Determinacy properties and conditions of equilibrium solution have been the subject of growing discussion and research in macroeconomics. Following in the footsteps of previous studies, we analyze determinacy in the baseline open-economy New Keynesian model developed by Gali and Monacelli in Rev Econ Stud 72(3):707–734 (Oxford University Press) (2005). We find that the open economy structure causes multifaceted behaviors in the system creating extra challenges for policy making. The degree of openness significantly affects determinacy properties of equilibrium under various forms and timing of monetary policy rules. Conditions for the uniqueness and local stability of equilibria are established. Determinacy diagrams are constructed to display the regions of unique and multiple equilibria. Numerical analyses are performed to confirm the theoretical results. Limit cycles and periodic behaviors are possible, but in some cases only for unrealistic parameter settings. Complex structures of open economies require rigorous policy design to achieve optimality.
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Uniqueness and multiplicity of equilibrium solutions have brought a rigorous discussion and research inspiration in the economics literature. Determinacy concerns the existence of a unique equilibrium path of a dynamic system. Equilibrium of a macroeconomic model is called determinate, as defined by McCallum (2009b), if it is locally unique and dynamically stable under relevant specifications of policy tools. Studying the determinacy issues in a wide range of models has been an important subject among macroeconomists. One of the reasons for high focus on the subject is the fact that uniqueness of the solution path plays an important role for policy makers and researchers. Determinacy is important to the monetary authority in designing policy to manage inflationary expectations and preventing self-fulfilling economic fluctuations. Indeterminacy permits existence of multiple solutions, which could be fundamental equilibria or nonfundamental sunspot equilibria, as suggested by McCallum (2003) and Bullard (2006), among other authors.
McCallum (2003) further divides the indeterminacy cases into two categories: failure of the model to determine the values of nominal variables (nominal indeterminacy) or of real variables (real indeterminacy). But Woodford (2003b) finds the distinction to be insignificant. He argues that both types of indeterminacy are quantitatively indifferent. Models with indeterminate solutions, as stressed by Bullard and Mitra (2002), are considered undesirable for macroeconomic analysis and policy design. In the presence of indeterminacy, as pointed out by McCallum (2003), Gauthier and Guesnerie (2005), Beyer and Farmer (2007) and others, non-fundamental shocks trigger extra variance and fluctuations in the economy. Then policy makers, unable to acquire full information, encounter unforeseen problems in monetary policy design. On the other hand, as Benhabib et al. (2001) point out, multiplicity of deterministic solutions might be informative about different regimes associated with certain equilibria of the model. Therefore, models with multiple equilibria can serve as a foundation for regime-switching models. Aruoba et al. (2018), for example, imposing a zero lower bound constraint on the interest rate rule, develop a New Keynesian model in which a sunspot shock moves the economy between a targeted-inflation regime and a deflation regime. Isakin and Ngo (2021) show that while the central bank pursues an active policy rule to stabilize both inflation and output, traditional New Keynesian models might result in three deterministic equilibria, associated with three stochastic regimes.
Following Clarida et al. (2000) seminal work, interest has been growing in the determinacy issue associated with certain types of monetary policy rules in the context of different model settings, with inflation being a primary concern. Studying determinacy of macroeconomic models, as suggested by Coibion and Gorodnichenko (2011), requires attention to trend inflation, and to the monetary authority’s policy responses to inflation rate, price level, and output gap or output growth. In particular, consideration must be given to forward or backward-looking approaches and complementary policy tools such as interest rate smoothing. In addition, macroeconomic models must consider sticky or flexible prices, closed or open economy, and other variables and structure potentially affecting the overall results.
Learnability and determinacy properties are often studied together in New Keynesian open economy models. Although McCallum (2003, 2007, 2009b) and Bullard (2006) argue that learnability is a more essential criterion than determinacy for a plausible rational expectations equilibrium, learnable equilibria are normally determinate, while the converse is not necessarily true.
