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S0261-5606(19)30142-1 https://doi.org/10.1016/j.jimonfin.2019.102098 JIMF 102098

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Journal of International Money and Finance

Please cite this article as: J.H. Choi, Capital Controls and Foreign Exchange Market Intervention, Journal of International Money and Finance (2019), doi: https://doi.org/10.1016/j.jimonfin.2019.102098

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Capital Controls and Foreign Exchange Market Intervention Jae Hoon Choi∗ July 29, 2019

Abstract High level of capital controls and volatile exchange rates in developing countries are considered suboptimal in classical trilemma frameworks. The paper presents a New Keynesian small open economy model that assesses middle-ground policies, such as partial capital controls and managed exchange regimes. The paper introduces a policy-endogenous risk premium, which creates financial frictions; breaking exchange rate peg exposes foreign investors to exchange rate risk and even signals the country’s economic instability, and financial sector responds by raising the country’s exchange rate risk premium. Contrary to the classical theory, the results suggest that implementing both capital controls and managed exchange regimes can be optimal, while maintaining domestic monetary policy sovereignty. JEL Classification: F3, F4, E-MMC Keywords: Capital Controls, Foreign Exchange Regime, Monetary Policy, Trilemma, Exchange Rate Risks, Policy-endogenous Risk Premium

∗

Department of Economics, Xavier University - Williams College of Business, 3800 Victory Parkway, Smith Hall 325, Cincinnati, OH 45207, [email protected] I thank the jury and participants of the Central Bank of the Republic of Turkey 2017 Research Awards. I thank the seminar participants at University of California Santa Cruz, Bates College, Wesleyan University, University of San Francisco, and Xavier University. I am especially grateful to Kenneth Kletzer, Carl Walsh, and Michael Hutchison for their advice and encouragements and to Johanna Francis, Pierre-Olivier Gourinchas, Grace Weishi Gu, Ajay Shenoy, and Amit Sen for their comments and suggestions.

1

Introduction

Since the financial crises, capital has been flowing back to emerging market economies (Ostry et al., 2010). Increases in capital inflows can allow financially less-developed countries to allocate the resource more efficiently and raise growth rates by providing financing for high-return investment. They can also foster a diversification of investment risk and intertemporal trades. Excessive capital inflows, however, raise concerns over undue appreciation pressure on the currency, which can reduce the competitiveness of the emerging market countries’ export sector and raise the possibility of a sudden reversal in shortterm inflows and concomitant risks to macroeconomic and financial stability. Such risks require appropriate policy responses. Capital controls restrict volatile movements of capital inflows and outflows. However, employing capital controls is constrained by the economy’s monetary policies and foreign exchange regime choices; Mundell’s trilemma (Mundell, 1963) states that a country cannot simultaneously have free capital flows, monetary sovereignty, and fixed exchange rates as described in diagrams in Figure 1. Recent studies find that although the countries’ international policy portfolios may be constrained by trilemma, they do not always reside in the corners of trilemma. Popper et al. (2013) and Klein and Shambaugh (2015) suggest that the choices of international policy portfolio may be on the lines of the triangle between one corner and another corner, called “middle-ground.” Especially, many developing countries adopt more middle-ground policies that are away from the corners of trilemma, and therefore, the diagrams in Figure 2 potentially provide a better description of developing countries’ policy portfolio and how the policies are constrained by the plane of trilemma. A brief comparison of policy portfolio between developing and advanced countries also suggests the same idea. Figure 3 presents the levels of capital controls imposed in 72 countries (36 developed economies and 36 developing economies) from 1995 to 2013 – the capital control index from Fern´andez et al. (2015) represents the intensity of capital controls on a scale of 0 to 1. The bar graphs suggest that developing countries have managed capital flows more

1

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Figure 1: Classical representation of the open economy policy trilemma Free Capital Flow

Free Capital Flow

Fixed Exchange Rates

Fixed Exchange Rates

Independent Monetary Policy

Independent Monetary Policy

Figure 2: Potential representation of developing countries’ policy trilemma

actively than developed countries have. Figure 4 shows that while the average level of capital controls in developing countries is consistently higher than the one in advanced countries, the exchange rates in developing countries, however, are not steady, and are in fact as volatile as ones in advanced countries. This does not confirm that developing countries have freely floated their exchange rates as the developed countries may have; however, it implies that more countries in emerging markets have not fixed their currencies or have adopted managed exchange regimes, while implementing partial capital controls. The empirical studies also support this observation. Klein and Shambaugh (2008) study 3,253 exchange rate spells of 104 developing countries from 1973 to 2004 and find that 44.19% of the peg spells have been broken within 2 years, and that only 26.78% of the peg spells stay pegged for more than 5 years. Batini et al. (2006) also report that the share of developing countries adopting fixed exchange rate regimes fell from 75% in 1985 to 55% in 2005. Fiess and Shankar (2009) study fundamental pressure on exchange rate regimes in 15 countries 2

Australia Austria Bahrain Belgium Brunei Darussalam Canada Cyprus Czech Republic Denmark Finland France Germany Greece Hong Kong Iceland Ireland Israel Italy Japan Korea Kuwait Latvia Malta Netherlands Norway Portugal Qatar Singapore Slovenia Spain Sweden Switzerland United Arab Emirates United Kingdom United States Uruguay

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0

.1

.2

.3 .4 .5 .6 .7 .8 mean of CapitalControls

.9

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0

.1

.2

.3 .4 .5 .6 .7 .8 mean of CapitalControls

.9

1

Figure 3: Average levels of capital controls (range from 0 to 1, 0 being zero capital controls and 1 being full capital controls) from 1995 to 2013 (Left: Developed countries / Right: Developing countries) (Data source: Fern´ andez et al. (2015))

from 1985 to 2004 and identify that countries frequently release the pressure with low intervention on foreign exchange markets. A large literature studies the mechanics of central banks’ foreign exchange market intervention process and furthermore suggest policy guidelines under managed exchange regimes.1 There is also a growing literature that conducts model-based assessments of the welfare costs and benefits of capital controls.2 However, the attempts to study the portfolio of both international policies in a theoretical framework have been rare; one of the potential reasons for the scarcity stems from the complex nature of employing both capital flow management and foreign exchange market decision in the existing theoretical frameworks. See Schmitt-Groh´e and Uribe (2012), Farhi and Werning (2012), and Farhi and Werning (2014). 2 See Bianchi (2010), Schmitt-Groh´e and Uribe (2012), Jeanne and Korinek (2010), Benigno et al. (2012), and Kitano (2011). 1

3

.5 .4 .3 .2

Capital controls

.1 0 1995m1

1998m1

2001m1

2004m1

Advanced

2010m1

2013m1

Advanced (Euro)

.04 .03 .02 .01

FX volatility (11 mo. moving std dev)

.05

Developing

2007m1

1995m1

1998m1

2001m1

2004m1

Developing

Advanced

2007m1

2010m1

2013m1

Advanced (Euro)

Figure 4: Average level of capital controls (top) and exchange rate volatility (bottom) of developing, advanced, and Eurozone countries (1995 - 2013) (Data source: Fern´ andez et al. (2015) and University of British Columbia – The Pacific Exchange Rate Service)

4

This paper contributes to the literature by providing a theoretical representation of middle-ground policies that are empirically identified and documented in Popper et al. (2013) and Klein and Shambaugh (2015). I introduce a novel and tractable approach to consider the trade-off between capital controls and foreign exchange intervention, so we can assess the developing countries’ policy portfolio trend: partial capital controls and managed exchange regimes. The model is constructed under the assumption that rational policymakers understand the trajectories of shadow exchange rates under different regimes and decide how much to float or peg by simply making series of regime choices. Furthermore, I introduce a financial friction representing exchange rate risk premium that is endogenous to foreign exchange regime choice; this financial friction differentiates this model from conventional small open economy models, where currency price adjustment yields the first-best outcome. The model is built upon the standard New Keynesian small open economy framework that is first constructed in Gal´ı and Monacelli (2005) and Gal´ı (2009) and that is later modified in Farhi and Werning (2014) by employing incomplete market to study the optimal level of capital controls. By minimizing the welfare loss function, the balanced path of optimal policy portfolio in response to the temporary regime-elastic risk premium shock is derived. The results suggest that partial capital controls and managed exchange regimes can be optimal for the countries with financial markets that are sensitive to foreign exchange risks. Furthermore, the model reflects “fear of floating” documented in Calvo and Reinhart (2002); the additional friction has a multiplying effect and it makes exchange rate stabilization become essential to prevent a further loss in welfare. It also captures the feature of capital controls that they indirectly manage exchange rate depreciation, which allows policymakers to put less resource to stabilize currency prices. Section 2 reviews the related literature. Section 3 sets up a model; the innovations of this model are highlighted in Section 3.1. Section 4 derives the equilibrium in a log-linearized form. Section 5 analyzes the optimal policy in the small open economy under a particular parameterization. Section 6 concludes. 5

2

Literature Review

This paper is related to a large literature on the volatility of international capital flows. The seminal study of Calvo (1998) sets a theoretical framework focusing on sudden stops, large and unexpected cutbacks in capital flows to a country. Gal´ı and Monacelli (2005) and Gal´ı (2009) set a standard New Keynesian small open economy framework to study monetary policies and their welfare implication in response to a temporary productivity shock, which creates volatile international capital movements. Later, the welfare implications of taxes has been explored in a context of open economies subject to volatile capital inflows. The research points to welfare-enhancing effects of taxes on capital inflows (Korinek, 2010) or on foreign debt (Bianchi, 2011). A related strand of literature emphasizes pecuniary externalities in the borrowing constraints. Jeanne and Korinek (2010) presents a simple model where a “financial accelerator” can be managed through the optimal Pigouvian tax. Increases in borrowing push up the price, and this raises the value of the collateral against which borrowing is secured. It then becomes easier to borrow to buy assets, which pushes prices even higher. This operates in reverse during a downturn. Falling prices erode the value of collateral, tightening credit and depressing demand. When they take on debt, borrowers fail to take into account the effect of their actions on the collateral constraints faced by others. To offset this, the authors propose a counter-cyclical tax on debt. Benigno et al. (2012), on the other hand, provides a different perspective that a commitment to a price support policy in the event of crisis welfare-dominates prudential capital controls. By considering a broader set of policy instruments within the same theoretical framework (instead of assuming that a tax on debt is the only instrument), the authors show that capital are not necessary because they are the second-best instrument, while other tools can achieve the first-best allocation. The authors adopt a similar model economy as in Bianchi (2011) where a tax on borrowing is the only policy tool, and introduce two other distortionary policy instruments, a tax on nontradable consumption and a tax on tradable consumption.

