3.1 Firms and households

As in the closed economy, domestic real output is produced using a linear technology: $y=h$ . Under the assumption that the labor market is perfectly competitive, profit maximization of the representative firm leads to: $w=1$ .

Let F denote the stock of nominal foreign currency, and $f\equiv F/P^{\ast }$ is real foreign currency balances. The subutility function v shown in (2) is modified as follows: ^{Footnote 9}

(14) \begin{equation}v=\left\{\begin{array}{c}\frac{\left( c^{a}l^{1-a}\right) ^{1-\sigma }-1}{1-\sigma } \\a\log c+\left( 1-a\right) \log l\end{array}\right. ,\,\ \text{if \ }\begin{array}{c}\sigma \neq 1 \\\sigma =1\end{array}, \end{equation}

where l represents real liquidity services that are a composite of real domestic currency, m, and real foreign currency, f, in the constant-elasticity-of-substitution (CES) functional form [Chen et al. (Reference Chen, Tsaur and Chou1981); Miles (Reference Miles1978, Reference Miles1981); Imrohoroglu (Reference Imrohoroglu1996)]:

(15) \begin{equation}l=\left[ \delta m^{\theta }+\left( 1-\delta \right) f^{\theta }\right] ^{\frac{1}{\theta }}, \end{equation}

where the parameter $\delta \in \left( 0,1\right) $ is the share of domestic currency in total liquidity. When $\delta =1$ , I recover the subutility function v for the closed economy, as shown in (2). In this case, there is no incentive for households to hold the foreign currency since they derive zero utility from it. The parameter $\delta $ therefore captures the degree of openness in terms of diversified currency holdings, where a lower value of $\delta $ represents a higher degree of openness.

The parameter $\theta <1$ presenting in (15) measures the degree of currency substitution. Specifically, the elasticity of substitution between m and f equals $\frac{1}{1-\theta }>0$ . A larger value of $\theta $ thus represents a higher degree of substitution between the currencies. When $\theta \rightarrow -\infty $ , the elasticity of substitution approaches 0, and l approaches the Leontief form: $l=\min\left( m,f\right) $ ; in this case, m and f display perfect complementarity. When $\theta \rightarrow 0$ , the elasticity of substitution approaches 1, and l approaches the Cobb-Douglas form: $l=m^{\delta}f^{1-\delta }$ ; in this case, m and f are imperfect substitutes. When $\theta \rightarrow 1$ , the elasticity of substitution approaches infinite, and l is linear in m and f: $l=\delta m+\left( 1-\delta \right) f$ ; and m and f are therefore perfect substitutes.

The instantaneous utility function (1) under the subutility function (14) is increasing and strictly concave with respect to c, $m$ , f, and h. When $\sigma =1$ , the household’s preference exhibits additive separability between consumption and real liquidity services; hence, the marginal utility of c is independent of m and f: $u_{cm}=u_{cf}=0$ . When $\sigma >\left( <\right) $ 1, not only c and m but also c and f are Edgeworth substitutes (complements): $u_{cm}<\left(>\right) 0$ and $u_{cf}<\left( >\right) 0$ , if $\sigma >\left( <\right) $ 1. Finally, if $\theta >\left( <\right) \Sigma -\sigma $ , then m and f display Edgeworth substitutability (complementarity): $u_{mf}<\left(>\right) 0$ , if $\theta >\left( <\right) \Sigma -\sigma $ .

In the presence of diversified currency holdings, the budget constraint faced by the representative household, (3), is modified as follows:

(16) \begin{equation}\dot{m}+\dot{f}=wh-c-\pi m-\pi ^{\ast }f+\tau ,\,\,\,f_{0}>0\,\,\,\text{given.} \end{equation}

It is straightforward to derive the following first-order conditions for the representative household and the associated TVC:

(17) \begin{eqnarray}c &:&\overbrace{a\left( c^{a}l^{1-a}\right) ^{1-\sigma }/c}^{u_{c}}=\lambda , \end{eqnarray}

(18) \begin{eqnarray}h &:&h^{\gamma }/\lambda =w, \end{eqnarray}

(19) \begin{eqnarray}m &:&\overbrace{\frac{\left( 1-a\right) \alpha c\lambda }{am}}^{u_{m}}-\pi \lambda =\rho \lambda -\dot{\lambda}, \end{eqnarray}

(20) \begin{eqnarray}f &:&\overbrace{\frac{\left( 1-a\right) \left( 1-\alpha\right) c\lambda }{af}}^{u_{f}}-\pi ^{\ast }\lambda =\rho \lambda -\dot{\lambda}, \end{eqnarray}

(21) \begin{eqnarray}\text{TVC} &:&\lim_{t\rightarrow \infty }e^{-\rho t}\lambda m=\lim_{t\rightarrow \infty }e^{-\rho t}\lambda f=0, \end{eqnarray}

where $\alpha \equiv \delta \left( \frac{m}{l}\right) ^{\theta }\in \left(0,1\right) $ . With (17), equations (19) and (20) lead to the no arbitrage condition between holdings of domestic and foreign currencies:

(22) \begin{equation}\overbrace{\frac{\left( 1-a\right) \alpha c}{am}}^{\frac{u_{m}}{u_{c}}=R}-\pi =\overbrace{\frac{\left( 1-a\right) \left( 1-\alpha \right) c}{af}}^{\frac{u_{f}}{u_{c}}}-\;\pi^{\ast }, \end{equation}

from which the equilibrium domestic inflation rate is determined as $\pi=\pi ^{\ast }+\frac{\left( 1-a\right) c}{a}\left( \frac{\alpha }{m}-\frac{1-\alpha }{f}\right) $ . In turn, the law of one price (13) determines the depreciation rate of the domestic currency $\varepsilon =\pi-\pi ^{\ast }$ .

