2.1. Firms

Suppose that the final good $Y$ is produced through the use of a range of differentiated intermediate inputs $y_{i}$ . Following Bénassy (Reference Benassy1996), final output is produced by a perfectly competitive firm with the following technology:Footnote ⁴

(1) \begin{equation} Y=N^{\gamma +1-\frac{1}{\lambda }}\left(\int _{0}^{N}y_{i}^{\lambda }\,di\right)^{\frac{1}{\lambda }}, \end{equation}

where $N$ is the number of intermediate goods, the parameter $\lambda \in (0,1)$ measures the degree of monopoly power, and the parameter $\gamma$ ( $\geq$ 0) measures the extent of production specialization.

One point should be mentioned here. If all intermediate goods are produced in the same quantities, then the final output is expressed by $Y=N^{\gamma +1}y$ . It turns out that in the presence of production specialization (i.e., $\gamma \gt 0$ ), a rise in the number of firms increases the production of final output more than proportionally, and this is referred to as the production-enhancing effect; see, example, Chang et al. (Reference Chang, Lai and Chang2018).

Assuming that the final good is the numéraire and that $p_{i}$ is the price of the $i$ th intermediate good, the profit-maximization problem for the final good firm can then be expressed as:

\begin{equation*} \underset{y_{i}}{Max}\,\,\pi ^{f}=Y-\int _{0}^{N}p_{i}y_{i}\,di . \end{equation*}

Accordingly, the solution for the first-order condition leads to the following expression:

(2) \begin{equation} p_{i}=N^{\lambda \left(\gamma +1-\frac{1}{\lambda }\right)}y_{i}^{\lambda -1}Y^{1-\lambda } .\end{equation}

Eq. (2) is the inverse demand function for the $i$ th intermediate good, and the function is characterized by a price elasticity $1/(1-\lambda )$ . A larger $\lambda$ leads to a higher price elasticity, implying that the intermediate goods sector is more competitive. Therefore, $\lambda$ measures the degree of monopoly power in the markets for intermediate products.

Intermediate good producers use capital and labor to produce a differentiated product, which they sell to the final good producers at the profit-maximizing price. The production technology for the $i$ th intermediate good producer is given by

(3) \begin{equation} y_{i}=Ak_{i}^{a}h_{i}^{1-a}-{\Gamma}, \end{equation}

where $A\gt 0$ is a constant technology parameter, $k_{i}$ and $h_{i}$ , respectively, denote the capital and labor hired by the $i$ th intermediate good producer, $1\gt a\gt 0$ is the share of capital, and ${\Gamma}$ is the overhead cost.

To highlight the difference between the literature on belief-driven fluctuations and this paper, the overhead cost in the model is expressed as follows:

(4) \begin{equation} {\Gamma} =f\hspace{0pt}N^{\sigma },\,\,f\gt 0,\sigma \in ({-}1,\infty ). \end{equation}

This general form allows us to consider three distinctive scenarios for the overhead cost. The first scenario in association with $\sigma =0$ denotes the fixed effect. In this case, the overhead cost is fixed (i.e., ${\Gamma} =f$ ), and almost all existing studies including Chang et al. (Reference Chang, Hung and Huang2011), Pavlov and Weder (Reference Pavlov and Weder2012), Chang et al. (Reference Chang, Lai and Chang2018) and Chen and Guo (Reference Chen and Guo2022) adopt this specification. The second scenario in association with $\sigma \in (0,\infty )$ captures the congestion effect due to competitive differentiation, and denotes the viewpoint proposed by Kim (Reference Kim2004) and Feng et al. (Reference Feng, Li and Swenson2017): the greater the number of firms is associated with a higher overhead cost. The third scenario in association with $\sigma \in ({-}1,0)$ captures the spillover effect from industry clusters, and reveals the Peretto and Connolly (Reference Peretto and Connolly2007) and Kamal and Sundaram (Reference Kamal and Sundaram2016) viewpoint: the greater the number of firms is associated with a lower overhead cost.Footnote ⁵

Let $w$ and $r$ respectively denote the market wage and capital rental rate. Based on the inverse demand function reported in Eq. (2) and the production function reported in Eq. (3), the optimization problem of the $i$ th intermediate good producer can be expressed as

(5) \begin{align} & \underset{h_{i}\,k_{i}}{Max}\quad \,\pi _{i}^{m}=p_{i}y_{i}-w\enspace h_{i}-r\enspace k_{i},\\& s.t.\ \text{Eqs. (2) and (3)}. \nonumber \end{align}

