Dealing with the Trilemma: Optimal CapitalControls with Fixed Exchange Rates

Emmanuel FarhiHarvard University

Iván WerningMIT

April 2012Preliminary and Incomplete

Mundell’s trilemma states the incompatibility of free capital flows, independent mon-etary policy and a fixed exchange rate. We lay down a standard macroeconomicmodel of a small open economy in a currency union and study optimal capital con-trols. We provide sharp analytical and numerical characterizations. We find that cap-ital controls are employed to respond to some shocks but not others. We discuss howthe solution depends on the degree of nominal rigidity and the openness of the econ-omy. We find that capital controls are much more effective stabilization tools thangovernment spending. We also find that capital controls may be optimal even if theexchange rate is not fixed if wages, in addition to prices, are sticky. Finally, we com-pare the single country’s optimum to a coordinated solution for the union as a whole.Our results show a very limited need for coordination. In particular, the noncoopera-tive equilibrium features the same capital controls as the coordinated solution.

1 Introduction

Mundell’s trilemma states that a country cannot simultaneously have free capital flows,independent monetary policy, and a fixed exchange rate. Countries often maintain a fixedexchange rate, either by independent choice or as members of a currency union. Howthen should these countries cope with macroeconomic shocks? To what degree shouldthey give up on free capital mobility to regain monetary policy? Although the Interna-tional Monetary Fund has recently sided more sympathetically with the use of capitalcontrols (Ostry et al., 2010), we still lack a theoretical framework to answer these ques-tions. Our goal is to fill in this gap by studying optimal capital control policy in a standardopen economy model with fixed exchange rates.

Our model, which builds on Galí and Monacelli (2005, 2008), introduces capital con-trols in a standard open economy model with nominal rigidities. Goods are differentiated

1

by country and variety. Countries experiences different macroeconomic shocks. Theseshocks can be of several kinds, as we allow for productivity shocks, foreign demand forexports shocks, foreign consumption shocks and shocks to wealth—the latter capturesfluctuations in the price of commodities. We analyze different price setting assumptions:flexible prices, rigid prices, one-period in advance sticky prices, and Calvo price setting.We characterize the way a country should optimally use capital controls to maximize wel-fare in response to shocks. We show that this depends crucially on the nature of the shock,on the stickiness of prices, and on the openness of the economy. We also show that theirare gains from coordination. We characterize how countries would optimally use capitalcontrols if they could perfectly coordinate.

We first start with the case of flexible prices. Even with flexible prices, optimal capitalcontrols are generally nonzero, a point explored in detail in Costinot et al. (2011). Inpresent context, with a small open economy there is no ability to affect the world interestrate. However, each country still has some monopoly power over their terms of trade.Without trade barriers capital controls emerge as an imperfect tool to manipulate terms oftrade. By reallocating spending intertemporally a country can raise their export prices insome periods and lower them in others. This effect is not the focus of this paper. Indeed,we isolate a few cases where this effect is not at play: when shocks are permanent, orin the Cole-Obstfeld parametrization (unitary inter- and intra-temporal elasticities). Butmore generally, the case with flexible prices acts as a benchmark to compare our resultswith nominal rigidities.

We first contrast the case of flexible prices with its polar opposite: perfectly rigidprices. Thus, we assume that the exchange rate and prices are fixed forever. As before,capital controls are not employed in response to permanent shocks. In response to transi-tory shocks, however, capital controls are typically the reverse of what would be optimalwith flexible prices. A useful example is the Cole-Obstfeld case with a trend in produc-tivity. In this case, optimal capital controls are zero when prices are flexible, but they takethe form of a tax on inflows (or a subsidy on outflows) when prices are completely rigid.With flexible prices, the country’s price index decreases over time. This expected defla-tion raises the real interest rate and increases the growth rate of consumption. With rigidprices, the real interest rate is fixed and hence the growth rate of consumption is too low,inducing a boom relative to the flexible price allocation. By taxing inflows (or subsidizingoutflows) the country can increase its nominal interest rate, cooling off the economy. Thegrowth rate of output is also increased, but by less, moving the trade balance into surplus.In contrast, with flexible prices trade is always balanced in the Cole-Obstfeld case. Thisunderscores that capital controls are a second best tool, allowing the country to regain

2

some monetary autonomy and therefore some control over the intertemporal allocationof spending. However, this reallocation is costly since it introduces a wedge betweenthe intertemporal prices for home and foreign households. Moreover, capital controlscannot affect the composition of spending between home and foreign goods, which isdetermined by relative prices that are unchanged since the exchange rate and prices arefixed.