6

These studies provide a rationale for controls on capital movements to prevent over-borrowing; however, because the models in these papers are in real terms, the optimal capital controls become independent of the choice of the exchange rate regimes. A few attempts have been made to assess capital flow management assuming nominal rigidity. Schmitt-Groh´e and Uribe (2012) study second-best policies in economy with downward wage rigidity and a fixed exchange rate regime and finds that optimal capital controls are prudential and achieve large reduction in unemployment and increases in welfare. The authors characterize a wedge in the model with a innocuous and rather more realistic assumption on the labor market. However, the wedge is present only with fixed exchange rate, and switching to a float regime always achieves the first best. More recently, Farhi and Werning (2012, 2014) extend the framework designed by Gal´ı and Monacelli (2005) to derive the optimal path of capital controls. The authors set an environment of imperfect risk sharing and introduce the international consumption wedge, which can be controlled by a tax on international capital flows. The results suggest that optimal level of capital controls is nonzero even under floating regime. This paper also relates to the literature on sterilized foreign exchange intervention in the presence of market friction. The earlier literature includes Weber (1986) that considers the case in which the impossible trinity is not impossible, provided that bonds denominated in different currencies are not perfectly interchangeable. The model by Evans and Lyons (2002) employs the portfolio-balance channel under a microstructure approach. The authors estimate a partial equilibrium model in which the trading process reveals information contained in order flows. Maggiori and Gabaix (2015) build an analytically tractable 2-period general equilibrium model where constrained international financiers intermediate capital flows across countries. They provide a novel micro-foundation to the portfolio balance channel and analyze the welfare effects of heterodox policies such as foreign intervention. Cavallino (2019) employs the framework developed in Maggiori and Gabaix (2015) and studies the effects of exchange rate fluctuations driven by capital flows and characterize the optimal foreign exchange intervention. 7

3

Model

The model builds on the standard new Keynesian small open economy framework first introduced in Gal´ı and Monacelli (2005) and Gal´ı (2009) and further developed in Farhi and Werning (2014). The key difference between the two models is that Gal´ı and Monacelli (2005) and Gal´ı (2009) posit complete markets for securities traded internationally, while Farhi and Werning (2014) create an environment in which international financial markets are incomplete and therefore risk-sharing between countries is limited. This imperfect risk-sharing between home and foreign households adds distortions in the Gali-Monacelli framework, and thus creates a room for interventions. Using a similar methodology, this paper introduces an additional distortion: exchange rate risk premiums. This additional distortion creates a more realistic environment where a policymaker now considers optimal combination of two policies – level of capital controls and exchange rate regimes. Given the equilibrium conditions developed in this section and the section 4, the policymaker, who is a social planner, identifies the optimal path of time-varying Pareto weight, which provides the optimal levels of capital controls and foreign exchange market intervention over time.

3.1 3.1.1

Two innovations of the model Exchange rate regime decision: Managed exchange regime

In most open economy models, characterized as trilemma, capital controls become unnecessary under floating exchange rate regimes. Therefore, the models that assess the validity of capital controls implicitly assume fixed exchange rate regimes. However, only few countries, such as Hong Kong or Iran, fix their exchange rates, and most developing countries adopt de facto managed peg or floating regimes (Klein and Shambaugh, 2008). To account for this, the paper sets a more realistic environment by allowing policymakers to change regimes by controlling the levels of intervention on exchange rates. This modification is achieved through setting the dynamics of exchange rates as the geomet-

8

ric average of the change in floating exchange rates and the change in fixed exchange rates, which can be expressed as following: Et+1 ≡ Et

A

f Et+1 Etf

Bλt A

E¯t+1 E¯t

B1−λt

,

(1)

where Et is the effective nominal exchange rate at time t, Etf is the nominal exchange rate under a floating exchange rate regime, E¯t is the nominal exchange rate under a fixed exchange rate regime, and λt ∈ [0, 1] denotes the policymaker’s foreign exchange regime choice or the level of managed float/peg. The policymaker first finds the expected growth of exchange rates when there is no intervention in the foreign exchange market (Etf ∀t). Then she finds the optimal path of exchange rates (Et ∀t) and determines the level of managed float/peg in each period (λt ∀t). Therefore, when λt is high and close to 1, the level of foreign exchange intervention is low and the dynamic of the exchange rate is closer to the one under a floating regime. On the other hand, a lower λt implies that the economy requires a higher level of foreign exchange market intervention for more stabilized exchange rates. The log-linearized form of (1) is et+1 . ∆et+1 = λt ∆eft+1 + (1 − λt )∆¯ Since ∆¯ et+1 = 0 (or E¯t+1 /E¯t = 1) under a fixed exchange rate regime, the expression is simplified to ∆et+1 = λt ∆eft+1 , which provides a clear implication of λt . For example, suppose that it is optimal for exchange rates to depreciate 3% between t and t + 1 (∆et+1 ). If a policymaker finds the exchange rates will depreciate 10% when any interventions are not implemented during the period (∆eft+1 ), her optimal choice of λt will be 0.3 to suppress its depreciation rate from 10% to 3%.3 3

To present a simple example, this example assumes that changes in λt do not affect

9

For a tractability of the model, I further assume that the country’s foreign reserve is sufficient enough to intervene in exchange markets without altering money supply, so exchange market intervention does not constrain its central bank’s other monetary policy decisions.4 3.1.2

Exchange rate risk premium

Equation (1) is a convenient way to introduce exchange rate regime selection into the model, compared to the attempts made by previous studies to introduce foreign exchange market intervention in models. However, if a policymaker controls the exchange rate regime under a standard small open economy framework setting, she will always choose a freely-floating regime. This is mainly due to the assumption made in most small open economy frameworks that the country is small enough that its policy or behavior do not alter world prices, interest rates, or incomes. Therefore, under this environment, the country’s decision to break the exchange rate peg shall induce only minimal costs within the system, and compared to the benefits from stabilization via adjustment in exchange rates, its cost is insignificant. However, such environments do not reflect one of the crucial factors of exchange rates and exchange rate regimes: exchange rate risks. Flexible exchange rates may protect real economy from external shocks, as suggested in conventional open economy models.5 On the other hand, floating exchange regimes allow volatile currency prices, and the volatility of currency prices creates potential risks of loss in cross-border transactions. Such risks bring in additional economic costs because foreign investors require extra return for the risks, called exchange rate risk premium. Additional risk premium tightens budget constraints of local borrowers and eventually lowers the welfare of economy. These costs may outweigh the benefits of floating exchange other factors in the economy. In the model, the choice of λt affects the price of foreign loans (see section 3.1.2), which changes the dynamics of optimal outcomes. 4 In Section 6, I discuss a potential extension of the model by including international reserve constraints as suggested in Aizenman (2013). 5 As seen in Gal´ı and Monacelli (2005), where autonomous monetary policy with floating exchange rates perfectly secure the economy from external shocks.

10

rates in some countries, particularly a majority of developing countries whose economies are dependent on foreign trades and investments. Therefore, the countries may prefer stabilizing exchange rates by adopting fixed (or managed peg or floating) exchange rate regimes to lower exchange rate risks and attract more foreign investments and loans, especially when they cannot borrow abroad in their own currencies. The relationship between exchange rate regime choice and exchange rate risks – often characterized as “fear of floating” – is well-documented in previous studies, including Calvo and Reinhart (2002), Hausmann et al. (2002), and Levy-Yeyati and Sturzenegger (2005). Moreover, some studies find that foreign investors may even consider switching regimes from fixed to floating as a signal of economic instability (regardless of whether or not it is true), which raises the risk premium of foreign loans, because breaking pegs can be considered as breaking “promises” of stable exchange market, and a government would not break promises unless its economy is in dire straits. Alesina and Wagner (2006) find that switching exchange rate regimes from fixed to floating brings external costs into the economy. They study the countries whose de jure and de facto exchange rate regimes are different to argue that wide exchange rate fluctuations (especially devaluations) are taken by markets as indication of poor economic management and that therefore breaking pegs signal instability of the economy. Carmignani et al. (2008) study various reasons of breaking promises of exchange pegs and its negative consequences. Choi and Limnios (2019) perform event study and synthetic control analysis on daily and monthly prices of assets indicating country risk premium, and provide robust evidence of signaling effect from breaking pegs. To account for exchange rate risks described above, this study assumes that foreign investors charge additional risk premium depending on country’s regime choice. The friction is employed in the household’s budget constraint in a form of a risk premium that is endogenous to central bank’s exchange regime policy.6 While a country keeps its peg, foreign investors are not exposed to The format of this friction is similar to the exogenous currency risk premium employed in McKinnon and Pill (1999) and to the UIP deviation term, whose variance is endogenous to the variance of exchange rates, introduced in Devereux and Engel (2002). 6

11

exchange rate risks, and therefore, exchange rate risk premium should be zero. When the peg is broken, foreign investors become exposed to exchange rate risks (and some may consider this policy change as a negative signal), and they raise the price of lending to domestic households. Moreover, I assume that there is a linear relationship between the risk premium and the level of foreign exchange intervention; the less the policymaker intervenes in foreign exchange market (the more flexible exchange rates), the higher risk premiums are charged for foreign loans to compensate higher exchange rate risks. The friction introduces the trade-off of breaking exchange pegs between benefits of currency (and commodity) price adjustments and costs of additional risk premiums for breaking promises of stable foreign exchange market. The idea can be simplified in a form of the linearized uncovered interest parity condition 5

6

rate rt = rt∗ + Et {∆et+1 } + λt · Exchange risk premium ,

which indicates that the domestic household pays additional exchange rate risk premiums for foreign loans and that the level of risk premium is endogenous to central bank’s regime choice λt ∈ [0, 1]. For example, foreign loans become less expensive, if a central bank adopts a fixed exchange rate regime λt = 0, because it eliminates the exchange rate risk premium. On the other hand, less managed exchange rate regime λt ∈ (0, 1] becomes costly because exchange rate risk premium is now added to the price of foreign loans. Furthermore, the policymaker finds the optimal path of λt (∀t) to maximize domestic household’s welfare by minimizing the economy’s welfare loss function in each period. Section 3.2.2 further discusses the details of this friction in the model; foreign interest rates in (8) and (20) are modeled based on this assumption, and the uncovered interest parity condition in (24) and (25) describes how exchange rate risk premium friction is employed upon the model developed in Farhi and Werning (2014).

12

3.2 3.2.1

Households Households problem

The world economy is modeled as a continuum of small open economies represented by the unit interval; i ∈ [0, 1]. The focus is on the behavior of a single country called “home” and its interaction with the world economy. A representative household seeks to maximize E0

∞ Ø

β

C

t

t=0

Ct1−σ N 1+φ − t 1−σ 1+φ

D

(2)

where Nt is labor, and Ct is a composite consumption index defined by 5

η−1 η

1 η

η−1 η

1 η

Ct ≡ (1 − α) CH,t + α CF,t

η 6 η−1

(3)

where α corresponds to the share of domestic consumption allocated to imported goods, and CH,t is an index of consumption of domestic goods given by the constant elasticity of substitution (CES) function CH,t ≡

3Ú 1 0

CH,t (j)

Ô−1 Ô

dj

Ô 4 Ô−1

(4)

where j ∈ [0, 1] denotes the good variety. CF,t is an index of imported goods given by γ CF,t ≡

3Ú 1 0

γ−1 γ

4 γ−1

Ci,t di

(5)

where Ci,t is an index of the quantity of good imported from country i and consumed by domestic households, and is given by a CES function Ci,t ≡

3Ú 1 0

Ci,t (j)

Ô−1 Ô

dj

Ô 4 Ô−1

.

