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On the theory of sterilized foreign exchange intervention Michael Kumhof International Monetary Fund, Modeling Unit, Research Department, Room 10-548E, 700 19th Street NW, Washington, DC 20431, USA

a r t i c l e in fo

abstract

Article history: Received 31 August 2009 Accepted 12 March 2010 Available online 18 April 2010

Standard theory ﬁnds that, given uncovered interest parity, sterilized foreign exchange intervention should not affect equilibrium prices and quantities. This paper shows that when, as in the data, taxation is not sufﬁciently ﬂexible in response to spending shocks, uncovered interest parity is replaced by a monotonically increasing relationship between the stock of domestic currency government debt and domestic interest rates. Sterilized intervention then becomes a second independent monetary policy instrument that affects portfolios, interest rates, exchange rates and consumption. It should be most effective in developing countries, where ﬁscal spending volatility is large and domestic currency government debt is small. & 2010 Elsevier B.V. All rights reserved.

JEL classiﬁcation: E42 F41 Keywords: Uncovered interest parity Imperfect asset substitutability Portfolio balance models Sterilized foreign exchange intervention

1. Introduction This paper presents a general equilibrium monetary portfolio choice model of a small open economy. The model emphasizes the critical roles of government debt and of government policies, and is therefore closely related to the portfolio balance literature of the 1980’s, which has since been criticized for its partial equilibrium approach. The paper applies the model to an analysis of the effects of government balance sheet operations on the currency composition of private sector portfolios, prices and allocations. The paper makes two key contributions. First, it shows that under plausible and empirically supported conditions on ﬁscal policy the currency composition of nominal bond portfolios remains determinate in general equilibrium, that is after taking account of the government budget constraint. In other words, domestic currency denominated government bonds become imperfect substitutes for foreign currency denominated assets in private sector portfolios, and the uncovered interest parity arbitrage condition is replaced by a portfolio balance relationship between interest rates that also depends on outstanding stocks of government debt. Speciﬁcally, in equilibrium the domestic interest rate increases monotonically, but at a decreasing rate, with the stock of domestic currency government debt. Second, the paper provides a detailed analysis of the monetary policy implications of a general equilibrium portfolio balance relationship. The implication for monetary policy in general is that it can make use of two independent policy instruments. It can, as usual, affect the path of the exchange rate and price level, via a target path for the nominal anchor. But in addition it can affect interest rates via balance sheet operations, with interest rates in turn affecting real allocations. The emphasis in this paper is on government balance sheet operations in debt instruments that leave base money unchanged, which are generally referred to as sterilized foreign exchange interventions. The implication of a portfolio balance relationship for balance sheet operations is that, unlike in standard open economy models, sterilized intervention Tel.: + 1 202 623 6769; fax: +1 202 623 8291.

E-mail address: [email protected] 0165-1889/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jedc.2010.04.005

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is not irrelevant for prices and allocations. The paper ﬁnds that it can have sizeable effects on optimal private sector portfolios, interest rates, exchange rate volatility and, to a lesser extent, on consumption and the level of the exchange rate. The paper is motivated by a tension between economic theory and practice on the question of sterilized foreign exchange intervention. Central bankers routinely intervene in foreign exchange markets with offsetting operations in domestic currency debt, with the intention of affecting exchange rates, interest rates and real activity without changing the money supply and therefore inﬂation. Their thinking, as we will show below, does receive some support from the more recent empirical literature, while in terms of theory it may reﬂect older versions of portfolio balance theory such as Henderson and Rogoff (1982), Kouri (1983), and Branson and Henderson (1985).1 But while the portfolio balance literature did stress the central role of cross-border holdings of nominal government debt, which has to be part of any model of ofﬁcial intervention, its partial equilibrium models did not incorporate the government budget constraint. The economics profession, such as Obstfeld and Rogoff (1996), has therefore challenged the validity of such models, and with it the effectiveness of sterilized foreign exchange intervention. We begin by summarizing the empirical literature and the theoretical critique, and then develop our model. The early empirical literature on the portfolio balance channel, especially Jurgensen (1983) and the survey in Edison (1993), concluded that sterilized interventions did not have a signiﬁcant impact on the behavior of exchange rates. However, the apparent success of concerted interventions following the September 1985 Plaza Accord led to renewed academic interest. The resulting literature is surveyed in Sarno and Taylor (2001), who conclude that studies done in the 1990s, a key one being Dominguez and Frankel (1993), have been generally supportive of a portfolio effect on the level, volatility and risk premium of exchange rates. An important problem for this literature has traditionally been that intervention data have not been disclosed publicly by the authorities. This has however recently started to change, so that Fatum and Hutchison (2003) and Ito (2002) were able to conduct thorough studies of the effects of ofﬁcial interventions in the US$/DM and US$/Yen markets.2 They use an event study approach that is better suited to the study of ofﬁcial interventions than a time-series approach, given that interventions are very sporadic and intense while exchange rates change continuously over time due to a multitude of reasons. Both papers ﬁnd strong evidence that sterilized intervention systematically affects the behavior of exchange rates. The most comprehensive theoretical critique of the portfolio balance literature is made in an important paper by Backus and Kehoe (1989) on sterilized foreign exchange intervention.3 Using only an arbitrage condition, they show that under complete asset markets, or under incomplete asset markets and a set of spanning conditions, changes in the currency composition of government debt require no offsetting changes in monetary and ﬁscal policies to satisfy both the government’s and households’ budget constraints. Consequently this ‘strong form’ of intervention is irrelevant for equilibrium allocations and prices. Weaker forms of government intervention in asset markets that do not satisfy sufﬁcient spanning conditions generally do require offsetting changes in monetary and/or ﬁscal policies to meet the government budget constraint. But because the impact of such ‘weak form’ interventions can as easily be attributed to these monetary and/or ﬁscal changes as to the intervention per se, sterilized intervention cannot be considered a separate, third policy instrument. This is a powerful theoretical argument. But to obtain it one needs to make the very strong assumption that arbitrary monetary and ﬁscal policies are available following asset market interventions, while in practice these policies are much more likely to either be exogenous or to follow rules that are completely independent of interventions. Our paper represents an exploration of such rules. This exercise can be interpreted as imposing additional constraints on the form of ‘weak form’ interventions. We can then ask how sterilized intervention affects equilibrium allocations and prices conditional on the form of these rules. In other words, we ask whether sterilized intervention can be effective as a second independent instrument of monetary policy, taking as given ﬁscal policy. Several papers such as Obstfeld (1982) and Grinols and Turnovsky (1994) have given a negative answer even to that narrower question. The latter show that while stochastic money growth gives rise to currency risk in partial equilibrium, this disappears once the ﬁscal use of stochastic seigniorage has been accounted for. In the general equilibrium of their model, domestic and foreign currency denominated assets are perfect substitutes, and a version of uncovered interest parity holds. Therefore, once a monetary policy rule is speciﬁed, sterilized intervention has no further effects on asset market equilibria. The particular form of the ﬁscal policy rule used by these authors, full lump-sum redistribution of all government net revenue, is critical for this result. However, while this is a convenient and frequently used assumption, it is extreme as a description of actual government behavior. In this paper we therefore move away from that assumption, by introducing ﬁscal spending shocks that are exogenous in that they are not automatically ﬁnanced by offsetting tax changes, but that instead induce exchange rate movements that revalue the government’s nominal liabilities. The evidence presented in Click (1998) strongly suggests that such shocks are an important feature of the data. In a large cross-section of countries he ﬁnds that most permanent government spending is ﬁnanced by conventional tax revenue. But transitory government spending (which has a high standard deviation in developing countries) is ﬁnanced mainly by seigniorage.

1 2 3

This theory was recently also used by Blanchard et al. (2005). See also Humpage (1999). See also Sargent and Smith (1988) on the irrelevance of open market operations in foreign currencies.

