MODEL

Our model derives from the open-economy sticky-wage textbook model of Walsh (2003). To it, we add uncertainty in the form of measurement errors and time variation in the elasticity parameters of aggregate supply and demand. The presence of uncertainty forces central banks to form and update beliefs about the elasticities in the model. We assume Bayesian updating of beliefs, so central banks are rational in their learning. They are, however, hampered by the measurement errors that cloud the signals in observed variables. The final component of the model is a definition of central bank objectives.

Structure of the Open Economy

The structure of the open economy is represented by AS-AD-UIP equations (1)–(5), taken directly from Section 6.3.1 in Walsh (2003). Equations (1) and (2) are open-economy aggregate supply curves, where outputs y_t and y_t^ast in the home and foreign countries are determined by the real exchange rate ρ_t (the relative price of home and foreign output expressed in terms of the home currency), unexpected inflation, and aggregate supply shocks e_t and e_t*. A real exchange rate depreciation (ρ_t↑) contracts supply because it increases the price of imported materials and raises consumer prices relative to producer prices, which as a result of wage rigidity increases the real wage in terms of producer prices. The aggregate supply shocks are uncorrelated i.i.d. Gaussian distributed with mean zero and variance σ_e²:

Aggregate demand curves (3) and (4) link demand to the real exchange rate, real interest rates r_t and r_t*, foreign output, and aggregate demand shocks u_t and u_t*. A real exchange rate depreciation (ρ_t↑) stimulates demand through improved competitiveness as domestic goods become cheaper relative to foreign goods. The presence of foreign output in the aggregate demand curves reflects the direct spillover that arises when an increase in output in one country raises demand for goods produced in the other. The aggregate demand shocks are uncorrelated i.i.d. Gaussian with mean zero and variance σ_u². Equation (5) is the uncovered interest rate parity condition. Written as r_t*−r_t=ρ_t−E_tρ_t+1 it shows that any real interest rate differential must be matched by an offsetting expected depreciation or appreciation in the real exchange rate. Real interest rates in the model are linked to nominal interest rates through the Fisher equations r_t=i_t−E_tπ_t+1 and r_t*=i_t*−E_tπ_t+1*. We assume that nominal interest rates are the instruments of monetary policy.

Uncertainty and Timing Protocol

To introduce uncertainty into the Walsh (2003) model, we allow some of the elasticity parameters to be unobserved and time-varying, while making output and inflation subject to measurement errors. Because our focus is on uncertainty and learning in the open economy, we restrict unobservability and time variation to the parameters b_1t and a_1t governing the elasticities of aggregate supply and demand with respect to the real exchange rate. The other elasticity parameters (b₂, a₂, a₃) are assumed to be time-invariant and observable. For simplicity of the learning process, the time-varying parameters are further restricted to follow a hidden two-state Markov process. In other words, (b_1t, a_1t) switches between

and

and there is time variation in the absolute and relative magnitude of real exchange rate effects on aggregate supply and demand.²

The timing of the model is shown in Figure 1. At the beginning of the period, private agents form time t−1 dated expectations of inflation in period t. The aggregate supply and demand shocks are then revealed to both central banks, after which private agents form time t dated expectations of inflation and the real exchange rate in period t+1.³

Timing protocol.

Our timing protocol retains the feature of the original textbook model that all expectations terms are zero in equilibrium. As long as monetary policy strategies do not inject systematic biases then the i.i.d. Gaussian nature of shocks is sufficient to guarantee

and

. With regard to the real exchange rate, when shocks are i.i.d. Gaussian and expectations are rational it is possible to write ρ_t=θE_tρ_t+1+v_t, where 0<θ<1 and v_t is white noise. It follows that E_tρ_t+1=0 in any no-bubbles solution.⁴

Equilibrium

A convenient representation of equilibrium can be obtained using the uncovered interest rate parity condition to substitute out for the real exchange rate in the aggregate supply and demand curves. Because all expectations terms are zero in equilibrium, the result is a structural relationship linking observed output and inflation to nominal interest rates, shocks, and measurement errors. Appendix A derives the state-space form (6) of the structural relationship, where Y_t, R_t, ξ_t and η_t are vectors of observed outputs and inflations, nominal interest rates, shocks and measurement errors respectively. The matrix A_t switches between

and

as the elasticities in the aggregate supply and demand curves switch between

and

, whereas matrix B is time-invariant:

The definition of equilibrium is completed by adding monetary policy strategies to the structural relationship (6) and the stochastic processes for A_t, ξ_t and η_t. Monetary policy strategies specify nominal interest rate choices R_t by central banks, typically as functions of beliefs, shocks and nominal interest rates set by other central banks. We specify separate monetary policy strategies with and without policy coordination in Sections 3 and 4.

