beta_1 y_1}{k_1^1} = p_2 \frac{\gamma_1 y_2}{k_2^1}, \quad r_2 = \frac{\alpha_2 y_0}{k_0^2} = p_1 \frac{\beta_2 y_1}{k_1^2} = p_2 \frac{\gamma_2 y_2}{k_2^2}, \end{gathered} \quad (17) $$

we can define the network structure of the economy through the input-output matrix:

$$ \mathcal{A}(\omega, p) = \begin{pmatrix} a_{00}(w) & a_{01}(w, p_1) & a_{02}(w, p_2) \\ a_{10}(r_1) & a_{11}(r_1, p_1) & a_{12}(r_1, p_2) \\ a_{20}(r_2) & a_{21}(r_2, p_1) & a_{22}(r_2, p_2) \end{pmatrix} \quad (18) $$

where $p = (1, p_1, p_2)'$, $\omega = (w, r_1, r_2)'$, and each coefficient $a_{ij}$ represents the amount of "good" $i$, that is, labor or intermediate capital good 1 or 2, that it takes to produce one unit of good $j$

$$ \begin{align} a_{00}(w) &= \frac{l^0}{y_0} = \frac{1-\alpha_1-\alpha_2}{w}, & a_{10}(r_1) &= \frac{k_1^0}{y_0} = \frac{\alpha_1}{r_1}, & a_{20}(r_2) &= \frac{k_2^0}{y_0} = \frac{\alpha_2}{r_2} \notag \\ a_{01}(w, p_1) &= \frac{l^1}{y_1} = \frac{p_1(1-\beta_1-\beta_2)}{w}, & a_{11}(r_1, p_1) &= \frac{k_1^1}{y_1} = \frac{p_1\beta_1}{r_1}, & a_{21}(r_2, p_1) &= \frac{k_2^1}{y_1} = \frac{p_1\beta_2}{r_2} \tag{19} \\ a_{02}(w, p_2) &= \frac{l^2}{y_2} = \frac{p_2(1-\gamma_1-\gamma_2)}{w}, & a_{12}(r_1, p_2) &= \frac{k_1^2}{y_2} = \frac{p_2\gamma_1}{r_1}, & a_{22}(r_2, p_2) &= \frac{k_2^2}{y_2} = \frac{p_2\gamma_2}{r_2} \notag \end{align} $$

As stated in Lemma 6.1 and Lemma 6.2, using the results of Benhabib and Nishimura (1979a), it is straightforward to show that the factor-price frontier and the factor market-clearing equations depend on the matrix of input coefficients $\mathcal{A}(\omega, p)$ as given by (18).

**Lemma 6.1.** $p = \mathcal{A}'(\omega, p)\omega$ and $dp = \mathcal{A}'(\omega, p)d\omega$.

<footnote n="13">¹³ The time variable $(t)$ is dropped to save notations.</footnote>
**Lemma 6.2.** Denote $x = (\ell, k_1, k_2)'$ and $y = (y_0, y_1, y_2)'$. Then $\mathcal{A}(\omega, p)y = x$ and

$$ \mathcal{A}(w, p)dy + \begin{pmatrix} \left( \frac{\partial a_{00}}{\partial w} y_0 + \frac{\partial a_{01}}{\partial w} y_1 + \frac{\partial a_{02}}{\partial w} y_2 \right) dw + \frac{\partial a_{01}}{\partial p_1} y_1 dp_1 + \frac{\partial a_{02}}{\partial p_2} y_2 dp_2 \\ \left( \frac{\partial a_{10}}{\partial w} y_0 + \frac{\partial a_{11}}{\partial w} y_1 + \frac{\partial a_{12}}{\partial w} y_2 \right) dr_1 + \frac{\partial a_{11}}{\partial p_1} y_1 dp_1 + \frac{\partial a_{12}}{\partial p_2} y_2 dp_2 \\ \left( \frac{\partial a_{20}}{\partial w} y_0 + \frac{\partial a_{21}}{\partial w} y_1 + \frac{\partial a_{22}}{\partial w} y_2 \right) dr_2 + \frac{\partial a_{21}}{\partial p_1} y_1 dp_1 + \frac{\partial a_{22}}{\partial p_2} y_2 dp_2 \end{pmatrix} = dx. $$

Denoting $\mathbf{k} = (k_1, k_2)$, $\mathbf{p} = (p_1, p_2)$ and $\mathbf{z} = (\mathbf{k}, \mathbf{p}, \ell)$, Lemma 6.1 implies wage rate and rental rates are functions of the output prices only, that is, $w = w(\mathbf{p})$, $r_1 = r_1(\mathbf{p})$ and $r_2 = r_2(\mathbf{p})$ whereas Lemma 6.2 implies that outputs are functions of the capital stocks, the output prices and labor, $y_j = \tilde{y}_j(\mathbf{z})$, $j = 0, 1, 2$. Then, considering $w = w(\mathbf{p})$ and $c_0 = \tilde{y}_0(\mathbf{z})$ with (16), and solving for $\ell$, we express the labor supply as a function of capital stocks and output prices, $\ell = \ell(\mathbf{k}, \mathbf{p})$, and we derive $y_j = y_j(\mathbf{k}, \mathbf{p}) = \tilde{y}_j(\mathbf{k}, \mathbf{p}, \ell(\mathbf{k}, \mathbf{p}))$, $j = 1, 2$, and $c_0 = c_0(\mathbf{k}, \mathbf{p}) = y_0(\mathbf{k}, \mathbf{p}) = \tilde{y}_0(\mathbf{k}, \mathbf{p}, \ell(\mathbf{k}, \mathbf{p}))$. Using (15), we have:

$$ c_1 = c_1(\mathbf{k}, \mathbf{p}) = c_0(\mathbf{k}, \mathbf{p}) \frac{1-\theta}{\partial p_1}. \qquad (20) $$