(6)

α ∈ [0, 1] can be interpreted as a measure of openness of a country. Ô > 1 is the elasticity of substitution between varieties produced within any given country. η > 0 is the substitutability between domestic and world goods in

13

home country, and γ > 0 measures the substitutability between domestic and world goods in different foreign countries. 3.2.2

Household budget constraints

Households maximizes (2) subject to a sequence of budget constraints of the form Ú 1 0

PH,t (j)CH,t (j)dj+

Ú 1Ú 1 0

0

Pi,t (j)Ci,t (j)djdi + Dt+1 +

≤ Wt Nt + Πt + Tt + Rt−1 Dt + where

Ú 1

Ú 1 0

0

i Ei,t Dt+1 di

i Rt−1 Ei,t Dti di

(7)

1+ξλt

˜ ti 1 + τt Ψt Rti ≡ R 1 + τti Ψit

(8)

for t = 0, 1, 2, . . . , where all terms are expressed in domestic currency. PH,t (j) is the price of domestic variety j, Pi,t (j) is the price of variety j imported from country i, Dt is the home bond holdings of home agents, and Dti is bond holdings of country i of home agents. Wt is the nominal wage, and Tt denotes lump-sum transfers/taxes. Capital controls are introduced as a tax (or a subsidy) on foreign borrowing/lending. Home and foreign risk premium Ψt and Ψit create wedges between local and foreign investors. The external shock will be on risk premium that borrowers pay, and ψt ≡ log Ψt and ψti ≡ log Ψit i follow AR(1) processes ψt = ρψ ψt−1 + Ôψ,t and ψti = ρψ ψt−1 + Ôiψ,t . τt is a tax on net capital flows in the home country and τti is a tax on net capital flows in country i. The taxes collected here are rebated lump-sum to the households. The prices of bonds are determined by the interest rate Rt in the home country and the interest rate Rti of country i. The interest Rti that home agents pay for their loans from country i is determined by the risk free rate ˜ i that lenders of the country i charge to all agents including their local R t t /Ψit , and the net capital controls borrowers, the net level of risk premium Ψ1+ξλ t (1 + τt )/(1 + τti ). The link between exchange regime policy decision and risk premium is characterized as the risk premium Ψt and its exponential term (1 + ξλt ). λt ∈ 14

[0, 1] denotes the level of exchange market intervention as discussed in (1). ξ > 0 is a multiplier on the level of foreign exchange market intervention, and this parameter reflects the investors’ currency risk sensitivity.Suppose domestic investors are international net borrowers (Dti < 0). As a home policymaker loosens (or loses) controls over exchange rates (λt > 0), foreign lenders is exposed to higher currency risks and increase the price of their loans (1 + ξλt > 1).7 If home country’s economy is more sensitive to volatile exchange rates (high ξ), it is subject to face even higher risk premium shock (1 + ξλt becomes higher given the same value of λt ). Introducing this friction creates an environment where floating exchange regime may be a suboptimal exchange rate policy because it can magnify the shock to the economy and therefore can tighten households’ budget constraints, which lowers the level of total consumption and eventually lowers the level of welfare. Price indices are defined as following. Home consumer price index (CPI) is defined as è é 1 1−η 1−η 1−η + αPF,t . (9) Pt ≡ (1 − α)PH,t The terms of trade are defined by

St ≡

PF,t Et Pt∗ = PH,t PH,t

and the real exchange rate is Qt ≡

Et Pt∗ . Pt

See section 3.1.2 for more detailed explanation. Another way to interpret this form of t risk premium is to decompose the risk premium Ψ1+ξλ or its log-linearized form (1 + ξλt )ψt t into two parts: a general risk premium (Ψt or ψt ) and an exchange rate risk premium t (Ψξλ or ξλt ψt ). Under fixed exchange rates regime (λt = 0), exchange rate risks do not t t exist (Ψξλ = 1 and ξλt ψt = 0). As a central bank relaxes its control over exchange rates t t (λt > 0), foreign investors face higher exchange rate risks (Ψξλ > 1 and ξλt ψt > 0). t 7

15

Then log-linearized CPI around the steady state with PH,t = PF,t yields pt ≡ (1 − α)pH,t + αpF,t = pH,t + αst ,

(10)

where the (log) terms of trade is st ≡ pF,t − pH,t = et + p∗t − pH,t

(11)

and the real exchange rate is qt ≡ pF,t − pt = (1 − α)st .

(12)

From (10), the following relationship between the domestic inflation and CPI inflation is derived (13) πt = πH,t + α∆st , where the domestic inflation is defined as πH,t ≡ pH,t+1 −pH,t and CPI inflation is defined as πt ≡ pt+1 − pt . Home’s producer price index (PPI) is defined as PH,t ≡

5Ú 1 0

1−Ô

PH,t (j)

dj

1 6 1−Ô

,

(14)

and the price index for imported goods is PF,t ≡

5Ú 1 0

1 6 1−γ

1−γ Pi,t di

,

(15)

where a country i’s PPI is defined as Pi,t ≡

5Ú 1 0

1−Ô

Pi,t (j)

16

dj

1 6 1−Ô

.

(16)

Then the optimal allocation of any given expenditure within each category of goods yields the demand functions8 CH,t (j) =

A

PH,t (j) PH,t

B−Ô

and Ci,t =

CH,t ; A

Pi,t PF,t

Ci,t (j) = B−γ

A

Pi,t (j) Pi,t

B−Ô

Ci,t

CF,t .

(17)

(18)

Assume further that the foreign countries are identical and also that there are no risk premium shocks in the foreign countries and foreign countries do not impose capital controls, then we can re-express the household budget constraints 9 as ∗ ∗ Pt Ct + Dt+1 +Et Dt+1 ≤ Wt Nt + Πt + Tt + Rt−1 Dt + Rt−1 Et Dt∗ ,

(19)

where t ˜ t∗ (1 + τt )Ψ1+ξλ . Rt∗ = R t

(20)

The world variables are denoted with a star. The wedges that risk premium shocks and capital controls create are introduced to the model in a similar way. The key difference between two is that risk premium shocks affect both the interest rates that home agents perceive they can borrow and lend to the world and the interest rates that agents can borrow and lend to the world. On the other hand, capital controls affect only the interest rates that home agents perceive because the tax collected from capital controls is rebated to the home agents; the lump-sum rebate takes the following form 1+ξλt−1

∗ ˜ t−1 Tt = −τt−1 R Ψt−1

Et∗ Dt∗ + τL Wt Nt ,

(21)

where τL is a constant labor tax. In other words, capital controls divert home agent’s foreign borrowing/lending to domestic lenders/borrowers by changing 8 9

See Appendix A.1 for derivation. See Appendix A.2 for derivation.

17

the perceived world interest rates. Therefore it is clear that not only should the optimal level of capital controls offset the risk premium shocks but it should also account for home agent’s deviation from the optimal borrowing/lending allocation and corresponding nominal changes.10 3.2.3

Optimality conditions of home and world households

The optimality conditions of households yield the standard Euler’s equation with respect to the domestic return 1 = βEt Rt

I3

Ct+1 Ct

4−σ A

Pt Pt+1

BJ

(22)

and another Euler’s equation with respect to the world return 1 = βEt t ∗ ˜ Rt (1 + τt )Ψ1+ξλ t

I3

Ct+1 Ct

4−σ A

Pt Pt+1

B3

Et+1 Et

4J

.

(23)

Combining (22) and (23), we have the uncovered interest parity (UIP) condition, ; < t ˜ t∗ Et Et+1 (1 + τt )Ψ1+ξλ , (24) Rt = R t Et where the risk premium shock Ψt introduce a wedge, which may be magnified by loosening controls on exchange rates (λt > 0), in the UIP condition. The capital controls τt are expected to lean against the wind to close the gap. The log-linearized form of (24) is rt = r˜t∗ + Et {∆et+1 } + log(1 + τt ) + ü

ûú

Capital controls

ý

ψ

t üûúý

Risk premium

+

ξλt ψt ü ûú ý

.

(25)

Exchange rate risk premium

This condition summarizes the loan compensation structure of this model. When a local borrower uses a foreign loan, she has to pay the currency adjusted return (with exchange rate rates), the capital control tax, the exogenous This provides the reason why capital controls simply offsetting risk premium shocks (τt = Ψ−1 − 1) are not optimal in this framework. 10

18

country risk premium, and the exchange rate risk premium, which is endogenous to the country’s regime choice λt . The optimal labor supply policy is Wt = Ctσ Ntφ . Pt

(26)

Assuming all foreign countries are identical, when the world households solve their problems, the optimality conditions yield the symmetric Euler’s equation as (22), A B−σ A B C∗ Pt∗ 1 t+1 . (27) ∗ ˜ t∗ = βEt Ct∗ Pt+1 R

Then, combining (23) and (27), together with the definition of the real exchange rate, yields the following expression of incomplete international risk sharing between home and world households A

Ct Ct∗

B1/σ

Qt = Et

A C

t+1 ∗ Ct+1

B1/σ

t Qt+1 (1 + τt )Ψ1+ξλ . t

(28)

After iteration, (28) yields the Backus-Smith condition 1/σ

Ct = Θt Ct∗ Qt ,

(29)

where Θt is a relative Pareto weight11 whose evolution is given by î

t Θt = Et {Θt+1 } (1 + τt )Ψ1+ξλ t

ï−1/σ

.