Beliefs

Central banks cannot observe the time-varying elasticity parameters in A_t directly—they can only form a belief about their current values. Because the elasticities are restricted to switch according to a two-state Markov process, beliefs can be represented parsimoniously by a single variable,

; the belief at time t that A_t is currently equal to

. If μ_t=1 the home central bank is certain that

. Conversely, if μ_t=0 there is certainty that

. The beliefs of the foreign central bank can similarly be summarized as

. The symmetry of information in the model means that beliefs of both central banks always coincide and

for all t.

Learning

Beliefs are not static in the model but are updated whenever new information becomes available. The central bank learning process updates beliefs according to which set of elasticity parameters is more consistent with new observations of output and inflation. Mathematically, the central bank has to infer whether

or

in the structural equilibrium relationship (6). The central bank observes (R_t, ξ_t) and is assumed to know all variances, so the question is which of predicted distributions (7) and (8) is most likely to have generated observed output and inflation Y_t. The role of policy activism in learning is reflected in the nominal interest rate choices R_t determining how easy it is for the central bank to distinguish between observations from the two predicted distributions. If policy is extremely cautious with R_t=(0 0)′ then the predicted distributions coincide exactly and it is impossible for a central bank to learn:

A simple application of Bayes rule describes the mechanics of the home central bank updating its beliefs and learning. Equation (9) shows how initial beliefs μ_t are updated to

on the basis of new observations of output and inflation. Under such Bayesian learning,

depends on the relative probability of observing Y_t under the two sets of elasticity parameters:

The updated belief

represents optimal inference on the values of the elasticity parameters at the end of period t. From this, the home central bank forms a prediction μ_t+1 of which elasticities will apply in period t + 1, taking into account that the elasticities may switch before then. In equation (10), the prediction is calculated as a function of the probability of having

at time t (and not switching from it) and the probability of having

at time t (but switching from it). The switching probabilities ω₁₁ and ω₂₁ are assumed exogenous and known by the central bank, so that

Equations (9) and (10), when combined with the predicted distributions (7) and (8) for Y_t, define a nonlinear learning process (11) for updating the beliefs of the central bank:

According to equation (11), updated beliefs are a function of current beliefs, shocks, nominal interest rates and observations of output and inflation.

represents the Bayesian operator modified to take account of Markov-switching effects. The symmetric nature of information means that beliefs of the foreign central bank,

, are updated using exactly the same Bayesian formula. In the model, central banks always learn together and beliefs are updated simultaneously and identically. With such joint learning

for all t.

Central Bank Objectives

The final component of the model is a definition of central bank objectives. We assume that central banks attempt to minimize a weighted average of output variability and inflation variability, following Svensson (1999) and Walsh (2003). Such an objective ensures there are no systematic biases to policy and all expectations terms are zero in i.i.d. equilibrium. Equations (12) and (13) show the algebraic form of the central bank loss functions, with λ measuring the relative weight of output variability and β being a discount factor between 0 and 1. The central banks have identical preferences:

Central bank loss functions (12) and (13) are used to derive optimal monetary policy strategies with and without policy coordination in Sections 3 and 4.

POLICY WITH COORDINATION

Our analysis of coordinated monetary policy strategies is based on deriving optimal policy under two alternative assumptions about how central banks take learning into account. With the passive learning policy, the central banks learn but make no conscious effort to influence the speed of learning. This forms our reference case because, although both central banks learn, neither takes into account that the degree of activism or caution in policy affects learning. In contrast, with an active learning policy the central banks do internalise the consequences of activism or caution for learning. The difference between monetary policy strategies under passive and active learning measures the degree of activism or caution induced by active learning.