In the spirit of Benhabib and Nishimura (1979a), all of these functions are $C^1$ with respect to all their arguments. Finally, rearranging equations (3), (9)-(14) and (20) provides the system of differential equations (8) in Proposition 1 with:

$$ E_{\eta, \chi}^{\theta} \equiv \theta + \eta \left[ 1 - \theta - \frac{1}{c_0} \left( p_1 \frac{\partial c_0}{\partial p_1} + p_2 \frac{\partial c_0}{\partial p_2} \right) \right] \qquad (21) $$

and

$$ F_1(\mathbf{k}, \mathbf{p}) = \eta \left[ \frac{\partial c_0}{\partial p_2} \frac{r_1(\mathbf{p})p_2 - r_2(\mathbf{p})p_1}{c_0} + \frac{p_1}{c_0} \left( y_1(\mathbf{k}, \mathbf{p}) - gk_1 - c_1(\mathbf{k}, \mathbf{p}) \right) + \frac{\partial c_0}{\partial k_2} \left( y_2(\mathbf{k}, \mathbf{p}) - gk_2 \right) \right] $$

$$ F_2(\mathbf{k}, \mathbf{p}) = \frac{[r_2(\mathbf{p})p_1 - r_1(\mathbf{p})p_2] \left( \eta \frac{\partial c_0}{\partial p_1} + (1-\theta)(1-\eta) \frac{c_0}{p_1} \right)}{c_0} + \eta \frac{p_2}{c_0} \left( y_1(\mathbf{k}, \mathbf{p}) - gk_1 - c_1(\mathbf{k}, \mathbf{p}) \right) + \frac{\partial c_0}{\partial k_2} \left( y_2(\mathbf{k}, \mathbf{p}) - gk_2 \right) $$

Some proof of the necessity of the transversality condition is provided in Michel (1982) and Kamihigashi (2001). ☐

## 6.2. Proof of Proposition 2

Considering Proposition 1, a steady state is defined by a vector $(c_1^*, \mathbf{k}^*, \mathbf{p}^*)$ solution of

$$ y_1(\mathbf{k}, \mathbf{p}) = gk_1 + c_1 = gk_1 + c_0(\mathbf{k}, \mathbf{p}) \frac{1-\theta}{\partial p_1}, \quad y_2(\mathbf{k}, \mathbf{p}) = gk_2 \qquad (22) $$

$$ r_1(\mathbf{p}) = (\delta + g)p_1, \quad r_2(\mathbf{p}) = (\delta + g)p_2. $$

The derivation of the unique steady state proceeds as follows. A first step derives the expressions of $w$, $r_1$ and $r_2$, and then solves for the steady state values of $p_1$ and $p_2$. Using $\mathcal{A}(\omega, p)y = x$ in Lemma 6.2, a second step expresses the steady state values of $k_1$ and $k_2$ as a function of $y_1$ and thus $c_1$. A third step gets the expressions of $c_0$ and $c_1$. A last step provides the steady states values of $\ell$, $k_1$ and $k_2$. All derivations are available in the online Appendix (Section 9.1). ☐

## 6.3. Proof of Proposition 3

Linearizing the dynamical system around $(c_1^*, \mathbf{k}^*, \mathbf{p}^*)$ gives the Jacobian matrix $J$:

$$ J = \begin{pmatrix} \frac{\partial y_1}{\partial k_1} - g - \frac{\partial c_1}{\partial k_1} & \frac{\partial y_1}{\partial k_2} - \frac{\partial c_1}{\partial k_2} & \frac{\partial y_1}{\partial p_1} - \frac{\partial c_1}{\partial p_1} & \frac{\partial y_1}{\partial p_2} - \frac{\partial c_1}{\partial p_2} \\ \frac{\partial y_2}{\partial k_1} & \frac{\partial y_2}{\partial k_2} - g & \frac{\partial y_2}{\partial p_1} & \frac{\partial y_2}{\partial p_2} \\ \frac{\eta}{E_{\eta, \chi}^{\theta}} \frac{p_1}{c} \mathbf{B}_1 & \frac{\eta}{E_{\eta, \chi}^{\theta}} \frac{p_1}{c} \mathbf{B}_2 & J_{33} & J_{34} \\ \frac{\eta}{E_{\eta, \chi}^{\theta}} \frac{p_2}{c} \mathbf{B}_1 & \frac{\eta}{E_{\eta, \chi}^{\theta}} \frac{p_2}{c} \mathbf{B}_2 & J_{43} & J_{44} \end{pmatrix} \equiv \begin{pmatrix} J_1 & J_2 \\ J_3 & J_4 \end{pmatrix} \qquad (23) $$