We then study the same model under the intermediate assumption that prices are setone period in advance, just as in Obstfeld and Rogoff (1995). This case is more com-plex for the following reason. As explained above, capital controls lead to intertemporalwealth transfers. Wealth transfers, in turn, lead to terms of trade and real exchange ratechanges—familiar at least since Keynes (1929) and Ohlin (1929). Taking these effects intoaccount complicates the analysis, yet we are able to provide tight characterizations forthe use of capital controls in this context, both for transitory and for permanent shocks ofdifferent kinds.

We then consider the case of Calvo price setting. In this setting, capital controls affectinflation dynamics in more complex way. This allows us to capture a prudential role ofcapital controls, where the government is concerned about the effect of temporary shockson absolute and relative prices. A temporary capital inflow, for example, may heat up theeconomy and affect prices in a way that is harmful once the flows are reverted.

With Calvo pricing, the planning problem is more involved because there is a now acost to inflation due to the induced price dispersion. Nevertheless, we are able to pro-vide analytical characterization in the limit of a closed economy and explore the solutionnumerically away from this limit case. In general, capital controls are employed counter-cyclically. One important insight is that the more closed the economy, the more effectiveare capital controls.

There is a growing theoretical literature in international macro demonstrating, amongother things, how restrictions on international capital flows may be welfare-enhancing inthe presence of various credit market imperfections; see e.g. Calvo and Mendoza (2000),Caballero and Lorenzoni (2007), Aoki et al. (2010), Jeanne and Korinek (2010), and Martinand Taddei (2010). An older literature emphasized the trilemma, see for example McK-innon and Oates (1966). To the best of our knowledge ours is the first paper to studyoptimal capital controls in a micro-founded model with nominal rigidities.

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2 A Small Open Economy

We build on the framework by Galí and Monacelli (2005, 2008), who develop a modelcomposed of a continuum of open economies. Our main focus is on policy in a singlecountry, which we call Home, taking as given the rest of the world, which we call Foreign.However, we also explore the joint policy problem for the entire world when coordinationis possible. We make a few departures from Gali and Monacelli’s model to adapt it to thequestions we address. First and foremost, they studied monetary and fiscal policy, so wemust extend the model to include capital controls. Second, in contrast to their simplifyingassumption of complete markets, we prefer to assume international financial markets areincomplete. No risk sharing between countries is allowed, only risk free borrowing andlending. Given this assumption, to keep the analysis tractable, we limit our attention toone-time unanticipated shocks to the economy.1 Third, Gali and Monacelli confine theirnormative analysis to the Cole-Obstfeld parameter specification, where various elastici-ties of substitution are unity. This case is more tractable and we will make extensive useof this fact here too. However, we also include some results outside of this parameterconfiguration. Finally, we also consider a larger variety of shocks to the economy andconsider extensions where we include wage rigidity alongside price stickiness.

2.1 Households

There is a continuum measure one of countries i ∈ [0, 1]. We focus attention on a singlecountry, which we call Home, and can be thought of as a particular value H ∈ [0, 1]. Inevery country, there is a representative household with preferences represented by theutility function

∞

∑t=0

βt

[C1−σ

t1− σ

− N1+φt

1 + φ

]

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

Ct =

[(1− α)

1η C

η−1η

H,t + α1η C

η−1η

F,t

] ηη−1

.

1Relative to the literature, this is not a limitation since most studies, including Gali-Monacelli, workwith linearized equilibrium conditions, so that the response to shocks is unaffected by the presence offuture shocks.

4

where CH,t is an index of consumption of domestic goods given by

CH,t =

(ˆ 1

0CH,t(j)

ε−1ε dj

) εε−1

,

where j ∈ [0, 1] denotes an individual good variety. Similarly, CF,t is a consumption indexof imported goods given by

CF,t =

(ˆ 1

0Λ

1γ

i,tCγ−1

γ

i,t di

) γγ−1

where Ci,t is, in turn, an index of the consumption of varieties of goods imported fromcountry i, given by

Ci,t =

(ˆ 1

0Ci,t(j)

ε−1ε dj

) εε−1

.

