The Federal Reserve Board eagle logo links to home page

PPP Rules, Macroeconomic (In)stability and Learning

Luis-Felipe Zanna*

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.

Abstract:

Governments in emerging economies have pursued real exchange rate targeting through Purchasing Power Parity (PPP) rules that link the nominal depreciation rate to either the deviation of the real exchange rate from its long run level or to the difference between the domestic and the foreign CPI-inflation rates. In this paper we disentangle the conditions under which these rules may lead to endogenous fluctuations due to self-fulfilling expectations in a small open economy that faces nominal rigidities. We find that besides the specification of the rule, structural parameters such as the share of traded goods (that measures the degree of openness of the economy) and the degrees of imperfect competition and price stickiness in the non-traded sector play a crucial role in the determinacy of equilibrium. To evaluate the relevance of the real (in)determinacy results we pursue a learnability (E-stability) analysis for the aforementioned PPP rules. We show that for rules that guarantee a unique equilibrium, the fundamental solution that represents this equilibrium is learnable in the E-stability sense. Similarly we show that for PPP rules that open the possibility of sunspot equilibria, a common factor representation that describes these equilibria is also E-stable. In this sense sunspot equilibria and therefore aggregate instability are more likely to occur due to PPP rules than previously recognized.

Keywords: Small Open Economy, PPP rules, Multiple Equilibria, Sunspot Equilibria, Indeterminacy, Expectational Stability and Learning

JEL Classification: C62, D83, E32, F41

*Email address: luis-felipe.zanna@frb.gov. This paper is based on a chapter of my Ph.D. dissertation at the University of Pennsylvania. I am grateful to Martín Uribe, Stephanie Schmitt-Grohé and Frank Schorfheide for comments on early versions of this paper. I also thank David Bowman, George Evans and Dale Henderson for comments and suggestions. All errors remain mine. The views in this paper are solely the responsibility of the author and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. Return to text

1   Introduction

It has been claimed that the real exchange rate is perhaps the most popular real target in developing economies. The reason is that policy makers in these economies are always concerned about avoiding losses in competitiveness in foreign markets, or similarly, about maintaining purchasing power parity (PPP). In order to achieve the real exchange target policy makers often follow PPP rules. Such rules link the nominal rate of devaluation of the domestic currency to the deviation of the real exchange rate from its long run level or to the difference between the domestic inflation rate and the foreign inflation rate. For instance, Calvo et al. (1995) argued that Brazil, Chile and Colombia followed such rules in the past.

The characterization of the channels through which real exchange rate targeting affects the business cycles in emerging economies is a central issue in the design and implementation of the PPP rules. The theoretical literature about PPP rules has tried to disentangle these channels.1 One of these important attempts is made by Uribe (2003) who analyzes a PPP rule whereby the government increases the devaluation rate when the real exchange rate is below its steady-state level. He pursues a determinacy of equilibrium analysis and argues that PPP rules may lead to aggregate instability in the economy by inducing endogenous fluctuations due to self-fulfilling expectations.

From the economic policy-design perspective, this result has important implications. It states that the aforementioned rules may open the possibility of sunspot equilibria and lead the economy to equilibria with undesirable properties such as a large degree of volatility. This implication in turn suggests that a determinacy of equilibrium analysis can be used to differentiate among rules favoring those that at least avoid sunspot equilibria by guaranteeing a unique equilibrium with a lower degree of volatility.2 Although appealing this argument is still far from complete and may suffer from some drawbacks. The reason is that in the typical determinacy of equilibrium analysis, it is implicitly assumed that agents can coordinate their actions and learn the equilibria (unique or multiple) induced by the rule. But relaxing this assumption may have interesting consequences for the design of PPP rules. On one hand, if agents cannot learn the unique equilibrium targeted by the rule then the economy may end up diverging from this equilibrium. But if this is the case then it is clear that there are some rules that although guaranteeing a unique equilibrium, do not insure that the economy will reach it.3 On the other hand, if agents cannot learn sunspot equilibria then one may doubt about the relevance of characterizing rules that lead to multiple equilibria as ``bad'' ones. After all, if agents cannot learn sunspot equilibria then they are less likely to occur.

Therefore, it seems clear that a determinacy of equilibrium analysis of PPP rules should in principle be accompanied by a learnability of equilibrium analysis. Both analyses would help policy makers to distinguish and design PPP rules satisfying two requirements: uniqueness and learnability of the equilibrium. The first requirement would prevent the economy from achieving sunspot equilibria with undesirable properties such as a large degree of volatility. Whereas, the second requirement would guarantee that agents can indeed coordinate their actions on the equilibrium the policy makers are targeting.

The present paper is motivated by the interest of studying if particular representations of the equilibria (unique or multiple) induced by PPP rules are learnable in the Expectational - Stability (E-Stability) sense proposed by Evans and Honkapohja (1999, 2001).4 5 In fact our purpose in the present paper is three-fold. First we study and disentangle the structural conditions of an open economy under which an Uribe-type PPP rule may generate multiple equilibria (real indeterminacy).6 We use a small open economy model with traded and non-traded goods. We assume flexible prices for the former and sticky prices for the latter. Under this set-up we show how the aforementioned conditions depend not only on the responsiveness of the rule to the real exchange rate but also on some important structural parameters of the economy. For instance we find that ceteris paribus, given the sensitivity of the rule to the real exchange rate, the lower the degree of openness of the economy (the lower the share of traded goods), the more likely that the rule will induce aggregate instability in the economy by generating multiple equilibria. In addition, keeping the rest constant, the lower (the higher) the degree of price stickiness (the degree of monopolistic competition) in the non-traded sector, the more likely that the rule will lead to real indeterminacy.

The second goal of this paper consists of showing that under real determinacy the fundamental solution that describes the unique equilibrium induced by the PPP rule is learnable in the E-stability sense.7 In addition we use the recent work by Evans and McGough (2003) to prove that under real indeterminacy some common factor representations of stationary sunspot equilibria are also E-stable.8 This result suggests that under some reasonable assumptions agents can learn and coordinate their actions to achieve sunspot equilibria, making them ``more likely'' to occur under PPP rules. In this sense these equilibria should not be perceived as mere mathematical and theoretical curiosities.

The natural question that arises from these results is whether under a different timing of the PPP rule, it is possible for policy makers to design a simple rule that avoids sunspot equilibria but still induces a unique equilibrium whose characterization is learnable. In accord with the findings in the interest rate rule literature, we find that a PPP rule that is backward-looking in the sense of being defined in terms of the (deviation of the) past real exchange rate (from its long run level) satisfies these two requirements.