with
$$J_{33} = \frac{1}{E_{\eta, \chi}^{\theta}} \left[ \left( \delta + g - \frac{\partial r_1}{\partial p_1} \right) E_{\eta, \chi}^{\theta} - \eta \frac{p_1 \left( A_1 + \frac{\partial c_0}{\partial k_1} \frac{\partial c_1}{\partial p_1} \right)}{c_0} + (1-\theta)(1-\eta) \left( \delta + g - \frac{\partial r_1}{\partial p_1} \right) \right]$$

$$J_{44} = \frac{1}{E_{\eta, \chi}^{\theta}} \left[ \left( \delta + g - \frac{\partial r_2}{\partial p_2} \right) E_{\eta, \chi}^{\theta} - \eta \frac{p_2 \left( A_2 + \frac{\partial c_0}{\partial k_1} \frac{\partial c_1}{\partial p_2} \right)}{c_0} - (1-\theta)(1-\eta) \frac{p_2}{p_1} \frac{\partial r_1}{\partial p_2} \right]$$

$$J_{34} = -\frac{1}{E_{\eta, \chi}^{\theta}} \left[ \frac{\partial r_1}{\partial p_2} E_{\eta, \chi}^{\theta} + \eta \frac{p_1 \left( A_2 + \frac{\partial c_0}{\partial k_1} \frac{\partial c_1}{\partial p_2} \right)}{c_0} + (1-\theta)(1-\eta) \frac{\partial r_1}{\partial p_2} \right]$$

$$J_{43} = -\frac{1}{E_{\eta, \chi}^{\theta}} \left[ \frac{\partial r_2}{\partial p_1} E_{\eta, \chi}^{\theta} + \eta \frac{p_2 \left( A_1 + \frac{\partial c_0}{\partial k_1} \frac{\partial c_1}{\partial p_1} \right)}{c_0} - (1-\theta)(1-\eta) \frac{p_2}{p_1} \left( \delta + g - \frac{\partial r_1}{\partial p_1} \right) \right]$$

and

$$A_1 = \frac{\partial c_0}{\partial p_1} \left( \frac{\partial r_1}{\partial p_1} - g - \delta \right) + \frac{\partial c_0}{\partial p_2} \frac{\partial r_2}{\partial p_1} - \frac{\partial c_0}{\partial k_1} \frac{\partial y_1}{\partial p_1} - \frac{\partial c_0}{\partial k_2} \frac{\partial y_2}{\partial p_1}$$

$$A_2 = \frac{\partial c_0}{\partial p_1} \frac{\partial r_1}{\partial p_2} + \frac{\partial c_0}{\partial p_2} \left( \frac{\partial r_2}{\partial p_2} - g - \delta \right) - \frac{\partial c_0}{\partial k_1} \frac{\partial y_1}{\partial p_2} - \frac{\partial c_0}{\partial k_2} \frac{\partial y_2}{\partial p_2}$$

$$B_1 = \frac{\partial c_0}{\partial k_1} \left( \frac{\partial y_1}{\partial k_1} - g - \frac{\partial c_1}{\partial k_1} \right) + \frac{\partial c_0}{\partial k_2} \frac{\partial y_2}{\partial k_1}, \quad B_2 = \frac{\partial c_0}{\partial k_1} \left( \frac{\partial y_1}{\partial k_2} - \frac{\partial c_1}{\partial k_2} \right) + \frac{\partial c_0}{\partial k_2} \left( \frac{\partial y_2}{\partial k_2} - g \right)$$

Given that the optimization program (6) has a Hamiltonian structure, as initially demonstrated by Kurz (1968) and Levhari and Liviatan (1972), if $\lambda$ is a characteristic root, then $\bar{\lambda}$, $\delta - \lambda$, and $\delta - \bar{\lambda}$ are also characteristic roots.

Let us denote $\mathcal{T}$ the sum of minors of order one, $S_{\eta, \chi}^{\theta}$ the sum of minors of order two, $\Sigma_{\eta, \chi}^{\theta}$ the sum of minors of order three and $D_{\eta, \chi}^{\theta}$ the determinant of $J$. The degree-four characteristic polynomial of the Jacobian (evaluated at the steady state) is given by:

$$P_{\eta, \chi}^{\theta}(\lambda) = \lambda^4 = \lambda^4 - \mathcal{T} \lambda^3 + S_{\eta, \chi}^{\theta} \lambda^2 - \Sigma_{\eta, \chi}^{\theta} \lambda + D_{\eta, \chi}^{\theta}$$

By evaluating all partial derivatives in the Jacobian matrix and calculating the sum of the principal minors of orders 1 through 4, as detailed in Section 9.2 of the online Appendix, we obtain:

$$P_{\eta, \chi}^{\theta}(\lambda) = \lambda^4 - \lambda^3 2\delta + \lambda^2 S_{\eta, \chi}^{\theta} - \lambda \delta (S_{\eta, \chi}^{\theta} - \delta^2) + D_{\eta, \chi}^{\theta} \quad (24)$$

We can easily observe from $P_{\eta, \chi}^{\theta}(\lambda)$ and the expressions of the partial derivatives provided in Lemma 9, available in the online Appendix, that if the utility function is linearly homogeneous with respect to the two consumption goods (i.e., $\eta = 0$), then the local stability properties of the steady state do not depend on the parameter $\chi$, which determines the elasticity of labor supply.