Bullard and Mitra (2002), using the methodology of Evans and Honkapohja (1999, 2001), evaluate the Taylor-type monetary policy rules based on determinacy, expectational stability, and learnability. They find that monetary policy rules satisfying the Taylor Principle usually produce both determinate and learnable equilibria. They conclude that the determinacy settings of the equilibrium depend not only upon the monetary policy rule but also on the overall configuration of the economic structure. Llosa and Tuesta (2008) investigate under which conditions rule-based policies could generate a determinate and learnable rational expectational equilibrium in a New Keynesian open economy model. Similar to the findings by Zanna (2003), De Fiore and Liu (2005) and Bullard and Schaling (2006), Llosa and Tuesta (2008) find that the effects of trade openness depend on the elasticity of substitution between tradable domestic and imported goods. Moreover, with contemporaneous data, the Taylor principle satisfies necessary and sufficient conditions for determinate and learnable equilibria. The monetary authority could achieve unique equilibrium by targeting either the CPI or domestic inflation, while a managed exchange rate policy could ease the extent of response to inflation. With forecasted data on the other hand, openness makes it harder to achieve a determinate and learnable equilibrium. Then the monetary authority should employ an aggressive response to domestic inflation to prevent indeterminacy.
Nevertheless, substantial additional analytical and empirical research remains to be done in exploring determinacy issues of New Keynesian open economy models. Different model configurations provide valuable insights into possible indeterminacy problems. Carlstrom et al. (2006) have found that regardless of what price index the monetary authority considers, the Taylor Principle holds in a multi-sector economy in which sectors differ in price stickiness.
Taylor Principle has often been considered the right policy tool to achieve determinacy in equilibrium. But under certain circumstances, such as the presence of trend inflation, the Taylor Principle fails to guarantee determinacy. Giannoni (2014) and Ambler and Lam (2015) argue that interest rate rules that respond to fluctuations in price-level are less prone to equilibrium indeterminacy than interest rate rules, such as Taylor rules, which respond to fluctuations in inflation rate. Coibion and Gorodnichenko (2011), Huang and Thurston (2012), and Hirose et al. (2020) argue that the Taylor Principle does not guarantee determinacy of New Keynesian models, especially in the presence of positive trend inflation. Hirose et al. (2020), in their study of Great Inflation of 1970’s in the US, conclude that to achieve determinacy within a generalized New Keynesian framework, an active interest rate policy should be run along with a lower trend inflation (lower inflation target), or with diminished policy response to the output gap, or with firmer response to output growth. Kiley (2007) shows that as trend inflation increases, the determinacy region shrinks. He suggests a moderately active interest rate policy accompanied by a slightly positive response to output gap to ensure determinacy.
Gerko and Sossounov (2015) studied the effects on determinacy of positive trend inflation within a New Keynesian model with Calvo-type price setting and capital accumulation. While they verify the previous research findings on active monetary policy not necessarily guaranteeing determinacy of equilibrium, they also find that the parameters of monetary policy which ensure determinacy mainly depend upon the level of trend inflation. As the level of trend inflation increases, the regions of determinacy under Taylor–type rules shrink. While the Taylor-type rule might lead to indeterminacy within a large region of the parameter space, Gerko and Sossounov (2015) advocate employing strict price level targeting along with responses to current output gap to ensure determinacy of equilibrium. Fanelli (2012), on the other hand, argues that if the monetary authority does not react to inflation by an aggressive increase in the nominal interest rate, a unique equilibrium may not be achieved. Adding to the previous conclusions, Coibion and Gorodnichenko (2011) show that in the presence of positive trend inflation, an endogenous monetary policy based on the Taylor Principle may not be sufficient. Components of the policy, such as interest rate smoothing along with price-level targeting, can be needed to ensure determinacy.
Dupor (2001) shows that including endogenous investment in the neoclassical imperfect competition model with sticky prices reverses the effects of interest rate rules on determinacy. While passive policy rules lead to locally unique equilibria, active rules do not. Carlstrom and Fuerst (2005) explore a Calvo-type sticky price model including capital and investment spending. They find that monetary authority aggressive response to the current inflation rate is the only way to achieve local determinacy. Under forward-looking interest rate rules, the determinacy region shrinks considerably, and local indeterminacy arises. Tesfaselassie and Schaling (2016) explore Blanchard and Galí (2010) New Keynesian model incorporating labor market frictions. They find that determinacy depends not only on the policy rule but also on inflation and unemployment expectations as well as hiring costs. Under the policy rules reacting to current inflation and unemployment, the indeterminacy region of the parameter space widens together with hiring costs. Under policy rules based on inflation and unemployment expectations, the indeterminacy region shrinks with hiring costs, while too much or too little reaction may still lead to indeterminacy. Assuming that the steady state is known, lack of enough reaction to inflation and unemployment can also lead to indeterminacy.