(30)

The log-linearized form of (29), using (12) is ct = θt + c∗t +

1−α st , σ

(31)

In Gal´ı and Monacelli (2005) and Gal´ı (2009), by assuming the complete securities markets, Θt is set as constant for all t, while Farhi and Werning (2014) relaxes the assumption to introduce capital controls in the model. 11

19

where the movement of θt is found to be θt = Et {θt+1 } −

3.3

1 {τt + (1 + ξλt )ψt }. σ

(32)

Firms

A firm in the home country produces a differentiated good with a linear technology represented by the production function Yt (j) = At Nt (j),

(33)

The real marginal cost is common across domestic firms and given by M Ct = (1 + τL )

Wt , At PH,t

(34)

where τL is an employment subsidy to have steady-state output equal to the efficient level,12 and the nominal marginal cost is Wt , At

(35)

mcnt = −ν + wt − at ,

(36)

M Ctn = (1 + τL ) and its log-linearized form is given by

where ν ≡ − log(1 + τL ). ès é Ô Ô−1 Assuming the aggregate output is defined as Yt ≡ 01 Yt (j) Ô dj Ô−1 as similarly assumed for the aggregate consumption, from Nt ≡

Ú 1 0

Nt (j)dj =

Yt

s 1 Yt (j)dj 0

At

Yt

,

The employment subsidy is the result of a balancing act between offsetting the monopoly distortion of individual producers and exerting some monopoly power as a country. Farhi and (1−α)(η−1)+γ Werning (2012, 2014) find that the labor subsidy is constant. (τL = Ô−1 Ô (1−α)(η−1)+γ−α −1). 12

20

we have an aggregate relationship in a log-linearized form yt = at + nt . 3.3.1

(37)

Price setting

This paper focuses on Calvo price setting; in every period, a randomly selected fraction 1 − δ of firms resets their prices. The firms resetting their prices solve the following problem13 to choose the newly set domestic prices P H,t max

∞ Ø

P H,t k=0

k

δ Et

IA k Ù

h=1

1 Rt+h

B

è

1

Yt+k P H,t −

n M Ct+k

2é

J

subject to the demand constraints Yt+k ≤

A

P H,t PH,t+k

B−Ô

∗ d (CH,t+k + CH,t+k ) ≡ Yt+k (P H,t ).

Then, the log-liniearized optimal price-setting strategy is given by p¯H,t = µ + (1 − βδ)

∞ Ø

k=0

1

î

ï

(βδ)k Et mcnt+k ,

(38)

2

Ô where µ ≡ log Ô−1 , the log of the markup in the steady state. The world firms face the same price setting problem, and the optimal strategy is analogous to (38),

p¯∗t

∗

= µ + (1 − βδ )

∞ Ø

k=0

î

ï

(βδ ∗ )k Et mcn∗ t+k ,

(39)

and for simplicity, I assume the degree of price stickiness in the world economy δ ∗ is identical to that in home country δ. See Chapter 3 of Gal´ı (2009) and Gal´ı and Monacelli (2005) for Calvo price setting derivation. 13

21

3.4

Market clearing

From (17),(18), and (68) we have A

Pi,t (j) Ci,t (j) = α Pi,t and

B−Ô A

Pi,t PF,t

A

PH,t (j) CH,t (j) = (1 − α) PH,t

B−γ 3

B−Ô 3

PF,t Pt

PH,t Pt

4−η

4−η

Ct

(40)

Ct .

(41)

The goods market clearing condition in the home country requires Yt (j) = CH,t (j) +

Ú 1 0

i CH,t (j)dj,

(42)

i where CH is the country i’s demand for home goods and takes the following form

i (j) CH,t

A

PH,t (j) =α PH,t

B−Ô A

PH,t i Ei,t PF,t

B−γ A

Combining (41), (42), and (43) and assuming Yt ≡ rewrite the goods market clearing condition as 3

PH,t Yt = (1 − α) Pt

4−η

Ct + α

Ú 1A 0

PH,t i Ei,t PF,t

i PF,t Pti

B−η

ès

Yt (j)

B−γ A

1 0

i PF,t Pti

Cti .

B−η

Ô−1 Ô

(43)

dj

é

Cti di.

Ô Ô−1

, I can

(44)

Labor market clearing condition is A B−Ô Yt Ú 1 PH,t (j) Nt = dj At 0 PH,t

where Nt =

s1 0

Nt (j)dj.

22

(45)

4

Equilibrium

4.1

The demand side: Consumption and output determination

4.1.1

World consumption and output

The log-linearization of world Euler equation (27) is c∗t = Et {c∗t+1 } −

1 ∗ ∗ } − ρ), (˜ r − Et {πt+1 σ t

(46)

where ρ ≡ − log β, and combined with the market clearing condition yt∗ = c∗t , it yields 1 ∗ ∗ ∗ } − (˜ } − ρ). (47) rt − Et {πt+1 yt∗ = Et {yt+1 σ 4.1.2

Home consumption and output ès

1

é

Ô

The first order approximation14 of home aggregate output Yt ≡ 01 Yt (i)1− Ô di Ô−1 is written as ωα (48) st + (1 − α)θt , yt = yt∗ + σ where ωα ≡ 1 + α(2 − α)(ση − 1) > 0. Combining (31) and (48) to substitute out for st , we have ct = Φa yt + (1 − Φa ) yt∗ + {1 − (1 − α)Φa } θt ,

(49)

> 0. where Φa = 1−α ωα Then we can combine (48), (49) with (13), and home consumer’s Euler equation to derive a difference equation for domestic output in terms of domestic real interest rates, world output, and relative Pareto weight: ωα ∗ }+(ωα −1+α)Et {∆θt+1 }, (rt −Et {πH,t+1 }−ρ)+(ωα −1)Et {∆yt+1 σ (50) ∗ ∗ ≡ yt+1 − yt and ∆θt+1 ≡ θt+1 − θt .

yt = Et {yt+1 }− ∗ where ∆yt+1 14

See Appendix A.3 for derivation.

23

4.1.3

The trade balance

The net exports nxt are denoted in terms of domestic output, expressed as a fraction of steady state output Y , 1 nxt ≡ Y

A

B

Pt Yt − Ct , PH,t

and its first-order approximation yields nxt = yt − ct − pt + pH,t , which combined with (48) and (49) implies 3

4

αΛ 1−α nxt = θt , (yt − yt∗ ) − 1 − ωα ωα

(51)

where Λ ≡ (2 − α)(ση − 1) + (1 − σ).

4.2

The supply side: Marginal cost and inflation dynamics

4.2.1

World marginal cost and inflation dynamics

Combining the optimal price setting equation (39) with the log-linearized equation of the evolution of aggregate price level, we have ∗ πt∗ = βEt {πt+1 }+

(1 − δ)(1 − βδ) ∗ äct , m δ

(52)

äc∗t ≡ mc∗t + é denotes the log real marginal cost, expressed as a where m deviation from its steady state value (−é). Assuming the symmetric home and world economies, as implied in (36), the world log real marginal cost is given by

mc∗t = −ν ∗ + (σ + φ)yt∗ − (1 + φ)a∗t

24

(53)

where ν ∗ ≡ − log(1 + τL∗ ). 4.2.2

Home marginal cost and inflation dynamics

The dynamics of domestic inflation of home are described by an equation analogous to the world’s; (1 − δ)(1 − βδ) äct . m δ

πH,t = βEt {πH,t+1 } + The home log real marginal cost is

mct = −ν + φyt + σyt∗ + st + σθt − (1 + φ)at .

(54)

(55)

Using (48) to substitute for st , (55) can be rewritten as following 3

4

4

3

σ 1 y ∗ + (1 − σ + α)θt − (1 + φ)at . (56) + φ yt + σ 1 − mct = −ν + ωα ωα t

4.3

Equilibrium dynamics

I define the output gap xt as the deviation of log output yt from its natural level y t , where the latter is in turn defined as the equilibrium level of output in the absence of nominal rigidities: xt ≡ y t − y t ,

(57)

and the counterpart for world is, x∗t ≡ yt∗ − y ∗t . 4.3.1

(58)

World equilibrium dynamics

Under flexible prices, real marginal costs in the world economy will be constant over time, and given by mc∗ ≡ −é, the level that would obtain under flexible

25

prices. Then using (53), the natural level of world output can be expressed as y ∗t =

ν ∗ + (−é) 1 + φ ∗ + a. σ+φ σ+φ t

(59)

The relationship between real marginal cost and the output gap can be derived as the following equation yt∗

−

y ∗t

A

ν ∗ + mc∗t ν ∗ + (−é) 1 + φ ∗ 1+φ ∗ = + at − + a σ+φ σ+φ σ+φ σ+φ t

B

äc∗t = (σ + φ)x∗t , ⇒ m

which, combined with (52), gives the New Keynesian Phillips curve: ∗ πt∗ = βEt {πt+1 }+

(1 − δ)(1 − βδ)(σ + φ) ∗ xt . δ

(60)

(47) can be also re-written in terms of the world output gap: x∗t = Et {x∗t+1 } − 4.3.2

2 11 ∗ ∗ r˜t − Et {πt+1 }−ρ . σ

(61)

Home equilibrium dynamics

The steady state value of log real marginal cost is − µ = −ν +

3

4

3

4

1 σ y ∗ + (1 − σ + α)θ¯t − (1 + φ)at , (62) + φ y¯t + σ 1 − ωα ωα t

which provides the natural level of output in home country y¯t =

ωα (ν − µ) σ(1 − ωα ) ∗ ωα (1 − σ + α) ¯ ωα (1 + φ) + y − θt + at , σ + ωα φ σ + ωα φ t σ + ωα φ σ + ωα φ

(63)

then the relationship between output gap and real marginal cost is derived as following ωα ωα (1 − σ + α) ˆ äct − (64) xt = m θt . σ + ωα φ σ + ωα φ

26

The New Keynesian Phillips Curve can be derived using (54) and (64): πH,t

3

4

σ = βEt {πH,t+1 } + κ + φ xt + ακθˆt , ωα

(65)

. where κ ≡ (1−δ)(1−βδ) δ Using (50), the IS equation can be expressed in terms of output gap: xt = Et {xt+1 } −

ωα (rt − Et {πH,t+1 } − rrt ) + (ωα − 1 + α)Et {∆θˆt+1 }, (66) σ

where the home country’s Wicksellian interest rate rrt is defined as rrt ≡ ρ +

5

σ(1 + φ) σφ(1 − ωα ) ∗ } Et {∆at+1 } − Et {∆yt+1 σ + ωα φ σ + ωα φ I J ωα − 1 + α 1 − σ + α − +σ Et {∆θ¯t+1 }. ωα σ + ωα φ

(67)

Optimal Policy

The optimal policy path is derived in perspective of a social planner who minimizes welfare loss with the resource constraints that are characterized in equilibrium conditions. Following the literature, I focus on the Cole-Obstfeld case σ = η = γ = 1 to derive a tractable second-order approximation of the welfare function.

5.1

Summarizing the economy

The demand block is summarized by the following equations: xt = xt+1 − (rt − Et {πH,t+1 } − rrt ) + αEt {∆θˆt+1 }, rt = r˜∗ + λt Et {∆eft+1 } + Et {∆θˆt+1 } + (1 + ξλt )ψt t

rrt = ρ − Et {∆at+1 } + ∞ Ø

αφ (1 + ξλt )ψt , and 1+φ

β t θˆt = 0,

t=0

27

representing the Euler equation with goods market clearing condition and the Backus-Smith condition, the uncovered interest parity condition, Wicksellian interest rate, and the budget constraint, respectively. The New-Keynesian Philips Curve summarizes the supply block, πH,t = βEt {πH,t+1 } + κ(1 + φ)xt + ακθˆt .