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Because the risk of exogenous ﬁscal spending shocks is undiversiﬁable at the aggregate level, and because foreign currency assets are not subject to the same risk, domestic currency government bonds and foreign currency assets become imperfect substitutes even in general equilibrium. Their portfolio share is determined by a portfolio balance equation that features the excess return of domestic currency government bonds over foreign currency assets in the numerator, and ﬁscally induced exchange rate volatility in the denominator. In this environment, sterilized intervention becomes an effective second instrument of monetary policy. Speciﬁcally, a purchase of foreign exchange that is sterilized through domestic government bond issuance, leaving the money supply unchanged, is associated with a higher domestic interest rate and a depreciated nominal exchange rate. The extent of the depreciation however depends on the interest elasticity of money demand, and can therefore be quite small for a cash-inadvance economy. We conclude this introduction by relating our paper to two other important literatures. One is the literature on interest rate risk premia. As shown in Lewis (1995), empirical risk premia have been both large in absolute value and highly variable in industrialized countries, and they are even larger in developing countries. An attempt at explaining that fact has to take into account both default and currency risk, but as is well known, it is very difﬁcult to empirically disentangle the effects of these two types of risk. The focus of this paper is theoretical, and it deals exclusively with the currency risk portion of risk premia.4,5 Engel (1992) and Stulz (1984) show that in ﬂexible price monetary models monetary volatility per se will not give rise to any currency risk premium. And Engel (1999), using the frameworks of Obstfeld and Rogoff (1998, 2000) and Devereux and Engel (1998), shows that sticky prices are required to generate a risk premium. But the source of the risk premium in such models is a hedging term that is due to the covariance of consumption and the exchange rate. This makes it difﬁcult to rationalize large absolute-value risk premia because consumption is not very variable. The recent theoretical literature has made great strides in incorporating portfolio theory into state of the art monetary dynamic general equilibrium models. Papers include Devereux and Sutherland (2007), Tille and van Wincoop (2008), Engel and Matsumoto (2009), and Coeurdacier et al. (2007). These papers take a different focus to the earlier portfolio balance literature, in that their agents only choose portfolios of private equity with country-speciﬁc return characteristics, or private debt whose return characteristics depend on these country-speciﬁc returns. While there is a role for government in setting nominal interest rates, there is no role for government asset market operations in nominal debt. In such settings imperfect asset substitutability is a given. The problem in the older portfolio balance literature however was that, for nominal government debt stocks in different currencies, imperfect asset substitutability cannot be taken as a given. Establishing the conditions under which this type of imperfect asset substitutability can hold is therefore an important task. The rest of the paper is organized as follows. Section 2 presents the model. Section 3 calibrates a baseline economy. Section 4 discusses the effects of sterilized foreign exchange intervention. Section 5 concludes. Mathematical details are presented in three appendices. 2. The model We consider a small open economy composed of a continuum of identical inﬁnitely lived households and a government. We use a continuous time stochastic monetary portfolio choice model to derive households’ optimal consumption and portfolio decisions.6,7 Government behavior is characterized by a set of ﬁscal and monetary policy rules. 2.1. Uncertainty We ﬁx a probability space ðO, ,PÞ: A stochastic process is a measurable function O½0,1Þ : /R. The value of a process X at time t is the random variable written as Xt. a r 0 There are four sources of risk in this economy. We deﬁne a three-dimensional Brownian motion Bt ¼ ½BM t Bt Bt , a to the growth rate of velocity a , and to the growth rate of the nominal money supply M , shocks B consisting of shocks BM t t t t shocks Brt to the growth rate of an international investment index Rbt . We also deﬁne a one-dimensional Brownian motion includes every event based on the history of the Wt that represents shocks to government spending dGt.8 The tribe BW t above four Brownian motion processes up to time t. We complete the probability space by assigning probabilities to 4 There is a well-established and growing literature on default risk. The early contributions include Eaton and Gersovitz (1981) and Aizenman (1989). More recent contributions include Kletzer and Wright (2000), Kehoe and Perri (2002), Uribe and Yue (2006) and Arellano (2008). 5 Our result that the elasticity of interest rates with respect to bond stocks is decreasing in bond stocks has to be seen in this light. The common intuition that the opposite should be true comes from considerations to do with endogenous default risk, that is with default risk that increases with the bond stock, and at an increasing rate. 6 Useful surveys of the technical aspects of stochastic optimal control are contained in Chow (1979), Fleming and Rishel (1975), Malliaris and Brock (1982), Karatzas and Shreve (1991), and Dufﬁe (1996). The seminal papers using this technique to analyze macroeconomic portfolio selection are Merton (1969, 1971) and Cox et al. (1985). 7 The main advantage of working in continuous time is that this approach yields very intuitive closed-form expressions for optimal portfolio shares. 8 Purely in theoretical terms, money supply shocks and government spending shocks would be sufﬁcient to derive the main results of the paper. The other two shocks are added for realism in calibration and simulation.

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subsets of events with zero probability. We deﬁne t to be the tribe generated by the union of BW and the null sets. This t leads to the standard ﬁltration ¼ f t : t Z0g. The difference between Bt-shocks and Wt-shocks is critical for our results. Speciﬁcally, all shocks affect the real returns on domestic currency denominated government debt through the exchange rate. They therefore affect the government budget constraint. The difference between Bt-shocks and Wt-shocks is in whether ﬁscal or monetary policy is exogenous, or active in the sense of Leeper (1991). For Bt-shocks we adopt the conventional assumption that money supply is exogenous while the ﬁscal policy response is endogenous, both in the sense that the government redistributes the resulting net ﬁscal revenue back to households instantaneously via lump-sum transfers, and in the sense that it balances its budget thereafter by way of the anticipated or drift component of lump-sum transfers. In contrast, for Wt-shocks ﬁscal policy is exogenous in the sense that the exchange rate adjusts instantaneously, and permanently, to balance the government’s budget following unanticipated shocks to government spending. This in turn implies that money is endogenous, speciﬁcally that the money supply is adjusted instantaneously to accommodate the exchange rate movement necessitated by ﬁscal balance. However, after this exchange rate jump ﬁscal policy is again assumed to be endogenous, or passive, in that it adjusts anticipated or drift taxation to ensure solvency. Money supply: The nominal money supply follows a geometric Brownian motion, with a drift process mt that is determined by the inﬂation target of monetary policy. There is an endogenous diffusion sgM,t with respect to Wt-shocks, and a r o process, Mt is a constant, exogenous three dimensional diffusion sM ¼ ½sM M sM sM with respect to Bt-shocks. Being an Itˆ continuous, which ensures exchange rate determinacy. We have dMt ¼ mt dt þ sM dBt þ sgM,t dW t : Mt

ð1Þ

We index endogenous drift and diffusion terms by time if they represent possibly time-varying monetary policy choices, or if they are functions of such choices. Velocity of money: The process for velocity is similar to (1), except that velocity does not respond to ﬁscal shocks dat

at

¼ n dt þ sa dBt :

ð2Þ

International investment returns: All assets other than domestic money and domestic currency government bonds are for simplicity treated as a homogenous asset class that we will refer to as internationally tradable assets, and which is denominated in foreign currency. Conceptually this includes domestic equity (whose nominal value moves one for one with the exchange rate), net foreign direct investment in equity, and international borrowing and lending in foreign currency. The real return on internationally tradable assets follows the process b

dr t ¼

b

dRt ¼ r dt þ sr dBt : Rbt

ð3Þ

It is assumed that the stochastic processes dlogðMt Þ, dlogðat Þ and dlogðRbt Þ are correlated with variance–covariance matrix S. Fiscal spending shocks: Government spending follows an Itˆo process with zero drift9 dGt ¼ sgg dW t , at

ð4Þ

where at denotes aggregate household wealth. Fiscal spending shocks affect the resources available for private consumption. In order for this to represent a risk to households in general equilibrium, it must be true that government consumption is an imperfect substitute for private consumption. We choose the simplest and most tractable assumption under which this is true, namely that government spending does not enter household utility.10 Exchange rates: The drift et and diffusions sE,t and sgE,t of the nominal exchange rate process Et are endogenously determined in general equilibrium as a function of the preceding four processes and of monetary and ﬁscal policy choices. All goods are tradable and the international price level is normalized to one. Assuming purchasing power parity, domestic goods prices Pt therefore satisfy Pt =Et. Thus, while our discussion will be in terms of the exchange rate, this is everywhere interchangeable with the price level. We have dEt ¼ et dt þ sE,t dBt þ sgE,t dW t : Et

ð5Þ

RT We assume and later verify that the endogenous drift and diffusion terms are adapted processes satisfying 0 jet jdt o 1, RT x 2 0 ðsE,t Þ dt o1 (x ¼ M, a,r,g) almost surely for each T. The corresponding conditions for all exogenous or policy determined drift and diffusion processes hold by assumption—the exogenous terms are constants and policy choices are assumed to be bounded. We use the following shorthand notation for diffusion terms, choosing terms relating to exchange rates and 9 A nonzero drift would affect feasible choices for the inﬂation target. But because this does not affect the presence or transmission mechanism of a portfolio channel, we ignore it without loss of generality. 10 This assumption would not seem to require an apology, as it is still the dominant choice in dynamic business cycle models.