Note that (22) holds in equality as I concentrate on the interior equilibrium whereby both currencies are held. In this situation, the representative household chooses its holdings of m and f such that it derives the same levels of net (after-inflation tax) marginal benefits from holding domestic and foreign currencies: $\frac{u_{m}}{u_{c}}-\pi =\frac{u_{f}}{u_{c}}-\pi ^{\ast }$ . However, when domestic and foreign currencies are displayed as Edgeworth substitutes, there can be an equilibrium where either f or m is zero. For example, a sufficiently high level of $\pi $ ( $\pi ^{\ast }$ ) will make the left-hand side of (22) lower (higher) than the right-hand side of (22), and consequently, the household will not hold the domestic (foreign) currency. ^{Footnote 10}

3.2 Government and the central bank

Under a regime of flexible exchange rates, foreign reserves are constant over time. Following AgÉnor and Montiel (Reference AgÉnor and Montiel2015, pp. 357–359), I normalize the constant level of reserves to zero. Accordingly, by denoting $D $ as the nominal stock of domestic credit that grows at the constant rate $\mu =\dot{D}/D\neq 0$ , changes in the real money balances are equal to changes in the real domestic credit stock, that is,

(23) \begin{equation}\dot{m}/m=\dot{d}/d=\mu -\pi , \end{equation}

where $d(\equiv D/P)$ is the real stock of domestic credit. As in Section 2, the government distributes seigniorage to the representative household as a lump-sum transfer. The flow budget constraint of the government is thus given by: $\tau =\mu m$ .

By putting together the household’s budget constraint (16), the evolution of real money balances (23), the government’s budget constraint $\tau =\mu m$ , the production technology $y=h$ , and the firm’s optimization $w=1$ , the SOE’s consolidated budget constraint is obtained as follows:

(24) \begin{equation}\dot{f}=y-c-\pi ^{\ast }f, \end{equation}

which equates net financial outflows driven by holdings of the foreign currency to current account surplus.

3.3 Analysis of dynamics

It is straightforward to show that the model possesses a unique interior steady state given by (49)–(55). In the neighborhood of the steady state, the model’s equilibrium conditions can be approximated by the following log-linear dynamical system:

(25) \begin{equation}\left[\begin{array}{c}\overset{\cdot }{\hat{m}} \\\overset{\cdot }{\hat{f}} \\\overset{\cdot }{\hat{\lambda}}\end{array}\right] =\underbrace{\left[\begin{array}{ccc}\Gamma _{1} & \Gamma _{2} & \frac{\mu -\pi ^{\ast }}{\Sigma } \\\frac{\Xi \Pi \left( \sigma -\Sigma \right) }{\left( 1+\Xi \right) \Sigma }& \frac{\Pi \left( \sigma -\Sigma \right) }{\left( 1+\Xi \right) \Sigma }-\pi ^{\ast } & \frac{\pi ^{\ast }+\Pi }{\gamma }+\frac{\Pi }{\Sigma } \\\frac{\rho +\mu }{1+\Xi }\left( 1-\theta +\frac{\sigma \Xi }{\Sigma }\right)-\Gamma _{1} & \frac{\rho +\mu }{1+\Xi }\left( \theta -1+\frac{\sigma }{\Sigma }\right) -\Gamma _{2} & \frac{\pi ^{\ast }+\rho }{\Sigma }\end{array}\right] }_{\mathbf{J}}\left[\begin{array}{c}\hat{m}-\hat{m}^{s} \\\hat{f}-\hat{f}^{s} \\\hat{\lambda}-\hat{\lambda}^{s}\end{array}\right] ,\,\,\ \hat{f}_{0}\,\text{given,} \end{equation}

where $\Xi =\left[ \frac{\delta }{1-\delta }\left( \frac{\pi ^{\ast }+\rho }{\mu +\rho }\right) ^{\theta }\right] ^{\frac{1}{1-\theta }}>0$ , $\Pi =\frac{a\left( \pi ^{\ast }+\rho \right) \left( 1+\Xi \right) }{1-a}>0$ , and

\begin{eqnarray*}\Gamma _{1} &\equiv &\frac{\left( 1-\theta +\frac{\sigma \Xi }{\Sigma }\right) \mu +\left( 1-\theta \right) \left( 1+\Xi \right) \rho -\Xi \left(\theta -1+\frac{\sigma }{\Sigma }\right) \pi ^{\ast }}{1+\Xi }, \\\Gamma _{2} &\equiv &\frac{\left( \theta -1+\frac{\sigma }{\Sigma }\right)\mu -\left( 1-\theta \right) \left( 1+\Xi \right) \rho +\left[ \frac{\sigma}{\Sigma }+\Xi \left( 1-\theta \right) \right] \pi ^{\ast }}{1+\Xi }.\end{eqnarray*}

It follows that the determinant and trace of the model’s Jacobian matrix $\mathbf{J}$ are

(26) \begin{equation}Det\left( \mathbf{J}\right) =\frac{\left( \theta -1\right) \left( \rho +\mu\right) \left( \rho +\pi ^{\ast }\right) \left( \frac{\sigma }{\gamma }+1\right) \left( \Pi +\pi ^{\ast }\right) }{\Sigma }<0, \end{equation}

and

(27) \begin{equation}Tr\left( \mathbf{J}\right) =\underbrace{\left( 1-\theta +\frac{\sigma \Xi }{\Sigma }\right) }_{\left( +\right) }\frac{\mu -\pi ^{\ast }}{1+\Xi }+\left(2-\theta \right) \rho +\left( 1-\theta \right) \pi ^{\ast }\gtreqqless 0.\end{equation}