Finally the third goal of this paper is associated with the original work by Dornbusch (1980, 1982) that studies how a PPP rule whereby the nominal exchange rate is linked to the (deviation of the) current domestic price level (from its long-run level), may affect the output price-level stability trade-off by playing a role as an absorber of fundamental shocks.9 We analyze a rule motivated by Dorbunsch's works assuming that the nominal devaluation rate is positively linked to the difference between the domestic and foreign CPI-inflation rates. In fact this specification tries to capture the previously mentioned stylized facts about PPP rules in Brazil, Colombia and Chile. As before we state the conditions under which this rule leads to real indeterminacy. We also show that the common factor representation of stationary sunspot equilibria as well as the fundamental solution that describes the unique equilibrium induced by the rule are learnable in the E-stability sense.

The remainder of this paper is organized as follows. Section 2 presents the set-up of a sticky-price model with its main assumptions. Section 3 pursues the determinacy of equilibrium analysis for a PPP rule defined in terms of the current real exchange rate. Section 4 deals with the learnability analysis for the aforementioned rule. Section 5 pursues all the previous analyses for a PPP rule defined in terms of the CPI-inflation rate. Finally Section 6 concludes.

2   A Sticky-Price Model

2.1   The Household-Firm Unit

Consider a small open economy inhabited by a large number of identical household-firm units indexed by $ j\in\lbrack0,1]$. The household-firms live infinitely and the preferences of the representative agent $ j$ can be described by the intertemporal utility function10

$\displaystyle E_{0}\underset{t=0}{\overset{\infty}{\sum}}\beta^{t}\left[ A(c_{t... ...2}\left( \frac{P_{t}^{N}(j)}{P_{t-1}^{N}(j)}-1-\bar{\pi}^{N}\right) ^{2}\right]$ (1)

$\displaystyle A(c_{t}^{T}(j),c_{t}^{N}(j),h_{t}^{T}(j),h_{t}^{N}(j))=\alpha\log... ...j))+(1-\alpha)\log(c_{t}^{N}(j))+\psi\left[ 1-h_{t}^{T}(j)-h_{t} ^{N}(j)\right]$ (2)

where $ \alpha,\ \beta\in(0,1),$ $ \psi$, $ \phi>0;$ $ c_{t}^{T}(j)$ and $ c_{t}^{N}(j)$ denote the consumption of traded and non-traded goods respectively, $ h_{t}^{T}(j)$ and $ \ h_{t}^{N}(j)$ are the labor allocated to the production of the traded good and the non-traded good. $ E_{t}$ denotes the expectational operator.11 Equations (1) and (2) imply that the representative agent derives utility from consuming traded and non-traded goods, and from not working in either sector.

We assume that the non-traded good is a composite good. We introduce monopolistic competition in the model by assuming that the household-firm unit $ j$ can choose the price of the non-traded good it supplies, $ P_{t}^{N}(j)$, subject to a particular demand constraint described by

$\displaystyle y_{t}^{N}(j)\geq c_{t}^{N}\left( \frac{P_{t}^{N}(j)}{P_{t}^{N}}\right) ^{-\mu}$ (3)

where $ c_{t}^{N}=\left[ \int_{0}^{1}c_{t}^{N}(j)dj\right] $ represents the aggregate demand for the non-traded good and $ \mu>1.$

We also assume that there are sticky prices in the production of the non-traded good. This assumption is useful to understand the last term of the intertemporal utility function (1). Following Rotemberg (1982) we suppose that the household-unit dislikes having its price of non-traded goods grow at a rate different from $ \bar{\pi}^{N}$, the steady-state level of the non-traded goods inflation rate.12 We introduce sluggish adjustment in prices not only because this will enrich our analysis, but also because, as Uribe (2003) points out, one of the main reasons that explains and motivates the real exchange targeting through PPP rules, is that policy makers believe in eliminating the real rigidities imposed by a fixed exchange rate system in an environment with nominal rigidities.

The production of traded and non-traded goods only requires labor and uses the following technologies

$\displaystyle y_{t}^{T}(j)=z_{t}^{T}\left( h_{t}^{T}(j)\right) ^{\theta_{T}}\qquad and\qquad y_{t}^{N}(j)=z_{t}^{N}\left( h_{t}^{N}(j)\right) ^{\theta_{N} }$ (4)

where $ \theta_{T},\theta_{N}\in(0,1),$ $ z_{t}^{T}$ and $ z_{t}^{N}$ are random productivity parameters that satisfy
$\displaystyle \hat{z}_{t}^{T}=\varrho^{T}\hat{z}_{t-1}^{T}+\zeta_{t}^{T}\qquad and\qquad \hat{z}_{t}^{N}=\varrho^{N}\hat{z}_{t-1}^{N}+\zeta_{t}^{N}$ (5)

where $ \hat{z}_{t}^{k}=\log(z_{t}^{k}),$ $ \zeta_{t}^{k}\backsim N(0,\sigma _{\zeta}^{2})$ and $ \varrho^{k}\in(0,1)$ for $ k=T,N.$ For simplicity in the analysis we assume no correlation between the productivity shocks.

The law of one price holds for the traded good and to simplify the analysis we normalize the foreign price of the traded good to one. Therefore, the domestic currency price of traded goods ($ P_{t}^{T}$) is equal to the nominal exchange rate ( $ \mathcal{E}_{t}$). This simplification in tandem with (1) and (2) can be used to derive the consumer price index (CPI)

$\displaystyle p_{t}=\frac{\left( \mathcal{E}_{t}\right) ^{\alpha}\left( P_{t}^{N}\right) ^{1-\alpha}}{\alpha^{\alpha}(1-\alpha)^{1-\alpha}}$ (6)

Using equation (6) and defining the nominal devaluation rate as
$\displaystyle \epsilon_{t}=\mathcal{E}_{t}/\mathcal{E}_{t-1}-1$ (7)

it is straightforward to derive the CPI-inflation rate, $ \pi_{t}$, as a weighted average of the nominal depreciation rate, $ \epsilon_{t},$ and the inflation of the non-traded goods, $ \pi_{t}^{N}=P_{t}^{N}/P_{t-1}^{N}-1$; that is
$\displaystyle 1+\pi_{t}=\left( 1+\epsilon_{t}\right) ^{\alpha}(1+\pi_{t}^{N})^{(1-\alpha )}$ (8)

We define the real exchange rate ($ e_{t}$) as the ratio between the price of traded goods and the aggregate price of non-traded goods

$\displaystyle e_{t}=\mathcal{E}_{t}/P_{t}^{N}$ (9)

From this definition of the real exchange rate we deduce that

$\displaystyle e_{t}=e_{t-1}\left( \frac{1+\epsilon_{t}}{1+\pi_{t}^{N}}\right)$ (10)

We assume that in each period $ t\geq0$ the representative agent can purchase two types of financial assets: fiat money $ M_{t}^{d}(j)$ and nominal state contingent claims, $ D_{t+1}(j),$ that pay one unit of currency in a specified state of period $ t+1$ and that are traded internationally. There exists a set of these state contingent claims that completely spans the fundamental (intrinsic) uncertainty associated with productivity shocks. Moreover following Kimbrough (1986), we suppose that money reduces the transaction costs in goods markets. These costs measured in terms of the traded good can be described by

$\displaystyle g_{t}(j)=A\left( c_{t}^{T}(j)+\frac{c_{t}^{N}(j)}{e_{t}}\right) ^{1+\gamma }\left( \frac{M_{t}^{d}(j)}{\mathcal{E}_{t}}\right) ^{-\gamma}$ (11)

where $ A,\gamma>0$.