Passive Learning

With the passive learning policy, central banks optimally account for current uncertainty but fail to realize that policy actions affect future uncertainty through the learning process. Learning is not internalized. Because learning is the only source of dynamics in the model, the coordinated policy problem reduces to that of minimising the sum of the one-period central bank loss functions each period, subject to equilibrium structural relationship (6), shocks

and beliefs μ_t. The loss minimization problem with policy coordination is defined in (14):

In the objective, Y_t is a vector of observations of

subject to measurement errors η_t. Λ is a 4×4 matrix of preference weights with (λ 1 λ 1) on the leading diagonal and zeros elsewhere, giving equal weight to each central bank in coordinated equilibrium. The choice variable

is a vector of nominal interest rates, the instrument of monetary policy. The loss minimization problem (14) is a simple modification of the linear-quadratic control problems studied by, for example, Sargent (1987). Standard techniques give the solution in terms of an optimal coordinated monetary policy strategy (15):

As expected in a linear-quadratic framework, optimized monetary policy strategy has nominal interest rates reacting linearly to the shocks. The extent of the reaction depends on beliefs μ_t, structural parameters

and preferences Λ. In general, passive learning policies tend to be cautious in order to avoid the increase in uncertainty that strong policy actions create. This call for caution under parameter uncertainty, most closely associated with Brainard (1967), resurfaces in our model as the reaction coefficient in (15) is less than its certainty equivalent value.⁵

Active Learning

The passive learning policy is not fully optimal because it does not internalize the costs and benefits of learning. This forms the basis for the closed-economy argument that monetary policy strategy should be more activist. Activism is beneficial because strong policy reactions create valuable information about the state of the economy and help the central banks to learn. To assess this argument in an open-economy coordinated policy context, we calculate the active learning policy followed by central banks when they take all learning costs and benefits into account. By definition, the coordinated active learning policy solves dynamic loss-minimization problem (16), where the expected net present value of losses is minimized subject to the equilibrium structural relationship, shocks, beliefs, and the learning process (11) by which central banks update their beliefs. The minimization problem is intertemporal because future beliefs depend on current actions:

The problem has a recursive nature so coordinated active learning policy must satisfy the Bellman equation (17):

It is not possible to derive a closed-form solution to this problem because of the nonlinearity in the learning process for updating beliefs. The presence of the nonlinear learning constraint also complicates any proof of existence and uniqueness of equilibrium. However, Wieland (2000a) applies results from Kiefer and Nyarko (1989) to show that a unique equilibrium policy does exist, so standard dynamic programming algorithms can be used to obtain a numerical solution to the Bellman equation and active learning policy.⁶

Numerical Example

Our baseline numerical example is designed to best highlight the incentives for activism and caution in monetary policy strategies for the open economy. To this end, we choose parameter values that emphasize the real exchange rate channel and its time variation. Table 1 shows our baseline parameterization. In words, the parameterization implies that the real exchange rate channel of monetary policy transmission switches between acting through aggregate supply and acting through aggregate demand. When elasticities (b_1t, a_1t) take the values (1, 0) a real exchange rate depreciation contracts aggregate supply but leaves aggregate demand unchanged. With the values (0, 1) the opposite is true and real exchange rate depreciations stimulate aggregate demand while having no supply-side effects. The Markov-switching process determining time variation in the elasticities is assumed to be symmetric with a 5% probability of switching each period. Elasticities that do not switch are less central to our analysis. We set a₂ to a small number so the direct effect of nominal interest rates on aggregate demand is weak and central banks are forced to exploit the real exchange rate channel to stabilize their economies. a₃ is set to zero to shut down the direct spillover of output in one country to aggregate demand in the other, as this spillover is not part of the real exchange rate channel. The elasticity of aggregate supply with respect to surprise inflation b₂ is normalized to unity to give a 45^o Phillips curve slope.

The parameters of the stochastic processes for shocks and measurement errors are chosen to maintain the essence of the model while reducing computational complexity. To keep the dimension of the state space manageable we normalize

to unity and follow Walsh (2003) by setting

to zero, implying that only aggregate supply is subject to shocks.⁷

in Table 1 is the variance of measurement errors in observed output. The final parameters concern the preferences of the central bank. We set λ large because otherwise there would be too much incentive for policy to react to what are effectively inflation shocks in our model. The discount factor β is close to one as usual.