Thus, ε is the elasticity between varieties produced within a given country, η the elasticitybetween domestic and foreign goods, and γ the elasticity between goods produced indifferent foreign countries. An important special case obtains when σ = η = γ = 1. Wecall this the Cole-Obstfeld case, in reference to Cole and Obstfeld (1990). This case is moretractable and has some special implications that are worth highlighting. Thus, we devotespecial attention to it, although we will also derive results away from it.

The parameter α indexes the degree of home bias, and can be interpreted as a measureof openness. Consider both extremes: as α → 0 the share of foreign goods vanishes; asα → 1 the share of home goods vanishes. Since the country is infinitesimal, the lattercaptures a very open economy without home bias; the former a closed economy barelytrading with the outside world.

We have included a taste shifter Λi,t, which is always normalized so that´

Λi,tdi =

1, that affects the utility of imports from country i. All countries experience the sametaste configuration {Λi,t}. Thus, variations in ΛH,t allow us to consider variations in thedemand for Home’s exports.

Households seek to maximize their utility subject to the sequence of budget con-

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straints

ˆ 1

0PH,t(j)CH,t(j)dj +

ˆ 1

0

ˆ 1

0Pi,t(j)Ci,t(j)djdi + Dt+1 +

ˆ 1

0Ei,tDi

t+1di

≤WtNt + Πt + Tt + (1 + it−1)Dt +

ˆ 1

0

1 + τt−1

1 + τit−1

Eit(1 + ii

t−1)Ditdi

for t = 0, 1, 2, . . . In this inequality, PH,t(j) is the price of domestic variety j, Pi,t is theprice of variety j imported from country i, Wt is the nominal wage, Πt represents nom-inal profits and Tt is a nominal lump sum transfer. All these variables are expressed indomestic currency. The portfolio of home agents is composed of home and foreign bondholding: Dt is home bond holdings of home agents, Di

t is bond holdings of country i ofhome agents. The returns on these bonds are determined by the nominal interest ratein the home country it, the nominal interest rate ii

t in country i, and the evolution of thenominal exchange rate Ei,t between home and country i. Capital controls are modeled asfollows: τt is a tax on capital inflows and subsidy on capital outflows in the home country,and similarly τi

t is a tax on capital inflows and subsidy on capital outflows in country i.The proceeds of these taxes are rebated lump sum to the households at Home and countryi, respectively. The Home country taxes inflows to make the after tax (net of any subsidypaid by their own country of origin) return to foreign investors (1 + it−1)/(1 + τt−1) indomestic currency.2

We can re-express the household budget constraint as

PtCt + Dt+1 +

ˆ 1

0Ei,tDi

t+1di

≤WtNt + Πt + Tt + (1 + it−1)Dt +

ˆ 1

0

1 + τt−1

1 + τit−1

Eit(1 + ii

t−1)Ditdi,

2In other words, if we take after tax return to be (1− ϑt−1)(1 + it−1) then this represents a tax equal toϑt−1 = τt−1/(1 + τt−1). To see why this is required assume τi

t−1 = 0. Absence of arbitrage by domesticresidents requires

1 + it−1 = (1 + τt−1)Ei

tEi

t−1(1 + ii

t−1),

while absence of arbitrage for residents of country i requires

(1− ϑt−1)Ei

t−1

Eit

(1 + it−1) = 1 + iit−1.

The two are compatible if and only if ϑt−1 = τt−1/(1 + τt−1).

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using price indices

Pt = [(1− α)P1−ηH,t + αP1−η

F,t ]1

1−η ,

PH,t =

(ˆ 1

0PH,t(j)1−εdj

) 11−ε

,

PF,t =

(ˆ 1

0Λi,tP

1−γi,t di

) 11−γ

,

Pi,t =

(ˆ 1

0Pi,t(j)1−εdj

) 11−ε

.

Here Pt is the Home consumer price index (CPI), PH,t is the Home producer price index(PPI), PF,t is a price index of imported goods at Home, while Pi,t is country i’s PPI.