Using the previous assumptions the representative agent's flow constraint each period can be written as13

$\displaystyle M_{t}^{d}(j)+E_{t}Q_{t,t+1}D_{t+1}(j)\leq W_{t}(j)+\mathcal{E}_{t... ..._{t}-\mathcal{E}_{t} c_{t}^{T}(j)-P_{t}^{N}c_{t}^{N}(j)-\mathcal{E}_{t}g_{t}(j)$ (12)

where $ Q_{t,t+1}$ refers to the period-$ t$ price of a claim to one unit of currency delivered in a particular state of period $ t+1$ divided by the probability of occurrence of that state and conditional of information available in period $ t$. Hence $ E_{t}Q_{t,t+1}D_{t+1}(j)$ denotes the cost of all contingent claims bought at the beginning of period $ t$. Constraint (12) says that the total end-of-period nominal value of the financial assets can be worth no more than the value of the financial wealth brought into the period, $ W_{t}$, plus non-financial income during the period net of the value of taxes, $ \mathcal{E}_{t}\tau_{t\text{ }}$, the value of consumption spending and the value of the liquidity transaction costs.

To derive the period-by-period budget constraint of the representative agent, it is important to notice that total beginning-of-period wealth in the following period is given by

$\displaystyle W_{t+1}(j)=M_{t}^{d}(j)+D_{t+1}(j)$ (13)

and that $ E_{t}Q_{t,t+1}$ corresponds to the price at period $ t$ of a claim that pays one unit of currency in every state in period $ t+1$ and represents the inverse of the risk-free gross nominal interest rate, $ 1+i_{t}$; that is
$\displaystyle E_{t}Q_{t,t+1}=\frac{1}{1+i_{t}}$ (14)

Then we can use equations (12), (13) and (14) to derive the budget constraint of the representative agent as
$\displaystyle E_{t}Q_{t,t+1}W_{t+1}(j)$ $\displaystyle \leq W_{t}(j)+\mathcal{E}_{t}y_{t}^{T} (j)+P_{t}^{N}(j)y_{t}^{N}(j)-\mathcal{E}_{t}\tau_{t}-\frac{i_{t}}{1+i_{t} }M_{t}^{d}(j)$ (15)
  $\displaystyle -\mathcal{E}_{t}c_{t}^{T}(j)-P_{t}^{N}c_{t}^{N}(j)-\mathcal{E}_{t} g_{t}(j)$    

The agent is also subject to a Non-Ponzi game condition described by

$\displaystyle \underset{s\rightarrow\infty}{\lim}E_{t}q_{t+s}W_{t+s}(j)\geq0$ (16)

at all dates and under all contingencies, where $ q_{t}$ represents the period-zero price of one unit of currency to be delivered in a particular state of period $ t$ divided by the probability of occurrence of that state, given information available at time $ 0.$ It is given by
$\displaystyle q_{t}=Q_{1}Q_{2}.....Q_{t}$ (17)

with $ q_{0}\equiv1$.

Under this sticky-price set-up the problem of the representative agent is reduced to choose the sequences { $ c_{t}^{T}(j),c_{t}^{N}(j),h_{t} ^{T}(j),h_{t}^{N}(j),M_{t}^{d}(j),W_{t+1}(j),P_{t}^{N}(j)\}_{t=0}^{\infty} $ in order to maximize (1) subject to (2), (3), (4), (11), (15) and (16), and given $ W_{0}(j),$ and $ \bar{\pi}^{N}$ and the time paths for $ i_{t} $, $ \mathcal{E}_{t}$, $ P_{t}^{N},$ $ c_{t}^{N},$ $ Q_{t+1}$, $ \tau_{t}$ and $ z_{t}^{T}$ and $ z_{t}^{N}.$ Note that the utility function specified in (1) and (2) implies that the preferences of the agent display non-sasiation. This means that constraints (15) and (16) both hold with equality.

The first order conditions correspond to (15) and (16) both with equality and

$\displaystyle \frac{\alpha}{c_{t}^{T}(j)}=\lambda_{t}(j)\left[ 1+\Gamma\left( \frac{i_{t} }{1+i_{t}}\right) ^{\frac{\gamma}{1+\gamma}}\right]$ (18)

$\displaystyle \frac{\alpha c_{t}^{N}(j)}{(1-\alpha)c_{t}^{T}(j)}=e_{t}$ (19)

$\displaystyle \left( \frac{P_{t}^{N}(j)}{P_{t}^{N}}-\frac{\delta_{t}(j)e_{t}}{\... ...a_{N}-1)}}{e_{t}}=z_{t}^{T}\theta_{T}\left[ h_{t}^{T}(j)\right] ^{\theta_{T}-1}$ (20)

$\displaystyle \frac{M_{t}^{d}(j)}{\mathcal{E}_{t}}=\left[ (A\gamma)\left( 1+\fr... ...ht] ^{\frac{1}{1+\gamma}}\left( c_{t}^{T}(j)+\frac {c_{t}^{N}(j)}{e_{t}}\right)$ (21)

$\displaystyle \frac{\lambda_{t}(j)}{\mathcal{E}_{t}}Q_{t,t+1}=\frac{\lambda_{t+1} (j)}{\mathcal{E}_{t+1}}\beta$ (22)

$\displaystyle \beta\phi E_{t}\left\{ \left[ \frac{P_{t+1}^{N}(j)}{P_{t}^{N}(j)}... ...\pi}^{N}\right] \frac{P_{t+1}^{N}(j)}{\left[ P_{t}^{N}(j)\right] ^{2} }\right\}$ $\displaystyle =\phi\left[ \frac{P_{t}^{N}(j)}{P_{t-1}^{N}(j)}-1-\bar{\pi} ^{N}\right] \frac{1}{P_{t-1}^{N}(j)}+$ (23)
  $\displaystyle \frac{\lambda_{t}(j)z_{t}^{N}\left[ h_{t}^{N}(j)\right] ^{\theta_... ...u c_{t}^{N}}{P_{t}^{N}}\left[ \frac{P_{t}^{N}(j)}{P_{t}^{N}}\right] ^{-(1+\mu)}$    

where $ \lambda_{t}(j)/\mathcal{E}_{t}$ corresponds to the multiplier of the budget constraint, $ \delta_{t}(j)$ is the multiplier associated with the demand constraint (3) and $ \Gamma=A(1+\gamma)(A\gamma )^{^{-\frac{\gamma}{1+\gamma}}}$. The interpretation of the first order conditions is straightforward. In particular, equation (18) is the usual intertemporal envelope condition that makes the marginal utility of consumption of traded goods equal to the marginal utility of wealth ( $ \lambda_{t}(j)$) multiplied by the intertemporal price of consuming traded goods.14Condition (19) implies that the marginal rate of substitution between traded and non-traded goods must be equal to the real exchange rate. In addition, condition (20) equalizes the marginal revenue products of labor among sectors. Equation (21) represents the demand for real balances of money as an increasing function of consumption expenditure and a decreasing function of the risk-free nominal interest rate. And finally condition (22) implies a standard pricing equation for one-step-ahead nominal contingent claims. Note that in each period $ t$ there is one condition of this type for each possible state in period $ t+1.$

Finally we postpone the explanation of condition (23). The reason is that it will be used to derive the augmented Phillips curve for non-traded goods, that is actually one of the relevant equations for the determinacy and learnability of equilibrium analyses.