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14

19

96 19 97 Q1 19 98 Q1 19 99 Q1 20 00 Q1 20 01 Q1 20 02 Q1 20 03 Q1 20 04 Q1 20 05 Q1 20 06 Q1 20 07 Q1 20 08

M1 to GDP Ratio

Q1

19 99 20 00 Q 1 20 01 Q 1 20 02 Q 1 20 03 Q 1 20 04 Q 1 20 05 Q 1 20 06 Q 1 20 07 Q 1 20 08 1

1

Q

Q

12 11.5 11 10.5 10 9.5 9 8.5 8 7.5 7

Peso Bonds to GDP Ratio

Q1

18 17 16 15 14 13 12 11 10 9 8

40

Share of Foreign Holdings of Peso Bonds

1407

12

35

10

30

1-month Interest Rate

25

8

20 6

15

4

98 19 99 Q1 20 00 Q1 20 01 Q1 20 02 Q1 20 03 Q1 20 04 Q1 20 05 Q1 20 06 Q1 20 07 Q1 20 08

19

Q1

Q1

19

96 19 Q1

Q1

08

07

nJa

06

nJa

05

nJa

04

nJa

03

nJa

02

nJa

01

nJa

nJa

nJa

nJa

00

0

99

5

0

97

10

2

Fig. 1. Mexican debt and interest rates, 1996Q1–2008Q2.

money as the example 2 a 2 r 2 ðsE,t Þ2 ¼ ðsM E,t Þ þ ðsE,t Þ þðsE,t Þ , M a a r r sM sE,t ¼ sM M sE,t þ sM sE,t þ sM sE,t , M a a r r ðsM sE,t Þ ¼ ½ðsM M sE,t Þ ðsM sE,t Þ ðsM sE,t Þ:

2.2. Households Preferences: The representative household has time-separable logarithmic preferences11 that depend on his expected lifetime path of tradable goods consumption fct g1 t¼0 Z 1 bt E0 e lnðct Þ dt, 0 o b o 1, ð6Þ 0

where E0 is the expectation at time 0, and b is the rate of time preference. Trading: Household consumption ct is ﬁnanced from the returns on three types of ﬁnancial assets: (1) domestic currency denominated money Mt with a zero nominal return, (2) domestic currency denominated government bonds Qt with a nominal return iqt dt, and (3) internationally tradable, foreign currency denominated assets bt with a real return drbt . We denote the real stocks of money and domestic bonds by mt =Mt/Et and qt =Qt/Et, and total private ﬁnancial wealth by at =mt +qt +bt. Portfolio q shares of money and domestic bonds will be denoted by nm t ¼ mt =at and nt ¼ qt =at . In order to determine the equilibrium portfolio share of domestic currency denominated assets in a small open economy, we follow Grinols and Turnovsky (1994) in assuming that these bonds are held exclusively by domestic residents. This is a realistic assumption for many developing countries, where the vast majority of claims held by foreigners tends to be denominated in dollars. The bottom left panel of Fig. 1 illustrates this for the case of Mexico, where in recent years only around 10% of all Peso-denominated government bonds have been held by foreigners, while in the earlier part of this decade that share was at a negligible 2%. Taxation: Households are subject to a lump-sum tax dTt levied as a proportion of wealth at. This tax follows an Itˆo process with adapted drift process tt and diffusion process sT,t dT t ¼ tt dt þ sT,t dBt : at

ð7Þ

11 Logarithmic preferences are commonly used in the open economy asset pricing and portfolio choice literature for their analytical tractability, see e.g. Stulz (1984, 1987) and Zapatero (1995).

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The drift and diffusion terms will be determined in equilibrium from a balanced budget requirement for the government. RT RT We assume that 0 jtt j dt o1 and 0 ðsT,t Þ2 dt o 1 almost surely for each T, and will later verify that this is satisﬁed in equilibrium. Note that taxes respond to Bt-shocks but not to Wt-shocks. Budget constraint: The household budget constraint is given by m

q

b

q q m dat ¼ at ½nm t dr t þnt dr t þ ð1nt nt Þ dr t ct dtat ½tt dt þ sT,t dBt ,

ð8Þ

where drit is the real rate of return on asset i. Using Itˆo’s lemma we can derive the real returns in terms of tradable goods on money and domestic currency bonds as (see Appendix A) m

dr t ¼ ðet þðsE,t Þ2 þ ðsgE,t Þ2 Þ dtsE,t dBt sgE,t dW t ,

ð9Þ

q

dr t ¼ ðiqt et þðsE,t Þ2 þðsgE,t Þ2 Þ dtsE,t dBt sgE,t dW t :

ð10Þ

Note that the exchange rate affects these returns in two ways. First, a depreciation such as sE,t dBt 40 reduces the realized (ex-post) real return in terms of tradables. Second, by Jensen’s inequality, larger exchange rate volatility ðsE,t Þ2 increases their expected (ex-ante) real return, because the real value of nominal assets is convex in the exchange rate. Cash constraint: Monetary portfolio choice models often introduce money into the utility function separably because this preserves the separability between portfolio and savings decisions found in Merton (1969, 1971). However, as pointed out by Feenstra (1986), without a positive cross partial between money and consumption the existence of money cannot be rationalized through transactions cost savings. We therefore use a cash constraint instead, and show that it is nevertheless possible to obtain tractable, and in fact elegant, analytical solutions. Speciﬁcally, consumers are required to hold real money balances equal to a time-varying multiple at of their consumption expenditures ct. We have ct ¼ at mt ¼ at nm t at :

ð11Þ

The now very common treatment of the cash-in-advance constraint in Lucas (1990) has two aspects, a cash requirement aspect and an in-advance aspect. Our own treatment goes back to the earlier Lucas (1982), which uses only the cash requirement aspect. This is due to the difﬁculty of implementing the in-advance timing conventions in a continuous-time framework. In the continuous time stochastic ﬁnance literature, Bakshi and Chen (1997) have used the same device. Portfolio problem: The household’s portfolio problem is to maximize present discounted lifetime utility by the q 1 appropriate portfolio choice fnm t ,nt gt ¼ 0 max

q 1 fnm t ,nt gt ¼ 0

s:t:

E0

Z

1 0

ebt lnðat nm t at Þ dt

g 2 g q q 2 2 dat ¼ at fðrtt Þ dt þ nm t ½ðat ret þ ðsE,t Þ þ ðsE,t Þ Þ dtsE,t dBt sE,t dW t þnt ½ðit ret þ ðsE,t Þ q þðsgE,t Þ2 Þ dtsE,t dBt sgE,t dW t þ ð1nm t nt Þsr dBt sT,t dBt g:

ð12Þ

We will solve this optimal portfolio problem recursively using a continuous time Bellman equation, as in Merton (1969, 1971). Let Vðat Þ ¼ ebt Jðat Þ 2 C 2 be a solution of the portfolio problem, and let Ja and Jaa be the ﬁrst and second derivatives of J(at). Then the Hamilton–Jacobi–Bellman equation solves n g 2 q q g 2 2 2 m bJ ¼ sup lnðat nm t at Þ þ Ja at ½ðrtt Þ þ nt ðat ret þ ðsE,t Þ þ ðsE,t Þ Þ þnt ðit ret þ ðsE,t Þ þ ðsE,t Þ Þ q nm t ,nt

1 q 2 g 2 q 2 q q 2 2 2 2 m m m Jaa a2t ½ðnm t þnt Þ ððsE,t Þ þ ðsE,t Þ Þ þ ð1nt nt Þ ðsr Þ þðsT,t Þ 2ðnt þ nt ÞsE,t sr þ 2ðnt þ nt Þ sE,t sr 2 q q m þ 2ðnm t þ nt ÞsE,t sT,t 2ð1nt nt Þsr sT,t ,

þ

ð13Þ

with boundary condition lim E0 ½ebt jJðat Þj ¼ 0:

ð14Þ

t-1

nqt

nm t

and are The ﬁrst order necessary conditions for optimality of Ja ðiq ret þ ðsE,t Þ2 þ ðsgE,t Þ2 Þ þðsr Þ2 þ sE,t sr sE,t sT,t sr sT,t Jaa at t q m nt þ nt ¼ , ððsE,t Þ2 þðsgE,t Þ2 þ ðsr Þ2 þ 2sE,t sr Þ 1 iqt : ¼ J 1 þ a at nm at t at

ð15Þ

ð16Þ

We will revisit these optimality conditions in Section 2.4 after solving for equilibrium taxation sT,t and the value function J.