Having parameterized our numerical example, we are able to solve for optimal monetary policy strategies that map shocks and beliefs to nominal interest rates under passive and active learning. In our case, the monetary policy strategies are a complex multidimensional mapping from

(two shocks and one belief) to

(two nominal interest rates). Our discussion therefore focuses on a representative part of the mapping, asking how nominal interest rates react to the combination of a positive aggregate supply shock in the home country and a negative aggregate supply shock in the foreign country. In the absence of any policy reaction, the shocks would cause inflation to fall below target in the home country and rise above target in the foreign country. The optimal policy response to such a situation depends on whether the real exchange rate is currently believed to act through the aggregate supply or the aggregate demand side of the economy. If the supply-side effect dominates, there is a strong incentive for the home country to reduce its nominal interest rate and the foreign country to raise its nominal interest rate. The resulting real exchange rate depreciation (as determined by the uncovered interest rate parity condition) affects aggregate supply and so suitably offsets the aggregate supply shocks in both countries. If the demand-side effect dominates, then the incentives for central banks to react are much weaker. Engineering a real exchange rate depreciation is now less desirable because it acts on aggregate demand, which can only imperfectly offset the aggregate supply shocks.

Figure 2 presents numerically solved optimal monetary policy strategies in the baseline parameterization of the model. The first panel shows the reaction of nominal interest rates to a unit positive aggregate supply shock in the home country and unit negative aggregate supply shock in the foreign country. As expected, under both passive and active learning the home central bank reduces its nominal interest rate i_t and the foreign central bank raises its nominal interest rate

. Again as expected, the reactions of nominal interest rates are stronger if the central banks believe that the real exchange rate acts more through aggregate supply than aggregate demand.⁸

Policy function and unconditional value functions withpolicy coordination.

Comparing monetary policy strategies under the different learning assumptions, it is apparent that the active learning policy involves slightly stronger policy reactions than the passive learning policy.⁹

Although the policies look only slightly different in Figure 2, the benefits to pursuing an active learning policy are significant. Small differences in policy quickly cumulate into substantial differences in the dynamic behavior of the economy. The small differences here are already sufficient to raise the correlation between beliefs and the true values of the time-varying elasticities from 0.237 to 0.260, representing a 10% improvement in the ability of central banks to track the true structure of the economy. To quantify the benefit of faster learning, the third panel of Figure 2 plots the unconditional value functions as defined by their expected values at the beginning of the period before shocks are observed.¹⁰

Sensitivity Analysis

To establish the robustness of our results, we show they are valid for a wide range of alternative parameterizations of the model. It is computationally demanding to calculate the active learning policy for each parameterization, so we take a short cut and identify only the incentives for activism in optimal monetary policy strategy, accepting our results if we can show there are incentives to increase policy activism over and above the level found in the passive learning policy. The advantage of this approach is that incentives can be identified without iterating the Bellman equation of the active learning policy. The incentives themselves depend on whether the unconditional value function is convex or concave under passive learning. If the unconditional value function is concave then central banks strictly prefer an activist policy that reduces uncertainty.¹¹

is also concave under passive learning, a sufficient condition for concavity of the unconditional value function itself.

In the remaining rows of Table 2, we allow parameters

to change whilst keeping other model parameters at their baseline values. The fact that the expected return function is always concave implies that the unconditional value function remains concave and optimal monetary policy strategy is activist in each case. We take this as strong support for the robustness of our result. The incentives for policy activism survive a relative strengthening or weakening of the real exchange rate channel as represented by changes in a₂ in the second and third rows. Similarly, changing the relative magnitude of aggregate supply and demand elasticities in the fourth and fifth rows does not destroy the incentives for activism. Finally, it is possible to remove time-variation in either elasticity in the sixth and seventh rows without overturning our result.