5.2

The planning problem in gaps

The optimal allocation must maximize the utility of consumption and minimize the disutility of labor; that is, it must maximize the present value of the sum q t of future consumer utility ∞ t=0 β {U (Ct ) − V (Nt )}. This can be expressed in the form of welfare loss function, which is a second-order approximation of the welfare function, and a social planner solves the following planning problem15 : min

{πH,t ,xt ,θˆt ,rt ,λt }

E0

∞ Ø

β

t

t=0

;

<

1 1 1 2 απ πH,t + x2t + αθ (θˆt + θ¯t )2 , 2 2 2

1

2

Ô α 2−α and αθ = 1+φ + 1 − α , subject to the following equiwhere απ = κ(1+φ) 1−α librium conditions derived in the previous sections:

πH,t = βEt {πH,t+1 } + κ(1 + φ)xt + ακθˆt , xt = xt+1 − (rt − Et {πH,t+1 } − rrt ) + αEt {∆θˆt+1 }, rt = r˜∗ + λt Et {∆eft+1 } + Et {∆θˆt+1 } + (1 + ξλt )ψt , t

rrt = ρ − Et {∆at+1 } +

αφ (1 + ξλt )ψt , 1+φ

Et {∆θ¯t+1 } = (1 + ξλt )ψt , ∞ Ø

β t θˆt = 0, and

t=0

λt ∈ [0, 1]. (32) shows that the difference between home and foreign consumptions is 15

See Appendix A.4 for derivation.

28

widened by the risk premium shock but also can be narrowed by the capital controls. That is, the policymaker controls the path of θˆt and therefore the capital controls are characterized as τt = ∆θˆt+1 .

5.3

Optimal policies

To gain an intuition of the effects of each policy, I test the economy with AR(1) risk premium shock (ψt = ρψ ψt−1 + εψ,t ) and compare how the systems response to the shocks under four different cases of international policy portfolios: (1) the system under the home inflation-based Taylor rule with zero foreign exchange market intervention and zero capital controls (Section 5.3.1); (2) the system allowing foreign exchange market intervention only (Section 5.3.2); (3) the system allowing capital controls only (Section 5.3.3); and (4) the system under both policies (Section 5.3.4). Although it is possible to solve and present the optimal allocations of each case in closed forms, the expressions quickly become difficult to handle. Especially, the foreign exchange market intervention policy λt is not continuous but is defined to be within a range of zero and one, and this condition requires discontinuous and non-linear solution sets. Therefore, the study performs numerical simulations to provide impulse responses to AR(1) shock.16 Numerical simulations can be summarized to the following steps. The series of exchange rates are derived first from minimizing the loss functions in the absence of either capital controls or foreign exchange intervention Et {∆eft+1 } = −

αφ (1 + ξ)ψt ; 1+φ

the path of exchange rates solved here are used as the contingent paths of freely floating exchange rates eft in following sections.17 The series of domestic inflation rates πH,t and output gap xt are solved in terms of freely floating exchange rates and risk premium shocks by minimizing the loss function for 16 This simulation is meant to be an example and should not be thought of as a calibration exercise because the model is probably too stylized. 17 For the case in Section 5.3.1, exchange rates based on Taylor rule are used.

29

each case. The optimal levels of policies are solved in terms of floating exchange rates, domestic inflation rates, and risk premium shocks, and they are solved by using the values found in the previous steps. Due to the discontinuous nature of λt , all solutions should be solved in three cases: (1) when λt is not restricted by the range of [0, 1], (2) when λt = 0, and (3) when λt = 1. The solution sets are found without restrictions on λt first: the case of (1). In events where the suggested optimal level of λt is below zero, λt is set to zero and the existing solution set is replaced with the one solved given (2) λt = 0. Similarly, when the suggested optimal level of λt is above one, λt is set to one and the existing solution set is replaced with the one solved given (3) λt = 1. The other state variables are derived given the series of effective exchange rates (the exchange rates with interventions, ∆et+1 = λt ∆eft+1 ), capital controls, domestic inflation rates, and output gap. Each case is simulated with two levels of the exchange rate risk sensitivity, ξ = 1 and ξ = 0.05. Since the risk premium shock is characterized as (1 + ξλt )ψt , under ξ = 1, switching exchange rate regime from fixed to freely floating results in 100% increase in the risk premium shock, and it reflects the results of event study in Choi and Limnios (2019), where the risk premium shows 85.7–108.5% abnormal returns in its trend when countries switch regimes. The setting of ξ = 0.05 provides the benchmark case to illustrate how exchange rate risk premium affects the economy. For other parameter values, I follow Gal´ı and Monacelli (2005) by setting α = 0.4, β = 0.99, ρ = 0.04, Ô = 6, and φ = 3 and Farhi and Werning (2014) by setting δ = 0.68 and the initial risk premium shock to be 5 percent, which diminishes at the rate of ρψ = 0.9.

30

5.3.1

Under floating exchange rate regime and zero capital controls with home inflation-based Taylor rule

This case presents a base case of home inflation-based Taylor rule.18 When the policymaker does not intervene in the foreign exchange market or levy taxes on international capital flows, we have λt = 1 and θˆt = 0 ∀t, then the planning problem simplifies to min E0

{πH,t ,xt }

∞ Ø t=0

β

t

;

<

1 1 2 + x2t , απ πH,t 2 2

subject to πH,t = βEt {πH,t+1 } + κ(1 + φ)xt , xt = xt+1 − (rt − Et {πH,t+1 } − rrt ), rt = r˜t∗ + Et {∆eft+1 } + (1 + ξ)ψt , ∆rt = ρ + φπ πH,t , and αφ (1 + ξ)ψt , rrt = ρ + 1+φ where Taylor rule coefficient φπ = 1.5 following Gal´ı and Monacelli (2005). Figure 5 displays the impulse responses of the system to a risk premium shock under the different levels of exchange rate risk premium sensitivity. This case provides a benchmark to examine how the system works with different sets of policies. In addition, it is useful to consider this case to examine how exchange rate risk premium can affect the economy in general; the impulse Taylor rule case is considered as a baseline case in this study because it provides a meaningful baseline response of the economy to different levels of exchange rate risk sensitivity ξ. In the economy with optimally chosen real interest rates, fluctuations of domestic inflation and output gap (two components of the loss function) are muted to zero, as seen in the special case in Gal´ı and Monacelli (2005), simply by setting rt = rrt . Therefore, such environment is not suitable to describe how an economy without international monetary policies responds to different levels of exchange rate risk sensitivity. 18

31

!

#

1

1

e

0.4

0.5

0.3

0

0.2

-0.5

0.1

0.5 Low " High "

0

-0.5

-1 0

10

20

30

$H

0.05

0 0

10

20

30

x

0.5

0

10

0.4

0.015

0.03

0.3

0.01

0.02

0.2

0.005

0.01

0.1

0

30

$

0.02

0.04

20

nx

0 -0.01 -0.02

0

0 0

10

20

30

r

0

-0.005 0

10

20

30

0.5

-1.5 10

20

30

20

30

Utility

-0.2

-0.04

-0.4

-0.06

-0.6

-0.08

-0.8

10

20

30

10

20

30

Accumulated Welfare

0

-0.1 0

0

-0.02

0 0

10

0

1

-1

-0.04 0

s

1.5

-0.5

-0.03

-1 0

10

20

30

0

10

20

30

Figure 5: Impulse response function of the system under zero foreign exchange market intervention and zero capital controls: Low signaling effect (blue) and high signaling effect (red)

response functions (IRFs) with a low ξ (blue line) represent the classical small open economy model with risk premium shock that is almost inelastic to the exchange rate regime change, while the IRFs with a high level of ξ (red line) depict how the regime-elastic risk premium changes the dynamics of classical models. When the risk premium shock hits the economy under the floating exchange rate regime, the depreciation of the exchange rate is higher when the economy is more sensitive to the exchange rate volatility; the risk premium shock is amplified under the floating regime, and without other policies, the price of local currency has to be rearranged to mitigate the shock. The amplified risk premium shock and the floating exchange rates together bring in even higher

32

e

0.03

1

1

0.5 0.02

0.5

Low High

0 0

0.01 -0.5

-0.5

-1 0

10

20

30

H

0.008

0 0

10

30

x

0

0.006

20

0

10

20

30

nx

0

0.003

-0.02

-0.005 0.002

0.004

-0.04

-0.01 0.001

0.002

-0.06

0

-0.015

-0.08 0

10

20

30

r

0

0 0

10

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30

s

0.15

-0.02 0

10

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Utility

0

10

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30

Accumulated Welfare -0.004 -0.006

-0.001

-0.02

0

0.1

-0.008 -0.002 -0.01

-0.04

0.05 -0.003

-0.06

0 0

10

20

30

-0.012 -0.014

-0.004 0

10

20

30

0

10

20

30

0

10

20

30

Figure 6: Impulse response function of the system under managed exchange rates and zero capital controls: Low signaling effect (blue) and high signaling effect (red)

impact on price levels and output. The home inflation under higher ξ deviates from its steady state much more than the one under lower ξ does (4.25 vs. 0.79 percent at peaks) and the output gap deviation is six times larger under high ξ (42.9 vs. 7.02 percent at impact). As a result, a welfare loss is almost 30 times bigger (9.2 vs. 0.25 percent at impact), while the risk premium shock under the high ξ is only about twice of that under the low ξ (10 vs. 5.25 percent). The figures indicate that volatile exchange rates bring multiplying effects when ξ is high, and therefore the optimal policies derived in the following sections are expected to stabilize the exchange rates.

33

5.3.2

Under managed exchange rates and zero capital controls

When policymakers are able to manage the exchange rates without capital controls, where θˆt = 0 ∀t, then the problem becomes min

{πH,t ,xt ,rt ,λt }

E0

∞ Ø t=0

βt

;

<

1 1 1 2 + x2t + αθ θ¯t2 , απ πH,t 2 2 2

subject to πH,t = βEt {πH,t+1 } + κ(1 + φ)xt , xt = xt+1 − (rt − Et {πH,t+1 } − rrt ), rt = r˜t∗ + λt Et {∆eft+1 } + (1 + ξλt )ψt , αφ (1 + ξλt )ψt , rrt = ρ + 1+φ Et {∆θ¯t+1 } = (1 + ξλt )ψt , ∞ Ø

β t θˆt = 0, and

t=0

λt ∈ [0, 1]. Solving the planning problem, the optimal path of foreign exchange market intervention can be derived in terms of home inflation, shadow floating exchange rates, and the risk premium shock as following: Proposition 1. (Managed exchange rates under zero capital controls) Suppose the policymaker intervenes in the foreign exchange market without capital controls. The optimal level of managed exchange regime is given by C

