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Persistent Appreciations and Overshooting: A Normative Analysis

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Abstract

Most economies experience episodes of persistent real exchange rate appreciations, when the question arises whether there is a need for intervention to protect the export sector. This paper presents a model of irreversible destruction where exchange rate intervention may be justified if the export sector is financially constrained. However, the criterion for intervention is not whether there are bankruptcies or not, but whether these can cause a large exchange rate overshooting once the factors behind the appreciation subside. The optimal policy includes ex-ante and ex-post interventions. Ex-ante (that is, during the appreciation phase) interventions have limited effects if the financial resources in the export sector are relatively abundant. In this case the bulk of the intervention takes place ex-post, and is concentrated in the first period of the depreciation phase. In contrast, if the financial constraint in the export sector is tight, the policy is shifted toward ex-ante intervention and it is optimal to lean against the appreciation.

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Notes

  1. The figure shows real exchange rate indices normalized to 100 at date 0. The latter corresponds to June 1997 for the Asian Crisis, November 1994 for Mexico, and October 1991 for Finland. The Asian crisis real exchange rate is a simple average of the indices for Indonesia, Korea, Malaysia, Philippines, and Thailand. Data Source: IFS (Effective Real Exchange Rate) for Finland, Indonesia, Malaysia, and Philippines. Real Exchange Rate from Hausmann, Panizza, and Rigobon (2006) for Mexico, Korea, and Thailand.

  2. In practice, the drop in the relative price of nontradables and real wages often takes the form of a sharp nominal depreciation which is not matched by a rise in the nominal price of nontradables and wages. See, for example, Goldfajn and Valdes (1999) and Burnstein, Eichenbaum, and Rebelo (2005).

  3. The NBER Working Paper was circulated in 2007.

  4. See the review of Korinek (2011) in a previous issue of this journal.

  5. For the moment, we suppose that the prices {pt), qt)} are such that the entrepreneur's expected utility is finite. We will check later, case by case, that this condition is satisfied in equilibrium.

  6. Formally, by “A phase” we mean all the histories of the form θt=〈θ A ,...,θ A 〉. By “D phase” all those of the form θt=〈θ A ,...,θ A D ,...,θ D 〉.

  7. The parameters to generate this figure are β=0.97, δ=0.2, f=3, θ A =2.1. We choose k−1=k D fb and p−1=p D fb as conventional initial conditions. These initial conditions arise if the economy makes an unexpected transition from the D state to the A state in period 0. We plot an appreciation lasting five periods, since that is its expected duration when δ=0.2.

  8. As long as we have φ A >1 and so c A is set to zero (see the proof of Proposition 2).

  9. The parameters are the same as those used for Figure 2, plus a0=0.5.

  10. A “fundamental” view of constraint (equation (1)) is that it is impossible to extract payments from entrepreneurs, whether in the form of financial payments or in the form of taxes.

  11. See the Appendix for a detailed setup of the planner problem. In the Appendix we show that the second best allocation shares the following features with the competitive equilibrium: the consumption of tradables is constant and equal to c A T and c D T, respectively, in the A phase and in the D phase, and the consumption of nontradables is constant and equal to c A N in the A phase and only depends on j in the D phase. Therefore, also in the perturbation argument we focus directly on allocations with these features.

  12. Note that the consumer's labor income is pt) while its expenditure on nontradables is pt)(1−kt)). Thus, net income is pt)kt).

  13. The parameters are the same as those used for Figures 2 and 3. As in those figures, we plot the realized path when the appreciation lasts five periods. The optimal paths are computed using the characterization result in the Appendix (proof of Proposition 6).

  14. It is possible to complete this argument by deriving the optimal value of μ at the optimum. The proof of Proposition 6 in the Appendix reaches the same conclusion using a different approach. The reason for the different approach is that, given the nonconcavity of the problem, the perturbation arguments derived here only yield necessary conditions for an optimum, while the argument in the Appendix can be used to show sufficiency as well.

  15. The characterization can be easily extended to the case φ A ce>(f−1)/(βf). In that case, the optimal path may involve a real exchange rate equal to zero for the first period(s) of the D phase. The overshooting is still frontloaded to the early periods of D, but may last more than one period.

  16. See discussion on p. 15, following Proposition 3. The condition holds in all the examples presented (and in all reasonable parametrizations we have looked at).