2.2 Firms

Technology. A typical firm in the home economy produces a differentiated good with alinear technology given by

Yt(j) = AtNt(j) (1)

where At is productivity in the home country. We denote productivity in country i byAt(i).

We allow for a constant employment tax 1 + τL, so that real marginal cost deflated byHome PPI is given by

MCt =1 + τL

At

Wt

PH,t. (2)

We take this employment tax to be constant in our model. We explain below how it isdetermined.

Price-setting assumptions. We will consider a variety of price setting assumptions: flex-ible prices, one-period in advance sticky prices, and sticky prices a la Calvo. Across allspecifications we will maintain the assumption that the Law of One Price (LOP) holdsso that at all times, the price of a given variety in different countries is identical once ex-pressed in the same currency. This assumption is sometimes known as Producer CurrencyPricing (PCP).

First, consider the case of flexible prices. Firm j optimally sets its price PH,t(j) to max-

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imizemaxPH,t(j)

(PH,t(j)Yt|t − PH,tMCtYt|t)

where

Yt|t =(

PH,t(j)PH,t

)−ε

Ct+k

Second, consider the case where prices are set one period in advance. Since we con-sider only one time-unanticipated shocks around the symmetric deterministic steadystate, this simply means that prices are fixed at t = 0 and flexible for t ≥ 1.

Third, consider Calvo price setting, where in every period, a randomly selected frac-tion 1− δ of firms can reset their prices. Those firms that get to reset their price choose areset price Pr

t to solve

maxPr

t

∞

∑k=0

δkΠkh=1

11 + it+h

(Prt Yt+k|t − PH,tMCtYt+k|t)

where

Yt+k|t =(

Prt

PH,t+k

)−ε

Ct+k

2.3 Market Clearing

Defining aggregate output to be Yt =(´ 1

0 Yt(j)ε−1

ε dj) ε

ε−1 we obtain the goods marketclearing condition

Yt = (1− α)

(PH,t

Pt

)−η

Ct + αΛH,t

ˆ 1

0

(PH,t

Ei,tPiF,t

)−γ(Pi

F,t

Pit

)−η

Citdi. (3)

Labor market clearing in the home economy is then

Nt =Yt

At

ˆ 1

0

(Pt(j)

Pt

)−ε

dj (4)

where Nt =´ 1

0 Nt(j)dj.

2.4 Terms of Trade, Nominal and Real Exchange Rates

Let Ei,t be nominal exchange rate between home and i (an increase in Ei,t is a deprecia-tion of the home currency). Because the Law of One Price holds, we can write Pi,t(j) =

Ei,tPii,t(j) and Pi,t = Ei,tPi

i,t where Pii,t(j) is country i’s price of variety j expressed in its own

8

currency, and Pii,t =

(´ 10 Pi

i,t(j)1−ε) 1

1−ε is country i’s domestic PPI (in terms of country i’scurrency, as opposed to Pi,t which is expressed in Home currency). We therefore have

PF,t = EtP∗t

where

P∗t =

(ˆ 1

0Λi,tP

i1−γi,t di

) 11−γ

is the world price index and

Et =

(´ 10 Λi,tE

1−γi,t Pi1−γ

i,t di) 1

1−γ

(´ 10 Pi1−γ

i,t di) 1

1−γ

is the effective nominal exchange rate.The effective terms of trade are defined by

St ≡PF,t

PH,t=

(ˆ 1

0Λi,tS

1−γi,t di

) 11−γ

(5)

where Si,t = Pi,t/PH,t is the terms of trade of home versus i. The terms of trade can thenbe expressed as

St =EtP∗tPH,t

. (6)

The terms of trade can be used to rewrite the home CPI as

Pt = PH,t[1− α + αSηt ]

11−η . (7)

Finally we can define the real exchange rate between home and i as Qi,t = Ei,tPit /Pt.

Similarly let the effective real exchange rate be

Qt =EtP∗t

Pt.