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2.3. Government Monetary policy: Monetary policy is characterized by two policy variables and by a technical condition on the government budget. First, primary control over the level of depreciation/inﬂation is achieved through a target path for the nominal anchor consistent with an inﬂation target. In our model this is simply a target path fmt g1 t ¼ 0 for money in Eq. (1). Second, we will show that control of the volatility of depreciation/inﬂation, and control of interest rates, can be achieved by setting a target path for the stock of nominal government debt fQt g1 t ¼ 0 . Furthermore, under our assumptions there will be a monotonically increasing relationship between Qt and iqt for all iqt 4 0, so that there is an equivalent target path for the q nominal interest rate on government debt fiqt g1 t ¼ 0 . To show that the government can indeed control Qt (or it ) independently q of mt requires that we ﬁnd a determinate portfolio demand share for domestic currency bonds nt in Eq. (15), after endogenizing ﬁscal policy. We refer to operations in Qt as part of monetary policy because in practice they typically take the form of central bank balance sheet operations. Finally, we need a technical condition on the government budget. Discrete unanticipated policy changes will generally result in discontinuous jumps of the nominal exchange rate on impact,12 denoted E0 E0 . Here 0 stands for the instant before the announcement of a new policy at time 0. At such points the government could either spend the associated net seigniorage revenue, or it could fully redistribute it through a one-off net transfer of internationally tradable assets. We assume the latter, to ensure that private ﬁnancial wealth remains continuous upon the impact of any new policy. We denote internationally tradable assets held by the government by ht, and will refer to them by the more familiar term of foreign exchange reserves. We denote asset transfers to compensate exchange rate jumps by Dh0 ¼ Db0 .13 Then we can formalize the above as 1 1 ¼ Db0 ¼ Dh0 : ðM0 þQ0 Þ ð17Þ E0 E0 Fiscal policy: The exogenous, spending component of ﬁscal policy is speciﬁed in (4) and the endogenous, lump-sum tax component in (7). We assume that the latter meets three requirements. First, the expected budget balance is always zero. Second, the budgetary effects of shocks to money, velocity and international interest rates (Bt-shocks) are instantaneously offset by lump-sum taxes. Third, endogenous lump-sum taxes do not adjust to react to exogenous ﬁscal spending shocks (Wt -shocks). Instead, the budget balancing role in response to such shocks falls to the exchange rate. These assumptions take on board Click’s (1998) ﬁndings on the ﬁnancing of ﬁscal spending, while otherwise retaining the traditional lumpsum redistribution assumption as in Grinols and Turnovsky (1994). The government’s budget constraint is b

m

q

at tt dt þ at sT,t dBt þht dr t ¼ mt dr t þ qt dr t þ at sgg dW t :

ð18Þ

The assumption of expected budget balance, combined with instantaneous ﬁnancing of shocks through taxes or the exchange rate, implies that the government’s net wealth ht mt qt does not change over time. Therefore we have dht ¼ dmt þ dqt . For simplicity we also assume the initial condition h0 ¼ m0 þ q0 , which implies together with (17) that ht ¼ mt þqt

8t^0:

ð19Þ

Condition (19) therefore states that the government’s net domestic currency denominated debt, that is its bonds plus money, is fully backed by foreign exchange reserves.14 To determine the drift tt and diffusion sT,t of the tax process dTt, and the diffusion sgE,t of the exchange rate process, we equate terms in (18) using (3), (9), (10) and (19), and we use our three requirements for lump-sum taxes. We obtain the following q g 2 2 tt ¼ nqt iqt ðnm t þ nt Þðr þ et ðsE,t Þ ðsE,t Þ Þ,

ð20Þ

q sT,t ¼ ðnm t þ nt ÞðsE,t þ sr Þ,

ð21Þ

sgE,t ¼

g g : m ðnt þ nqt Þ

s

ð22Þ

The ﬁrst condition ensures that the expected budget balance is always zero. The second condition represents the response of lump-sum taxes to Bt-shocks. The third condition is critical. It represents the response of the exchange rate that is required to maintain budget balance following ﬁscal spending shocks (Wt-shocks). Fiscally induced exchange rate volatility is increasing in the volatility of the ﬁscal shocks themselves, but it is decreasing in the amount of nominal government debt 12

Subsequent discontinuous exchange rate jumps are ruled out by arbitrage. Note that Dh0 need not equal h0 h0 , because the policy itself may in addition involve the purchase or sale of foreign exchange reserves against domestic money or bonds at the new exchange rate E0. 14 This formulation treats government issued and central bank issued domestic currency bonds as perfect substitutes, so that qt could represent either debt class. Note that qt could be negative and represent government claims on the private sector. In that case ht could also be negative. 13

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held in household portfolios. This is because a larger stock of nominal debt that can be revalued by nominal exchange rate movements represents a larger base of the stochastic inﬂation tax.15 Deﬁnition 1. A feasible government policy is an initial net compensation Dh0 and a list of stochastic processes g m q 1 fmt ,Qt , tt , sT,t g1 t ¼ 0 such that, given a list of stochastic processes fet , sE,t , sE,t ,nt ,nt gt ¼ 0 , initial conditions b0 ,h0 ,M0 ,Q0 and E0 , and an initial exchange rate jump E0 E0 , the conditions (20), (21), (22) and (17) are satisﬁed at all times. In all our policy experiments in Section 4 we will assume that fmt ,Qt g1 t ¼ 0 are deterministic sequences. 2.4. Equilibrium, current account and interest rate differential

Deﬁnition 2. An equilibrium is a set of initial conditions b0 ,h0 ,M0 ,Q0 and E0 , exogenous stochastic processes 1 fBt ,Wt g1 t ¼ 0 , an allocation consisting of stochastic processes fct ,bt ,ht ,Qt ,Mt gt ¼ 0 , a price system consisting of an initial , and a feasible government policy such that, given the exchange rate jump E0 E0 and stochastic processes fet , sE,t , sgE,t g1 t¼0 initial conditions, the exogenous stochastic processes, the feasible government policy and the price system, the allocation solves households’ problem of maximizing (6) subject to (8) and (11). The condition ht ¼ mt þqt 8t ensures that at ¼ bt þ ht 8t, so that private assets at any point are equal to the economy’s net internationally tradable assets. Then the current account can be derived by consolidating households’ and the government’s budget constraints (8) and (18) dat ¼ ðrat ct Þdt þ at sr dBt at sgg dW t :

ð23Þ

Value function: We can now derive a closed form expression for the household value function. The solution proceeds by ﬁrst substituting (15), (16), (20), (21) and (22), which contain the terms Ja and Jaa , back into the Hamilton–Jacobi–Bellman equation (13). That equation is then solved for J by way of a conjecture and veriﬁcation. Given the logarithmic form of the utility function a good conjecture is Jðat Þ ¼ x½lnðat Þ þ lnðXðwt ÞÞ,

ð24Þ

where x and Xðwt Þ are to be determined in the process of verifying the conjecture, and where wt is a set of exogenous policy variables, parameters and shock processes. The veriﬁcation is presented in Appendix C. We show there that x¼b

1

:

ð25Þ

Using (25) and the equations deﬁning tax policy (20) and (21), the ﬁrst-order conditions (15) and (16) therefore become ct ¼

bat , ð1 þ ðiqt =at ÞÞ

nqt þ nm t ¼

ð26Þ

iqt ret þðsE,t Þ2 þ ðsgE,t Þ2 þ ðsr Þ2 þ sE,t sr ðsgE,t Þ2

:

ð27Þ

Eq. (26) is a standard condition in this model class. It states that consumption is proportional to wealth and, because of the cash constraint, negatively related to nominal interest rates. It implies the following money demand Mt bat ¼ : Et ðat þ iqt Þ

ð28Þ

The general equilibrium portfolio balance Eq. (27) is, together with (22), the key equation of this paper. It shows that the portfolio share of domestic currency denominated assets is determinate even after taxes have been endogenized.16 The following paragraphs discuss the economic intuition behind (27). Interest rate differential: Assume for the moment that the volatility of exogenous ﬁscal spending is zero (sgg ¼ 0). Then (27) in conjunction with (22) would imply that iqt ¼ r þ et ðsE,t Þ2 ðsr Þ2 sE,t sr :

ð29Þ

Endogenizing taxes in the complete absence of exogenous ﬁscal spending therefore results in a relationship between interest rates that is independent of asset stocks. Household risk aversion is reﬂected in the presence in (29) of Jensen’s inequality terms relating to exchange rate and interest rate volatility. Hedging terms are absent because of the assumption q of log utility. Because the portfolio share (nm t + nt ) is indeterminate, the currency composition of the government’s balance sheet is irrelevant. This is a version of the result of Grinols and Turnovsky (1994) and Backus and Kehoe (1989). 15 This result is consistent with the empirical results of Reinhart et al. (2003), who ﬁnd that inﬂation is consistently more variable in countries with a high degree of liability dollarization, including government liability dollarization. 16 We can ignore portfolio shares of domestic currency assets above one, because this would correspond to an economy where the government owns more than the economy’s entire means of production, with a government debt to GDP ratio of around 3000%. A realistic calibration typically is in the q neighborhood of nm t þ nt o 0:01.