We conclude that, due to the structure of the characteristic roots, one of the following cases must hold for any $\eta \ge 0$, $\chi \ge 0$, and $\theta \in (0, 1]$:

i) the four roots are real and distincts,

ii) the four roots are given by two pairs of non-real complex conjugates,

iii) there are two real double roots. $\square$

## 6.4. Proof of Theorem 1

Let $\Theta = (\alpha, \beta, \gamma)$ denote the technology parameters with $\alpha = (\alpha_1, \alpha_2)$, $\beta = (\beta_1, \beta_2)$, and $\gamma = (\gamma_1, \gamma_2)$, and $\Gamma_1, \Gamma_2, \bar{\Gamma}_2$ and $\Gamma_3$ denote the sets of feasible production technologies that satisfy respectively the following constraints:

$$\Gamma_1 = \left\{ \Theta : \frac{[\alpha_2(1-\gamma_1)-\gamma_2(1-\alpha_1)+\beta_1(1-\alpha_2)-\alpha_1(1-\beta_2)]^2}{4[\alpha_2(1-\beta_1)-\beta_2(1-\alpha_1)][\gamma_1(1-\alpha_2)-\alpha_1(1-\gamma_2)]} < 1 \right\} \quad (25)$$

$$\Gamma_2 = \left\{ \Theta : \frac{\alpha_2 - \gamma_2 + \alpha_1 - \beta_1 + \gamma_2 \beta_1 - \gamma_1 \beta_2}{\beta_1(\alpha_2 - \gamma_2) + \beta_2(\gamma_1 - \alpha_1) + \gamma_2 \alpha_1 - \gamma_1 \alpha_2} \in (-1, 1) \right\} \quad (26)$$

$$\bar{\Gamma}_2 = \left\{ \Theta : \frac{\alpha_2 - \gamma_2 + \alpha_1 - \beta_1 + \gamma_2 \beta_1 - \gamma_1 \beta_2}{\beta_1(\alpha_2 - \gamma_2) + \beta_2(\gamma_1 - \alpha_1) + \gamma_2 \alpha_1 - \gamma_1 \alpha_2} < -1 \right\} \quad (27)$$

$$\Gamma_3 = \left\{ \Theta : \frac{\beta_2(1-\gamma_1)-\gamma_2(1-\beta_1)}{\beta_1(\alpha_2-\gamma_2)+\beta_2(\gamma_1-\alpha_1)+\gamma_2\alpha_1-\gamma_1\alpha_2} > 0 \right\}. \quad (28)$$

It can be easily shown that if Assumptions 1 and 2 hold, then $\Theta \in \Gamma_1 \cap \Gamma_2 \cap \Gamma_3$, while if Assumptions 1 and 3 hold, then $\Theta \in \Gamma_1 \cap \bar{\Gamma}_2 \cap \Gamma_3$.

The general formulation of Theorem 1 is as follows:

**Theorem 1.** Let $\eta = 0$. Then for any nonnegative value of the inverse of the labor supply elasticity, $\chi \ge 0$:

A - 1 When $\theta = 1$, the characteristic roots are complex if and only if $\Theta \in \Gamma_1$. If $\Theta \in \Gamma_1 \cap \Gamma_2$, then there exists a value of the discount factor, denoted by $\delta^* > 0$, such that the steady state is saddle-point stable with oscillating convergence when $\delta \in [0, \delta^*)$ and totally unstable when $\delta > \delta^*$. Moreover, $\delta^*$ is a Hopf bifurcation value generically giving rise to non-constant saddle-point stable (or unstable) closed orbits around the steady state in a right (or left) neighborhood of $\delta^*$.
A - 2 When $\theta \in (0, 1)$, there exists $\theta \in (0, 1)$ such that the characteristic roots are complex if and only if $\Theta \in \Gamma_1$ and $\theta \in (\theta, 1]$. Furthermore, suppose that $\Theta \in \Gamma_1 \cap \Gamma_2 \cap \Gamma_3$. Then, if $\delta \in [0, \delta^*)$, there exist two values of the nondurable consumption share, denoted by $\theta^* \in (\theta, 1)$ and $\hat{\theta} \in (\theta^*, 1)$, such that the steady state is saddle-point stable with oscillating convergence when $\theta \in (\hat{\theta}, 1)$, totally unstable when $\theta \in (\theta^*, \hat{\theta})$ and again saddle-point stable when $\theta \in (\theta, \theta^*)$. Moreover, the nondurable consumption share $\theta^*$ (respectively, $\hat{\theta}$) is a Hopf bifurcation value generically giving rise to non-constant saddle-point stable (or unstable) closed orbits around the steady state in a right or left neighborhood of $\theta^*$ (respectively, of $\hat{\theta}$).