Since both the initial foreign price of the single traded good, denoted as $P_{0}^{\ast }$ , and the foreign inflation rate $\pi ^{\ast }$ , are exogenously given to the SOE, the foreign price level $P^{\ast }\left(t\right) =P_{0}^{\ast }e^{\pi ^{\ast }t}$ is constrained to adjust continuously. It thus follows that the stock of real foreign currency $f(=F/P^{\ast })$ is a nonjump predetermined variable, and that the initial real stock of foreign currency held by domestic residents, $f_{0}$ , is an exogenously given initial condition. On the contrary, since both the domestic inflation rate $\pi $ and the initial domestic price level, denoted as $P_{0}$ , are endogenously determined, the domestic price level $P\left( t\right) =P_{0}e^{\pi t}$ can jump instantaneously. As a result, the stock of real domestic currency $m\equiv M/P$ is a jump variable without an initial condition. Note from (22) that, with real consumption c and the domestic inflation rate $\pi $ both instantly adjusting, no arbitrage between holdings of domestic and foreign currencies by the agent can be achieved under a freely-jumping m and a predetermined f.

Given that the first-order dynamical system (25) possesses one predetermined variable $\hat{f}$ , and both $\hat{m}$ and $\hat{\lambda}$ are nonpredetermined jump variables, the economy displays saddle-path stability and equilibrium uniqueness if and only if one of the eigenvalues of $\mathbf{J}$ has a negative real part, and two of the remaining have positive real parts. When two or more eigenvalues have negative real parts, the steady state is a locally indeterminate sink that can be exploited to generate endogenous business cycle fluctuations driven by agents’ self-fulfilling expectations or sunspots. The steady state becomes a source when all eigenvalues have positive real parts. In the remainder of this section, I examine the local dynamics of the model’s steady state in two parametric configurations.

3.3.2 When $\pi ^{\ast }> \mu $

For this parametric configuration, the foreign inflation rate is higher than the long-run domestic inflation rate. I rewrite (27) as

(28) \begin{equation}Tr\left( \mathbf{J}\right) =\underbrace{\left( 1-\theta +\frac{\sigma \Xi }{\Sigma }\right) \frac{\mu }{1+\Xi }+\left( 2-\theta \right) \rho }_{\left(+\right) }+\underbrace{\frac{\Xi \left( 1-a\right) \left( 1-\theta \right)\pi ^{\ast }}{\left( 1+\Xi \right) \Sigma }}_{\left( +\right) }\cdot \Theta ,\end{equation}

where

(29) \begin{equation}\Theta \equiv \frac{\left( 1-a\right) \left( 1-\theta \right) -\left[1-a\left( 1-\theta \right) \right] \sigma }{\left( 1-a\right) \left(1-\theta \right) }\gtreqqless 0. \end{equation}

It is straightforward to show that, when $\theta <\underline{\theta }\equiv\frac{a-1}{a}<0$ , the bracket $\left[ 1-a\left( 1-\theta \right) \right] $ in the numerator of $\Theta $ , given by (29), has a negative sign. It follows that $\Theta >0$ , and hence the Jacobian matrix $\mathbf{J}$ possesses a positive trace $\left( Tr\left( \mathbf{J}\right) >0\right) $ . Adding this with the fact that $Det\left( \mathbf{J}\right) <0$ , I find that in the present case where $\theta <\underline{\theta }$ , the model’s steady state displays only one stable (negative) eigenvalue. Hence, equilibrium uniqueness is ensured. This case is illustrated as Zone I of Figure 2.

Figure 2. Regions of Equilibrium (In)determinacy: SOE and $\pi^{*}>\vec{\mu.}$

I then turn to the case where $\theta >\underline{\theta }$ . Here, a positive $\left[ 1-a\left( 1-\theta \right) \right] $ is derived, and hence the numerator of $\Theta $ has an ambiguous sign. I then derive from (29) that $\Theta \gtreqqless 0$ , if $\sigma \lesseqqgtr \underline{\sigma }\equiv \frac{\left( 1-a\right) \left( 1-\theta \right) }{1-a\left(1-\theta \right) }$ . Thus, when $\theta >\underline{\theta }$ and $\sigma <\underline{\sigma }$ , positive values of both $\Theta $ and $Tr\left(\mathbf{J}\right) $ are guaranteed, implying that the model possesses a determinate steady state. This case is illustrated as Zone II of Figure 1. ^{Footnote 11}

Zone III in Figure 1 represents the case where $\theta >\underline{\theta }$ and $\sigma >\underline{\sigma }$ (hence, $\Theta <0$ ), and the steady state displays $Det\left( \mathbf{J}\right) <0$ and $Tr\left( \mathbf{J}\right) \gtreqqless 0$ . Since structural and policy parameters enter the Jacobian matrix J in highly nonlinear manners, I am unable to analytically identify the number of (un)stable roots. In what follows, I quantitatively investigate the local stability properties of the steady state locating in Zone III.