λt = Et C

î

∆eft+1

ï

ξ(1 + φ − αφ) + ψt 1+φ

D−1

1 + φ − αφ ψt − {κ(1 + φ)απ − 1}Et {πH,t+1 } 1+φ 34

D

where λt ∈ [0, 1].19 Proposition 1 implies that it is preferred to stabilize the exchange rates (λt → î ï 0), when the floating exchange rates are expected to be volatile (Et ∆eft+1 ↑) or the investors are more sensitive to broken pegs (ξ ↑) to avoid another wave of risk premium shock. The level of intervention does not totally lean against the wind, and it indicates the trade-off between the benefits of currency priceadjustment and the cost of impact on terms of trade and furthermore the local consumers’ purchasing power. IRFs in Figure 6 show that when there are no capital controls, exchange rates shall be actively managed; λt does not deviate much from zero at impact and quickly converges back to zero under both levels of ξ; especially under high level of ξ, it is essential to stabilize the exchange rates to avoid the additional increase in risk premium, and managed regime lives only for two periods. As a result, price and output levels become stabilized, and the welfare loss has significantly decreased compared to the first case (9.2 percent under floating exchange rate regime to 0.34 percent under managed exchange rate regime at impact). 5.3.3

Under floating exchange rate regime and capital controls

When the policymaker can use capital controls under the floating exchange rate regime, where λt = 1 ∀t, the problem becomes min

{πH,t ,xt ,θˆt ,rt }

E0

∞ Ø t=0

β

t

;

<

1 1 1 2 απ πH,t + x2t + αθ (θˆt + θ¯t )2 , 2 2 2

If the optimal value of λt is found to be less than 0 or greater than 1, the effective λt takes the value of 0 or 1 respectively. 19

35

0

1

e

0.06

-0.02 0.04

0.5

-0.04

-0.08 -0.5

Low High

0.02

-0.06

0

0

-0.1 0

10

20

30

H

0.015

0

10

20

30

0

x

0.08

10

20

30

nx

0.008

0.06

0.006

0.04

0.004

0.02

0.002

0.0015

0.01

0.001

0.005

0

0 0

10

20

30

r

0

0.0005

0 0

10

20

30

s

0.3

0 0

10

30

Utility

0

-0.02

20

-0.06 0.1

-0.08

10

20

30

Accumulated Welfare

0

-0.0005 0.2

-0.04

0

-0.005

-0.001

-0.01

-0.0015

-0.015

-0.002

-0.02

-0.1 0 0

10

20

30

-0.0025 0

10

20

30

-0.025 0

10

20

30

0

10

20

30

Figure 7: Impulse response function of the system under floating exchange rates and capital controls: Low signaling effect (blue) and high signaling effect (red)

subject to πH,t = βEt {πH,t+1 } + κ(1 + φ)xt + ακθˆt , xt = xt+1 − (rt − Et {πH,t+1 } − rrt ) + αEt {∆θˆt+1 }, rt = r˜∗ + Et {∆eft+1 } + Et {∆θˆt+1 } + (1 + ξ)ψt , t

rrt = ρ +

αφ (1 + ξ)ψt , 1+φ

Et {∆θ¯t+1 } = (1 + ξ)ψt , and ∞ Ø

β t θˆt = 0

t=0

Solving the planning problem brings to the following proposition:

36

Proposition 2. (Capital controls under floating exchange rate regime) Suppose the policymaker does not intervenes in the foreign exchange market. The optimal capital controls can be characterized by τt = −(1 + ξ)ψt −

ακαπ Et {πH,t+1 } . αθ

Optimal capital controls are proportional to the current risk premium shock. The tax τt has the opposite sign from the risk premium ψt – the policy leans against the wind when the home price level is rigid. High home inflation is expected due to volatile exchange rates, and therefore the optimal level of capital controls will increase accordingly. Figure 7 shows that the capital controls are relatively more effective in mitigating the sudden stop; the welfare losses under capital controls are 0.051 and 0.18 percent, while the ones under managed exchange rates are 0.21 and 0.34 percent. The risk premium shock is transmitted to the local economy through the change in international capital flows; as it becomes more expensive to make loans internationally, the households face tighter budget constraints. By levying taxes on international capital flows, it directly counteracts the effect of the shock. Furthermore, the model captures that the capital controls indirectly affect the foreign exchange market as well. Regulating the currency movements prevents a further depreciation of the local currency, and the example in Figure 7 shows that exchange rates depreciate 3.1 and 5.9 percent under capital controls, while in the first case (Figure 5) they depreciate 18.3 and 35 percent without capital controls. This implies that capital controls can be utilized as a partial substitute of foreign exchange market intervention to maximize consumer’s utility, and the next case validates this.

37

5.3.4

Under managed exchange rates and capital controls

The policymaker now solves the full problem laid out in the beginning of this section: min

{πH,t ,xt ,θˆt ,rt ,λt }

E0

∞ Ø

β

t=0

t

;

<

1 1 1 2 + x2t + αθ (θˆt + θ¯t )2 , απ πH,t 2 2 2

subject to πH,t = βEt {πH,t+1 } + κ(1 + φ)xt + ακθˆt , xt = xt+1 − (rt − Et {πH,t+1 } − rrt ) + αEt {∆θˆt+1 }, rt = r˜∗ + λt Et {∆eft+1 } + Et {∆θˆt+1 } + (1 + ξλt )ψt , t

rrt = ρ + ∞ Ø

αφ (1 + ξλt )ψt , 1+φ

β t θˆt = 0, and

t=0

λt ∈ [0, 1].

The optimal policy portfolio can be characterized in the following proposition: Proposition 3. (Managed exchange rates and capital controls) Suppose the policymaker is able to intervene in the foreign exchange market and levy taxes on international capital flows. The policy portfolio of λt and τt is characterized by the following equations: C

λt = Et C

î

∆eft+1

ï

ξ(1 + φ − αφ) + ψt 1+φ

D−1

D

1 + φ − αφ ψt − (1 − α)τt − {κ(1 + φ)απ − 1}Et {πH,t+1 } , 1+φ

and τt = −(1 + ξλt )ψt −

38

ακαπ Et {πH,t+1 }, αθ

1

e

0.06

0 -0.02

0.04

-0.04

0.5

-0.08

0

Low High

0.02

-0.06

0

-0.1 0

10

20

30

H

0.015

0

10

20

30

x

0.08

0

10

20

30

nx

0.008

0.06

0.006

0.04

0.004

0.02

0.002

0

0

0.001 0.0008

0.01 0.0006 0.0004

0.005

0 0

10

20

30

r

0

0

10

0.2

-0.1

0.1

-0.15

30

10

20

30

10

20

30

Utility

0

0 0

0 0

s

0.3

-0.05

20

0.0002

-0.0002

-0.004

-0.0004

-0.005

-0.0006

-0.006

10

20

30

10

20

30

Accumulated Welfare

-0.003

-0.0008 0

0

-0.007 0

10

20

30

0

10

20

30

Figure 8: Impulse response function of the system under managed exchange rates and capital controls: Low signaling effect (blue) and high signaling effect (red)

where λt ∈ [0, 1]. Proposition 3 illustrates the interaction between two policy tools. The first equation indicates that with a higher level of capital controls (τt ↓; as seen in the previous cases, τt < 0 since we assume positive risk premium shock, capital flows are controlled with a subsidy on inflow or a tax on outflow), the optimal level of λt is higher; that is, the optimal exchange rate regime becomes closer to freely floating regime with a higher level of capital controls. This agrees with a theoretical evidence in the previous section that policymaker can partially mitigate the depreciation of exchange rates by controlling international capital flows. It suggests that capital controls can be utilized as a partial substitute of exchange rate policy, and this relationship further implies that a policymaker may be able to achieve the same or a better welfare level with less 39

capital controls and less intervention in the foreign exchange markets than the previous cases. According to the second equation, on the other hand, the lower foreign exchange market intervention (λt → 1) shall bring in additional wave of risk premium shock that capital controls are to handle, which is a tradeoff for implementing foreign exchange market intervention policy. Therefore, solving two equations should provide us the balanced path of the optimal policy portfolio against the risk premium shock and exchange rate risk premium. Figure 8 depicts the optimal policy paths of managed exchange rates and capital controls and the corresponding responses of the economy. When ξ is low, it is optimal to float the exchange rates (λt close to 1) and instead to utilize the capital controls, which can partially manage the exchange rate depreciation as well as mitigate the shock by leaning against the wind. Similarly, even when the exchange rate risk sensitivity ξ is high, it is optimal to have a managed floating exchange rate regime by letting exchange rates to depreciate 50 percent of that under freely floating exchange rate regime (λt = 0.5 at impact), while controlling international capital flows at the same time. The length of nonfixed regime has been extended from 2 periods to 9 periods, which may explain developing countries’ relatively persistent managed exchange regime and, as a result, their exchange rate volatility that is comparable to the ones in advanced countries. The effect of this policy implementation is evident when comparing price and output levels under different levels of ξ. The gaps between home inflations with different values of ξ have been brought down as well as the gaps between output gaps; the difference in home inflations is 0.13 percent at peaks and the difference in output gaps is 0.42 percent at impact. One can also notice a significant improvement in terms of trade. By allowing to float the exchange rates partially, the country can avoid a bigger risk premium shock, while alleviating the effect of the shock to its economy by taxing international capital flows. An improvement in terms of trade leads to a higher purchasing power of home consumers, and as a result, the difference in the welfare levels becomes minimal (0.02 percent at impact). Figure 9 compares the welfare losses under different policy portfolio avail40

0

Instantaneous Utility

! 10-3

Accumulated Welfare

0

-0.5

-0.005

-1 -1.5

-0.01

-2

-0.015

-2.5

Both policies Capital controls only FX intervention only

-3

-0.02

-3.5

-0.025 0

5

10

15

20

25

30

0

5

10

15

20

25

30

Figure 9: Welfare level comparison among different policy portfolios

ability. It shows that the welfare losses are minimum when the full set of policies are utilized, despite that the set of policies is still constrained by trilemma. This result is worth noting. A proper use of the middle-ground policies can neutralize the potential unfavorable effect of exchange rate regime choice; as capital controls has become available as a policy tool to mitigate the shock, the policymakers in the countries that are sensitive to exchange rate regimes can now put less resources to stabilize exchange rates and still minimize the welfare loss from the shock.

6

Conclusion

I consider a New Keynesian small open economy model where international financial markets are not perfect. I introduce a novel and tractable approach to employ managed exchange regime by assuming that a rational policymaker simply decides the level of intervention in the foreign exchange markets. Therefore, a policymaker in the model makes a series of middle-ground policy portfolio choices of capital controls and exchange rate regimes to minimize the loss in welfare in response to the risk premium shock, which can be amplified by the exchange rate risk premium. The results suggest that the coexistence of active capital controls and managed exchange regimes may be optimal as documented in Klein and Shambaugh (2015), and this represents one of the

41

robust features of developing countries’ policy portfolio decisions in the last decades. Using simple characterization of international policy choices – capital controls as a tax on international capital flows and foreign exchange market intervention as a choice of degree of exchange rate regimes, the model successfully captures major traits of the relationship between international monetary policies. A policymaker manages the perceived price of the international loans and therefore becomes able to indirectly manage exchange rate depreciation. It suggests that for an economy whose country risk is sensitive to its exchange rate regime choice, the role of capital controls becomes clear as it allows one to put less resource to stabilize the foreign exchange market.