B - When $\Theta \in \Gamma_1$ and $\theta \in (\theta, 1]$, suppose now that $\Theta \in \Gamma_1 \cap \bar{\Gamma}_2 \cap \Gamma_3$. Then for any given $\delta > 0$, there exist two values of the nondurable consumption share $\theta^* \in (\theta, 1)$ and $\hat{\theta} \in (\theta^*, 1)$, such that the steady state is saddle-point stable with oscillating convergence when $\theta \in (\hat{\theta}, 1]$, totally unstable when $\theta \in (\theta^*, \hat{\theta})$ and again saddle-point stable when $\theta \in (\theta, \theta^*)$. Moreover, $\theta^*$ (respectively, $\hat{\theta}$) is a Hopf bifurcation value generically giving rise to non-constant saddle-point stable (or unstable) closed orbits around the steady state in a right or left neighborhood of $\theta^*$ (respectively, of $\hat{\theta}$).

The proof of Theorem 1 follows these steps:

✓ Step 1: Derive the degree-4 characteristic equation of $\mathcal{J}$ evaluated at the steady state under the specified parametric restrictions (e.g., $\eta = 0$);

✓ Step 2: Identify the occurrence of a pair of complex eigenvalues by examining the determinant of the characteristic equation and verify that it is negative under the technological constraints (e.g., $\Gamma_1$);

✓ Step 3: Analyze the real part of the complex characteristic roots and demonstrate that there exists at least one bifurcation value at which the sign of the real part changes, following a continuity argument;

✓ Step 4: Detail the stability properties of the periodic orbit in a right or left neighborhood of the Hopf bifurcation value(s).

In this respect, the proof of case A-1 of Theorem 1 proceeds as follows.

**Step 1: characteristic polynomial** If $\eta = 0$, $\theta = 1$ and thus $u(c_0, c_1) = c_0$, we derive from Lemma 9 available in the online Appendix that

$$ \frac{\partial y_1}{\partial k_1} = \frac{\partial r_1}{\partial p_1}, \quad \frac{\partial y_1}{\partial k_2} = \frac{\partial r_2}{\partial p_1}, \quad \frac{\partial y_2}{\partial k_1} = \frac{\partial r_1}{\partial p_2}, \quad \frac{\partial y_2}{\partial k_2} = \frac{\partial r_2}{\partial p_2}, \qquad (29) $$

The characteristic polynomial (24) then simplifies as follows:

$$ P_0^1(\lambda) = \left[ \lambda^2 - \lambda \left[ 2(\delta + g) - \frac{\partial r_1}{\partial p_1} - \frac{\partial r_2}{\partial p_2} \right] + \left( \delta + g - \frac{\partial r_1}{\partial p_1} \right) \left( \delta + g - \frac{\partial r_2}{\partial p_2} \right) - \frac{\partial r_1}{\partial p_2} \frac{\partial r_2}{\partial p_1} \right] \\ \times \left[ \lambda^2 - \lambda \left( \frac{\partial r_1}{\partial p_1} + \frac{\partial r_2}{\partial p_2} - 2g \right) + \left( \frac{\partial r_1}{\partial p_1} - g \right) \left( \frac{\partial r_2}{\partial p_2} - g \right) - \frac{\partial r_1}{\partial p_2} \frac{\partial r_2}{\partial p_1} \right] \equiv Q_1^1(\lambda)Q_2^1(\lambda). \qquad (30) $$

and does not depend on $\lambda$.

**Step 2: existence of complex roots** Denoting $\Delta_i^1$ the discriminant of polynomial $Q_i^1(\lambda)$, $i = 1, 2$, one has:

$$ \Delta_1^1 = \Delta_2^1 = \Delta^1 = \left[ \frac{\partial r_1}{\partial p_1} - \frac{\partial r_2}{\partial p_2} \right]^2 + 4 \frac{\partial r_1}{\partial p_2} \frac{\partial r_2}{\partial p_1} \\ = \left[ 2(\delta + g) - \frac{\partial r_1}{\partial p_1} - \frac{\partial r_2}{\partial p_2} \right]^2 - 4 \left[ \left( \delta + g - \frac{\partial r_1}{\partial p_1} \right) \left( \delta + g - \frac{\partial r_2}{\partial p_2} \right) - \frac{\partial r_1}{\partial p_2} \frac{\partial r_2}{\partial p_1} \right]. \qquad (31) $$

Complex characteristic roots are therefore obtained if and only if $\Delta^1 < 0$. Moreover, the four characteristic roots are given by:

$$ \lambda_{1,2} = \frac{2(\delta+g) - \frac{\partial r_1}{\partial p_1} - \frac{\partial r_2}{\partial p_2} \pm i\sqrt{-\Delta^1}}{2}, \quad \lambda_{3,4} = \frac{\frac{\partial r_1}{\partial p_1} + \frac{\partial r_2}{\partial p_2} - 2g \pm i\sqrt{-\Delta^1}}{2}. \qquad (32) $$