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2.5 First-Order Conditions

First-order conditions for the household problem. The first-order conditions for thehousehold problem include the labor-leisure condition

Cσt Nφ

t =Wt

Pt, (8)

the Euler equation

βC−σ

t+1

C−σt

Pt

Pt+1=

11 + it

, (9)

and the arbitrage conditions for all i ∈ [0, 1]

1 + it =1 + τt

1 + τit(1 + ii

t)Ei,t+1

Ei,t. (10)

This last equation indicates that capital control introduce a wedge in the Uncovered Inter-est Parity (UIP) equation—an observation that will play an important role in our analysis.Combining (9) and (10), we can derive a modified version of the Backus-Smith conditionin the presence capital controls:

Ct = ΘitC

itQ

1σi,t, (11)

whereΘi

t+1

Θit

=

(1 + τt

1 + τit

)1σ .

First-order conditions for the firm problem. The first-order conditions for the firmproblem depend on the price setting assumption.

If prices are flexible, then firm j sets its price PH,t(j) equal to a constant markup M =ε

ε−1 over nominal marginal cost 1+τL

AtWt so that

MCt =1M

.

If prices are set one period in advance, then prices are simply fixed at t = 0, and setflexibly as above for t ≥ 1.

If prices are set a la Calvo, then a firm that gets to reset its price at t chooses

Prt = M

∑∞k=0 δkΠk

h=11

1+it+hYt+k|t(PH,t+k MCt+k)

∑∞k=0 δkΠk

h=11

1+it+h

. (12)

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2.6 Equilibrium Conditions with Symmetric Rest of the World

We now summarize the equilibrium conditions. For simplicity of exposition, we focuson the case where all foreign countries are identical. Moreover, we assume that foreigncountries do not impose capital controls. We denote foreign variables with a star. Takingforeign variables as given, equilibrium in the home country can be described by the fol-lowing equations. We find it convenient to group these equations into two blocks, whichwe refer to as the demand block and the supply block.

The demand block is independent of the nature of price setting. It is composed of theBackus-Smith condition

Ct = ΘtC∗tQ1σt ,

the equation relating the real exchange rate to the terms of trade

Qt =[(1− α) (St)

η−1 + α] 1

η−1 ,

the goods market clearing condition

Yt = C∗t

[(1− α)ΘtQ

1σt

(Qt

St

)−η

+ αΛH,tSγt

],

the labor market clearing condition

Nt =Yt

At∆t

where ∆t is an index of price dispersion ∆t =´ 1

0

(PH,t(j)

PH,t

)−ε, the Euler equation

1 + it = β−1 Cσt+1

Cσt

Πt+1

where Πt =Pt+1

Ptis CPI inflation, the arbitrage condition between home and foreign bonds

Θσt+1

Θσt

=1 + it

1 + i∗t

Et

Et+1,

and the country budget constraint

NFAt = −C∗−σt

(S−1

t Yt −Q−1t Ct

)+ β(1 + τ∗t )NFAt+1

where NFAt is the country’s net foreign assets at t, which for convenience, we measure

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in the foreign price at home PF,t as the numeraire, and which we adjust by the foreignmarginal utility of consumption C∗−σ

t . The country budget constraint is derived fromthe consumer’s budget constraint after substituting out the lump-sum transfer. Undergovernment budget balance the transfer equals the sum of the revenue from the labor taxand the tax on foreign investors, net of the revenue lost to subsidize domestic residents’investments abroad.3

The supply block varies with the nature of price setting. With flexible prices, it boilsdown to the following condition which combines the household labor leisure condition(8) and the optimal markup condition

C−σt S−1

t Qt = M1 + τL

AtNφ

t

together with the no price dispersion assumption ∆t = 1. With one period in advanceprice stickiness, the only difference is that at t = 0, all price are fixed. This means thatS0 = E0

P∗0PH,0

where P∗0 and PH,0 are fixed. Finally with Calvo price setting, the supply blockis more complex. It is composed of the equations summarizing the first-order conditionfor optimal price setting

1− δΠε−1H,t

1− δ=

(Ft

Kt

)ε−1

,

Kt =ε

ε− 11 + τL

AtYtN

φt Πε

H,t + δβKt+1,

Ft = YtC−σt S−1

t QtΠε−1H,t + δβFt+1,

together with an equation determining the evolution of price dispersion

∆t = h(∆t−1, ΠH,t)

where h(∆, Π) = δ∆Πε + (1 − δ)(

1−δΠε−1

1−δ

) εε−1 , and the equation relating CPI and PPI

inflation

Πt = ΠH,t[(1− α) + αS1−η

t ]1

1−η

[(1− α) + αS1−ηt−1 ]

11−η

.