Although it has rarely been considered in macroeconomics research, the open economy framework makes the determinacy analysis substantially more complicated. According to Clarida et al. (2001), considering the fact that open economy form is isomorphic to closed economy model in New Keynesian tradition, it is the degree of openness parameter which substantially affects the parameters of the model and the extent of interest rate response to inflation. De Fiore and Liu (2005) and Karagiannides and Liambas (2019) find that whether a policy rule leads to unique equilibria depends upon the degree of openness to trade. Karagiannides and Liambas (2019) argue that as trade openness increases, Taylor-rule based monetary policies should put more weight on output gap and less on price stability. But as openness decreases, price stability should take priority. In open economies, De Fiore and Liu (2005) explain transmission mechanisms which could lead to determinacy and transmission mechanisms that would not lead to determinacy. Unlike closed economies, in which the transmission mechanism operates through the substitution effect between consumption and leisure or savings, in open economies a rise in real interest rates affects the exchange rates and thus terms of trade between foreign and domestic goods. Improving terms of trade, depending on the level of openness, creates incentives for the household to substitute consumption for leisure or saving.
In the open economy case, as the real interest rate is increased by the monetary authority, the domestic currency appreciates. The domestic goods’ price index rises relative to the consumer price index. Domestic inflation then differs from CPI growth, producing complications for policy making. There are other studies that incorporate different parameters or variables into the standard New Keynesian model to analyze open economy structures. While Clarida et al. (2001) and Taylor (2001) argue that responding to the real interest rate turns out to be ineffective and even destabilizing in monetary policy, Guender (2005) and Froyen and Guender (2017) emphasize the critical role of the real exchange rate in conducting monetary policy. Mihailov et al. (2011) evaluate the external determinants of inflation dynamics in OECD countries and find that expected relative changes in terms of trade play a bigger role in inflation than contemporaneous domestic output gap. Rhee and Turdaliev (2012), adding a direct exchange rate channel to domestic inflation in an open economy New Keynesian model, find that CPI inflation targeting causes a lower volatility in output than domestic inflation targeting.
Our study seeks to enlighten some aspects of the determinacy problem in the New Keynesian open economy literature. For that purpose, we employ Gali and Monacelli (2005) model of a small open economy in the New Keynesian tradition. Gali and Monacelli’s (2005) model represents a small open economy as part of the world economy, which is itself a continuum of small open economies, identical in terms of preferences, technology, and Calvo-type staggered price setting. Under various policy regimes, the model seamlessly reveals the trade-offs between the stabilizations of inflation, output gap, and exchange rate. Since its publication, Gali and Monacelli (2005) model has attracted the attention of many researchers and policy makers and has become one of the most influential models in macroeconomic analysis. It has been used as a baseline model in a wide range of research and policy analysis. Footnote 1
Using Gali and Monacelli’s (2005) model, we investigated the determinacy conditions under a variety of alternative monetary policy rules. The conditions for the uniqueness and local stability of the equilibria are established for each model and are evaluated using numerical analysis. We reestablish the determinacy conditions in the open economy New Keynesian structures. Determinacy diagrams are constructed to show the regions of unique and multiple equilibria. Numerical analyses are performed to confirm the theoretical results. The numerical simulations show that limit cycles and periodic behaviors are possible, but on some occasions, only at unlikely parameter settings. We have found that in a broad class of open economy New Keynesian models, the degree of openness has a significant role in equilibrium determinacy under various forms and timing of monetary policy rules. The open economy framework creates substantial complications within the dynamic structure of the system. The resulting broad range of qualitative behaviors requires meticulous policy responses.
In this study, we use Gali and Monacelli (2005) model of a small open economy in the New Keynesian tradition. The model consists of the following three equations: the IS curve, which represents the demand side; the aggregate supply curve, often called the New Keynesian (NK) Phillips curve; and a simple (i.e., non-optimized) monetary policy rule.
The IS curve is:
$$where \(_\) is the gap between actual output and flexible-price equilibrium output, \(<\pi >_\) is the inflation rate, \(_\) is the nominal interest rate, \(\overline_>\) is the small open economy’s natural rate of interest, and β is the discount factor. Then \(_=\sigma <\left(1-\alpha +\alpha \omega \right)>^\) and \(\omega =\sigma \gamma +\left(1-\alpha \right)\left(\sigma \eta -1\right)\) are composite parameters, where \(_\) is the terms of trade, which is a function of the degree of openness \(\alpha \in [\mathrm]\) and the elasticity of substitution between domestic and foreign goods η > 0, σ is the elasticity of intertemporal substitution, while γ measures the substitutability between goods produced in different foreign countries, while \(_\) is the expectation operator. Lowercase letters denote the logs of the respective variables.