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With exogenous ﬁscal spending shocks (sgg 4 0) we obtain a very different result. A spending shock dW t 4 0 is a net resource loss to households because government spending does not enter private utility. The government passes this loss on to holders of domestic currency denominated government debt through exchange rate movements, and this exchange rate risk is the source of imperfect asset substitutability. Eq. (29) becomes iqt ¼ r þ et ðsE,t Þ2 ðsr Þ2 sE,t sr ðsgg Þ2

q ð1nm t nt Þ q 2 ðnm t þ nt Þ

:

ð30Þ

This relationship features an additional portfolio risk term whose size depends on exogenous ﬁscal spending volatility sgg , q but also on the endogenous portfolio share of domestic currency assets nm t þ nt . The latter can, in this environment, be controlled through a second instrument of monetary policy. The portfolio risk term is a discount because more volatile exchange rates make domestic currency denominated nominal assets more attractive ceteris paribus, see the discussion following Eqs. (9)–(10). Second policy instrument: To understand the transmission channel of monetary policy in this economy, let us ﬁrst think of the interest rate iqt as the government’s policy instrument.17 There are two complementary effects of raising iqt . First, by Eq. (27) a higher iqt raises (ceteris paribus) the mean return on domestic government bonds, and therefore the portfolio q share ðnm t þ nt Þ. Second, by Eq. (22), a higher portfolio share further reduces the risk of holding domestic bonds, thereby reinforcing the effect of the higher mean return. The result is a monotonically increasing relationship between iqt and domestic currency government debt, and a monotonically decreasing relationship between iqt and exchange rate risk. But q we can say more, by noting that the portfolio discount in (30) is concave in ðnm t þ nt Þ, meaning it becomes smaller at a decreasing rate as domestic currency government debt increases. To understand this, note that in a typical calibration q g 2 ðnm t þ nt Þ o0:01, so that the portfolio discount is approximately equal to ﬁscally induced exchange rate volatility ðsE,t Þ , q þ n Þ. This is because there are decreasing returns, in which by Eq. (22) is decreasing and convex in the portfolio share ðnm t t the sense of lower exchange rate volatility, to expanding the base of the stochastic inﬂation tax. If we now think not of interest rates but of balance sheet operations in Qt as the policy instrument, the foregoing says that they are highly effective at changing interest rates at low levels of domestic currency government debt, and much less effective at high levels of debt. In other words, the relationship between domestic currency government debt and domestic interest rates is concave. Equilibrium system of equations: We now solve for the economy’s complete equilibrium system of equations, which will be used in Section 4 to simulate the effects of sterilized foreign exchange intervention. The interesting part of our model’s dynamics is fully captured by its instantaneous impulse responses, that is by its diffusions. Beyond that dynamic paths are not informative because the underlying state variables, most importantly household wealth, evolve as nonstationary Itˆo processes. We therefore instead perform an analysis of equilibria for a given set of state variables, which conveys all the necessary information because optimal portfolio shares and marginal propensities to consume out of wealth are independent of household wealth.18 We derive the ﬁrst part of the system of equations, which determines the drift and diffusions of the exchange rate process, in Appendix C. Denoting state variables by a bar above the respective variables, we have q b it mt þ net þ ðsE,t Þ2 þðsgE,t Þ2 sM sE,t sgM,t sgE,t þ sa sM sa sE,t ¼ r ð n þ s s s s Þ ð31Þ þ a M a E,t , ð1 þ ðiqt =a ÞÞ a þiqt

sM þ sa sr sE,t ¼ sgM,t sgE,t þ sgg ¼ 0, sgE,t ¼

g g m nt þ nqt

s

,

a M ¼ Et ct , ct ¼

ba , ð1 þðiqt =a ÞÞ

nm t ¼

M , Et a

iqt

a þiqt

sa ð3 equationsÞ,

ð32Þ

ð33Þ

ð34Þ

ð35Þ

ð36Þ

ð37Þ

17 With imperfect asset substitutability the policy problem can always be described as either ﬁxing the quantity of bonds and allowing the interest rate to clear the market, or as ﬁxing interest rates and then supplying as many bonds as the market demands at that interest rate. 18 Recall that condition (17) ensures that household wealth will be continuous at the time of implementation of a new policy.

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nqt ¼

Qt , Et a

q nm t þ nt ¼

ð38Þ iqt ret þðsE,t Þ2 þ ðsgE,t Þ2 þ ðsr Þ2 þðsr sE,t Þ ðsgE,t Þ2

:

ð39Þ

g g q a r m This set of 11 equilibrium Eqs. (31)–(39) determines the 11 endogenous variables et , sM E,t , sE,t , sE,t , sM,t , sE,t , Et , ct , nt , nt and iqt , given policy variables mt and Qt , state variables a, M, a , and exogenous diffusions sM , sa , sr , sgg . This system has unique solutions for all calibrations we have considered. This veriﬁes the regularity conditions posited earlier for et , sE,t , sgE,t , tt and sT,t .

3. Calibration To analyze the properties of the model we now turn to a calibrated example. We use quarterly Mexican data from the ﬁrst quarter of 1996 through the second quarter of 2008 (50 observations) to obtain empirically reasonable magnitudes for baseline money-to-GDP and debt-to-GDP ratios, and for the shock processes Mt and at . But the overall calibration exercise should not be interpreted as a full quantitative application of the model to Mexico, as several caveats mentioned below will make clear. For money supply we use data for M1, and for velocity the ratio of domestic absorption to M1. For the return on internationally tradable assets we use a Morgan Stanley MSCI Total Return Index for world stock markets, which measures price performance plus income from dividend payments. The speciﬁc index chosen assumes maximum possible dividend reinvestment and hedging into US dollars. This is consistent with thinking of the model’s exchange rate E as the Peso/$ rate. The fourth shock process, exogenous government spending dGt, cannot be estimated directly for two reasons. First, dGt is not modeled as a geometric Brownian motion, but instead as an Itˆo process that relates spending to aggregate wealth at, for which there is no easily identiﬁable counterpart in the data. Second, our theoretical concept of government spending excludes both a drift or positive mean spending, and spending volatility that induces an endogenous tax response instead of an exchange rate response. A meaningful decomposition of actual government spending along the lines of our model is beyond the scope of this paper.19 Instead we calibrate sgg based on an assumption about the fraction of money volatility that is driven by government spending, and we then perform sensitivity analysis by varying that fraction. Estimation of shock processes: Note that the three Bt shock processes can be rewritten, using Itˆo’s Law, as 1 1 ð40Þ dlogMt ¼ mt ðsM Þ2 ðsgM,t Þ2 dt þ sM dBt þ sgM,t dW t , 2 2 1 dlogat ¼ n ðsa Þ2 dt þ sa dBt , 2

ð41Þ

1 dlogRbt ¼ r ðsr Þ2 dt þ sr dBt : 2

ð42Þ

The model-consistent variance–covariance matrix between these shocks is 3 2 ðsM Þ2 þðsgM,t Þ2 sM sa sM sr 7 6 S¼6 sM sa ðsa Þ2 sa sr 7 5: 4 sM sr sa sr ðsr Þ2

ð43Þ

We want to use the maximum-likelihood estimates of this matrix to separately identify the nine diffusion processes sM , sa and sr .20 The difﬁculty is of course that the tenth component of this matrix, ðsgM,t Þ2 , is not separately identiﬁed. For our baseline calibration we therefore make the additional assumption ðsgM,t Þ2 ¼ ðsM Þ2 . This says that the accommodation of ﬁscal shocks accounts for 50% of the observed variability in money growth. We impose the further identifying restrictions a that the real interest rate is completely exogenous (sM r ¼ sr ¼ 0), and that the money supply responds instantaneously to M velocity shocks but not vice versa (sa ¼ 0). Finally the estimated diffusions, from (40)–(42), can be used to recover the drift a processes m, n and r. We obtain the following results: m ¼ 0:0713, n ¼ 0:0054, r ¼ 0:0343, sM M ¼ 0:0002, sM ¼ 0:0476, g r a r r sM ¼ 0:0233, sa ¼ 0:0521, sa ¼ 0:0116, sr ¼ 0:0703 and sM,t ¼ 0:0507. Our identiﬁcation assumptions therefore imply that virtually the entire money growth variance not accounted for by ﬁscal shock accommodation is due to an endogenous money supply response to velocity (mainly) and real interest rate changes. The signs of the responses are intuitive, with 19 This is closely related to the problem of distinguishing ﬁscally dominant and monetary dominant policy regimes in the data, see Canzoneri et al. (2001). 20 Identifying the shock processes requires estimating continuous-time diffusion processes from a discrete sample. Aı¨t-Sahalia (2002) shows that this can be quite complex in general settings, but Campbell et al. (1997, Ch. 9) show that geometric Brownian motion processes can be estimated in a straightforward fashion by maximum likelihood.