Per the parameterization that is commonly adopted in the RBC-based indeterminacy literature, the time discount rate $\rho $ is set equal to $0.01$ . Recall that one needs a strictly positive value of the inverse of the intertemporal elasticity of substitution in labor supply, $\gamma $ , for guaranteeing the existence of labor market equilibrium; $\gamma $ is thus calibrated to be 2 [Chetty et al. (Reference Chetty, Guren, Manoli and Weber2011, Reference Chetty, Guren, Manoli and Weber2012)]. The domestic inflation rate is set at 4% per year, implying that $\mu $ equals $0.04/4$ . Finally, under the calibrated value of $\gamma $ , Figure 1 shows that equilibrium indeterminacy is possible in Section 2’s closed economy only when $a<\tilde{a}=\frac{\gamma }{1+\gamma }=\frac{2}{3}$ . To test the power of diversified currency holdings in mitigating belief-driven cyclical fluctuations, the preference parameter a is chosen to be $0.5$ ( $<\frac{2}{3}$ ). It follows that the minimum level of $\sigma $ such that indeterminacy arises in the closed economy equals $\tilde{\sigma}=7$ . In addition, the critical value of $\underline{\theta }=\frac{a-1}{a}$ equals $-1$ ; hence, only the cases associated with $\theta >-1$ need to be examined.

Given the baseline parameter values, Figure 3 plots for various combinations of $\sigma $ and $\delta $ the minimum value of the annual foreign inflation rate $\pi ^{\ast }$ needed for equilibrium indeterminacy as a function of the degree of currency substitution, $\theta $ , when the model’s steady state is located in Zone III of Figure 1. The two rows of Figure 3, respectively, illustrate the case of $\sigma =10$ and 8. In particular, while the literature finds that the estimate of the coefficient of relative risk aversion, $\sigma $ , can range from 0 to around 30, the general consensus is that it is below 10. ^{Footnote 12} Under the calibrated values of $\rho $ , $\gamma $ , $\mu $ , and a, belief-driven cyclical fluctuations are impossible when $\sigma $ is below 7. I therefore show in the figure the cases of $\sigma =10$ and $8 $ only. On the other hand, a lower value of the share of domestic currency in total liquidity, $\delta $ , represents a higher degree of openness in terms of diversified currency holdings. The three columns of Figure 3 thus correspond to the situations in which the degree of openness is low ( $\delta=0.8$ ), medium ( $\delta =0.5$ ), and high ( $\delta =0.2$ ), respectively.

Figure 3. Requisite $\pi$ * for Indeterminacy: Zone III of Figure 2.

It is obvious that each of the panels in Figure 3 displays a negative nexus between the requisite level of $\pi ^{\ast }$ for indeterminacy and the degree of currency substitution, $\theta $ . Moreover, for any combination of $\sigma $ , $\delta $ , and $\theta $ , the value of $\pi ^{\ast }$ above which equilibrium indeterminacy occurs is too high to square with data. In particular, in each of the panels in Figure 3, the lowest requisite level of $\pi ^{\ast }$ for indeterminacy is observed when $\theta\rightarrow 1$ . I numerically derive that, when $\theta \rightarrow 1$ , this level of $\pi ^{\ast }$ equals 42% (97%) when $\sigma =10$ (8), for $\delta =0.8$ , $0.5$ , and $0.2$ . This means that, when $\theta \rightarrow 1$ , the minimum level of $\pi ^{\ast }$ that guarantees equilibrium indeterminacy does not change to variations in $\delta $ , but increases when $\sigma $ falls.

When $\theta <1$ , the requisite level of $\pi ^{\ast }$ for indeterminacy does change to variations in $\delta $ and/or $\theta $ . First, under a given level of $\sigma $ , each of the rows of Figure 3 illustrates that the requisite level of $\pi ^{\ast }$ for indeterminacy rises faster as $\theta $ declines, if $\delta $ takes on a lower value. Based on the related literature’s estimation, Airaudo (Reference Airaudo2014) sets a degree of currency substitution that implies $\theta =0.75$ . When taking $\sigma =10$ and $\theta =0.75$ as an example, the top panel of Figure 2 shows that the model’s steady state turns from a saddle into a locally indeterminate sink when $\pi ^{\ast }\geq 49\%$ , 50%, or 108%, if $\delta =0.8$ , $0.5$ , or $0.2$ . Since an increase in the degree of openness in terms of diversified currency holdings raises the minimum level of $\pi ^{\ast }$ that guarantees equilibrium indeterminacy, it makes belief-driven fluctuations more difficult to occur.

Next, under a given level of $\delta $ , each of the columns of Figure 3 shows that the requisite level of $\pi ^{\ast }$ for indeterminacy rises faster as $\theta $ declines, if $\sigma $ takes on a lower value. Taking $\delta =0.8$ and $\theta =0.75$ as an example, the requisite level of $\pi^{\ast }$ for indeterminacy is 49% (113%) if $\sigma =10$ (8). Recall that belief-driven cyclical fluctuations are impossible when $\sigma $ is below 7. Because both a high $\pi ^{\ast }$ and a high $\sigma $ are sources of equilibrium indeterminacy, when $\sigma $ is above 7, the minimum level of $\pi ^{\ast }$ that guarantees equilibrium indeterminacy decreases when $\sigma $ becomes higher. The above results are summarized in the following proposition.

Proposition 3 For a small-open monetary economy where diversified currency holdings are allowed, if the inflation rate of the foreign country is higher than the long-run domestic inflation rate, then:

1. (i) belief-driven cyclical fluctuations originally present in the domestic country are entirely removed when $\theta <\underline{\theta }$ , or when $\underline{\theta }<\theta <1$ and $\sigma <\underline{\sigma }$ , where $\underline{\theta }=\frac{a-1}{a}<0$ and $\underline{\sigma }=\frac{\left( 1-a\right) \left( 1-\theta\right) }{1-a\left( 1-\theta \right) }<1$ ;

2. (ii) when $\underline{\theta }<\theta <1$ and $\underline{\sigma }<\sigma $ , indeterminacy occurs at very high foreign inflation rates and the coefficient of relative risk aversion; in this case, the higher the degree of openness in terms of diversified currency holdings and/or the lower the coefficient of relative risk aversion is, the higher the requisite level of the foreign inflation rate will be.