The first-order conditions with respect to $h_{i}$ and $k_{i}$ are

(6) \begin{equation} w=\frac{\lambda \left(1-a\right)p_{i}\left(y_{i}+fN^{\sigma }\right)}{h_{i}}, \end{equation}

(7) \begin{equation} r=\frac{\lambda ap_{i}\left(y_{i}+fN^{\sigma }\right)}{k_{i}}. \end{equation}

Then we obtain the profit of the $i$ th intermediate good producer by substituting Eqs. (6) and (7) into (5):

(8) \begin{equation} \pi _{i}^{m}=p_{i}\left[\left(1-\lambda \right)y_{i}-\lambda f\hspace{0pt}N^{\sigma }\right]. \end{equation}

The analysis is confined to a symmetric equilibrium under which $p_{i}=p, k_{i}=k, h_{i}=h$ , and $y_{i}=y$ for all i. Let $K$ and $H$ denote the aggregate capital stock and labor inputs, respectively. We then have: $K=Nk$ and $H=Nh$ .

The zero-profit condition of perfect competition in the final good sector yields the following result:

(9) \begin{equation} p=N^{\gamma } .\end{equation}

Moreover, free entry guarantees zero profits for the intermediate goods sector. As a result, the production of each intermediate good and the number of firms in equilibrium are given by

(10) \begin{equation} y=\frac{\lambda fN^{\sigma }}{1-\lambda }, \end{equation}

(11) \begin{equation} N=\left(\frac{\left(1-\lambda \right)AK^{a}H^{1-a}}{f}\right)^{\frac{1}{1+\sigma }}. \end{equation}

By inserting Eqs. (10) and (11) into (1), we can further obtain the aggregate production function as follows:

(12) \begin{equation} Y=\lambda \left(\frac{1-\lambda }{f}\right)^{\frac{\gamma }{1+\sigma }}A^{\frac{1+\sigma +\gamma }{1+\sigma }}K^{\frac{a\left(1+\sigma +\gamma \right)}{1+\sigma }}H^{\frac{\left(1-a\right)\left(1+\sigma +\gamma \right)}{1+\sigma }}. \end{equation}

Based on Eq. (12), the following condition should be imposed to ensure that the externality is not sufficiently strong to generate sustained growth:Footnote ⁶

Condition NSG [the Nonsustained Growth Condition].

(12)' \begin{equation} 0\lt \frac{a\left(1+\sigma +\gamma \right)}{1+\sigma }\lt 1. \end{equation}

One point regarding Eqs. (10)–(12) should be stressed.Footnote ⁷ To compare the current endogenous overhead costs model with the fixed overhead costs model, we consider an extreme case of the fixed overhead costs scenario (i.e., $\sigma =0$ ). In this extreme case, we find that the individual firm’s output is fixed in equilibrium, and the number of firms is homogeneous of degree one in capital and labor. As a result, the aggregate production exhibits increasing returns to scale when $\gamma \gt 0$ . By contrast, this paper considers the feature of endogenous overhead costs. In this scenario, the equilibrium individual firm’s output is no longer fixed, and the number of firms is homogeneous of degree $1/(1+\sigma )$ in capital and labor. As a result, the aggregate production exhibits increasing returns to scale when $\gamma /(1+\sigma )\gt 0$ . Equipped with this knowledge, the analysis finds that the endogenously-determined overhead costs are closely related to the number of firms and therefore give rise to the congestion effect and the spillover effect, thereby governing the impact of the number of firms on aggregate production.

2.2. Households

Consider an economy consisting of a unit measure of infinitely-lived households. The households maximize their lifetime utility $U$ by deriving utility from consumption $C$ but incurring disutility from labor supply $H$ . Thus, the lifetime utility of the households can be expressed as

(13) \begin{equation} U=\int _{0}^{\infty }\left[\ln \,C_{t}-\zeta H_{t}\right]e^{-\rho \enspace t}dt, \rho \gt 0, \zeta \gt 0, \end{equation}

where $\rho$ represents the rate of time preference, $\zeta$ measures the disutility from labor supply, and $t$ is the time index.

The households face the following budget constraint:Footnote ⁸

(14) \begin{equation} \dot{K}=wH+rK+{\Pi} -C,^8 \end{equation}

where ${\Pi} (=\int _{0}^{N}\pi _{i}^{m}\,di)$ refers to the aggregate profits transferred from intermediate good producers.