For most of the paper, we will be concerned with fixed exchange rate regimes (eitherpegs or currency unions) in which case we have the additional restriction that Et = E0 for

3Of course, we do not not require budget balance but since Ricardian equivalence holds here, all othergovernment financing schemes have the same implications.

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all t ≥ 0 where E0 is predetermined.

2.7 Labor Tax

We assume that each country chooses their labor tax optimally assuming a symmetricdeterministic steady state with flexible prices.

To solve for this optimal labor subsidy it is useful to consider the following relaxedproblem, which allows taxes to vary over time. As it turns out, at a steady state, theoptimal tax is constant. Assume that the world is at a symmetric deterministic steadystate. Each country takes the rest of the world as given and uses a time-varying tax oncapital inflow and subsidy on capital outflow controls τt and labor tax τL,t to maximizethe welfare of its households. For example, the home country solves

max{Ct,Yt,Nt,Θt,Qt,St}

∞

∑t=0

[C1−σ

t1− σ

− N1+φt

1 + φ

]

subject to

Ct = ΘtC∗Q1σt ,

Yt = C∗[(1− α)Q

1σ−ηt Sη

t Θt + αSγt

],

Qt =[(1− α)Sη−1

t + α] 1

η−1 ,

Nt =Yt

A,

0 =∞

∑t=0

βtC∗1−σ(

S−1t Yt −Q−1

t Ct

).

We can then back out the optimal labor tax using and the optimal capital controls using

S−1t Θ−σ

t C∗−σ = M1 + τL

tA

Nφt ,

Θt+1

Θt= (1 + τt)

1σ .

Using this program we can prove the following.

Proposition 1 (Steady State Tax). At a symmetric deterministic steady state with flexible priceswhere each country can set time-varying labor subsidies and capital controls, the optimal labor tax

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is given by τLt = 1

Mη+γ−1

(1−α)η+γ−1 − 1 and optimal capital controls are equal to zero.

Importantly, since the optimal labor tax is constant, this also solves the problem weare interested in: where the labor tax is required to be constant.

From each country’s perspective, the labor tax is the result of a balancing act betweenoffsetting the monopoly distortion of individual producers and exerting some monopolypower as a country. The two terms in the optimal tax formula reflect the two legs of thistradeoff. Reflecting the former leg of the tradeoff, the optimal labor tax is decreasing inthe degree of monopoly power of individual firms (M). Reflecting the second leg of thetradeoff, the optimal labor tax is increasing in the degree of openness (α), and decreasingin the elasticity of substitutions between home and foreign goods (η) as well as in theelasticity of substitution between goods of different countries (γ).

3 Perfectly Flexible and Perfectly Rigid Prices

In this section, we focus on two extreme cases: perfectly flexible or perfectly rigid prices.To simplify the exposition, in this section and in the next we assume that the economy isinitially at the deterministic symmetric steady state. This assumption plays no materialrole in our results.

We consider shocks of four kinds: productivity shocks (shocks to {At}), export de-mand shocks (shocks to {ΛH,t}), foreign consumption shocks (shocks to {C∗t }) and wealthshocks (shocks to NFA0). The only shocks that require some elaboration are the wealthshocks. In the small open economy context, there are many interpretations to the shockto initial net foreign asset position NFA0. It may capture the return on investments inrisky assets, the default on debt held abroad, etc. Another interesting interpretation isthe following. Suppose in addition to the goods described previously, the Home countryowns an endowment of a commodity good Xt (e.g. oil, copper, soybeans, etc.) that it doesnot consume and exports to international markets taking the price PX,t as given in foreigncurrency. Thus, the budget constraint becomes

PtCt + Dt+1 +

ˆ 1

0Ei,tDi

t+1di

≤WtNt + Πt + E∗t P∗X,tXt + Tt + (1 + it−1)Dt +

ˆ 1

0

1 + τt−1

1 + τit−1

Eit(1 + ii

t−1)Dit−1.

where E∗t is the exchange rate against the reference country for which the price P∗X,t isquoted. The only new element is the presence of E∗t P∗X,tXt on the right hand side of this

14

constraint. The model is then summarized by the same equilibrium conditions as beforeif we reinterpret NFAt as also capturing the present value (discounting using the foreigninterest rate) of the revenue from exports of this commodity. Under this interpretation, ashock to the price path {P∗X,t} or the endowment path {Xt} can be captured by it’s impacton the present value ∑∞

t=0 βC∗−σt P∗X,tXt as a shock to NFA0.