3.3.3 Discussion

Under the calibrated values of $\rho $ , $\gamma $ , $\mu $ , and a, Proposition 1 states that the closed economy displays belief-driven business fluctuations when $\sigma $ is higher than the threshold level $\tilde{\sigma}=1+\frac{1}{\tilde{a}-a}=7$ . Proposition 2 then shows that, by allowing diversified currency holdings by agents, equilibrium indeterminacy becomes impossible when the foreign inflation rate is lower than the domestic long-run inflation rate. This is distinct from the indeterminacy result obtained in the literature. For instance, under an imperfect world capital market and a Taylor-type rule on the nominal government bonds rate, Airaudo’s (2014) quantitative analysis considering $\pi ^{\ast }=2\%<6.4\%=\pi $ shows that indeterminacy occurs.

Proposition 3 presents that even when $\pi ^{\ast }>\pi $ , equilibrium indeterminacy is impossible when the model’s steady state is located in Zone I or II of Figure 2, and it is extremely unlikely to occur when the steady state is located in Zone III of Figure 2 since the requisite level of $\pi^{\ast }$ for indeterminacy is too high to square with data.

The above discussion demonstrates that diversified currency holdings themselves create a stabilizing effect that eliminates the possibility of endogenous cyclical fluctuations. To understand the intuition behind the determinacy result under currency substitution, I present in what follows the evolution of the shadow value of wealth:

(30) \begin{equation}\overset{\cdot }{\hat{\lambda}}=\rho -\overbrace{\frac{\left( 1-a\right)\left( 1-\delta \right) }{a}\exp \left[ \frac{\log \left( a\right) +\left[\Sigma \left( 1-\theta \right) -\sigma \right] \hat{l}-\Sigma \left(1-\theta \right) \hat{f}-\hat{\lambda}}{\Sigma }\right] }^{u_{f}}+\pi ^{\ast}, \end{equation}

where the second term on the right-hand side is the nominal return on holding the foreign currency, that is, the marginal utility of the foreign currency, $u_{f}$ .

I start the economy from its steady state, where $\overset{\cdot }{\hat{\lambda}}=0$ and $\hat{\lambda}$ is at its steady-state value $\hat{\lambda}^{s}$ , and then consider a slight deviation caused by agents’ anticipation about a higher future shadow value of wealth. Acting upon this belief, households will decrease today’s consumption in exchange for the accumulation of wealth.

When $\sigma >1$ , consumption and the foreign currency are Edgeworth substitutes ( $u_{cf}<0$ ). Hence, when cutting $c_{t}$ , households will increase f. Since the marginal utility of the foreign currency exhibits diminishing returns, the nominal return on holding the foreign currency falls. This tends to raise the next period’s shadow value of wealth. However, currency substitution will offset this positive effect on $\overset{\cdot }{\hat{\lambda}}$ , thereby preventing agents’ anticipation from being self-fulfilling. Specifically, recall that m and f display substitutability (complementarity), if $\theta >(<)\Sigma $ . When $\theta>\Sigma $ , substitutability between m and f leads households to reduce $m $ when they increase f. Because $u_{mf}<0$ , the decline in $m_{t}$ raises the nominal return on holding f, which offsets the original decline in the nominal return on holding f. By contrast, when $\theta <\Sigma $ , complementarity between m and f leads households to raise m when they increase f. Since $u_{mf}>0$ , the rise in m increases the nominal return on holding f; the original decline in the nominal return on holding f is therefore offset.

When $\sigma <1$ , c, and f display complementarity ( $u_{cf}>0$ ). Hence, when c drops upon agents’ animal spirits, f falls as well. Diminishing returns of $u_{f}$ then lead to a higher nominal return on holding f, which in turn causes a decline in the next period’s shadow value of wealth. Again, this negative effect on $\overset{\cdot }{\hat{\lambda}}$ will be offset by currency substitution. Specifically, when $\theta >\Sigma $ , substitutability between m and f leads households to raises m when they reduce f. Because $u_{mf}<0$ , the increase in m shrinks the nominal return on holding f. By contrast, when $\theta <\Sigma $ , complementarity between m and f leads households to reduce m when they cut f. Since $u_{mf}>0$ , the drop in m reduces the nominal return on holding f.

Finally, when $\sigma =1$ , the household’s preference exhibits additive separability between c and f. Since f does not change upon agents’ animal spirits, it is clear from (30) that the nominal return on holding the foreign currency ( $u_{f}$ ) remains unchanged, and hence belief-driven business fluctuations will not occur.

4.1 Perfect world capital market

I first consider the case where the world capital market is perfect in the sense that domestic households of the SOE can freely borrow from or lend to the rest of the world at an exogenously given and constant risk-free interest rate $R^{\ast }=\bar{R}^{\ast }$ . The analysis below will show that indeterminacy and self-fulfilling expectations do not occur. This finding is in line with that obtained in Huang et al. (Reference Huang, Meng and Xue2018, Section 3.1) where diversified currency holdings are not allowed.