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Exchange Rate/Price Level

g

0.5 50

0

0 Initial Forex Intervention OMO

alpha M

−0.5

r Initial Forex Intervention OMO

0.8

6.699

9.7

9.65 Initial Forex Intervention OMO Domestic Bonds to GDP 20

0.6 All Shocks

6.698

9.75

E−Volatility x 100

Depreciation/Inflation 6.7

Domestic Money to GDP

E−Diffusions x 10

100

1413

15

0.4

6.697

0.2 G Shocks

6.696 Initial Forex Intervention OMO

10

Initial Forex Intervention OMO

Initial Forex Intervention OMO

Consumption

Domestic Assets to GDP

Nominal Interest Rates 9.8 9.6

0.02

uip

0.01

9.4

25

0 q

9.2

30

20

i

−0.01

Initial Forex Intervention OMO

Initial Forex Intervention OMO

Initial Forex Intervention OMO

Fig. 2. Foreign exchange intervention and open market operation.

money supply falling to accommodate higher velocity, and rising to accommodate the increase in consumption demand that follows higher investment returns. Fiscal spending volatility: To calibrate the key parameter sgg consistently with the model, we solve the calibrated baseline economy (31)–(39) taking sgM,t to be exogenous, and solve for sgg endogenously. We obtain sgg ¼ 0:0004. Our sensitivity analysis allows for higher and lower sgg to demonstrate the key importance of this parameter in creating imperfect asset substitutability and scope for balance sheet operations. To do so we ﬂip the system (31)–(39), holding sgg constant at the alternative value, and solving for sgM,t . This amounts to creating alternative economies in which the variance of money growth is higher or lower than in the baseline economy in order to accommodate different ﬁscal spending volatility. Benchmark calibration of state variables: To calibrate the state variables we ﬁrst normalize aggregate output to one by setting a ¼ 1=r. To ﬁx nominal asset stocks and velocity we normalize the nominal exchange rate to one and set the ratios to GDP of money and of domestic currency government bonds equal to their sample averages for Mexico, which equal 9.7% for money and 13.1% for bonds. Fig. 1 shows their time series.21 The implied velocity equals a ¼ 10:3, and the implied portfolio share of nominal assets is n =0.0078. Finally, we assume that the household discount rate is such that consumption equals output, which requires b ¼ 0:0347. The foregoing implies exchange rate depreciation (inﬂation) equal to e ¼ 0:0670, and a nominal interest rate of iq =0.0944. This is roughly equal to Mexican nominal interest rates over the second half of the sample period, as shown in Fig. 1. The ﬁrst half is clearly characterized by very high default risk premia following the Tequila crisis of 1994/1995. In our plots below we show the notional uncovered interest parity term r þ et ðsE,t Þ2 ðsr Þ2 sE,t sr of equations (29) and (30) as ‘‘uip’’, in order to illustrate the gap generated by the portfolio discount, which equals 0.25% in the baseline.

4. Sterilized foreign exchange intervention The two instruments available to monetary policy are ðmt ,Qt Þ or alternatively ðmt ,iqt Þ. We will hold mt constant at its estimated value, because the main subject of this paper is balance sheet operations, or alternatively interest rate policy that is independent of the target for the nominal anchor. Fig. 2 shows the effects of a sterilized foreign exchange intervention in detail. In each panel of this ﬁgure, we decompose the intervention into its two components, an unsterilized foreign exchange purchase and a subsequent domestic bond sale through an open market operation (OMO). The ﬁgure starts from our baseline calibration, holds all state variables, 21

The debt charts in Fig. 1 only start in 1999Q1 because consistent debt stock data were not available for the earlier period.

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parameters and exogenous diffusions including sgg at their baseline values throughout, and then solves the system of Eqs. (31)–(39) under different assumptions about the government balance sheet items h, Q and M. Asset stocks are shown in levels or as stock to GDP ratios, interest rates, inﬂation rates and volatilities are shown in annual percentage rates or annual percentage rate equivalents, and the exchange rate and consumption are shown in percent deviations from the baseline economy. The left half of each panel shows the effect of a doubling of the nominal money supply through an unsterilized purchase of foreign exchange from domestic households, speciﬁcally a purchase of internationally tradable assets h in exchange for money M. The economy’s net foreign asset position is unaffected by this transaction, so that the increase in h on the government’s balance sheet is matched by an equal decrease in b on households’ balance sheets. The right half of each panel shows how the economy reacts if this expansion of the money supply is reversed through a domestic open market sale to households, speciﬁcally a sale of domestic currency government bonds Q against money M. These two operations combined leave the money supply M constant while changing the currency composition of private portfolios. The unsterilized foreign exchange purchase causes a doubling of the exchange rate and price level. While this leaves the real money supply virtually unchanged, it reduces the real value of domestic bonds, so that the overall domestic currency government debt (bonds plus money) to GDP ratio falls from 22.8% to 16.2%. The equilibrium nominal (and real) interest rate drops by 0.25%, and because the government’s reduced domestic debt provides a smaller cushion against ﬁscal shocks, exchange rate volatility and therefore portfolio risk increases by almost half to 0.85% in annual interest equivalent terms.22 The lower nominal interest rate increases consumption through the cash constraint channel in (36), but this increase equals only 0.02%. The open market operation completely sterilizes the effects of the foreign exchange intervention on the money supply, so that the exchange rate depreciation is almost completely reversed. But other variables do not return to their baseline values. Most importantly, while real money balances are nearly unchanged after sterilization, the real quantity of outstanding domestic currency denominated government debt increases by almost half, from 22.8% to 32.3%. To place this additional debt the government must be willing to incur higher ﬁnancing costs, as the equilibrium interest rate increases by 0.13% above its pre-intervention value. At the same time the higher stock of domestic currency debt serves to reduce exchange rate volatility. To summarize, a sterilized intervention in our example has strong effects on the currency composition of private sector asset portfolios, sizeable effects on nominal and real interest rates, and fairly small but permanent effects on the exchange rate, which depreciates by 0.01%, and consumption, which experiences an overall contraction of 0.01%. We emphasize that the last two magnitudes are critically dependent on the assumed interest elasticity of consumption and of money demand. First, we note that the increasing relationship between domestic currency bond stocks and interest rates is largely independent of the form of money demand, as it depends mainly on the speciﬁcation of ﬁscal policy. But second, under a cash constraint any given interest rate change has only a very weak effect on consumption demand (26), and on money demand (28), where the latter in turn determines the response of the nominal exchange rate. The reason is that velocity a is very large relative to the interest rate ih. It is therefore useful to discuss how alternative speciﬁcations would impact the results. For consumption, replacing M1 with M2 in the calibration of velocity would make the effects of balance sheet operations on consumption considerably larger by resulting in a lower a, because M2 is much larger than M1. Note that the much stronger but transitory effects of real interest rates on consumption in conventional models depend on an intertemporal substitution channel. This is absent in our model, which features instantaneous adjustment to shocks without intrinsic dynamics. The changes in interest rates due to the portfolio channel are permanent rather than transitory, and their only effect on consumption operates through the cash constraint channel and is permanent. For the exchange rate, replacing the cash constraint with separable money in the utility function as in Grinols and Turnovsky (1994) (see however the discussion preceding (11)) would permit the calibration of a much higher interest elasticity of the demand for real balances. In that case the increase in interest rates following a sterilized foreign exchange purchase would, given an unchanged nominal money stock, translate into a much larger nominal exchange rate depreciation. Further exploring alternative speciﬁcations of money demand is therefore an important topic for future research. Figs. 3–5 illustrate the effects of sterilized intervention over a broader range, and without the decomposition of Fig. 2. This time we show the domestic currency government debt to GDP ratio along the horizontal axis, with a lower limit representing zero domestic currency bonds, so that the overall ratio equals the money to GDP ratio of 9.7%, and an upper limit representing just over 60% of domestic currency bonds and therefore a domestic currency debt to GDP ratio of 70%. We maintain this range in all three ﬁgures to facilitate comparison. The ﬁgures assume that, starting from our baseline calibration, which corresponds to the point 22.8 along the horizontal axis, the government makes a fully sterilized sale or purchase of nominal bonds against foreign exchange. All state variables, parameters and exogenous diffusions are held at their baseline values. Fig. 3 shows the results for our baseline calibration of ﬁscal volatility sgg ¼ 0:0004. The ﬁrst panel plots the relationship between the domestic currency debt to GDP ratio and the nominal stocks of money M and bonds Q, while the plot below

22

When reporting results we multiply variance terms by 100, like interest and inﬂation rates, to facilitate comparison.