  17. An alternative would be to look at the timing of the taxes used to implement the planner's allocation and distinguish ex-ante vs. ex-post intervention in terms of when nonzero taxes are imposed. Here we prefer to focus on the distortion of the real exchange rate as it closely matches the welfare rationale for intervention. In any event, as we shall see below, our setup also implies that a combination of ex-ante and ex-post taxes is, in general, necessary to implement the constrained efficient allocation.

  18. Parameters are the same as those used for Figure 5 except for a0=0.15 in panel (a), a0=0.5 in panel (b), a0=1.1 in panel (c).

  19. Note also that when export firms have abundant financial resources, there is a sort of Ricardian equivalence, in that any (at least small) intervention can be undone by the private sector (this is an exact result whenever φ A D,0).

  20. In an earlier draft we relaxed the complete markets assumption and studied the polar opposite case, where export firms only have access to a riskless bond. In this context, the export sector resources dwindle as the appreciation progresses. The main policy implication that follows from this modification is the timing of the exchange rate stabilization in the appreciation phase. Early on in the appreciation, the optimal policy is to postpone much of the intervention to the D phase. However, as the appreciation continues and the export sector's resources dwindle, the optimal policy shifts a larger share of the intervention to the appreciation phase (essentially, this amounts to a gradual leftward movement in Figure 8).

  21. Alternatively, we could introduce procyclical consumption (or short horizons) through nonrepresentative agents. The extreme version of this formulation is one where consumers live for only one period and must consume their income in that period. The social planner Pareto-weighs a generation t periods from the current one by βt. If no intergenerational transfer mechanism other than through the real exchange rate is available, then we are again in the situation just described. The constrained goal of the social planner is to reallocate consumption away from nontradables during the appreciation phase. Relative to the pure taste shock scenario, a larger share of the adjustment is done in the A phase, in order to reduce the burden of the adjustment on the first generation in the D phase.

  22. See Blanchard (2006b) for a more thorough discussion of wage rigidities and appreciations; and Blanchard (2006a) for an application to the case of Portugal.

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Authors

Additional information

*Ricardo J. Caballero is the Ford International Professor of Economics at the Massachusetts Institute of Technology. Guido Lorenzoni is Professor of Economics at Northwestern University. The authors are grateful to Manuel Amador, Gita Gopinath, Arvind Krishnamurthy, Pablo Kurlat, Enrique Mendoza, Klaus Schmidt-Hebbel, Carlos Végh and seminar participants at Berkeley, Harvard, IMF, MIT, IFM-NBER, WEL-MIT, the World Bank, San Francisco Federal Reserve, LACEA-Central Bank of Chile, Paris School of Economics, SED Meeting (Prague), EUI, Santa Cruz Conference on International Economics, University of Chicago for their comments, and to Nicolas Arregui, Jose Tessada, Lucia Tian, and, especially, Pablo Kurlat for excellent research assistance. Caballero thanks the NSF for financial support.

Appendix

Appendix

Proof of Proposition 1

The cutoff is given by

where p A fb,p D fb,k A fb. and k D fb are defined in the text. Let us conjecture and verify that if these prices and quantities form an equilibrium. Given the conjectured prices it is possible to show (by guessing and verifying) that V(a,kt)=a+qt)k (that is, ψ(θt)=0 and φ(θt)=1). Then, inspecting the entrepreneur’s optimality conditions (equations (9), (10) and (11)) shows that the entrepreneur is, at each θt, indifferent among all feasible choices of cT,et),kt), and If the entrepreneur begins with a0, he can consume the difference and then adopt the following rule: set kt)=k A fb, a(〈θt,D〉)=f(k D fbk A fb)−(1−p D fb)k D fb and , for each history θt={θ A , …, θ A }; set for each history θt A , …, θ A , θ D , …, θ D }. These decisions are consistent with labor market clearing. One final check, which we left aside in the main text, is that k A fb>0. This follows from substituting κfb in the definition of k A fb and using the inequalities 1−β(1−δ)<1, θ A /((1−β)θ A +δβ)<1, and 1/(1+δβf)<1.

Proof of Proposition 2

First, we establish a preliminary lemma.