The households maximize the intertemporal utility reported in Eq. (13) subject to the budget constraint reported in Eq. (14). Performing the optimization problem leads to the following first-order conditions:

(15) \begin{equation} \frac{1}{C}=\mu, \end{equation}

(16) \begin{equation} \zeta =\mu \,w, \end{equation}

(17) \begin{equation} \mu \,r=-\dot{\mu }+\mu \,\rho, \end{equation}

together with Eq. (14) and the transversality condition $\lim _{t\rightarrow \infty }\mu \,K\,e^{-\rho \,t}=0$ , where $\mu$ is the shadow price of physical capital.

Eq. (15) represents the households’ optimal consumption decision, Eq. (16) represents the households’ optimal decision regarding labor supply, and Eq. (17) is associated with the households’ optimal capital accumulation decision.

Combining Eq. (15) with (17) yields the familiar Keynes–Ramsey Rule:

(18) \begin{equation} \dot{C}=\left(r-\rho \right)C . \end{equation}

In addition, inserting Eqs. (3), (9), (11) and (12) into (6) and (7) yields:

(19) \begin{equation} w=\frac{\left(1-a\right)Y}{H}, \end{equation}

(20) \begin{equation} r=\frac{aY}{K} . \end{equation}

2.3. The Competitive Equilibrium

By substituting Eqs. (8), (19) and (20) into (14), we obtain the economy-wide resource constraint:

(21) \begin{equation} \dot{K}=Y-C . \end{equation}

Based on Eqs. (12), (15), (16) and (19), we can solve for employment for the instantaneous relationship:

(22) \begin{equation} H=H\left(K,C\right), \end{equation}

where $H_{K}=\frac{\partial H}{\partial K}=\frac{-a\left(1+\sigma +\gamma \right)H}{{\Omega} K}, H_{C}=\frac{\partial H}{\partial C}=\frac{\left(1+\sigma \right)H}{{\Omega} C}$ , and ${\Omega} =(1-a)(1+\sigma +\gamma )-(1+\sigma )$ .

3.1. The Condition for Equilibrium Indeterminacy

We are now in a position to examine how the feature of industry clusters affects the emergence of equilibrium indeterminacy. As addressed in the literature on dynamic rational expectations models, such as Liu and Turnovsky (Reference Liu and Turnovsky2005), Brito and Dixon (Reference Brito and Dixon2013), Lai and Chin (Reference Lai and Chin2013) and Chu et al. (Reference Chu, Liao, Liu and Zhang2021), there exists a unique perfect foresight equilibrium solution if the number of unstable roots equals the number of jump variables. Since $C$ is the only jump variable in this dynamic system, the steady-state equilibrium is locally determinate if only one of the roots is positive in the dynamic system, implying that the value of the determinant of the Jacobian is negative, that is, $Det(J)\lt 0$ . Based on this cognition, we formulate the following constraint for saddle-point stability:

(28) \begin{equation} {\Omega} =\left(1-a\right)\left(1+\sigma +\gamma \right)-\left(1+\sigma \right)\lt 0 . \end{equation}

However, as argued by Benhabib and Farmer (Reference Benhabib and Farmer1994), Aguiar-Conraria and Wen (Reference Aguiar-Conraria and Wen2008), Guo and Harrison (Reference Guo and Harrison2010) and Lai et al. (Reference Lai, Chin and Chen2017), there exists a continuum of equilibrium paths converging to the steady state if the dynamic system has two negative roots, that is, $Det(J)\gt 0\gt Tr(J)$ . Therefore, the necessary and sufficient condition for our macroeconomy to exhibit local indeterminacy is that the two roots are both negative. Accordingly, we formulate the following constraint for equilibrium indeterminacy:

(29) \begin{equation} {\Omega} =\left(1-a\right)\left(1+\sigma +\gamma \right)-\left(1+\sigma \right)\gt 0 . \end{equation}

The above discussion can be summarized by the following proposition:

It is clear that the condition for equilibrium (in)stability is closely related to the capital share $a$ , the degree of production specialization $\gamma$ , and the extent of industry clusters $\sigma$ . Therefore, we in turn examine how the possibility of local indeterminacy is related to each of these factors.