When $R^{\ast }=\bar{R}^{\ast }$ , the Euler equation, given by (32), is expressed as

(35) \begin{equation}\overset{\cdot }{\hat{\lambda}}=\rho -\bar{r}^{\ast }, \end{equation}

where $\bar{r}^{\ast }\equiv \bar{R}^{\ast }-\pi ^{\ast }$ is the exogenously given world risk-free real interest rate. With $\bar{r}^{\ast }$ and the rate of time preference $\rho $ both being exogenously given constants from the standpoint of the SOE, in order for (35) to imply a finite interior steady-state value for the shadow value of real net financial assets ( $\hat{\lambda}$ ), the following restriction has to be imposed: ^{Footnote 13}

(36) \begin{equation}\rho =\bar{r}^{\ast }\text{, \ \ }\forall t. \end{equation}

This further implies that $\overset{\cdot }{\hat{\lambda}}=0$ for all t, so that $\hat{\lambda}=\hat{\lambda}^{s}$ is constant over time. Moreover, as is widely known in the literature on SOE real business cycle models with a perfect world capital market in continuous time, (35) displays a zero eigenvalue. In the context of labor hours being the only production input, the constancy of $\hat{\lambda}$ in turn implies the constancy of hours worked ( $\hat{h}=\hat{y}^{s}$ ) over time.

The dynamics of the economy are governed by the following 2-dimensional system with one initial condition on $\hat{b}_{0}$ :

(37) \begin{eqnarray}\overset{\cdot }{\hat{m}} &=&\mu -\pi , \end{eqnarray}

(38) \begin{eqnarray}\overset{\cdot }{\hat{b}} &=&\exp [\hat{h}-\hat{b}]-\exp [\hat{c}-\hat{b}]-\pi ^{\ast }\exp [\hat{f}-\hat{b}]+\rho -\frac{f}{b}\frac{d\hat{f}}{d\hat{m}}\overset{\cdot }{\hat{m}}, \end{eqnarray}

where $\pi =\pi (\hat{m})$ , $\hat{c}=\hat{c}(\hat{m})$ , and $\hat{f}=\hat{f}(\hat{m})$ , with the partial derivatives $\frac{d\pi }{d\hat{m}}=\frac{\left(\pi +\rho \right) \left( \theta -1\right) \sigma }{\sigma \left( 1-\alpha\right) +\left( 1-\theta \right) \alpha \Sigma }<0$ , $\frac{d\hat{c}}{d\hat{m}}=\frac{\alpha \left( \Sigma -\sigma \right) \left( 1-\theta \right) }{\sigma \left( 1-\alpha \right) +\left( 1-\theta \right) \alpha \Sigma }$ , and $\frac{d\hat{f}}{d\hat{m}}=\frac{\alpha \left[ \left( 1-\theta \right)\Sigma -\sigma \right] }{\sigma \left( 1-\alpha \right) +\left( 1-\theta\right) \alpha \Sigma }$ . For the same reason as in the case of the stock of real foreign currency, since the foreign price level $P^{\ast }$ is constrained to adjust continuously, the stock of real net foreign debt, $b=B/P^{\ast }$ , is a nonjump predetermined variable with an initial condition $\hat{b}_{0}>0$ [see Turnovsky (Reference Turnovsky1997), among many others].

It is straightforward to show that the steady-state expressions of $\hat{m}^{s}$ , $\hat{y}^{s}$ , $\hat{c}^{s}$ , $\pi ^{s}$ , $\hat{\lambda}^{s}$ , and $\alpha ^{s}$ are the same as those given by (50)–(55), which are all functions of $\hat{f}^{s}$ . However, the model’s equilibrium conditions are unable to pin down the steady-state level of net foreign debt. By imposing $\overset{\cdot }{\hat{m}}=\overset{\cdot }{\hat{b}}=0$ on (37)–(38), I obtain that $\hat{f}^{s}$ is the solution to the following quadratic equation:

(39) \begin{equation}\hat{f}^{s}=F(\hat{f}^{s})\equiv \frac{\bar{r}^{\ast }\log (\hat{b}^{s})}{\Pi +\pi ^{\ast }}+\Omega (\hat{f}^{s})^{-\frac{\sigma }{\gamma }},\end{equation}

where $\Omega \,{\equiv}\, \frac{1}{\Pi +\pi ^{\ast }}\big\{ \frac{a}{\Pi^{\Sigma }}\big[ \delta \big( \frac{\Xi \bar{R}^{\ast }}{\mu +\rho }\big) ^{\theta }\,{+}\,1\,{-}\,\delta \big] ^{\frac{\Sigma -\sigma }{\theta }}\big\} ^{\frac{1}{\gamma }}\,{>}\,0$ , $F(\infty)\,{=}\,\frac{\bar{r}^{\ast }\log (\hat{b}^{s})}{\Pi +\pi ^{\ast }}\,{>}\,0$ , $F^{\prime }=-\frac{\sigma }{\gamma }\Omega (\hat{f}^{s})^{-\frac{\sigma }{\gamma }-1}<0$ , $F^{\prime \prime }=\frac{\sigma }{\gamma }\left( \frac{\sigma }{\gamma }+1\right) \Omega (\hat{f}^{s}){}^{-\frac{\sigma }{\gamma }-2}>0$ , and $F(0)\rightarrow \infty$ . Hence, under any arbitrarily specified value for the initial level of the stock of net foreign debt $\hat{b}^{s}=\hat{b}_{0}$ , Figure 4 illustrates that the equilibrium $\hat{f}^{s}$ is uniquely determined and located from the intersection of $F(\hat{f}^{s})$ and the 45-degree line. Such a property of multiplicity of steady states resulting from multiplicity of the equilibrium stock of net foreign debt is another widely known feature of SOE real business cycle models with a perfect world capital market [Lubik (Reference Lubik2007)].

Figure 4. Existence of a Steady State: SOE with Foreign Debt.

In terms of a steady state’s local stability properties, since the dynamical system (37)–(38) contains only one initial condition on $\hat{b}_{0}$ , a unique rational expectations equilibrium requires one root with a negative real part. More than one root with a negative real part implies equilibrium indeterminacy and sunspot equilibria, and all roots with positive real parts indicate nonexistence of a rational expectations equilibrium. I obtain from (37)–(38) that the two eigenvalues are $-\frac{d\pi }{d\hat{m}}>0$ and $\rho >0$ , respectively. Hence, the following proposition is established.