We characterize the optimal use of capital controls from an individual country’s per-spective in response to these different shocks, assuming that other countries do not usecapital controls.

3.1 Flexible Prices

We start with the case of flexible prices. We can write the corresponding planning problemas

maxΘt

∞

∑t=0

βtC∗1−σt

Θ1−σ

t Q1−σ

σt

1− σ− 1

1 + φ

1M(1 + τL)

Θ−σt

[(1− α)Q

1σ−ηt Sη−1

t Θt + αΛtSγ−1t

]

subject to

0 =∞

∑t=0

βtC∗1−σt α

(Sγ−1

t Λt −Q1σ−ηt Θt

),

1 = M(1 + τL)1

A1+φt

S1+φt C∗φ+σ

t Θσt

[(1− α)Q

1σ−ηt Sη−1

t Θt + αΛtSγ−1t

]φ

,

Qt =[(1− α)Sη−1

t + α] 1

η−1 .

In this problem if the sequence of productivity and export demand shocks {At, ΛH,t} areconstant, then the solution is constant, as long as the program is sufficiently convex. Wehave verified convexity in the Cole-Obstfeld case. To simplify, we assume it holds awayfrom this case.

Interestingly, even with flexible prices, when the paths for {At}, {ΛH,t} are not con-stant, it is generally optimal to use capital controls. Optimal capital controls can be in-ferred by taking the first-order conditions of the planning problem above and using thefact that Θt+1

Θt= (1 + τt)

1σ .

Proposition 2 (Flexible Prices). Suppose prices are flexible. Then in general, optimal capitalcontrols are non zero. Optimal capital controls are equal to zero for permanent shocks to produc-tivity At = A′ for all t ≥ 0 where A′ 6= A, for permanent export demand shocks ΛH,t = ΛH

15

for all t ≥ 0 where ΛH 6= 1, for permanent foreign consumption shocks C∗t = C∗′ for all t ≥ 0where C∗′ 6= C∗, and for wealth shocks W0 6= 0. In the Cole-Obstfeld case σ = η = γ = 1,optimal capital controls are equal to zero in response to arbitrary productivity shock {At} or for-eign consumption shocks {C∗t }, and optimal capital controls τt have the same sign as Λt+1 −Λt

in response to export demand shocks.

Proof. The first part of the proposition follows immediately from the planning problem,provided the program is sufficiently convex, as we assume. We therefore focus on theCole-Obstfeld case. Then the first order condition for the planning problem is

φ + α

1 + φ− (1− α)2φ

1 + φ

Θt

αΛt + (1− α)Θt+

(1− α)αΛt

1 + φ

1Θt− ΓΘt = 0

where Γ is chosen such that

α∞

∑t=0

βt(Θt −Λt) = NFA0.

For a given value of Γ, then Θt is independent of At and increasing in Λt. Since Θt+1Θt

=

(1 + τt)1σ the result follows.

The fact that capital controls are in general useful even though prices are flexible mightseem surprising given the fact that we are considering a small open economy, with noability to affect the world interest rate. This can be understood by noting that capitalcontrols, by allowing a country to reallocate demand intertemporally, allow this countryto manipulate its terms of trade, raising them in some periods and lowering them inothers. This margin is in general useful, unless of course the shocks are permanent, inwhich case no gain can be reached by engaging in this kind of intertemporal terms oftrade manipulation. Moreover, in the Cole-Obstfeld case, it is optimal not to use capitalcontrols in response to any path of productivity or foreign consumption shocks or towealth shocks.

3.2 Rigid Prices

After having considered the case of perfectly flexible prices, we now consider the oppositeextreme case where prices are entirely rigid (fixed for all t ≥ 0). This implies that St =

Qt = 1 for all t ≥ 0. The planning problem is now

16

max∞

∑t=0

βt

[Θ1−σ

tC∗1−σ

t1− σ

− (αΛt + Θt(1− α))1+φ A−(1+φ)t C∗1+φ

t1 + φ

]

subject to,

0 =∞

∑t=0

βtC∗1−σt α [Λt −Θt] .