**Step 3: real parts of characteristic roots and bifurcation value** Using the expressions of the input coefficients (19) and Lemma 9 available in the online Appendix, we easily derive that the discriminant (31) is negative if and only if $\Theta \in \Gamma_1$. Let us then assume that $\Theta \in \Gamma_1$. Using the expression (32), we derive the real parts of the complex characteristic roots such that

$$ \mathrm{Re}(\lambda_{1,2}) = -\frac{(\delta+g)\lambda}{2}, \quad \mathrm{Re}(\lambda_{3,4}) = \frac{\delta\mathcal{Y}+g\lambda\mathcal{X}}{2} \qquad (33) $$

with $\mathcal{X} = \mathcal{Z} - 1$, $\mathcal{Y} = \mathcal{Z} + 1$ and

$$ \mathcal{Z} = \frac{\alpha_2 - \gamma_2 + \alpha_1 - \beta_1 + \gamma_2 \beta_1 - \gamma_1 \beta_2}{\beta_1(\alpha_2 - \gamma_2) + \beta_2(\gamma_1 - \alpha_1) + \gamma_2 \alpha_1 - \gamma_1 \alpha_2} $$

Since $\mathcal{Y} > \mathcal{X}$, we immediately notice that if $\mathcal{Y} < 0$ and thus $\mathcal{X} < 0$ we get $\mathrm{Re}(\lambda_{1,2}) > 0$ and $\mathrm{Re}(\lambda_{3,4}) < 0$ for any $\delta \ge 0$. Similarly, if $\mathcal{X} > 0$ and thus $\mathcal{Y} > 0$ we get $\mathrm{Re}(\lambda_{1,2}) < 0$ and $\mathrm{Re}(\lambda_{3,4}) > 0$ for any $\delta \ge 0$. Therefore, if $\mathcal{X}\mathcal{Y} > 0$, or equivalently $\mathcal{Z}^2 > 1$, then the steady state is saddle-point stable for any $\delta \ge 0$.

In contrast, if $\mathcal{X} < 0$ and $\mathcal{Y} > 0$, or equivalently $-1 < \mathcal{Z} < 1$, i.e. condition $\Theta \in \Gamma_2$, then $\mathrm{Re}(\lambda_{1,2}) > 0$ and $\mathrm{Re}(\lambda_{3,4}) < 0$ for any $\delta \in [0, \delta^*)$ with $\delta^* = -g\mathcal{X}/\mathcal{Y}$ such that $\mathrm{Re}(\lambda_{3,4})|_{\delta=\delta^*} = 0$. Moreover, we obviously derive that
$$ \frac{d \mathrm{Re}(\lambda_{3,4})}{d \delta} \Big|_{\delta=\delta^*} > 0 \qquad (34) $$

with $\mathrm{Re}(\lambda_{3,4}) > 0$ for any $\delta > \delta^*$.

**Step 4: stability properties** It follows that if $\Theta \in \Gamma_1 \cap \Gamma_2$, $\delta^*$ is a Hopf bifurcation value generically giving rise to non constant saddle-point stable (or unstable) closed orbits around the steady state in a right (or left) neighborhood of $\delta^*$. The stability properties of the periodic orbit depend on the sign of a parameter which is obtained through a Taylor expansion of degree 3 of the dynamical system (8) on the dimension-2 center manifold associated to the two bifurcating eigenvalues. If this parameter is negative we are in the *supercritical* case, the periodic orbit occurs on the right neighborhood of $\delta^*$ and is stable on the dimension-2 center manifold, while if this parameter is positive, we are in the *subcritical* case, the periodic orbit occurs on the left neighborhood of $\delta^*$ and is unstable on the dimension-2 center manifold (see Guckenheimer and Holmes, 1983; Theorem 3.4.2, p.151-152).

The proof of cases A-2 and B (Theorem 1) follows a similar approach and are provided in the online Appendix.

Note that we can also show that endogenous fluctuations may arise in case A-2 when $\delta > \delta^*$, although there is only a single Hopf bifurcation value $\theta^*$. This case is not the focus of Theorem 1, as the bifurcation value $\delta^*$ is typically too large to align with empirically relevant estimates of the discount factor. Nevertheless, Benhabib and Rustichini (1990) demonstrate that $\delta^*$ can be arbitrarily close to 0, though this requires unusual values for the technological coefficients. ☐

## 6.5. Proof of Corollary 1

Building on the results of case A-1 in Theorem 1, there is also the possibility to compute the periodicity on the cycle. Indeed, as shown initially by Hopf (1942), and recalling that the bifurcating eigenvalues are such that $\mathrm{Im}(\lambda_{3,4}) = \sqrt{-\Delta^1}|_{\delta=\delta^*}$, we derive from (34) that for every $\delta$ with $|\delta - \delta^*|$ sufficiently small, the periodic orbit is characterized by a period $T = T(\delta)$ such that

$$ \lim_{|\delta-\delta^*| \to 0} T(\delta) = \frac{2\pi}{\sqrt{-\Delta^1}} \Big|_{\delta=\delta^*} = T^{\delta^*} $$

In other words, for $\delta$ very nearly equal to $\delta^*$, the period of the (emergent) periodic orbits of (8) nearly equals the period of the concentric periodic orbits of the linearized system characterized by the Jacobian matrix (23) with $\delta = \delta^*$, $\theta = 1$ and $\eta = 0$. Straight-forward computations based on the expressions of the input coefficients in (19) show that the more sector 2 is capital-intensive in capital good 1 compared to sector 1, i.e. $\frac{k_2^2}{l_2^2} \gg \frac{k_1^1}{l_1^1}$, and the more sector 0 is capital-intensive in capital good 2 compared to sector 2, i.e. $\frac{k_0^2}{l_0^2} \gg \frac{k_2^2}{l_2^2}$, the larger the value of $|\Delta^1|$.