Lemma A1

  • Define the function

the equation H(k)=0 has a unique solution k*∈(0,1), for each κ>0 and x>0. Moreover, H(k)>0 for each k>k*. The solution k* is increasing in x. If x=0 the equation can have one or two solutions, one of which is 0. In this case, the properties above apply to the largest solution.

Proof.

  • A solution exists because H is continuous in [0,1), H(0)=−x and limk→1H(k)=∞. Consider the case x>0. Let k* be a solution, then f−(1−κ/(1−k*))>0 must hold. If k>k*, H′(k)=f−(1−κ/(1−k))+κk/(1−k)2>0 follows from f−(1−κ/(1−k))>f−(1−κ/(1−k*))>0. This implies that H(k)>0 for each k>k*, and the solution is unique. The comparative statics result with respect to x follows from the implicit function theorem. When x=0 the solution k*=0 is trivial. If there is another solution k*>0, the properties stated can be proved following the steps of the case x>0.

The proof will proceed in three steps. First, we define a map T for the coefficient κ. Second, we derive some properties of this map. Finally, we show that this map has a unique fixed point. From this fixed point we can construct an equilibrium with the desired properties.

Step 1: Fix a value for κ∈⌊0,κfb⌋ and construct an equilibrium as follows.

Phase A. If

then set p A equal to p A fb, set k A =1−κθ A /p A fb and

Notice that k A >0. Since κ≤κfb we have 1−κθ A /p A fb≥1−κfbθ A /p A f>0, where the last inequality follows from assumption (A1).

If Equation (A.1) does not hold, then set p A equal to the solution of

(which has a unique solution in [1, p A fb]), set k A =1−κθ A /p A and aD,0=0. Notice that when p A =κθ A , the right-hand side of Equation (A.3) is zero, therefore p A ∈[κθ A ,p A fb] and k A ≥0.

Phase D. Define

Construct the sequence {k D,j } that satisfies:

until kD,J+1 is larger than From then on set

Letting x=aD,0+fk A , Lemma A1 ensures that Equation (A.4) has a solution for kD,0 (if aD,0+fk A =0, pick the solution with the largest kD,0). To show that kD,0k A consider the following: Either and the solution will be larger than . In this case the economy converges to immediately and

where the inequality in the middle follows from assumption (As3). If, instead then H(kD,0)=0. Notice that

where the inequality in the middle follows from

(the second inequality follows from Assumption (As3)). Therefore, Lemma A1 implies that kD,0>k A . In a similar way, it is possible to prove that Equation (A.5) implies k D,j kD,j−1 for each j.

From these two steps we obtain a sequence p A ,{p D,j }, which can then be substituted in expression (3), to obtain κ′. This defines a map T:⌊0,κfb⌋→[0,κfb].

Step 2: It can be shown that the map T is continuous. Furthermore, let us prove that

In the construction in Step 1, an increase in κ leads to a (weak) reduction in k A and k D,j for all j (for the initial conditions of phase D notice that if Equation (A.1) is satisfied, then, using the definition of p A fb, it is possible to show that aD,0+fk A is independent of κ; if Equation (A.1) is not satisfied, then an increase in κ leads to a decrease in k A ). But since k A =1−θ A κ/p A , k D,j =1−κ/p D,j , this implies that the prices p A and p D,j must increase less than proportionally than κ. Therefore, κ′ increases less than proportionally.

Step 3: Define the following map for z≡log(κ):

Step 2 shows that this map is continuous and has slope smaller than 1. Therefore this map has a unique fixed point (uniqueness is not needed for the statement of this proposition, but will be useful for the following results). Let κ be the fixed point and consider the prices and quantities constructed in Step 1. To ensure that they are an equilibrium, it remains to check that the sequence of prices and quantities are optimal for the entrepreneur. Derive the marginal utility of money at tD,0 from the recursion:

By construction we have p D,j p D fb, which implies that φ D,j ≥1. Moreover, entrepreneurs’ consumption and cash savings, aD,j+1, are zero until the point where φ D,j =1. To check optimality in phase A, notice that

and φ A D,0 iff p A <p A fb, and, by construction aD,0 is zero iff p A <p A fb. Notice that as long as either p A or some p D,j will be strictly below their first best value. Therefore, φ A >1.

Proof of Propositions 3 and 4

The following lemma provides a useful preliminary result.