2.2   The Government

The government issues two nominal liabilities: money, $ M_{t}^{s}$, and state contingent debt $ D_{t+1}^{s}.$ It also levies taxes, $ \tau_{t},$ pays interest on its debt, and receives revenues from seigniorage$ .$ Thus we can write the government budget constraint as

$\displaystyle E_{t}(Q_{t,t+1}W_{t+1}^{s})=W_{t}^{s}-\frac{i_{t}M_{t}^{s}}{1+i_{t} }-\mathcal{E}_{t}\tau_{t}$ (24)

where $ W_{t+1}^{s}=M_{t}^{s}+D_{t+1}^{s}.$ The government follows a Ricardian fiscal policy. That is, it picks the path of taxes, $ \tau_{t},$ satisfying the intertemporal version of (24) in conjunction with the transversality condition
$\displaystyle \underset{k\rightarrow\infty}{\lim}E_{t}(q_{t+k}W_{t+k}^{s})=0$ (25)

Finally we define the monetary policy as in Uribe (2003). The government follows a PPP rule whereby the government sets the nominal devaluation rate as a function of the deviation of the current real exchange rate ($ e_{t}$) with respect to its long-run level ($ \bar{e}$). That is

$\displaystyle \epsilon_{t}=\rho(e_{t}-\bar{e})\qquad with\qquad\qquad\rho_{e}=\frac{d\rho }{de_{t}}<0$ (26)

where $ \rho(.)$ is a continuous function that in steady state satisfies $ \bar{\epsilon}=\rho(0)$.

2.3   The Equilibrium

We will focus on a symmetric equilibrium in which all the household-firm units choose the same price for the good they produce. Therefore in equilibrium all agents are identical and we can drop the index $ j$. In equilibrium the money market and the non-traded goods market clear. Thus

$\displaystyle M_{t}^{d}=M_{t}^{s}$ (27)

and $ y_{t}^{N}=\left( h_{t}^{N}\right) ^{\theta_{N}}=c_{t}^{N}.$ As usual we ignore the wealth effects due to inflation by assuming that the transaction liquidity costs, $ g_{t}$, are rebated to the representative agent in a lump-sum fashion.

We also assume free capital mobility. This implies that the following non-arbitrage condition must hold

$\displaystyle Q_{t,t+1}^{\ast}=Q_{t,t+1}\frac{\mathcal{E}_{t+1}}{\mathcal{E}_{t} }$ (28)

where $ Q_{t,t+1}^{\ast}$ refers to the period-$ t$ foreign currency price of a claim to one unit of foreign currency delivered in a particular state of period $ t+1$ divided by the probability of occurrence of that state and conditional of information available in period $ t$. An equivalent condition to (22) holds for the foreign economy (rest of the world). That is,
$\displaystyle \frac{\lambda_{t}^{\ast}}{P_{t}^{T\ast}}Q_{t,t+1}^{\ast}=\frac{\lambda _{t+1}^{\ast}}{P_{t+1}^{T\ast}}\beta^{\ast}$ (29)

where $ \lambda_{t}^{\ast},$ $ P_{t}^{T\ast}$ and $ \beta^{\ast}$ represent the marginal utility of wealth, the price of traded goods and the subjective discount rate in the foreign economy respectively. Using (22), (28), (29), the law of one price for traded goods and the assumption that $ \beta^{\ast}=\beta$ we can derive that $ \frac{\lambda_{t+1} }{\lambda_{t}}=\frac{\lambda_{t+1}^{\ast}}{\lambda_{t}^{\ast}},$ that holds at all dates and under all contingencies.15 This equation implies that the domestic marginal utility of wealth is proportional to its foreign counterpart. Then $ \lambda_{t}=\xi\lambda_{t}^{\ast}$ where $ \xi$ refers to a constant parameter that determines the wealth difference across countries. Since we are dealing with a small open economy, $ \lambda _{t}^{\ast}$ can be taken as an exogenous variable. To simplify the analysis we assume that $ \lambda _{t}^{\ast}$ is constant and equal to $ \lambda^{\ast}$. This assumption implies that $ \lambda_{t}$ becomes a constant. Consequently
$\displaystyle \lambda_{t}=\lambda=\xi\lambda^{\ast}$ (30)

But this result of a constant marginal utility and conditions (14 ) and (22) imply that

$\displaystyle 1+i_{t}=\beta^{-1}\left[ E_{t}\left( \frac{1}{1+\epsilon_{t+1}}\right) \right] ^{-1}$ (31)

where $ E$ denotes the expectation operator. Note that condition (31) is very similar to the uncovered interest parity condition.

Utilizing (20), (23), $ \pi_{t}^{N}=P_{t}^{N}/P_{t-1} ^{N}-1,$ the symmetry in equilibrium, the equilibrium condition in the non-traded sector ( $ y_{t}^{N}=\left( h_{t}^{N}\right) ^{\theta_{N}} =c_{t}^{N}$) and (30), we obtain

$\displaystyle E_{t}\left[ (\pi_{t+1}^{N}-\bar{\pi}^{N})(1+\pi_{t+1}^{N})\right]... ...eta\theta_{N}}\left( \frac{c_{t}^{N}}{z_{t}^{N}}\right) ^{\frac{1}{\theta_{N}}}$ (32)

that corresponds to the augmented Phillips curve for the non-traded goods inflation.16

Furthermore applying the symmetry in equilibrium and recalling (30), we can rewrite (18), (19) and (21) as

$\displaystyle \frac{\alpha}{c_{t}^{T}}=\lambda\left[ 1+\Gamma\left( \frac{i_{t}}{1+i_{t} }\right) ^{\frac{\gamma}{1+\gamma}}\right]$ (33)

$\displaystyle \frac{\alpha c_{t}^{N}}{(1-\alpha)c_{t}^{T}}=e_{t}$ (34)

$\displaystyle \frac{M_{t}^{d}}{\mathcal{E}_{t}}=\left[ (A\gamma)\left( 1+\frac{... ...) \right] ^{\frac{1}{1+\gamma}}\left( c_{t}^{T}+\frac{c_{t}^{N} }{e_{t}}\right)$ (35)

We proceed giving the definition of a symmetric equilibrium for a government that pursues a Ricardian fiscal policy and follows a PPP rule that responds to the current real exchange rate as described by (26).