POLICY WITHOUT COORDINATION

It is well known that policy recommendations are often sensitive to the assumption of whether policies are coordinated or uncoordinated.¹²

Passive Learning

The loss minimization problem faced by the home passive-learning central bank is shown in (18). As in the coordinated policy case, learning is the only source of dynamics and the problem reduces to minimizing the one-period central bank loss function each period. However, compared to coordination there are two important differences. First, in the objective Λ₁ is a 4×4 matrix of zeros with (λ 1 0 0) on the leading diagonal because the home central bank only cares about output and inflation in its own country. Second, optimization is conditional on the monetary policy strategy of the foreign central bank, so the foreign nominal interest rate

is taken as given:

The constraint in the optimization problem is the same equilibrium structural relationship (6) as before, since it holds irrespective of whether monetary policy strategies are coordinated or uncoordinated. To distinguish between i_t as a choice variable and

as given, we decompose A_t in the equilibrium structural relationship into matrices A_1t and A_2t for i_t and

, respectively; Appendix A gives full details. The optimization problem again falls in the class of linear-quadratic control problems studied by Sargent (1987), so standard solution techniques produce an optimal uncoordinated monetary policy strategy (19) for the home central bank:

Optimal monetary policy strategy has the home nominal interest rate reacting linearly to shocks and the foreign nominal interest rate. The element of caution in coordinated policy is repeated here as uncertainty again dampens the reaction of the home nominal interest rate to shocks.¹³

The equilibrium under passive learning with no policy coordination can now be fully specified as the structural relationship (6), monetary policy strategies (19)–(20), and the stochastic processes for A_t, ξ_t and η_t.

Active Learning

When central banks follow active learning policies, they internalize learning in their monetary policy strategies. To analyze such policies when there is no policy coordination, we need to refine our concept of equilibrium. A natural candidate is to restrict our attention to Markov-perfect equilibria in which each central bank takes the current and future strategies of the other central bank as given. In practice, this translates into the home central bank taking the current foreign nominal interest rate as given but recognizing that future foreign nominal interest rates depend on future beliefs. The loss minimization problem of the home active-learning central bank is presented in (21):

The first two constraints are a simple restatement of the equilibrium structural relationship (6), as explained in the case of passive learning with coordination. The third constraint is the learning process by which the home central bank updates its beliefs, whereas the fourth constraint acknowledges that all learning is joint with the foreign central bank. The fifth constraint is the monetary policy strategy used by the foreign central bank to set nominal interest rates in the future. The nonlinearities inherent in central bank learning preclude any closed-form solution to the optimization problem and we again resort to numerical techniques to characterise equilibrium. In Markov-perfect equilibrium the monetary policy strategy of one central bank depends on the monetary policy strategy of the other, so we need to iterate twin Bellman equations (21)–(22) to obtain a numerical solution. We do this sequentially until convergence of active learning policies in both central banks:¹⁴

The equilibrium under active learning with no policy coordination is completely specified by the structural relationship (6), monetary policy strategies that solve Bellman equations (22)–(23), and the stochastic processes for A_t, ξ_t and η_t.

Numerical Example

The baseline numerical example of Section 3.3 is a natural reference point for analysing uncoordinated policies. We therefore retain the baseline parameter values in Table 1. To recap, the key feature of the baseline parameterization is the real exchange rate switching between having supply-side and demand-side effects. We also adopt the same focus as before by looking at the reaction of nominal interest rates to a unit positive aggregate supply shock in the home country and a unit negative aggregate supply shock in the foreign country. Figure 3 presents our numerical results.

Policy function and unconditional value functions withoutpolicy coordination.

The passive learning policy in the first panel of Figure 3 is very similar to that in Figure 2 for coordinated policies.¹⁵

The finding that cautious monetary policy strategies can be optimal runs counter to established wisdom that policy should be activist to generate information useful for learning. If anything, it appears that optimal uncoordinated monetary policy strategies are designed to avoid generating information and learning is better retarded than promoted. This is indeed the case as the correlation between beliefs and the true values of the time-varying elasticities falls from 0.236 under passive learning to 0.216 under active learning, a deterioration of 8% in the ability of central banks to track the structure of the economy. The reason why such a seemingly counterintuitive policy is optimal lies in the fact that central banks do not learn in isolation in the open economy, so any attempt at activist policy promotes the learning of both central banks. This is problematic when policies are not coordinated: an activist policy is desirable in that it helps a central bank to learn but undesirable in that it also helps the other central bank to learn. In our numerical example, the cost of allowing the other central bank to learn dominates so optimal monetary policy strategy is cautious. On balance, a central bank prefers not to learn how to manipulate the real exchange rate because in doing so the other central bank also learns and starts to manipulate the real exchange rate.¹⁶