Lemma A2

  • The equilibrium value of κ is nondecreasing in a0.

Proof.

  • Let T(κ;a0) be the mapping [0,κfb]→[0,κfb] defined in the proof of Proposition 2, indexed by the initial wealth a0. Choose two values a0′<a0″. Let κ′ and κ″ be the corresponding equilibrium values of κ. Now, fixing κ′ we want to show that T is monotone in a0, that is, T(κ′,a0″)≥T(κ′,a0′). If Equation (A.1) holds at a0′, then an increase in a0 leaves p A unchanged, and it increases aD,0 (from equation (A.2)) and leaves k A unchanged. If Equation (A.1) does not hold, an increase in aD,0 leads to an increase in p A , and an increase in aD,0+fk A , since aD,0 either remains zero or becomes positive and k A increases. In both cases, aD,0+fk A increases. This means that, in phase D, there will be a (weak) increase in k D,j for all j, and, thus, a (weak) increase in p D,j for all j. Therefore, T(κ′,a0″)≥T(κ′,a0′)=κ′. This implies that T(κ′,a0″) has a fixed point in [κ′,κfb]. Since T has a unique fixed point and T(κ″,a0″)=κ″, by construction, this implies κ″≥κ′. □

Now we can prove the two propositions. Consider first Proposition 3. Suppose that at a0 we have p A =p A fb in equilibrium. This means that Equation (A.1) holds at a0. Since κ″≥κ′, Equation (A.1) holds a fortiori for aD,0,κ″, it follows that at the new equilibrium p A =p A fb and aD,0>0.

Consider next Proposition 4. Suppose that at a0 we have pD,0=p D fb in equilibrium. This means that the following inequality holds

where . Now we want to prove the following inequality

If Equation (A.1) holds at a0 then some algebra (using the definition of p A fb) shows that

If Equation (A.1) does not hold at a0, then we have aD,0≥0=aD,0. Furthermore, we can show that k A k A . Notice that k″≥k′ holds because, on average, equilibrium prices are larger. However, p′=p D fb at a0, and p D, j ≤0=p D fb. This implies that

and since k A =1−κ/p A this implies k A k A . Therefore, Equation (A.8) holds in all cases. This implies that Equation (A.8) also holds at a0, and therefore pD,0=p D fb. Notice that if Equation (A.1) holds at a0 then, we can proceed as in the proof of Proposition 3 and show that aD,0>aD,0 and k A =k A .

Setup of the Optimal Policy Problem

The set of feasible allocations is defined as follows.

Definition A1

  • (Feasibility) The allocation {cTt),cNt),cT,et),kt)} is feasible iff there exists a sequence of tax rates {τTt),τNt)}, wealth levels {at)}, and prices {pt),qt)} such that the prices and quantities {pt),qt)} and {cTt),cNt),cT,et),kt),at)} constitute a competitive equilibrium under the tax rates {τTt),τNt)}.

We now derive three necessary conditions, (A.9)-(A.11) below, that any feasible allocation must satisfy. Then, we define the problem of a planner that chooses an allocation subject only to the consumer’s budget constraint, the entrepreneur’s budget constraint (equation (7)), and conditions (equations (A.9), (A.10) and (A.11)). This is a relaxed version of the original planning problem, given that this set of constraints is necessary but, in general, not sufficient for feasibility. We also perform a change of variables that makes the relaxed planning problem a concave problem and we derive first-order conditions which are sufficient for an optimum. In the proof of Proposition 6 we make use of these first-order conditions to find allocations that solve the relaxed planning problem.

First, notice that the entrepreneur’s optimality implies that a feasible allocation must satisfy the condition

To prove this inequality, multiply by kt) both sides of Equation (9), and use the complementarity condition to obtain

Moreover, the entrepreneur’s optimality condition for at) implies that φ(〈θtt+1〉)≥φ(θt) for all θt+1. Substituting in the equation above, gives Equation (A.9).

Second, recall that q(〈θtt+1〉)≤f which implies

Finally, define the function

for a generic variable x, and notice that equilibrium in the adjustment sector implies that, for all θt and θt+1,

We perform a change in variables, defining

for all consecutive histories θt−1 and θt.