Two points related to the condition for local indeterminacy should be mentioned. First, in previous studies, Chang et al. (Reference Chang, Lai and Chang2018) and Chen and Guo (Reference Chen and Guo2022) focus on the situation where the feature of industry clusters is absent (i.e., $\sigma =0$ ). As a result, their analysis finds that local indeterminacy can occur, provided that the degree of production specialization is relatively large. Second, our analytical result further indicates that the condition for local indeterminacy crucially depends upon the effect of industry clusters, in addition to the returns to production specialization proposed in the existing literature.Footnote ⁹

We first discuss the linkage between the possibility of equilibrium indeterminacy and the extent of production specialization $\gamma$ . Our intuitive explanation for this result is borrowed from Benhabib and Farmer (Reference Benhabib and Farmer1994). The condition for equilibrium indeterminacy requires that the equilibrium wage-hours locus be positively sloped and steeper than the labor supply curve. Accordingly, the condition for equilibrium indeterminacy can be expressed asFootnote ¹⁰

(30) \begin{equation} \frac{\left(1-a\right)\left(1+\sigma +\gamma \right)}{1+\sigma }-1\gt 0 . \end{equation}

When the households generate optimistic expectations as regards having a higher future return on capital, they will tend to increase their investment today, implying that the capital stock in the next period will increase. Given that labor and physical capital are complements in production, labor supply in the next period will increase, implying that more output will be produced, and this will in turn attract a greater number of firms to enter and produce in the market. In the presence of increasing returns to production specialization (i.e., $\gamma \gt 0$ ), a higher value of $\gamma$ further reinforces the labor productivity via the production-enhancing effect, which is reflected by the factor $\gamma$ on the left-hand side of Eq. (30). As a consequence, the economy is more susceptible to local indeterminacy when the magnitude of the production specialization is relatively large. In fact, this result is reminiscent of Chang et al. (Reference Chang, Lai and Chang2018) and Chen and Guo (Reference Chen and Guo2022).

We then deal with how the possibility of equilibrium indeterminacy is related to the effect of industry clusters $\sigma$ . As described above, when the households become optimistic, the number of firms will increase in response. When more firms are being set up, an incumbent firm will raise its overhead costs due to the presence of the congestion effect (i.e., $\sigma \gt 0$ ), and this will restrain firms from entering and producing in the market. To be specific, a higher value of $\sigma$ reduces the contribution of the production-enhancing effect of entry and therefore weakens the labor productivity, which is reflected by the factor $\sigma$ on the left-hand side of Eq. (30). As a result, the presence of the congestion effect mitigates the possibility of local indeterminacy. By contrast, in the presence of the spillover effect (i.e., $\sigma \lt 0$ ), an incumbent firm could reduce overhead costs when more firms are being set up, which may induce more firms to enter and produce in the market. As a result, the presence of the spillover effect enhances the contribution of the production-enhancing effect of entry and therefore reinforces the labor productivity, thereby increasing the possibility of local indeterminacy.

The above discussion can be summarized by the following proposition:

3.2. Quantitative Analysis

To perform further quantitative analysis for the emergence of local indeterminacy, in this subsection we provide a quantitative assessment by resorting to a numerical analysis. Figure 1 is drawn to highlight the effect of both $\gamma$ and $\sigma$ on the likelihood of equilibrium (in)determinacy. To compare our results with those of existing studies, the capital share is set to $a=0.3$ , a value used in Benhabib and Farmer (Reference Benhabib and Farmer1996), Chang and Lai (Reference Chang and Lai2017), Chen and Guo (Reference Chen and Guo2022) and many previous studies in the literature on belief-driven business cycles. We vary the extent of production specialization $\gamma$ from 0 to 0.5 and the industrial cluster effect $\sigma$ from −1 to 1 to highlight its importance. As can be seen from Figure 1, the $\sigma -\gamma$ space is divided into three areas. The white area at the lower right displays the region featured by saddle-point stability (i.e., ${\Omega} \lt 0$ ), the gray area at the upper middle depicts the region featured by equilibrium indeterminacy (i.e., ${\Omega} \gt 0$ ), while the white area at the upper left depicts the region that does not satisfy Condition NSG.

Figure 1. The saddle-point stability and equilibrium indeterminacy regions.

Three important results emerge from Figure 1. First, as indicated in Figure 1, a higher degree of production specialization matched with a higher degree of the spillover effect (a lower $\sigma$ ) or a lower degree of the congestion effect tends to increase the likelihood of local indeterminacy. Second, as in Chang et al. (Reference Chang, Lai and Chang2018) and Chen and Guo (Reference Chen and Guo2022), in the absence of the industrial cluster effect (i.e., $\sigma =0$ ), local indeterminacy can occur, provided that the degree of production specialization is relatively large. To be more specific, the minimum degree of production specialization required for local indeterminacy is $\gamma _{\min }=0.429$ . Third, Kasahara and Rodrigue (Reference Kasahara and Rodrigue2008) indicate that the estimated degree of production specialization can be as high as 0.3. Given this parameter value, we have the minimum level of industry clusters required to satisfy the condition for local indeterminacy. To be more specific, the minimum level of industry clusters required for local indeterminacy is $\sigma _{\min }=-0.3$ given that $\gamma =0.3$ .