Sensitivity Analysis

To check the robustness of our uncoordinated policy result, we subject it to the same sensitivity analysis as we did the coordinated policy result. We continue to take the short cut of identifying the incentives for increasing policy activism, as this proves particularly informative when policies are uncoordinated. To proceed, we substitute passive learning policies (19)–(20) and the equilibrium structural relationship (6) into the one period return in (12) to obtain the expected return of the home central bank

as a function of the beliefs of the home and foreign central banks. Writing the expected return function in terms of beliefs is useful because it admits a decomposition of incentives for activism according to which central bank is learning. The decomposition is based on the total second differential (24) of the expected return function of the home central bank (time subscripts on beliefs have been dropped for ease of notation):

The symmetric nature of the model means that the home and foreign central banks always learn together. We therefore evaluate the convexity or concavity of the expected return function along the locus of points where

. Equation (25) shows that the second derivative can then be decomposed into three components. The first two terms capture convexity or concavity with respect to the separate beliefs of the home and foreign central banks, whilst the third term explicitly accounts for central banks learning together:

The first row of Table 3 shows the decomposed second derivative in our baseline numerical example. The fact that

is negative implies that the expected return function is concave with respect to the beliefs of the home central bank. In this respect there is an incentive to increase the level of policy activism. However,

and

are both positive so overall the expected return function is convex and the incentives for policy activism are overturned once we factor in the learning of the foreign central bank. The decomposition confirms our explanation in Section 4.3 for why optimal uncoordinated monetary policy strategies are cautious. An activist policy is desirable in that it helps the home central bank to learn (

) but undesirable in that it also helps the foreign central bank to learn (

). Learning together is particularly problematic (

).

The remaining rows of Table 3 decompose the incentives for policy activism in alternative parameterizations of the model. In all cases, there is an incentive to increase policy activism if the home central bank considers its own learning in isolation, but once the learning of the foreign central bank is taken into account the optimal monetary policy strategy becomes cautious. We interpret this as evidence for the robustness of our result.

EQUILIBRIUM STRUCTURAL RELATIONSHIP

The structure of the open economy (1)–(5) describes equilibrium relationships linking real interest rates to the real exchange rate and output and inflation in each country. By adding Fisher equations and substituting out for the real exchange rate, we obtain an alternative equilibrium structural relationship in the form of equation (A.1), where observations Y_t of output and inflation

are a function of nominal interest rates

, shocks

and measurement errors η_t. Expectation terms drop out as they are zero in i.i.d. equilibrium with no systematic biases in policy:

The elements of matrix A_t are nonlinear functions of the elasticities (b_1t, b₂, a_1t, a₂, a₃) in the aggregate supply and demand curves. A_t switches between

and

as the time-varying elasticities switch between

and

. The nonlinear functions in A_t are given by equation (A.2):

The matrix B depends only on time-invariant elasticities (b₂, a₃) and is described by equation (A.3):

For the analysis with uncoordinated policies, it is useful to decompose the equilibrium structural relationship further to separate the roles of home and foreign nominal interest rates. This is done in equation (A.4):

Matrices A_1t and A_2t are defined in equation (A.5) and will switch between

and

as the underlying time-varying elasticities switch between

and

:

NUMERICAL SOLUTION OF ACTIVE LEARNING POLICY

The numerical solution to the coordinated active learning policy is obtained by iterating on the Bellman equation (17). This requires expressions for the expected one-period return

and continuation value E_tV_t+1 as functions of nominal interest rates R_t. The expected one-period return and continuation value can be calculated from the equilibrium structural relationship (6) and equation (B.1), respectively, where future beliefs μ_t+1 have been substituted out using the central bank learning process (11):

The expectation in equation (B.1) is formed before observing output, inflation, and next period shocks by evaluating the double integral in equation (B.2):

f(ξ_t+1) is the distribution of shocks ξ_t+1 and

is the predicted distribution of Y_t. They have independent multivariate distributions, normal and a mixture of normals respectively, as described by equations (B.3) and (B.4):

The computational algorithm starts by defining a grid of points in the state space (μ_t, ξ_t).¹⁸