Substituting the government budget balance in the consumer’s budget constraint (equation (21)), and using the market clearing condition cNt)=1−kt), we obtain the budget constraint

Substituting z and y and using Equation (A.11) on the right-hand side, we get

Substituting z and y in the entrepreneur’s flow of funds constraint gives

The relaxed planner problem is to choose sequences {cTt), cT,et), kt), at)} and {zt), yt)}, to maximize the consumer’s expected utility subject to Equations (22), (A.9), (A.10), (A.12), and (A.13). The relaxed problem is a concave problem, so the first-order conditions are sufficient for an optimum. It is possible to show that the solution is stationary in phase A and that, in phase D, quantities and prices only depend on the number of periods since the transition. To save space and help the interpretation, we write the problem directly in terms of variables indexed by A and (D,j). Moreover, we normalize the utility of the consumer, the budget constraint, and the entrepreneur’s participation constraint by the constant (1−β(1−δ)). Then, the planner maximizes

subject to

and conditions (equations (A.9) and (A.10)), which take the form

Next to each constraint we write the respective Lagrange multiplier.

Let us take first-order conditions with respect to k

with respect to y and z

and with respect to a and cT,e

Rearranging these conditions shows that a sufficient condition for an optimum is that there exist Lagrange multipliers λ≤ν A νD,0νD,1≤⋯≤μ, such that

and conditions (equations (A.17), (A.18), (A.19) and (A.20)) are satisfied.

Proof of Propositions 5 and 6

We prove Proposition 6 by giving a complete characterization of the optimal allocation. The proof is split in three steps. In the first step, we define two maps J and . In the second step, we use these maps to construct a candidate optimal allocation and we show that this allocation is indeed optimal. The proof of 5 is a side product of this step. In the third step, we derive the implications for the optimal path of the exchange rate and for the optimal tax in A.

Step 1: We define the two maps J and : Define the map J:[1+δβf A ce,p A fb]→ℝ as follows (notice that 1+δβf A ce<1+δβf=p A fb since φ A ce>1). For any p A ∈1+δβf A ce,p A fb] find the unique ξ that solves

where pD,0=1−f+δβ2f2/(φ A ce(p A −1)), and the sequence {k A ,{k D,j }} is given by

where

To show that such a ξ exists and is unique, notice that the right-hand side of Equation (A.24) is a continuous nonincreasing function of ξ, and ranges between a positive value, at ξ=0, and −∞ for ξ→∞. Set J(p A )=ξ (the function J is allowed to take negative values but we will see below that at the relevant values of p A , J(p A )>0). Combining the terms containing p A on the right-hand side of Equation (A.24) we obtain the expression

which is monotone decreasing in p A . Applying the implicit function theorem, it follows that J′(p A )<0.

Define the map as follows: For any k A ∈0,(1−(1−δ)β)a0/(p A fb−1)] find the unique positive ξ that solves

where pD,0=1−ff A ce, and the sequence {k D,j } is given by Equations (A.26) and (A.27). Again, it is easy to show that such a ξ exists and is unique. Set It is immediate to show that

Step 2: Define the function

From step 1, we know that L(p A ) is an increasing function of p A . Therefore, three mutually exclusive cases are possible. Either there exists a unique p A ∈1+δβf A ce,p A fb] that solves the equation L(p A )=0, or L(1+δβf A ce)>0, or L(p A fb)<0. We can construct an optimum for each of these cases. We will analyze in detail the first case, which correspond to the case depicted in Figure 5. Let p A * be such that L(p A *)=0. Set

the assumption φ A ce≤(f−1)/(βf) ensures that p*D,0≥0, given that p* A p A fb. In the case φ A ce>(f−1)/(βf) the argument needs to be amended to allow for a number of periods in which p D,j =0, this requires a slightly more involved definition of the function J, but otherwise the argument is analogous to the one for the case analyzed here. Set p* D,j =p D fb for all j≥1, q* A =0, and q* D,j =f for all j≥0. Let ξ*=J(p* A ) and set c A T=θ A ξ* and c D,j T=θ D ξ*. Set the sequence {k* A ,{k* D,j }} according to Equations (A.25) and (A.27). Finally, the values for the entrepreneur’s consumption are set as

Given the construction of the sequence {k* A ,{k* D,j }}, entrepreneur’s consumption is always nonnegative.