Discussion The focus of this paper is to introduce a simple and tractable small open economy framework to examine the trade-offs between international monetary policies, and the constant foreign exchange risk sensitivity parameter ξ serves its purpose. However, it is also fair to assume that foreign investors’ risk preference and the distribution of risk averse and risk loving investors in the market are in fact related to macroeconomic or microeconomic factors, such as the country’s economic outlook, market depth, and history of currency crises, and potential extensions can be made by endogenizing the parameter ξ related to such factors. One of potential approaches is to consider the relationship of exchange rate risk sensitivity with monetary policies. Bekaert et al. (2013) and Bekaert et al. (2019) study risk premiums and risk preferences (or risk appetite) and find that loose monetary policies decrease risk aversion in the market. The current model accommodates three international monetary policy tools of real interest rates, capital controls, and foreign exchange rate regimes, and building connections of these tools with the risk sensitivity and studying each channel will provide clear insights and theoretical bases of empirical findings. Another approach is to consider foreign reserves. Obstfeld (1986) suggests that the level of foreign reserves affects investor’s sentiment and the likeli-

42

hood of speculative attacks. Aizenman et al. (2010) and Aizenman (2013), suggest that controlling the level of international reserves has become one of the critical monetary policy choices and that international monetary policy constraints are moving from the trilemma to the quadrilemma with international reserves being the fourth dimension of quadrilemma. The current model presented in this paper assumes that the foreign exchange intervention decision is not constrained by the level of international reserves to simplify the dynamics of solutions. Despite the current trends of hoarding U.S. dollars, many developing countries do not hold enough foreign reserves to perfectly secure their currency markets from speculative attacks and relax other three monetary policies from trilemma. Therefore, maintaining the level of foreign reserves should be also considered in policy decisions, and connecting it with exchange rate risk sensitivity should create a reasonable linkage to embed the idea of quadrilemma and Obstfeld’s second-generation crisis model, which will enrich the trade-off dynamics between fixed and floating regimes; for example, if a central bank uses foreign reserves too much to protect the currency prices, investors will be concerned about lower foreign reserves and speculative attacks on currency markets, and as a result, they will respond to little changes in exchange rates, which is reflected in an increase of ξ.

43

References Aizenman, J., 2013. The impossible trinity—from the policy trilemma to the policy quadrilemma. Global Journal of Economics 2, 1350001. Aizenman, J., Chinn, M.D., Ito, H., 2010. The emerging global financial architecture: Tracing and evaluating new patterns of the trilemma configuration. Journal of international Money and Finance 29, 615–641. Alesina, A., Wagner, A.F., 2006. Choosing (and reneging on) exchange rate regimes. Journal of the European Economic Association 4, 770–799. Batini, N., Breuer, P., Kochhar, K., Roger, S., 2006. Inflation targeting and the imf. International Monetary Fund . Bekaert, G., Engstrom, E.C., Xu, N.R., 2019. The time variation in risk appetite and uncertainty. Technical Report. National Bureau of Economic Research. Bekaert, G., Hoerova, M., Duca, M.L., 2013. Risk, uncertainty and monetary policy. Journal of Monetary Economics 60, 771–788. Benigno, G., Chen, H., Otrok, C., Rebucci, A., Young, E.R., 2012. Capital controls or exchange rate policy? a pecuniary externality perspective. Federal Reserve Bank of St. Louis Working Paper Series . Bianchi, J., 2010. Credit externalities: Macroeconomic effects and policy implications. The American Economic Review , 398–402. Bianchi, J., 2011. Overborrowing and systemic externalities in the business cycle. American Economic Review 101, 3400–3426. Calvo, G.A., 1998. Capital flows and capital-market crises: the simple economics of sudden stops . Calvo, G.A., Reinhart, C.M., 2002. Fear of floating. The Quarterly Journal of Economics 117, 379–408. Carmignani, F., Colombo, E., Tirelli, P., 2008. Exploring different views of exchange rate regime choice. Journal of International Money and Finance 27, 1177–1197. Cavallino, P., 2019. Capital flows and foreign exchange intervention. American Economic Journal: Macroeconomics 11, 127–70. 44

Choi, J.H., Limnios, C., 2019. Cost of floating exchange rates: Event study and synthetic control estimation. Available at SSRN: https://ssrn.com/abstract=3215458 or http://dx.doi.org/10.2139/ssrn.3215458. Devereux, M.B., Engel, C., 2002. Exchange rate pass-through, exchange rate volatility, and exchange rate disconnect. Journal of Monetary Economics 49, 913–940. Evans, M.D., Lyons, R.K., 2002. Order flow and exchange rate dynamics. Journal of Political Economy 110. Farhi, E., Werning, I., 2012. Dealing with the trilemma: Optimal capital controls with fixed exchange rates. Technical Report. National Bureau of Economic Research. Farhi, E., Werning, I., 2014. Dilemma not trilemma? capital controls and exchange rates with volatile capital flows. IMF Economic Review 62, 569– 605. Fern´andez, A., Klein, M.W., Rebucci, A., Schindler, M., Uribe, M., 2015. Capital Control Measures: A New Dataset. Technical Report. National Bureau of Economic Research. Fiess, N., Shankar, R., 2009. Determinants of exchange rate regime switching. Journal of International Money and Finance 28, 68–98. Gal´ı, J., 2009. Monetary Policy, inflation, and the Business Cycle: An introduction to the new Keynesian Framework. Princeton University Press. Gal´ı, J., Monacelli, T., 2005. Monetary policy and exchange rate volatility in a small open economy. The Review of Economic Studies 72, 707–734. Hausmann, R., Panizza, U., Stein, E., 2002. Original sin, passthrough, and fear of floating. Financial Policies in Emerging Markets , 19–46. Jeanne, O., Korinek, A., 2010. Managing credit booms and busts: A Pigouvian taxation approach. Technical Report. National Bureau of Economic Research. Kitano, S., 2011. Capital controls and welfare. Journal of Macroeconomics 33, 700–710.

45

Klein, M.W., Shambaugh, J.C., 2008. The dynamics of exchange rate regimes: Fixes, floats, and flips. Journal of International Economics 75, 70–92. Klein, M.W., Shambaugh, J.C., 2015. Rounding the corners of the policy trilemma: sources of monetary policy autonomy. American Economic Journal: Macroeconomics 7, 33–66. Korinek, A., 2010. Regulating capital flows to emerging markets: An externality view. Available at SSRN 1330897 . Levy-Yeyati, E., Sturzenegger, F., 2005. Classifying exchange rate regimes: Deeds vs. words. European economic review 49, 1603–1635. Maggiori, M., Gabaix, X., 2015. International liquidity and exchange rate dynamics. Quarterly Journal of Economics 130. McKinnon, R.I., Pill, H., 1999. Exchange-rate regimes for emerging markets: moral hazard and international overborrowing. Oxford review of economic policy 15, 19–38. Mundell, R.A., 1963. Capital mobility and stabilization policy under fixed and flexible exchange rates. Canadian Journal of Economics and Political Science/Revue canadienne de economiques et science politique 29, 475–485. Obstfeld, M., 1986. Rational and self-fulfilling balance-of-payments crises. The American Economic Review 76, 72–81. Ostry, J.D., Ghosh, A.R., Habermeier, K., Chamon, M., Qureshi, M.S., Reinhardt, D., 2010. Capital inflows: The role of controls. Revista de Economia Institucional 12, 135–164. Popper, H., Mandilaras, A., Bird, G., 2013. Trilemma stability and international macroeconomic archetypes. European Economic Review 64, 181–193. Schmitt-Groh´e, S., Uribe, M., 2012. Prudential Policy For Peggers. Technical Report. National Bureau of Economic Research. Weber, W.E., 1986. Do sterilized interventions affect exchange rates? Quarterly Review , 14–23. Woodford, M., 2001. Inflation stabilization and welfare. Technical Report. National Bureau of Economic Research.

46

A

Appendix

A.1

Derivations of demand functions

Suppose CH,t (j) =

1

2 PH,t (j) −Ô PH,t

CH,t = = =

CH,t , then using (4), we have

Ô−1 B−Ô Ô Ú 1A P (j) H,t CH,t

PH,t

0

3

Ô−1 Ô

Ô−1 CH,t PH,t

Ô CH,t PH,t

Ú 1 0

3Ú 1

1−Ô

PH,t (j)

1−Ô

PH,t (j)

0

dj

dj

Ô Ô−1

dj

Ô 4 Ô−1

Ô 4 Ô−1

Since I previously defined in (14) that PH,t =

3Ú 1 0

3Ú 1

⇒

Ô PH,t

=

⇒

Ô PH,t

3Ú 1

Therefore CH,t (j) =

0

A

0

1−Ô

PH,t (j)

1−Ô

PH,t (j)

1−Ô

PH,t (j)

dj

dj dj

Ô 4 1−Ô

Ô 4 Ô−1

B−Ô

CH,t

B−Ô

Ci,t

PH,t (j) PH,t

1 4 1−Ô

=1

With a similar proof, Ci,t (j) =

A

Pi,t (j) Pi,t

47

1

Again, suppose Ci,t =

CF,t = = =

2 Pi,t −γ PF,t

CF,t , then using (5), we have

γ−1 B−γ γ Ú A 1 Pi,t CF,t

PF,t

0

3

γ−1 γ

γ−1 CF,t PF,t

γ CF,t PF,t

Ú 1

3Ú 1 0

0

di

γ 4 γ−1

1−γ Pi,t di

γ γ−1

γ 4 γ−1

1−γ Pi,t di

Since I previously defined in (15) that PF,t = ⇒ ⇒

0

=

γ PF,t

3Ú 1

Ci,t =

0

A

1 4 1−γ

1−γ Pi,t di

3Ú 1

γ PF,t

Therefore

A.2

3Ú 1

Pi,t PF,t

γ 4 1−γ

1−γ Pi,t di

0

γ 4 γ−1

1−γ Pi,t di

B−γ

CF,t

=1

Derivation of budget constraint

First term of (7) is Ú 1 0

PH,t (j)CH,t (j)dj = =

and (14) provides Ô PH,t

therefore

Ú 1 0

Ú 1 0

Ú 1 0

PH,t

A

Ô CH,t PH,t

PH,t (j) PH,t

Ú 1 0

B−Ô

PH,t (j)1−Ô dj

PH,t (j)1−Ô dj = PH,t

PH,t (j)CH,t (j)dj = PH,t CH,t

48

CH,t dj

Similarly,

Suppose that

s1 0

Ú 1 0

Pi,t (j)Ci,t (j)dj = Pi,t Ci,t

Pi,t Ci,t di = PF,t CF,t , then using (18), we have Ú 1 0

Pi,t

⇒ PF,t =

A

γ PF,t

1−γ ⇒ PF,t =

⇒ PF,t =

Pi,t PF,t

Ú 1 0

Ú 1

3Ú 1 0

B−γ

0

1−γ Pi,t di

1−γ Pi,t di 1 4 1−γ

1−γ Pi,t di

Since we defined in (15), we verify that the second term of (7) is Ú 1Ú 1 0

0

CF,t di = PF,t CF,t

s1 0

Pi,t Ci,t di = PF,t CF,t , and therefore

Pi,t (j)Ci,t (j)djdi = PF,t CF,t

The optimal allocation of expenditures between domestic and imported goods is given by CH,t

3

PH,t = (1 − α) Pt

4−η

Ct ;

3

PF,t =α Pt

CF,t

4−η

Ct .