Note, however, that the above proposition implies any trajectory that diverges away from the completely unstable steady state may settle down to a limit cycle or to some more complicated attracting sets.

4.2 Imperfect world capital market

The preceding subsection’s analysis shows that the model under a perfect world capital market suffers from problems of a zero eigenvalue and multiplicity of steady states. The literature has proposed various solutions for avoiding the zero-root problem. ^{Footnote 14} In contrast with the proceeding subsection’s analysis, I adopt here the approach of a debt-elastic interest rate risk premium for net foreign debt. As a result, domestic households of the SOE are now postulated to be able to borrow or lend in an imperfect world capital market at an interest rate $R^{\ast }$ that increases with the aggregate stock of net foreign debt [see, Turnovsky (Reference Turnovsky1997, p. 43), AgÉnor (Reference AgÉnor1997), Schmitt-GrohÉ and Uribe (Reference Schmitt-GrohÉ and Uribe2003), Lubik (Reference Lubik2007), and Huang et al. (Reference Huang, Meng and Xue2018), among others], that is,

(40) \begin{equation}R^{\ast }=\bar{R}^{\ast }+p\big(\bar{b}\big) ,\text{ \ \ }p^{\prime}\left( \cdot \right) >0, \end{equation}

where $\bar{R}^{\ast }$ is the same fixed world risk-free nominal interest rate shown previously, and $p\big(\bar{b}\big) $ is a country-specific interest rate risk premium that is taken as given by domestic households. ^{Footnote 15} Since agents are assumed to be identical, in equilibrium aggregate per capita stock of net foreign debt equals individual stock of net foreign debt; that is, $\bar{b}=b$ .

Under (40), the model’s equilibrium conditions can be collapsed into the following autonomous dynamical system:

(41) \begin{eqnarray}\overset{\cdot }{\hat{m}} &=&\mu -\pi , \end{eqnarray}

(42) \begin{eqnarray}\overset{\cdot }{\hat{b}} &=&\frac{\exp [\hat{h}-\hat{b}]-\exp [\hat{c}-\hat{b}]-\pi ^{\ast }\exp [\hat{f}-\hat{b}]+\rho -\frac{f}{b}\frac{\partial \hat{f}}{\partial \hat{m}}\overset{\cdot }{\hat{m}}-\left( 1+\frac{f}{b}\frac{\partial \hat{f}}{\partial \hat{\lambda}}\right) \overset{\cdot }{\hat{\lambda}}}{1+\frac{f}{b}\frac{\partial \hat{f}}{\partial \hat{b}}}, \end{eqnarray}

(43) \begin{eqnarray}\overset{\cdot }{\hat{\lambda}} &=&\rho -\bar{r}^{\ast }-p(b) .\end{eqnarray}

Here, $\pi =\pi (\hat{m},\hat{b},\hat{\lambda})$ , $\hat{h}=\hat{h}(\hat{\lambda})$ , $\hat{c}=\hat{c}(\hat{m},\hat{b},\hat{\lambda})$ , and $\hat{f}=\hat{f}(\hat{m},\hat{b},\hat{\lambda})$ , with the partial derivatives provided in the online appendix.

It is clear from (43) that, because of the debt-elastic interest rate, the shadow value of real financial assets ( $\hat{\lambda}$ ) now depends on an endogenous variable b. The zero root problem is thereby resolved. In addition, in the steady state where $\overset{\cdot }{\hat{\lambda}}=0$ , equation (43) with $p\left( \cdot \right) $ strictly increasing in b implies the existence of a unique steady-state level of net foreign debt:

(44) \begin{equation}b^{s}=p^{-1}\left( \rho -\bar{r}^{\ast }\right) . \end{equation}

With this $b^{s}$ , $F(\infty)\equiv\frac{\bar{r}^{\ast }\log (\hat{b}^{s})}{\Pi +\pi ^{\ast }}$ shown in Figure 4 is uniquely determined. As a result, the steady-state value of $\hat{f}^{s}$ determined by the intersection of $F(\hat{f}^{s})$ and the 45-degree line exists and is unique. The remaining endogenous variables $\{\hat{m}^{s},\hat{y}^{s},\hat{c}^{s},\pi ^{s},\hat{\lambda}^{s},\alpha ^{s}\}$ are then uniquely determined according to (50)–(55). Hence, a debt-elastic interest rate also resolves the problem of steady-state multiplicity.

In terms of the steady state’s local dynamics, I analytically compute the Jacobian matrix of the dynamical system (41)–(43) evaluated at $(\hat{m}^{s},\hat{b}^{s},\hat{\lambda}^{s})$ and find that it is hard to derive the analytical condition for (in)determinacy. ^{Footnote 16} Hence, I resort to a numerical method to show how the minimum value of the annual foreign inflation rate $\pi ^{\ast }$ needed for equilibrium indeterminacy depends on the degree of currency substitution ( $\theta $ ), the degree of openness in terms of diversified currency holdings ( $\delta $ ), and the elasticity of the risk premium with respect to net foreign debt ( $\varepsilon $ ).