Having defined a candidate optimal allocation, we can define the corresponding sequences for y* A ,{y* D,j } and z* A ,{z* D,j }, and show that we have found an optimum for the relaxed problem defined in the section “Setup of the Optimal Policy Problem.”. To do so, we need to find Lagrange multipliers λ*, v* A ,{v* D,j }, and μ* such that conditions (equations (A.17), (A.18), (A.19), (A.20), (A.21), (A.22) and (A.23)) are satisfied. Set λ*=1/ξ*. Notice that the condition L(p* A )=0 can be re-arranged to give

Moreover, by construction

with equality for j greater or equal than some J*. This implies that

with equality for jJ*. Then we can set v* A =λ and

By construction v D,j * will be constant for jJ* and we can set μ*=vD,j**. This confirms that Equation (A.20) is satisfied for all j and we can check that c D,j T,e*>0 only for jJ*, that is, when v D,j *=μ*.

Furthermore, we can check that the proposed allocation satisfies the consumer’s budget constraint and the entrepreneur’s participation constraint. The consumer’s budget constraint can be rewritten as

the construction of the functions L and J (in particular equation (A.24)) guarantees that this condition holds as an equality. Some lengthy but straightforward algebra, using the flow of funds constraints, shows that

Given the prices p* A and {p* D,j }, the right-hand side of this equation is equal to φ A cea0, which is equal to the entrepreneur’s expected utility U in the competitive equilibrium, since

This completes the argument that the candidate allocation solves the relaxed planning problem. It remains to show that this allocation is feasible. To do so, we first derive values for the φ* A and φ* D,j . We set φ* A A ce,

and φ* D,j =1 for all j≥1. These can be used to check that entrepreneur’s behavior is optimal, that is, that Equations (10) and (11) are satisfied. To check these conditions notice that φ* A ≥φ*D,0≥φ*D,1 and φ* D,j =1 for all j, while c A T,e*=cD,0T,e*=0 and a* D,j =0 for all j. Finally, the tax rates are set as follows: τ A T*=τT* D,j =0 for all j and τ A N* and {τN* D,j } are such that

Let us discuss briefly the cases where L(1+δβf A ce)>0 and L(p A fb)<0. In the first case, we have that condition (equation (A.29)) now holds as an inequality, and we have v* A >λ*. The rest of the construction is analogous to the one derived above. In the second case, we make use of the function to find the value of ξ*. In particular, define the function

and find an k* A ∈[0,(1−(1−δ)β)a0/(p A fb−1)] such that (notice that and ). Then, set and λ*=1/ξ*. In this case, Equation (A.29) holds as an equality. but we have k* A <(1−(1−δ)β)a0/(p A fb−1), so the entrepreneurs have positive financial savings when they enter phase D,

This is consistent with feasibility, given that P A =p A fb, so that φ* A =φ*D,0 (which implies that equation (11) is satisfied with a*D,0>0). The rest of the proof proceeds as in the baseline case.

Step 3. That p* A p A fb and p* D,j p D fb follows immediately from the construction of the optimal allocations. We want to show that p* A p A ce. Consider first the case where p* A =1+δβf A ce. In this case, it follows from Equation (A.30) and p D,j cep D fb that (p A ce−1)/(δβf A cej=0βf/(f−(1−p D,j ce))≥1. This implies that p A cep* A . Next, consider the case where p A ≥1+δβf A ce. In this case, we have, by construction L(p* A )≤0, which implies

where the first inequality follows because it is possible to show that J(p* A )≤κfb. If p A ce=p A fb then it immediately follows that p A cep* A . Therefore, consider the case p A ce<p A fb, where k A ce=(1−(1−δ)β)a0/(p A −1) (recall the construction of the equilibrium in Proposition 2). The assumption k A cek A fb and inequality (equation (33)) imply

giving the desired inequality.

Let us derive the optimal tax τ A when p*D,0<p D fb. By construction, p*D,0<p D fb implies p* A >1+δβf A ce. Also by construction, whenever p* A >1+δβf A ce the following condition holds as an equality 1−θ A ξ*/p A fb=k* A , which, together with Equation (A.31), implies that p* A (1+τA*)=p A fb.

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Caballero, R., Lorenzoni, G. Persistent Appreciations and Overshooting: A Normative Analysis. IMF Econ Rev 62, 1–47 (2014). https://doi.org/10.1057/imfer.2014.7

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