The New Keynesian Phillips curve is:
$$<\pi >_=\beta _<\pi >_+\mu \left(\frac+\varphi \right)_,$$where \(<\pi >_\equiv
_-
_\) is the CPI inflation with \(
_\equiv \mathrm_\) , μ denotes the optimal mark-up in a flexible price economy, φ denotes the elasticity of labor supply (the higher φ, the lower the elasticity), \(\mu =\frac<\left(1-\beta \theta \right)\left(1-\theta \right)>\) , and \(\omega =\sigma \gamma +\left(1-\alpha \right)\left(\sigma \eta -1\right)\) are composite parameters where θ is a measure of the degree of price rigidity a la Calvo (1983). The larger the parameter θ, the fewer the firms are able to adjust their prices each period and the longer the time period between price adjustments for the representative firm.
The monetary policy rule is:
where the coefficients \(<\phi >_>0\) and \(<\phi >_<\pi >>0\) measure the sensitivity of the nominal interest rate in the central bank’s response to changes in output gap and inflation rate, respectively. The policy rule, Eq. (3), is a version of the Taylor rule (Taylor 1993).
The first two equations are derived from the optimization of consumers and firms’ objective functions. Both equations are in log-linearized form. As for the monetary policy rule, we consider a variety of such simple policy rules. Footnote 2
The open economy is isomorphic to the closed economy version. Nevertheless, unlike the closed economy, the Gali and Monacelli (2005) model depends upon the open economy parameters, such as the degree of openness, the terms of trade, the substitutability among goods of different origin, and the world output, which is exogenously determined. Therefore, it is important to identify any influence of the open economy parameters on the determinacy of the model.
Following Barnett and Eryilmaz (2016), we consider varying the timing of the monetary policy rule to consider contemporaneous, forward and backward-looking policy rules as well as their hybrid combinations. We evaluate each model based on the determinacy criterion and establish the conditions for the determinacy of equilibria for each model. We derive analytical results and present numerical simulations. We use methodology based on the number of eigenvalues inside the unit circle, given the number of predetermined variables of the model.
By rearranging the Eqs. (1), (2), and (3), we first write the system in the form \(__=C_\) . For a two-equation first order stochastic difference equation system in terms of domestic inflation and output gap, the eigenvalues, \(<\lambda >_\) and \(<\lambda >_\) , of the Jacobian matrix C are computed by setting \(\mathrm\left(C-\lambda I\right)=0\) . This gives a second-order characteristic polynomial, \(p\left(\lambda \right)=<\lambda >^-_\lambda +_=0\) . The determinacy of the system, following Blanchard and Kahn (1980) and Gandolfo (1996), requires that both eigenvalues of the coefficient matrix C are outside the unit circle, so that the eigenvalues have modulus greater than one. This condition can be met, if and only if \(\left|_\right|<1\) and \(\left|_\right|<1+\left|_\right|\) . Then, following Bullard and Mitra’s (2002) methodology, we construct the propositions which establish the necessary and sufficient conditions for the matrix C to have both eigenvalues outside the unit circle in order to ensure determinacy of the system. Footnote 3
Following Bullard and Mitra’s (2002) approach, we use the calibrated values of the parameters as given in Gali and Monacelli (2005). Those values are \(\beta =0.99\) , \(\alpha =0.4\) , \(\sigma =\omega =1\) , \(\varphi =3\) , and \(\mu =0.086\) . For the three-equation case including policy Eq. (3), we set the policy parameters at \(<\phi >_=0.125\) , \(<\phi >_<\pi >=1.5\) , and \(<\phi >_=0.5\) . Footnote 4
Consider the model consisting of Eqs. (1), (2) and (3), in which the first two equations explain the economy, while the third equation is the monetary policy rule tracked by the central bank. In this setting, Eq. (3) describes the policy rule as a current-looking Taylor rule, in which the interest rate is determined according to the current inflation rate and the current output gap.
Rearranging the terms, the system can be written in the form \(__=C_\) as
To confirm that \(_=<\pi >_=0\) is the only solution, we need to check the determinacy properties of the system Eq. (4). Following Bullard and Mitra (2002), Proposition (1) establishes the necessary and sufficient conditions for the matrix C to have both eigenvalues outside the unit circle.
Given monetary policy based on the current-looking Taylor rule, the open economy New Keynesian model described by the system Eq. (4) has a unique stationary equilibrium, if and only if Footnote 5