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Nominal Debt and Money

E−Diffusions x 10 1

0.6

g

0.5

0.4

0.05

Q 0.2 0

Consumption 0.1

0 M

alpha M 0

10 20 30 40 50 60 70

r −0.5 10 20 30 40 50 60 70

Debt and Money to GDP

10 20 30 40 50 60 70

E−Volatility x 100

Exchange Rate/Price Level

60 1.5

0 All Shocks

40

1

Q/GDP

−0.05

20

0.5 M/GDP

G Shocks

10 20 30 40 50 60 70 Domestic Assets to GDP 70 60 50 40 30 20 10 10 20 30 40 50 60 70

−0.1 10 20 30 40 50 60 70

10 20 30 40 50 60 70 Nominal Interest Rates 9.5

uip

Depreciation/Inflation 6.704

iq

6.702 6.7

9

6.698 8.5

6.696

10 20 30 40 50 60 70

10 20 30 40 50 60 70 g g.

Fig. 3. Sterilized intervention—baseline s

Nominal Debt and Money

E−Diffusions x 10 0.5

0.6

0.02

0.4 Q

0

0.2 0

Consumption

M

alpha

g 0.01

M

0 −0.5

10 20 30 40 50 60 70

r 10 20 30 40 50 60 70

10 20 30 40 50 60 70

Debt and Money to GDP

E−Volatility x 100

Exchange Rate/Price Level

60 0.6 40 Q/GDP 20

0.4 0.2

−0.01 G Shocks −0.02

M/GDP 10 20 30 40 50 60 70 Domestic Assets to GDP 70 60 50 40 30 20 10 10 20 30 40 50 60 70

0

All Shocks

0 10 20 30 40 50 60 70

10 20 30 40 50 60 70

Nominal Interest Rates

Depreciation/Inflation

9.7

uip

iq

9.6 9.5 9.4

6.695 6.694 6.693 6.692

10 20 30 40 50 60 70 Fig. 4. Sterilized intervention—low s

10 20 30 40 50 60 70 g g.

1415

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0.4

2

0.6

0.3

0.4 Q

1

M

0

0.2 g

0.2 0

Consumption

E−Diffusions x 10

Nominal Debt and Money

M

0.1

alpha

0

10 20 30 40 50 60 70

r 10 20 30 40 50 60 70

10 20 30 40 50 60 70

Debt and Money to GDP

E−Volatility x 100

Exchange Rate/Price Level

60 5

0

4

40 Q/GDP

3

−0.1

All Shocks

−0.2

2

20 M/GDP 10 20 30 40 50 60 70 Domestic Assets to GDP 70 60 50 40 30 20 10 10 20 30 40 50 60 70

−0.3

1 G Shocks 10 20 30 40 50 60 70

−0.4 10 20 30 40 50 60 70 Depreciation/Inflation

Nominal Interest Rates 9

uip

q

i

6.72

8 6.71

7 6

6.7

5 10 20 30 40 50 60 70

10 20 30 40 50 60 70 g g.

Fig. 5. Sterilized intervention—high s

shows the corresponding ratios to GDP. Issuing more nominal bonds increases the real bond stock given that full sterilization keeps the nominal money stock constant, which keeps the nominal exchange rate nearly constant. The bottom middle panel plots the nominal interest rate against the domestic currency debt to GDP ratio. It shows the critical property of our model, a monotonically increasing and concave relationship between government debt and interest rates that arises endogenously. As the domestic debt ratio rises from 9.7% to 70%, the equilibrium interest rate iq that the government needs to pay to place this debt rises by 1.30%, from 8.38% to 9.68%. At the lowest domestic currency debt to GDP ratios a one percentage point increase in the government debt to GDP ratio leads to a 10 basis points increase in interest rates, while at the highest ratios it leads to a less than 1 basis point increase. Interestingly, over the relevant range these values are of the same order of magnitude as the empirical results, for the US, of Engen and Hubbard (2004), Laubach (2003) and Gale and Orszag (2004), who estimate that elasticity to be between 1 and 6 basis points. Of course for a developing country an expansion of domestic debt from 10% to 70% of GDP, even if backed by a simultaneous acquisition of foreign exchange reserves, would very likely lead to much larger increases in interest rates. But this would be due to higher perceived endogenous default risk, while our model isolates only the currency risk channel. The monotonically increasing relationship between government debt and interest rates is accompanied by a monotonically decreasing relationship between government debt and ﬁscally induced exchange rate volatility sgE (‘‘G Shocks’’ in the center panel). As the domestic currency government debt share rises from 9.7% to 70%, the base for the stochastic inﬂation tax expands, so that sgE falls from 1.34% in annual interest rate equivalent terms to almost zero. This lowers overall exchange rate volatility by about the same amount, because exchange rate volatility due to the three nonﬁscal shocks is nearly constant. As a result the portfolio discount in (30) declines to only 0.025% at the 70% debt share. We also note that at all debt shares considered, and especially at very low shares, the discount due to the portfolio channel is far larger than the Jensen’s inequality terms in (30).23 Fig. 3 shows that a higher interest rate also has a secondary effect on mean depreciation et , but this is negligible compared to the effect of the nominal anchor mt . Finally, the effects of higher interest rates on consumption and the nominal exchange rate are modest, with a 0.12% permanent drop in consumption (and depreciation) between the lowest and highest levels of government debt. Figs. 4 and 5 show how these results depend on the volatility of ﬁscal shocks, leaving all other model parameters unchanged. In Fig. 4 we see that, when the ﬁscal shock diffusion sgg is reduced by a factor of two to 0.0002, balance sheet

23 The absence of hedging terms in (30) is due to the assumption of log utility. However, as discussed in the Introduction, hedging terms are also generally small in size.

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operations over the same range as those reported in Fig. 3 have much smaller effects on interest rates and exchange rate volatility. Fig. 5 instead doubles sgg to 0.0008. This dramatically increases the effects of sterilized intervention, with interest rates increasing from 4.45% at the point of zero bond issuance to 9.60% at the maximum domestic currency debt to GDP ratio, and exchange rate volatility falling from 5.65% in annual interest equivalent terms to 0.45%. In this case exchange rate volatility at the actual government debt to GDP ratio is close to that in the Mexican data, which equals 2.5%.24 It also generates sizeable effects on consumption and the exchange rate, with consumption falling by 0.5% between the highest and lowest levels of government debt, and nominal exchange rates depreciating by the same amount. These magnitudes are signiﬁcant because they represent permanent changes. Fig. 4 suggests why empirical studies that focus on the price effects of sterilized intervention may have found little evidence for their effectiveness in industrialized countries. In such countries the ﬁscal situation is generally much more robust, and ﬁscal dominance is much less of a problem. Even if there was ﬁscal dominance, the ability of such countries to issue substantial amounts of domestic currency denominated debt means that the induced exchange rate volatility would be comparatively low. On the other hand, developing countries face the opposite scenario. As shown by Cata~ o and Terrones (2005), they face serious ﬁscal dominance problems. And the well-known work of Eichengreen and Hausmann (2005) documents that they have much more difﬁculty in issuing substantial stocks of domestic currency debt. In such countries sterilized intervention could be a more effective second tool of monetary policy.25 5. Conclusion We have studied a general equilibrium monetary portfolio choice model of a small open economy. The model emphasizes the importance of ﬁscal policy for the number and effectiveness of the instruments available to monetary policy, speciﬁcally for its ability to affect portfolios, prices and allocations through balance sheet operations known as sterilized foreign exchange intervention. Conventional theoretical results concerning the ineffectiveness of sterilized intervention depend on the strong assumption of full lump-sum redistribution of stochastic seigniorage income. We relax this assumption, by assuming that ﬁscal policy is not sufﬁciently ﬂexible in response to transitory government spending shocks, an assumption that is supported by empirical evidence. In such an environment the relationship between the rates of return on different currency assets is no longer correctly described by the uncovered interest parity arbitrage relationship but instead also depends on outstanding bond stocks. More precisely, a monotonically increasing and concave relationship between the outstanding stock of domestic currency government debt and nominal interest rates emerges as an equilibrium feature of the model. In this environment government balance sheet operations in domestic and foreign currency denominated assets become an effective second tool of monetary policy. They have sizeable effects on household portfolio choices, domestic interest rates and exchange rate volatility. If ﬁscal volatility is high enough and the outstanding stock of domestic currency government debt is low enough, they also have a signiﬁcant permanent effect on consumption and the level of the exchange rate, with a sterilized purchase of foreign exchange reducing consumption and depreciating the exchange rate.