Definition 1 Given, $ W_{0},$ $ \bar{\epsilon},$ $ \bar{\pi}^{N}$ and $ e_{0}$ and the exogenous stochastic processes $ \left\{ z_{t}^{N},z_{t}^{T}\right\} _{t=0}^{\infty},$ a Symmetric Equilibrium under a Ricardian fiscal policy is defined as a set of stochastic processes $ \{c_{t}^{T},$ $ c_{t}^{N},$ $ M_{t},$ $ \tau_{t},$ $ e_{t},$ $ Q_{t+1},\ q_{t},$ $ \mathcal{E}_{t},$ $ \epsilon_{t},$ $ \pi_{t}^{N} ,\ i_{t}\}_{t=0}^{\infty}$ satisfying conditions (32), (33), (34), (35), the intertemporal version of (24) together with (25), the PPP rule defined by (26), the money market clearing condition (27), definitions (7), (17) and equations (10), (14), and (31).

3   The Determinacy of Equilibrium Analysis

To pursue the determinacy of equilibrium analysis we reduce the model further. To do so we can use conditions (33) and (34) to obtain

$\displaystyle \frac{1-\alpha}{c_{t}^{N}}=\frac{\lambda}{e_{t}}\left[ 1+\Gamma\left( \frac{i_{t}}{1+i_{t}}\right) ^{\frac{\gamma}{1+\gamma}}\right]$ (36)

that together with the PPP rule (26) and equations (5), (10), (31) and (32), are the only equations necessary to pursue the determinacy of equilibrium analysis in our model. They help us to find the stochastic processes $ \{c_{t}^{N},$ $ e_{t},$ $ \epsilon_{t},$ $ \pi_{t}^{N},\ i_{t}\}_{t=0}^{\infty}.$ These set of equations are also useful to define the non-stochastic steady state. It corresponds to

$\displaystyle \bar{\pi}^{N}=\bar{\epsilon}\qquad\qquad\left( 1+\bar{\imath}\rig... ...rac{\bar{\imath}}{1+\bar{\imath} }\right) ^{\frac{\gamma}{1+\gamma}}\right] >0 $
where $ \left( \bar{c}^{N}\right) ^{\frac{1}{\theta_{N}}}=\frac {(1-\alpha)(\mu-1)\the... ...\frac{\bar {\imath}}{1+\bar{\imath}}\right) ^{\frac{\gamma}{1+\gamma}}\right] }$ and $ \bar{\epsilon}$ denotes the long-run nominal devaluation rate that is determined by the government.

We point out that we do not need to consider in the determinacy analysis equations (24) and (25). The reason is that under a Ricardian fiscal policy, the intertemporal version of the government's budget constraint in conjunction with its transversality condition will be always satisfied. Moreover the stochastic processes $ \{c_{t}^{T},$ $ M_{t},$ $ Q_{t+1},\ q_{t},$ $ \mathcal{E}_{t}\}_{t=0}^{\infty}$ can be recovered using (7), (14), (17), (27), (34) and (35).17

We can go further reducing and log-linearizing the model. Using equations (5), (10), (26), (31), (32), and (36) yields18

$\displaystyle E_{t}\hat{\pi}_{t+1}^{N}=\beta^{-1}\hat{\pi}_{t}^{N}-K\hat{e}_{t}+HE_{t} \hat{\epsilon}_{t+1}+K\hat{z}_{t}^{N}$ (37)

$\displaystyle \hat{e}_{t}=\hat{e}_{t-1}+\frac{\bar{\epsilon}}{1+\bar{\epsilon}}\left( \hat{\epsilon}_{t}-\hat{\pi}_{t}^{N}\right)$ (38)

$\displaystyle \hat{z}_{t}^{N}=\varrho^{N}\hat{z}_{t-1}^{N}+\zeta_{t}^{N}$ (39)

$\displaystyle \hat{\epsilon}_{t}=\frac{\rho_{e}\bar{e}}{\bar{\epsilon}}\hat{e} _{t}$ (40)

where $ \hat{x}_{t}=\log(x_{t})-\log(\bar{x})$, $ \bar{x}$ represents the steady-state level of $ x_{t}$,
$\displaystyle K=\frac{\mu\psi\left( \bar{c}^{N}\right) ^{\frac{1}{\theta_{N}}}}... ...\phi\bar{\epsilon}(1+\bar{\epsilon})}>0,\qquad\qquad H=(1-\theta_{N})\Omega K>0$ (41)

and $ \Omega=\left[ \frac{\bar{\epsilon}\Gamma\gamma\mu\psi}{\bar{\imath }(1+\bar{\e... ...) ^{\frac{\gamma }{1+\gamma}}\left( \bar{c}^{N}\right) ^{\frac{1}{\theta_{N}}}.$ For our future analyses it is important to observe that $ K>0$ and $ H>0.$ To see this recall our assumptions about the values that are feasible to assign to the structural parameters of the model and the definition of the steady-state.

As we mentioned before in this analysis we only study the possibilities of real indeterminacy or real determinacy of the equilibrium. By real indeterminacy we mean a situation in which the behavior of one or more (real) variables of the model are not pinned down by the model. This situation implies that there are multiple equilibria and opens the possibility of the existence of sunspot equilibria.

Before we analyze the conditions under which PPP rules may lead to real indeterminacy, it is worth constructing some intuition using the model of why these rules may induce equilibria in which expectations are self-fulfilled. In order to accomplish this task we can assume perfect foresight (no uncertainty). Then we rewrite equations (37) and (38) as

$\displaystyle \hat{\pi}_{t+1}^{N}=\beta^{-1}\hat{\pi}_{t}^{N}-K\hat{e}_{t}+H\hat{\epsilon }_{t+1}$ (42)

$\displaystyle \hat{e}_{t}=\hat{e}_{t-1}+\frac{\bar{\epsilon}}{1+\bar{\epsilon}}\left( \hat{\epsilon}_{t}-\hat{\pi}_{t}^{N}\right)$
In addition we can iterate forward both equations and derive19
$\displaystyle \hat{\pi}_{t}^{N}=\sum\limits_{k=0}^{\infty}\beta^{k}\left[ K\hat{e} _{t+k}-H\hat{\epsilon}_{t+1+k}\right]$ (43)

$\displaystyle \hat{e}_{t+1}-\hat{e}_{t}=\frac{\bar{\epsilon}}{1+\bar{\epsilon}}\left( \hat{\epsilon}_{t+1}-\hat{\pi}_{t+1}^{N}\right)$ (44)