The proofs of the two properties of the cycle periodicity in cases A-2 and B follow a similar approach and are provided in the online Appendix. ☐

## 6.6. Proof of Theorem 2

In this context, we introduce an additional technology constraint characterized by the set $\Gamma_1^*$ which is a subset of $\Gamma_1$, $\Gamma_1^* \subseteq \Gamma_1$:

$$ \Gamma_1^* = \left\{ \Theta : \frac{[\beta_2(\gamma_1-\alpha_1)+(1-\beta_1)(\gamma_2-\alpha_2)]^2+[\gamma_1(\beta_2-\alpha_2)+(1-\gamma_1)(\beta_1-\alpha_1)]^2}{2[\alpha_2(1-\beta_1)-\beta_2(1-\alpha_1)][\gamma_1(1-\alpha_2)-\alpha_1(1-\gamma_2)]} < 1 \right\}. \qquad (35) $$

Actually, it can be easily shown that if Assumptions 1 and 2 hold, then $\Theta \in \Gamma_1^* \cap \Gamma_2 \cap \Gamma_3$, while if Assumptions 1 and 3 hold, then $\Theta \in \Gamma_1^* \cap \bar{\Gamma}_2 \cap \Gamma_3$.

The general formulation of Theorem 2 is as follows:

**Theorem 2.** Let $\eta > 0$.

A - When $\chi = +\infty$ and $\theta \in (0, 1)$:

1. If $\Theta \in \Gamma_1^* \cap \Gamma_2 \cap \Gamma_3$, then there exist $\tilde{\theta} \in [\underline{\theta}, \theta^*)$ and $\tilde{\eta}_\theta > 0$ such that for any given $\theta \in (\tilde{\theta}, 1]$, the characteristic roots are complex if and only if $\eta \in [0, \tilde{\eta}_\theta)$. Moreover, if $\delta \in (0, \delta^*)$ and $\theta \in (\theta^*, \hat{\theta})$, there exists $\eta_\theta^* \in (0, \tilde{\eta}_\theta)$ such that the steady state is totally unstable when $\eta \in [0, \eta_\theta^*)$ and saddle-point stable with oscillating convergence when $\eta \in (\eta_\theta^*, \tilde{\eta}_\theta)$, with $\eta_\theta^*$ a Hopf bifurcation value giving rise to non-constant saddle-point stable (or unstable) closed orbits around the steady state in a left (right) neighborhood of $\eta_\theta^*$.

2. If $\Theta \in \Gamma_1^* \cap \bar{\Gamma}_2 \cap \Gamma_3$, $\sigma \in [0, \bar{\sigma}_\theta)$ and $\theta \in (\theta^*, \hat{\theta})$, then for any given $\delta > 0$, there exists $\eta_\theta^* \in (0, \tilde{\eta}_\theta)$ such that the steady state is totally unstable when $\eta \in [0, \eta_\theta^*)$ and saddle-point stable with oscillating convergence when $\eta \in (\eta_\theta^*, \tilde{\eta}_\theta)$, with $\eta_\theta^*$ a Hopf bifurcation value giving rise to non-constant saddle-point stable (or unstable) closed orbits around the steady state in a left (right) neighborhood of $\eta_\theta^*$.

B - When $\chi \in (0, +\infty)$ and $\theta \in (0, 1)$:

1. If $\Theta \in \Gamma_1^* \cap \Gamma_2 \cap \Gamma_3$ and $\eta \in [0, \eta_\theta^*)$, then there exists $\underline{\chi}_{\theta,\sigma} > 0$ such that the characteristic roots are complex if and only if $\chi \in (\underline{\chi}_{\theta,\sigma}, +\infty)$. Moreover, if $\delta \in (0, \delta^*)$ and $\theta \in (\theta^*, \hat{\theta})$, there exists $\chi_{\theta,\sigma}^* > \underline{\chi}_{\theta,\sigma}$ such that the steady state is totally unstable when $\chi \in (\chi_{\theta,\sigma}^*, +\infty)$ and saddle-point stable with oscillating convergence when $\chi \in (\underline{\chi}_{\theta,\sigma}, \chi_{\theta,\sigma}^*)$, with $\chi_{\theta,\sigma}^*$ a Hopf bifurcation value giving rise to non-constant saddle-point stable (or unstable) closed orbits around the steady state in a right (left) neighborhood of $\chi_{\theta,\sigma}^*$.
2. If $\Theta \in \Gamma_1^* \cap \bar{\Gamma}_2 \cap \Gamma_3$, $\eta \in [0, \eta_0^*)$, $\chi \in (\underline{\chi}_{\theta, \sigma}^*, +\infty)$ and $\theta \in (\theta^*, \hat{\theta})$, then for any given $\delta > 0$, there exists $\chi_{\theta, \sigma}^* > \underline{\chi}_{\theta, \sigma}$ such that the steady state is totally unstable when $\chi \in (\chi_{\theta, \sigma}^*, +\infty)$ and saddle-point stable with oscillating convergence when $\chi \in (\underline{\chi}_{\theta, \sigma}^*, \chi_{\theta, \sigma}^*)$, with $\chi_{\theta, \sigma}^*$ a Hopf bifurcation value giving rise to non-constant saddle-point stable (or unstable) closed orbits around the steady state in a right (left) neighborhood of $\chi_{\theta, \sigma}^*$.