Acknowledgments The views expressed here are those of the author, and do not necessarily reﬂect the position of the International Monetary Fund or any other institution with which the author is afﬁliated. The author thanks two referees of this journal, Gurdip Bakshi, Guillermo Calvo, Mick Devereux, Bernard Dumas, Ken Judd, Akito Matsumoto, Carmen Reinhart, Ken Rogoff, Tom Sargent, Martin Schneider, Stephen Turnovsky, Stijn van Nieuwerburgh, and seminar participants at Stanford University, the International Monetary Fund and the NBER for helpful comments. Appendix A. Returns on assets The return on real money balances is derived using Itˆo’s law to differentiate Mt =Et holding Mt constant 1 Mt 1 2Mt 2 m ¼ 2 Et ½et dt þ sE,t dBt þ sgE,t dW t þ mt dr t ¼ Mt d E ½ðsE,t Þ2 þ ðsgE,t Þ2 dt, Et 2 E3t t Et which yields the return m

dr t ¼ ðet þ ðsE,t Þ2 þ ðsgE,t Þ2 Þ dtsE,t dBt sgE,t dW t :

ðA:1Þ

24 Monetary business cycle models typically account for high exchange rate volatility using shocks to exogenous risk premia. Like endogenous default risk premia, these are absent from our model. 25 Of course, such government liability dollarization is otherwise not necessarily a blessing. Mishkin (2000) and Mishkin and Savastano (2001) discuss several problems that it creates for the conduct of monetary policy.

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The real return on the domestic bond is given by its nominal interest rate iqt , minus the change in the international value of domestic money as in (A.1). We have q

dr t ¼ ðiqt et þðsE,t Þ2 þðsgE,t Þ2 Þ dtsE,t dBt sgE,t dW t :

ðA:2Þ

The real return on internationally tradable assets is exogenous and given by (3), which is repeated here for completeness. b

dr t ¼ r dt þ sr dBt :

ðA:3Þ

Appendix B. The value function This Appendix veriﬁes the conjectured value function Vðat Þ ¼ ebt Jðat Þ ¼ ebt x½lnðat Þ þ lnðXðwt ÞÞ and derives closed form expressions for x and Xðwt Þ. Substitute the conjecture, the optimality condition (15) and (16), and the government policy rules (20), (21) and (22) into the Bellman Eq. (13). Then cancel terms to get x 2

bxlnðat Þ þ bxlnðXðwt ÞÞ ¼ lnðat ÞlnðxÞlnð1 þ ðiqt =at ÞÞ1=ð1 þ ðiqt =at ÞÞ þxr ½ðsr Þ2 þ ðsgg Þ2 : Equating terms on lnðat Þ yields x¼b

1

:

ðB:1Þ

This implies the ﬁrst-order conditions (27) and (26) shown in the paper. We are left with 1 2

blnðXðwt ÞÞ ¼ blnðbÞblnð1 þðiqt =at ÞÞb=ð1 þ ðiqt =at ÞÞ þ r ððsr Þ2 þ ðsgg Þ2 Þ:

ðB:2Þ

All terms on the right-hand side of (B.2) are, or can be expressed uniquely in terms of, exogenous policy variables, parameters or shock processes wt , as conjectured at the outset. Therefore X(.) is uniquely determined for each wt . Our approach has followed Dufﬁe’s (1996, chapter 9) discussion of optimal portfolio and consumption choice in that we have focused mainly on necessary conditions. This is because the existence of well-behaved solutions in a continuous-time setting is typically hard to prove in general terms. We have adopted the alternative approach of conjecturing a solution and then verifying it, and have found that our conjecture V(at) does solve the Hamilton–Jacobi–Bellman equation and is therefore a logical candidate for the value function. In the process of doing so we have also solved for the associated q m q m Þ are indeed optimal. feedback controls ðnt ,nt Þ and wealth process a t . We now verify that Vðat Þ and ðnt ,nt Note ﬁrst that our solutions solve the problem sup flnðat nm t at Þ þ DJðat Þg ¼ 0,

ðB:3Þ

q m q 2 1 DJðat Þ ¼ Ja ðat ÞFðat ,nm t ,nt Þ þ 2Jaa ðat Þ½Hðat ,nt ,nt Þ bJðat Þ:

ðB:4Þ

q nm t ,nt

where

The functions Fð:; :; :Þ and Hð:; :; :Þ are derived from the equilibrium evolution of wealth at in (23) given the conjectured form q m of the value function V(at) and given the associated feedback controls ðnt ,nt Þ. Let sa ¼ ½sr ðsgg Þ and Zt ¼ ½Bt Wt 0 . Then we have q m q dat ¼ Fðat ,nm t ,nt Þ dt þ Hðat ,nt ,nt Þ dZ t ,

where q m Fðat ,nm t ,nt Þ ¼ rat at nt at , q Hðat ,nm t ,nt Þ ¼ at sa , q 2 g 2 2 2 and where in line with our previous notation ½Hðat ,nm t ,nt Þ ¼ at ½ðsr Þ þ ðsg Þ . q ,n Þ be an arbitrary admissible control for initial wealth a Now let ðnm 0 and let at be the associated wealth process. By t t Itˆo’s formula, the stochastic integral for the evolution of the quantity ebt Jðat Þ can be written as Z t Z t q ebt Jðat Þ ¼ Jða0 Þ þ ebs DJðas Þ ds þ ebs cs dBs , where ct ¼ Ja ðat ÞHðat ,nm ðB:5Þ t ,nt Þ: 0

0

Rt We proceed to take limits and expectations of this equation. We show ﬁrst that E0 ð 0 ebs cs dBs Þ ¼ 0. To do so we need to Rt R t bs bs demonstrate that 0 e cs dBs is a martingale, which requires that e cs satisﬁes E0 ½ 0 ðebs cs Þ2 ds o 1,t 40. In our case, we have simply that ct ¼ sa =b. The condition is therefore satisﬁed, and we have that Z t Z 1 lim E0 febt Jðat Þg ¼ lim E0 Jða0 Þ þ ebs DJðas Þ ds ¼ Jða0 Þ þ ebs DJðas Þ ds: ðB:6Þ t-1

t-1

0

0

The transversality condition (14) can easily be veriﬁed because both wealth at and the term Xðwt Þ grow at most at an exponential rate. The left-hand side of (B.6) is therefore zero. Because the chosen control is arbitrary, (B.3)

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implies that DJðat Þ^lnðat nm t at Þ, and therefore Z Jða0 Þ^

1

0

ebs lnðas nm s as Þ ds:

ðB:7Þ

q m On the other hand, when we do the same calculation for our feedback controls ðnt ,nt Þ we arrive at (B.7) but with ^ replaced by an equality sign Z 1 m ebs lnðas ns as Þ ds: ðB:8Þ Jða0 Þ ¼ 0

We therefore conclude that Jða0 Þ dominates the value obtained from any other admissible control process, and that the q m controls ðnt ,nt Þ are indeed optimal. Appendix C. Equilibrium exchange rate dynamics To compute equilibrium exchange rate dynamics, we begin with the observation that consumption has to satisfy two equilibrium conditions. The ﬁrst is the cash constraint (11), and the second the consumption optimality condition (26). The latter in turn links the evolution of consumption and assets, and therefore needs to be analyzed in conjunction with Eq. (23). First, by Itˆo’s Law, (11) can be stochastically differentiated as dct ¼ at dmt þ mt dat þ dmt dat :

ðC:1Þ

Again by Itˆo’s Law, real money balances evolve as dmt ¼ mt ½mt et þðsE,t Þ2 þ ðsgE,t Þ2 sM sE,t sgM,t sgE,t dt þ mt ½sM sE,t dBt þ mt ½sgM,t sgE,t dW t :

ðC:2Þ

We substitute (2), (11) and (C.2) into (C.1) to obtain the evolution of consumption dct ¼ ½mt et þ n þðsE,t Þ2 þ ðsgE,t Þ2 sM sE,t sgM,t sgE,t þ sa sM sa sE,t dt þ ½sM sE,t þ sa dBt þ½sgM,t sgE,t dW t : ct

ðC:3Þ

Similarly, we stochastically differentiate the consumption optimality condition (26) and substitute (23) and (2). After simplifying, we obtain: q q

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