Equation (43) implies that current inflation of non-traded goods is determined by the discounted sum of the expected future real exchange rates and nominal depreciation rates. The first term inside of the parenthesis is associated with future real exchange rates. It captures the fact that higher expected future real exchange rates make non-traded goods become relatively cheaper than traded goods. This leads to a higher expected future excesses of demand for non-traded goods to which the firm-unit responds raising the current price of non-traded goods up and therefore increasing the current non-traded goods inflation rate. On the other hand, the second term in (43) that is associated with future nominal depreciation rates captures the effect of the intertemporal price of consumption on the determination of the current non-traded goods inflation. In essence, expectations of nominal appreciation (negative nominal depreciation rates) decrease the nominal interest rate provided that the uncovered interest parity condition holds under perfect foresight. But a decrease in the nominal interest rate pushes the liquidity transaction costs down, which in turn expands consumption of non-traded (and traded) goods. This increase in consumption lead to a positive excess of demand for non-traded goods and therefore to a higher current inflation.

Equation (44) simply describes the depreciation (or appreciation) of the real exchange rate as a difference between the nominal depreciation rate and the non-traded goods inflation rate.

With these two last equations, equation (42) and the PPP rule, $ \hat{\epsilon}_{t}=\frac{\rho_{e}\bar{e}}{\bar{\epsilon}}\hat{e}_{t},$ we can show that multiple equilibria are possible by constructing a self-fulfilling equilibrium. Assume that at time $ t-1,$ when the economy is in its steady state, private agents expect a real appreciation after time $ t$. More explicitly they expect the following path for the (deviation of the) real exchange rate.20 At time $ t-1,$ the real exchange rate is at the steady state level ( $ \hat{e}_{t-1}=0$). At times $ t$ and $ t+1$ the real exchange rate is above its steady state level ( $ \hat{e}_{t}>0$ and $ \hat{e}_{t+1}>0$) and satisfies $ \hat{e}_{t}>\hat{e}_{t+1},$ showing a real appreciation and convergence to the steady state level over time$ .$ Given this path of the real exchange rate and given the PPP rule, the government will induce a nominal appreciation ( $ \hat{\epsilon}_{t+k}=\frac{\rho_{e}\bar{e}}{\bar{\epsilon}}\hat{e}_{t+k}$ with $ \rho_{e}<0$ and $ k\geq0$) over time after time $ t$. Then using our interpretation of (43) we can infer that the expected path for the real exchange rate and the expected nominal appreciation will motivate the household-firm unit to raise the price of non traded goods in period $ t$. In other words inflation at time $ t$ will go up ( $ \hat{\pi}_{t}^{N}>0$). By equation (42) this effect will increase inflation of non traded goods in period $ t+1$ ( $ \hat{\pi}_{t+1}^{N}>0$), if it is strong enough to overcome the opposite effects that the assumed path for the real exchange rate ( $ \hat{e}_{t}>0$) and the rule-induced path for the nominal depreciation rate ( $ \hat{\epsilon}_{t+1}<0$) have over inflation of non-traded goods at period $ t+1$ ( $ \hat{\pi}_{t+1}^{N}$). Observe that this possibility is determined by the values of the structural parameters of the model that affect $ H$ and $ K$, and by the value of the nominal depreciation response coefficient to the real exchange rate ($ \rho_{e}$). But if inflation of non-traded goods goes up ( $ \hat{\pi}_{t+1}^{N}>0$) and people expect a nominal appreciation ( $ \hat{\epsilon}_{t+1}<0$) in period $ t+1$ accordingly with equation (44), we conclude that the real exchange rate will appreciate over time (( $ \hat{e}_{t+1}-\hat{e}_{t})<0$). Since all the variables of the system, including the real exchange rate, converge to their steady state level over time then the original expectations of a future real appreciation will be self-fulfilled.

Although this intuitive argument points out the possibility of self-fulfilling equilibria induced by a PPP rule, it is important to disentangle the conditions under which these equilibria are possible. The following proposition achieves this goal characterizing locally the equilibrium for the model described by equations (37)-(40).

Proposition 1   Suppose the government follows a PPP rule that is described by $ \epsilon _{t}=\rho(e_{t}-\bar{e})$ with $ \rho_{e}=\frac{d\rho}{de_{t}}<0.$ Let $ K$ and $ H$ be defined as in (41) and define
$\displaystyle \tilde{\rho}_{e}=\frac{\left[ 2(1+\bar{\epsilon})\left( 1+\frac{1... ...{\epsilon}\right] }{\left[ \bar{e}\left( 1+\frac{1}{\beta }-H\right) \right] } $
a) If $ 1+\frac{1}{\beta}<H$ and $ \rho_{e} <\tilde{\rho}_{e}$ then there is real indeterminacy.
b) If $ 1+\frac{1}{\beta}<H$ and $ \tilde{\rho} _{e}<\rho_{e}$ then there is real determinacy.
c) If $ H<1+\frac{1}{\beta}$ then there is real determinacy for any $ \rho_{e}<0$.
Proof. See Appendix. $ \blacksquare$

From Proposition 1 it is clear that conditions under which PPP rules lead to multiple equilibria do not simply depend on the response coefficient $ \rho _{e}.$ On the contrary some of the structural parameters of the model play a fundamental role in the determinacy of equilibrium. In essence all the parameters that affect $ H$ and $ K$ are relevant for the analysis. In order for the PPP rule to induce multiple equilibria two conditions must be satisfied. The first one constrains the possible values that the structural parameters may take ( $ 1+\frac{1}{\beta}<H$); the second one points out the importance of the PPP rule on inducing real indeterminacy. It sets a threshold for the nominal depreciation response coefficient that depends on the structural parameters of the model ( $ \rho_{e} <\tilde{\rho}_{e}$). Even more interesting is the result of part c) in the proposition. It says that for some values that the structural parameters may take and regardless of the value of the nominal devaluation response coefficient ($ \rho_{e})$, the model displays real determinacy. This result contrasts with the results of Uribe (2003) that claims that if the elasticity of the PPP rule is sufficiently large then a model with sticky-prices always displays real indeterminacy.

To understand the important role that some of the structural parameters of the model may play in the determinacy of equilibrium analysis, we study how the aforementioned threshold ( $ \tilde{\rho}_{e}$) varies with respect to some of these structural parameters. Specifically we consider the share of traded goods ($ \alpha $), the degree of monopolistic competition in the non-traded sector ($ \mu $) and the degree of price stickiness in the non-traded sector ($ \phi $).21 Note that the share of traded goods can be considered a measure of the degree of openness of the economy with $ \alpha\rightarrow0$ describing a very closed economy. The following corollary summarizes the main results.