The proof of Theorem 2 follows a similar approach to that of Theorem 1. The fundamental difference lies in the characteristic polynomial, which includes an additional term, denoted by $\tilde{P}_{\eta, \chi}^{\theta}$, when $\eta > 0$:

$$P_{\eta, \chi}^{\theta}(\lambda) = Q_{1}^{\theta}(\lambda)Q_{2, \eta, \chi}^{\theta}(\lambda) - \frac{\eta}{E_{\eta, \chi}^{\theta}c_{0}}\tilde{P}_{\eta, \chi}^{\theta}(\lambda),$$

where $Q_1^\theta$ is a degree-2 polynomials which only depends on the pure consumption share $\theta$, while $Q_{2, \eta, \chi}^\theta$ and $\tilde{P}_{\eta, \chi}^\theta$ are degree-2 polynomials dependent on the pure consumption share $\theta$, the curvature of the utility function $\eta$, and the (inverse of) labor supply elasticity $\chi$. This term explains that there is no separation result as in Theorem 1: endogenous fluctuations can emerge from labor supply elasticity effects.

Consequently, additional derivations and intermediate results are required for steps 2 and 3 in Theorem 1. Specifically, the key point is to begin with the conditions established in case A-1 (and respectively, in cases A-2 and B) of Theorem 1 and to locally examine the properties of $\tilde{P}_{\eta, \chi}^\theta$ under the given parametric restrictions. The corresponding proofs are provided in the online Appendix. $\square$

## 6.7. Proof of Proposition 4

Let us denote as $k(t) = (k_1(t), k_2(t))$ and $p(t) = (p_1(t), p_2(t))$ the solution of the dynamical system (8). We derive from the $C^1$ property of all the price, output and consumption functions derived in Proposition 1 that $k(t), p(t), y(t) = (y_1(t), y_2(t))$ and $c(t) = (c_0(t), c_1(t))$ are continuous functions. It follows that GDP as defined by $Y(t) = c_0(t) + p_1(t)y_1(t) + p_2(t)y_2(t)$ is a continuous function. Moreover, per GDP variables as given by $\tilde{x}(t) = x(t)/Y(t)$, with $x(t) = \{k(t), p(t), y(t), c(t)\}$, are also continuous functions. Let $T$ be the period of the closed orbit generated by the Hopf bifurcation, precisely identified in the case $\eta = 0$ in Corollary 1, such that $k(t+T) = k(t)$ and $p(t+T) = p(t)$. As a result of the continuity property, we get

$$y(t+T) = (y_1(k(t+T), p(t+T)), y_2(k(t+T), p(t+T))) = (y_1(k(t), p(t)), y_2(k(t), p(t))) = y(t)$$

$$c(t+T) = (c_0(k(t+T), p(t+T)), c_1(k(t+T), p(t+T))) = (c_0(k(t), p(t)), c_1(k(t), p(t))) = c(t)$$

Then that $y(t)$ and $c(t)$ have the same periodicity along the closed orbit. It follows also that $Y(t) = c_0(t) + p_1(t)y_1(t) + p_2(t)y_2(t)$, and any $\tilde{x}(t) = x(t)/Y(t)$ satisfy along the closed orbit

$$Y(t+T) = c_0(t+T) + p_1(t+T)y_1(t+T) + p_2(t+T)y_2(t+T) = c_0(t) + p_1(t)y_1(t) + p_2(t)y_2(t) = Y(t)$$

$$\tilde{x}(t+T) = \frac{x(t+T)}{Y(t+T)} = \frac{x(t)}{Y(t)} = \tilde{x}(t)$$

Therefore $Y(t)$ and any $\tilde{x}(t)$ are characterized by the same periodicity along the periodic orbit. $\square$

## CRediT authorship contribution statement

Kazuo Nishimura: Formal analysis. Florian Pelgrin: Formal analysis. Alain Venditti: Formal analysis.

## Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Alain VENDITTI reports financial support was provided by French National Research Agency (grants ANR-15-CE33-0001 and ANR-17-EURE-0020). Alain VENDITTI reports financial support was provided by AMIDEX. Kazuo Nishimura reports financial support was provided by Japan Society for the Promotion of Science (Grants-in-Aid for Scientific Research S # 20H05633 and B # 21H00717). If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Appendix A. Supplementary material

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.jet.2025.106010.

## Data availability

No data was used for the research described in the article.
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