I first follow Nason and Rogers (Reference Nason and Rogers2006) in postulating a linear interest rate risk premium function: $p\left( b\right) =\varepsilon b$ ; this is also the risk premium function adopted by Airaudo (Reference Airaudo2014) when carrying out the quantitative analysis. ^{Footnote 17} A larger coefficient $\varepsilon =p^{\prime }\left( \cdot\right) >0$ in the risk premium function indicates how more sensitively the risk premium responds to a deterioration in the borrower’s credit quality (i.e. an increase in b). It follows that the steady-state level of net foreign debt is: $b^{s}=\left( \rho -\bar{r}^{\ast }\right) /\varepsilon $ . Hence, an increase in the risk-free real interest rate $\bar{r}^{\ast }$ by raising the borrowing cost reduces the stock of net foreign debt. A higher elasticity of the risk premium with respect to net foreign debt ( $\varepsilon $ ) also reduces the amount borrowed, since it represents a stronger punishment to borrowers’ credit quality deteriorations.

I first present in Figure 5 the benchmark case where $\varepsilon $ takes on the value of $0.1$ [Airaudo (Reference Airaudo2014)]. In addition to the baseline calibration of $\rho =0.01$ , $\gamma =2$ , $\mu =0.04/4$ , and $a=0.5$ , the world risk-free nominal interest rate $\bar{R}^{\ast }$ is set equal to $0.04/4$ [Schmitt-GrohÉ and Uribe (Reference Schmitt-GrohÉ and Uribe2003); Huang et al. (Reference Huang, Meng and Xue2018)]. This value of $\bar{R}^{\ast }$ further restricts the foreign inflation rate to be lower than $0.04/4$ , so as to ensure a nonnegative risk-free real interest rate, $\bar{r}^{\ast }=\bar{R}^{\ast }-\pi ^{\ast }$ . The calibrated values of $\varepsilon $ , $\rho $ , and $\bar{R}^{\ast }$ , along with the highest possible value of $\pi ^{\ast }=0.04/4$ , imply an upper bound for the net foreign debt, denoted as $\bar{b}=10\%$ . As in Figure 3, Figure 5 contains two rows ( $\sigma =10$ and 8). However, since under the baseline calibration indeterminacy will not occur when $\delta <0.51$ , the three columns of Figure 4, respectively, present the cases where $\delta =0.8$ , $0.7$ , and $0.6$ .

Figure 5. Requisite $\pi^*$ for Indeterminacy: Foreign Debt under $\varepsilon=0.1\;(\bar{b}=10\%).$

As in Figure 3 where international borrowing and lending are absent, Figure 5 depicts that the requisite level of $\pi ^{\ast }$ for indeterminacy: (i) rises when the degree of currency substitution $\theta $ falls; (ii) rises when $\delta $ falls (meaning that the degree of openness in terms of diversified currency holdings increases); and (iii) slightly increases when the value of $\sigma $ declines from 10 to 8. In these cases, belief-driven cyclical fluctuations become more difficult to occur.

Figure 5 clearly reveals that, when the degree of currency substitution $\theta $ is sufficiently high, indeterminacy can occur under very low rates of foreign inflation. A comparison between Figures 3 and 5 thus highlights that, in line with the finding of Chen (Reference Chen2018), domestic agents’ access to an imperfect world capital market to borrow/lend internationally may increase the possibility of equilibrium indeterminacy and sunspot equilibria. The benchmark results shown in Figure 4 indicate that equilibrium indeterminacy is impossible when $\theta =0.75$ and when the share of domestic currency in total liquidity $\delta \leq 0.8$ .

I next present in Figure 6 cases where the elasticity of the risk premium with respect to foreign debt ( $\varepsilon $ ) takes on lower values, implying higher levels of the upper bound for the net foreign debt $\bar{b}$ . The top and bottom panels, respectively, illustrate the cases where $\varepsilon =0.02$ and $0.01$ , and their corresponding $\bar{b}$ are 50% and 100%. It is evident that as $\varepsilon $ declines and $\bar{b}$ rises, the minimum level of foreign inflation for belief-driven fluctuations to occur falls. This implies that setting a sufficiently low upper bound for the stock of net foreign debt $\bar{b}$ (or a sufficiently high $\varepsilon $ ) can help stabilize the domestic economy by mitigating cyclical fluctuations caused by agents’ animal spirits.

Figure 6. Requisite $\pi^*$ for Indeterminacy: Foreign Debt under $\sigma=8$ . Top Panel: $\sigma=0.02\;(\bar{b}=50\%)$ ; Bottom Panel: $\sigma=0.01\;(\bar{b}=100\%)$ .

The above finding on the stabilizing role of an increase in $\varepsilon $ is consistent with that obtained in Chen (Reference Chen2018). As I highlight in the top panel of Figure 6 where $\varepsilon $ reduces to $0.02$ and $\bar{b}$ increases to 50%, a value of $\theta =0.75$ allows for the occurrence of equilibrium indeterminacy only if the degree of openness in terms of diversifies currency holdings is low ( $\delta =0.8$ ) and if the annual foreign inflation rate is above $1.9\%$ . Likewise, the bottom panel depicts that, when $\varepsilon $ further reduces to $0.01$ and $\bar{b}$ increases to 100%, equilibrium indeterminacy occurs under $\theta =0.75$ when $\delta =0.8$ and $\pi ^{\ast }>1.23\%$ .

Finally, when $\delta \rightarrow 1$ such that domestic agents in the SOE do not hold the foreign currency, determinacy is impossible in the calibrated version of the SOE for all nonnegative values of the foreign inflation rate. Figure 5 illustrates that, as is true in Figure 5, when the degree of openness in terms of diversified currency holdings increases ( $\delta $ falls), it requires a higher foreign inflation rate for equilibrium indeterminacy to occur. This subsection’s analysis thus demonstrates that Section 3’s result, in which currency substitution operates as an automatic stabilizer that alleviates aggregate fluctuations in a SOE, is robust to the inclusion of an imperfect world capital market where domestic agents finance a shortage of funds or lend surplus funds.