Corollary 1 Suppose the government follows a PPP rule given by $ \epsilon _{t}=\rho(e_{t}-\bar{e})$ with $ \rho_{e}<0$. Let $ \tilde{\rho}_{e}$ be defined as in Proposition 1 and assume $ 1+\frac{1}{\beta}<H$, then a) $ \frac{\partial(\tilde{\rho }_{e}\bar{e})}{\partial\alpha}<0$; b) $ \frac{\partial(\tilde{\rho }_{e}\bar{e})}{\partial\mu}>0$ and c) $ \frac{\partial (\tilde{\rho}_{e}\bar{e})}{\partial\phi}<0$.

Proof. See Appendix. $ \blacksquare$

Using Proposition 1 and Corollary 1 we can understand the effects of varying some of the structural parameters and the semi-elasticity of the rule ( $ \rho_{e}\bar{e}),$ on the determinacy of equilibrium. In fact when $ 1+\frac{1}{\beta}<H$ we can conclude that given the semi-elasticity of the PPP rule, the less open the economy is (the lower $ \alpha $ is), the more likely that the PPP rule will induce aggregate instability in the economy by generating multiple equilibria. In addition, given the semi-elasticity of the PPP rule and keeping the rest constant, the higher the degree of monopolistic competition in the non-traded sector (the higher $ \mu $), the more likely that the rule will lead to real indeterminacy. Finally under ceteris paribus and given the semi-elasticity of the rule we find that the lower the degree of price stickiness in the non-traded sector (the lower $ \phi $), the more feasible that the rule will induce multiple equilibria.

Notwithstanding the relevance of these analytical results, it is crucial to investigate their quantitative importance. To accomplish this we rely on a specific parametrization of the model. Since this exercise is merely indicative we borrow some values of the parameters from previous studies about emerging and small open economies.22 Following Schmitt-Grohé and Uribe's (2001) study about Mexico we assign the following values to some of the relevant structural parameters of the model: $ \beta=0.98$ per quarter, $ \bar{\epsilon}=0.0157$ per quarter, $ \gamma=5.25$, $ A=0.55,$ $ \mu=10,$ and $ \theta_{N}=0.36$. We set $ \alpha=0.44,$ that corresponds roughly to the imports to GDP share in Mexico during the 90's. Finally we set $ \phi=2.80$, that corresponds to Dib's (2001) estimate of $ \phi $ for Canada in a model with only nominal rigidities.23 With these values, we will perform four exercises characterizing locally the equilibrium. We will vary the semi-elasticity of the rule, $ \rho_{e}\bar{e}$ and one and only one of the following structural and policy parameters: $ \alpha $, $ \mu,$ $ \phi $ and $ \bar{\epsilon}.$ We summarize the parametrization in the following table.

Table 1

$ \theta_{N}$ $ \phi $ $ \mu $ $ \gamma$ $ A$ $ \beta$ $ \alpha $ $ \bar{\epsilon}$
$ 0.36$ $ 2.8$ $ 10$ $ 5.25$ $ 0.55$ $ 0.98$ $ 0.44$ $ 0.0157$

The results of our exercises are presented in Figure 1, where ``I'' stands for real indeterminacy and ``D'' stands for real determinacy. As can be observed, this figure confirms the results in Proposition 1 and Corollary 1 showing how significant these results are in quantitative terms. Consider the top left panel. From this panel we can infer the following. Suppose that the government in response to a 1 per cent appreciation of the real exchange rate, devalues the nominal exchange rate by 2 percent. In other words, assume that the semi-elasticity of the PPP rule is -2. Whereas this PPP rule may induce multiple equilibria in an economy whose degree of openness is 0.2, the same rule leads to a unique equilibrium in an economy whose degree of openness is 0.6.

Similar inferences can be pursued from the top right and bottom left panels of Figure 1. That is although a rule with semi-elasticity of -2 guarantees a unique equilibrium in an economy with a degree of monopolistic competition of 5 (a degree of price stickiness of 5), the same rule induces multiple equilibria when the aforementioned degree corresponds to 15 (2).

Although it is not possible to derive an analytical result to see how varying the implied nominal depreciation target ( $ \bar{\epsilon}$) and the semi-elasticity of the rule ( $ \tilde{\rho}_{e}\bar{e}$) affects the determinacy of equilibrium, it is possible to evaluate this quantitatively as presented in the bottom right panel of Figure 1. This panel illustrates that given the semi-elasticity of the rule the lower the implied nominal depreciation target the more likely is that the PPP rule will induce real indeterminacy. In addition, it is important to observe that in all four panels of Figure 1 there are regions for which the model always displays real determinacy regardless of the semi-elasticity of the rule. This agrees with part c) of Proposition 1.

To finalize this section we want to point out that similar qualitative results to the ones presented in this section can be obtained if the PPP rule is defined in terms of the real depreciation rate. That is $ \epsilon_{t} =\rho(\Delta e_{t})$ where $ \Delta e_{t}=\frac{e_{t}}{e_{t-1}}.$24

4   The Learnability Analysis

The importance of the result from the previous section, that a PPP rule may induce aggregate instability by generating multiple equilibria in the economy, stems from the fact that such rule opens the possibility of expectations driven fluctuations in economic activity. In particular, the model may admit self-fulfilling rational expectations equilibria driven by extraneous processes known as sunspots.25

However the previous results, as the ones in Uribe (2003), do not discuss the attainability of these PPP rule induced sunspot equilibria. They do not even mention how attainable the unique equilibrium is. Strictly speaking, and regardless of real determinacy or real indeterminacy, it is not clear whether and how agents may coordinate their actions in order to achieve a particular equilibrium in the model. The purpose of this section is to address this issue. We want to study the potential of agents to learn the unique equilibrium characterized by the fundamental solution and sunspot equilibria described by a common factor solution.


Figure 1: This figure shows how the local determinacy of equilibrium varies with respect to the semi-elasticity of the rule ( $ \rho_{e}\bar{e}$), the share of traded goods ($ \alpha $), the degree of monopolistic competition in the non-traded sector ($ \mu $), the degree of price stickiness in the non-traded sector ($ \phi $) and the implied nominal depreciation rate target ( $ \bar{\epsilon}$). ``I'' stands for real indeterminacy (multiple equilibria) and ``D'' stands for real determinacy (a unique equilibrium). ``ES'' corresponds to E-Stability.

Figure 1 is described immediately below.