(68)

Combining (9) and (68), then we have PH,t CH,t + PF,t CF,t = Pt Ct ,

(69)

and the proof is given as following: Suppose PH,t CH,t + PF,t CF,t = Pt Ct , then using (68), we have 3

PH,t PH,t (1 − α) Pt

4−η

3

PF,t Ct + PF,t α Pt

1−η η 1−η η ⇒ (1 − α)PH,t Pt + αPF,t P t = Pt 1−η 1−η ⇒ (1 − α)PH,t + αPF,t = Pt1−η

è

1−η 1−η ⇒ Pt = (1 − α)PH,t + αPF,t

49

é

1 1−η

,

4−η

Ct = Pt Ct

and therefore we verify that PH,t CH,t + PF,t CF,t = Pt Ct . Since we treat the rest of the world as symmetric countries, Ú 1 0

Ú 1 0

i ∗ Ei,t Dt+1 di = Et∗ Dt+1 ;

i ∗ ˜ t−1 ˜ t−1 Ei,t R (1 + τt )(1 + ξλt )Ψt Dti di = Et∗ R (1 + τt )(1 + ξλt )Ψt Dt∗

With the assumptions of τti = 0 (∀t) and Ψit = 0 (∀t) we have the following budget constraints, ∗ ∗ Pt Ct + Dt+1 +Et Dt+1 ≤ Wt Nt + Πt + Tt + Rt−1 Dt + Rt−1 Et Dt∗ ,

where ˜ ∗ (1 + τt )(1 + ξλt )Ψt . Rt∗ = R t

A.3

Derivation of home output

∗ Let CH,t (i) denote the world demand for the domestic good i. Then the aggregate demand for the domestic good i is ∗ Yt (i) = CH,t (i) + CH,t (i)

and then using the demand functions (17) and (29) and assuming the degrees of openness of home and world economies are the same α = α∗ , ∗ Yt (i) = CH,t (i) + CH,t (i)

=

A

PH,t (i) PH,t

=

A

=

A

PH,t (i) PH,t

=

A

PH,t (i) PH,t

PH,t (i) PH,t

B−Ô

PH,t (i) CH,t + PF,t

B−Ô 3

PH,t Pt

B−Ô 3 B−Ô

A

PH,t Pt

B−Ô

∗ CH,t

A

PH,t (1 − α)Ct + Et Pt∗

4−η

PH,t 1/σ (1 − α)Θt Ct∗ Qt + Et Pt∗

4 3 PH,t −η 1/σ Y∗ (1 − α)Θt Qt t

B−η

4−η

Pt

50

A

A

αYt∗

B−η

PH,t + Et Pt∗

B−η

αYt∗

α .

(70)

Plugging (70) into the definition of aggregate output Yt ≡ we have Yt = =

5Ú 1 0

1− 1Ô

Yt (i)

ès

1 0

1

é

Yt (i)1− Ô di

Ô 6 Ô−1

Ô Ô−1

,

di

B−Ô 3 4−η Ú 1 A PH,t (i) 1/σ ∗ PH,t Yt (1 − α)Θt Qt PH,t Pt 0 è

1/σ

= Yt∗ Stη Q−η t (1 − α)Θt Qt 5

1−ση σ

= Yt∗ Stη (1 − α)Θt Qt

+ Stη α

é

6

+

A

PH,t Et Pt∗

B−Ô Ô−1 Ú 1 A Ô P (i) H,t · PH,t 0

+α .

ü

ûú

=1

B−η

1− 1 Ô α

Ô Ô−1

di

ý

Ô Ô−1

di

(71)

(71), up to a first order approximation, can be rewritten as yt = yt∗ +

ωα st + (1 − α)θt , σ

(72)

where ωα ≡ 1 + α(2 − α)(ση − 1) > 0.

A.4 A.4.1

Derivation of the Loss Function Utility of consumption

Backus-Smith condition is log-linearized as following 1

Ct = Θt Ct∗ Qtσ ⇒

log Ct = log Θt + log Ct∗ + σ log Qt

⇒ ct = θt + c∗t + σqt ⇒ ct = θt + c∗t + qt

51

(∵ σ = 1)

(73)

Goods market condition is 3

PH,t Yt = (1 − α) Pt A

4−η

Ct + α

B

A

Ú 1A 0

PH,t i Ei,t PF,t

B

B−γ A

Pt Pt∗ ⇒ Yt = (1 − α) Ct + α Et Ct∗ PH,t PH,t

i PF,t Pti

B−η

Cti di

(∵ σ = η = γ = 1)

1

and since Ct = Θt Ct∗ Qtσ , A

B

A

B

Pt Pt∗ Yt = (1 − α) Θt Ct∗ Qt + α Et Ct∗ PH,t PH,t ⇒ Yt = (1 − α)St Θt Ct∗ + αSt Ct∗ ⇒ Yt = St Ct∗ [(1 − α)Θt + α] log Yt = log St + log Ct∗ + log [(1 − α)Θt + α] 1 ⇒ st = yt − c∗t − (1 − α)θt − α(1 − α)θt2 2 ⇒

(74)

Combining (12), (73), and (74), we have 1 ct = αc∗t + (1 − α)yt + α(2 − α)θt − α(1 − α)2 θt2 2

(75)

and its natural allocation is 1 yt + α(2 − α)θ¯t − α(1 − α)2 θ¯t2 , c¯t = αc∗t + (1 − α)¯ 2

(76)

then expression of home consumption in gap is 1 cˆt = ct − c¯t = (1 − α)xt + α(2 − α)θˆt − α(1 − α)2 (θt2 − θ¯t2 ) 2 1 = (1 − α)xt + α(2 − α)θˆt − α(1 − α)2 (θt − θ¯t )(θt + θ¯t ) 2 1 = (1 − α)xt + α(2 − α)θˆt − α(1 − α)2 θˆt (θˆt + 2θ¯t ) (∵ θt = θ¯t + θˆt ) 2 (77)

52

As σ = 1, the utility of consumption becomes Ct1−σ ≈ log(Ct ) = ct = c¯t + cˆt 1−σ 1 = c¯t + (1 − α)xt + α(2 − α)θˆt − α(1 − α)2 θˆt (θˆt + 2θ¯t ) 2

U (Ct ) =

A.4.2

(78)

Disutility of labor

The disutility of labor can be expressed by the second-order approximation ;

Nt1+φ 1 ¯t n V (Nt ) = ≈ V¯t + V¯tÍ N n2t ˆ t + (1 + φ)ˆ 1+φ 2

<

Labor market clearing condition (45) is now A B−Ô Yt Ú 1 PH,t (j) Nt = dj A 0 PH,t

⇒

log Nt =

B−Ô Ú 1A (j) P H,t log Yt − log A + log dj 0

⇒ n t = y t − at + u t ,

where ut ≡

PH,t

B−Ô Ú 1A P (j) H,t log dj 0

PH,t

Since the productivity is assumed to be stable over time and the following holds at steady state B−Ô Ú 1A (j) P H,t log dj 0

PH,t

= 0,

we have n ˆ t = xt + ut .

(79)

Then I can rewrite the second-order approximation to the disutility of labor as ; < Nt1+φ 1 Í ¯ 2 ¯ ¯ V (Nt ) = ≈ Vt + Vt Nt xt + ut + (1 + φ)xt . (80) 1+φ 2

53

A.4.3

Loss function

¯t = Under the optimal subsidy scheme assumed, the optimality condition V¯tÍ N 1 − α holds for all t. Then, the period utility is U (Ct ) − V (Nt )

î

1 1 α(1 − α)2 θˆt (θˆt + 2θ¯t ) − V¯t − (1 − α) xt + ut + (1 + φ)x2t 2 2 î ï 1 1 α(2 − α) = −(1 − α) ut + (1 + φ)x2t + α(1 − α)2 θˆt (θˆt + 2θ¯t ) − + t.i.p., 2 2 1−α = c¯t + (1 − α)xt + α(2 − α)θˆt −

ï

where t.i.p. denotes terms independent of policy, which include constant terms. Woodford (2001) shows Ô ut = 1 + vari {pH,t (i)} + o(||a||3 ), 2 where o(||a||3 ) refers to terms of third or higher order. Under Calvo pricing assumption, we have ∞ Ø

β t vari {pH,t (i)} =

t=0

where κ ≡ function:

(1−δ)(1−βδ) . δ

∞ 1Ø 2 β t πH,t , κ t=0

Then I get the following expression for the objective

I

J

∞ 1−α Ø Ô 2 α(2 − α) ˆ E0 πH,t + (1 + φ)x2t + α(1 − α)θˆt (θˆt + 2θ¯t ) − θt . L=− βt 2 κ 1−α t=0

Now I use a second order approximation of the budget constraint to replace the linear term θˆt in the objective function. A second order approximation for nxt is 4 3 1 nxt = −α θt + θt2 2 then, a second order approximation for the budget constraint can be expressed as following: < ; ∞ Ø 1 E0 β t θˆt + θˆt (θˆt + 2θ¯t ) = 0, 2 t=0 so we can replace the linear term in the objective function as following: ;

3

4

<

∞ Ô 2 1−α Ø 2−α L=− βt E0 πH,t + (1 + φ)x2t + α + 1 − α θˆt (θˆt + 2θ¯t ) 2 κ 1−α t=0

54

or up to a constant ∞ î ï (1 − α)(1 + φ) Ø 2 E0 L=− β t απ πH,t + x2t + αθ (θˆt + θ¯t )2 , 2 t=0

1

2

α 2−α Ô where απ = κ(1+φ) and αθ = 1+φ + 1 − α , and therefore the objective 1−α function that the social planner minimizes is expressed as following:

E0

∞ Ø t=0

β

t

;

<

1 1 1 2 απ πH,t + x2t + αθ (θˆt + θ¯t )2 . 2 2 2

55

Highlights of “Capital Controls and Foreign Exchange Market Intervention” JIMF 2019 102

In this manuscript, I develop a New Keynesian small open economy general equilibrium model that assesses the “middle-ground international monetary policies” of which Klein and Shambaugh (2015) empirically document the trends among emerging market economies. Two main innovations of the model are (1) managed exchange rate regime as a policy choice and (2) exchange rate risk premium that is endogenous to the regime choice. The model captures major traits between international monetary policies, and shows that middle-ground policies can be optimal in countries whose market is sensitive to exchange rate risks.

1