Description of Figure 1

Figure 1 shows the combinations of the semi-elasticity of the rule and other structural parameters of the model under which there is a unique equilibrium (real determinacy) whose Minimal State Variable (MSV) representation is learnable in the E-stability sense. These combinations are denoted by `` D-ES''. The figure also shows the combinations of these parameters under which there are multiple equilibria (real indeterminacy) and sunspot equilibria whose Common Factor (CF) representation is learnable in the E-stability sense. We denote these combinations by `` I-ES''. Figure 1 has four panels. We proceed to describe each panel. The top-left panel shows the combinations for the semi-elasticity of the rule ( $ \rho_{e}\bar{e}$) and the share of traded goods ($ \alpha$) under which there is `` D-ES'' or `` I-ES''. The panel measures $ \rho_{e}\bar{e}$ on the vertical axis with $ \rho_{e}\bar{e}\in\lbrack-8,0]$ and $ \alpha$ on the horizontal axis with $ \alpha\in(0,1).$ For $ \alpha>0.8$ and regardless of $ \rho_{e}\bar{e}$ there is always `` D-ES''. For $ \alpha<0.8$ there is a frontier of combinations of $ \rho_{e}\bar{e}$ and $ \alpha$ that separates the combinations of `` D-ES'' from `` I-ES''. This frontier is concave and has a negative slope. It crosses the $ \rho_{e}\bar{e}$-axis at -1.5 and the $ \alpha$-axis at 0.7 approximately.(`` approximately'' since $ \alpha$ cannot be zero.) To the north-east of this frontier we have the combinations for $ \rho_{e}\bar{e}$ and $ \alpha$ under which there is `` D-ES''. To the south-west of this frontier we have the combinations for $ \rho_{e}\bar{e}$ and $ \alpha$ under which there is `` I-ES''.

The top-right panel shows the combinations for the semi-elasticity of the rule ( $ \rho_{e}\bar{e}$) and the degree of monopolistic competition ($ \mu$) under which there is `` D-ES'' or `` I-ES''. This panel measures $ \rho_{e}\bar{e}$ on the vertical axis with $ \rho_{e}\bar{e}\in\lbrack-5,0]$ and $ \mu$ on the horizontal axis with $ \mu\in(1,20).$ For $ \mu<4.3$ and regardless of $ \rho_{e}\bar{e}$ there is always `` D-ES''. For $ \mu>4.3$ there is a frontier of combinations of $ \rho_{e}\bar{e}$ and $ \mu$ that separates the combinations of `` D-ES'' from `` I-ES''. This frontier is concave and has a positive slope. It crosses the $ \mu$-axis.at 6 approximately. To the north-west of this frontier we have the combinations for $ \rho_{e}\bar{e}$ and $ \mu$ under which there is `` D-ES''. To the south-east of this frontier we have the combinations for $ \rho_{e}\bar{e}$ and $ \mu$ under which there is `` I-ES''.

The bottom-left panel shows the combinations for the semi-elasticity of the rule ( $ \rho_{e}\bar{e}$) and the degree of price stickiness ($ \phi$) under which there is `` D-ES'' or `` I-ES''. This panel measures $ \rho_{e}\bar{e}$ on the vertical axis with $ \rho_{e}\bar{e}\in\lbrack-5,0]$ and $ \phi$ on the horizontal axis with For $ \phi>7.5$ and regardless of $ \rho_{e}\bar{e}$ there is always `` D-ES''. For $ \phi<7.5$ there is a frontier of combinations of $ \rho_{e}\bar{e}$ and $ \phi$ that separates the combinations o$ \phi\in(0,20).$f `` D-ES'' from `` I-ES''. This frontier is concave and has a negative slope. It crosses the $ \rho_{e}\bar{e}$-axis at -0.8 and the $ \phi$-axis.at 4.75 approximately.(`` approximately'' since $ \phi$ cannot be zero.) To the north-east of this frontier we have the combinations for $ \rho_{e}\bar{e}$ and $ \phi$ under which there is `` D-ES''. To the south-west of this frontier we have the combinations for $ \rho_{e}\bar{e}$ and $ \phi$ under which there is `` I-ES''.

The bottom-right panel shows the combinations for the semi-elasticity of the rule ( $ \rho_{e}\bar{e}$) and the depreciation target ( $ \bar{\epsilon}$) under which there is `` D-ES'' or `` I-ES''. This panel measures $ \rho_{e}\bar{e}$ on the vertical axis with $ \rho_{e}\bar{e}\in\lbrack-5,0]$ and $ \bar{\epsilon}$ on the horizontal axis with $ \bar{\epsilon}\in(0,0.1).$ For $ \bar{\epsilon}>0.041$ and regardless of $ \rho_{e}\bar{e}$ there is always `` D-ES''. For $ \bar{\epsilon}<0.041$ there is a frontier of combinations of $ \rho_{e}\bar{e}$ and $ \bar{\epsilon}$ that separates the combinations of `` D-ES'' from `` I-ES''. This frontier is concave and has a negative slope. It crosses the $ \rho_{e}\bar{e}$-axis at 0 and the $ \bar{\epsilon}$-axis at 0.022 approximately. To the north-east of this frontier we have the combinations for $ \rho_{e}\bar{e}$ and $ \bar{\epsilon}$ under which there is `` D-ES''. To the south-west of this frontier we have the combinations for $ \rho_{e}\bar{e}$ and $ \bar{\epsilon}$ under which there is `` I-ES''.



As a criterion of ``learnability'' of an equilibrium we will use the concept of ``E-stability'' proposed by Evans and Honkapohja (1999, 2001). That is, an equilibrium is ``learnable'' if it is ``E-Stable''.26 Consequently we start by assuming that agents in our model no longer are endowed with rational expectations. Instead they have adaptive rules whereby agents form expectations using recursive least squares updating and data from the system. Then we derive the conditions for expectational stability (E-stability).

In our analysis we will focus on the expectational stability concept for the following reasons. First, in models that display a unique equilibrium (real determinacy models), Marcet and Sargent (1989) and Evans and Honkapohja (1999, 2001) have shown that under some general conditions, the notional time concept of expectational stability of a rational expectation equilibrium governs the local convergence of real time adaptive learning algorithms. Specifically they have shown that under E-stability, recursive least-squares learning is locally convergent to the rational expectations equilibrium. Second, Evans and McGough (2003) have numerically argued that under some assumptions about the parameters of a linear stochastic univariate model, with a predetermined variable, the same argument applies when this model displays sunspot equilibria. Formally they have stated that under a strict subset of the structural parameter space, there exist stationary sunspot equilibria that are locally stable under least square learning provided that agents use a common factor representation for their estimated law of motion.

We adapt the methodology of Evans and Honkapohja (1999, 2001) and Evans and McGough (2003) to pursue the learnability (E-stability) analysis. Accordingly we need to define the concept of E-stability. In order to define it we give an idea of the methodology we apply for the case of real determinacy.

To grasp the methodology, it becomes useful to reduce our model to the following linear stochastic difference equations system. Use (37), (38), (39) and (40) to rewrite the model as

$\displaystyle \hat{e}_{t}=\hat{\alpha}+\hat{\beta}E_{t}\hat{e}_{t+1}+\hat{\delt... ...^{N}\qquad and\qquad\hat{z}_{t}^{N} =\varrho^{N}\hat{z}_{t-1}^{N}+\zeta_{t}^{N}$ (45)

where