Proposition 3 (Rigid Prices). Suppose that prices are entirely rigid. Optimal capital controls τt

have the same sign as At+1 − At in response to productivity shocks. Optimal capital controls τt

have the opposite sign as Λt+1−Λt in response to export demand shocks. Optimal capital controlsτt have the opposite sign as C∗t+1−C∗t in response to foreign consumption shocks. Optimal capitalcontrols are zero in response to initial wealth shocks.

Proof. The problem is convex: it features a concave objective and linear constraint in Θt.Putting a multiplier Γ > 0 on the left-hand side of the budget constraint, the necessaryand sufficient first-order condition is

Θ−σt − (1− α) (αΛt + Θt(1− α))φ A−(1+φ)

t C∗σ+φt − Γα = 0.

The results in the proposition follow.

It is interesting to contrast the use of capital controls when prices are flexible withtheir use when prices are entirely rigid. There are both similarities and differences. Inboth cases, capital controls are not used in response to permanent productivity, exportdemand, or initial wealth shocks. But the use of capital controls in response to transitoryshocks is quite different. For example, at least in the Cole-Obstfeld case, in response toproductivity shocks or foreign consumption shocks, capital controls are zero when pricesare flexible, but have the opposite sign as C∗t+1 − C∗t or the same sign as At+1 − At whenprices are entirely rigid. In response to export demand shocks, optimal capital controlsτt have the same sign as Λt+1 − Λt when prices are flexible, but the opposite sign whenprices are entirely rigid.

A useful example is the Cole-Obstfeld case when when productivity is increasing overtime. In this case, optimal capital controls are zero when prices are flexible while they takethe form of a tax on inflows / subsidy on outflows when prices are rigid. With flexibleprices, the country’s price index decreases over time. This expected deflation raises thereal interest rate and increases the growth rate of consumption. With rigid prices, the realinterest rate is fixed and hence the growth rate of consumption is too low compared tothe flexible price allocation. By taxing inflows / subsidizing outflows the country can in-crease its nominal interest rate and reduce the growth rate of consumption. In the process,

17

the growth rate of output is also decreased, but less so, so that the county’s trade balancemoves towards surplus. This would not happen with flexible prices where trade wouldremain balanced. This underscores that capital controls are a second best tool. They allowthe country to regain some monetary autonomy and therefore some control over the in-tertemporal allocation of spending. However, this reallocation is costly since it introducesa wedge between the intertemporal prices for home and foreign households. Moreover,capital controls cannot affect the division of spending between home and foreign goodswhen prices are completely rigid.

4 One Period in Advance Price Stickiness

We use dynamic programming to break down the planning problem into two, at t = 0and after, with net foreign assets as the only endogenous state variable.

For t ≥ 1, the planning problem is

V(NFA1) = max{Ct,Θt,Nt,Yt,St,Qt}∞

t=1

∞

∑t=1

βt−1

[C1−σ

t1− σ

− N1+φt

1 + φ

]

subject to

Ct = ΘtC∗tQ1σt ,

Yt = C∗t

[(1− α)Q

1σ−ηt Sη

t Θt + αΛH,tSγt

],

Qt =[(1− α) (St)

η−1 + α] 1

η−1 ,

Nt =Yt

At,

C−σt S−1

t Qt = M1 + τL

AtNφ

t ,

for all t ≥ 1 and

NFA1 = −∞

∑t=0

βtC∗−σt

(S−1

t Yt −Q−1t Ct

).

The t = 0 planning problem is

maxC0,Θ0,N0,Y0,NFA1

(C1−σ

01− σ

− N1+φ0

1 + φ+ βV(NFA1)

)

18

subject toC0 = Θ0C∗0 ,

Y0 = C0

[(1− α) + αΛ0Θ−1

0

],

N0 =Y0

A0,

NFA0 = −C∗−σ0 (Y0 − C0) + βNFA1,

where we have used the fact that S0 and Q0 are given with S0 = Q0 = 1.With one-period in ahead sticky prices, we are able to provide tight results for tem-

porary productivity and export demand shocks, as well as for permanent productivity,export demand, and wealth shocks in the Cole-Obstfeld case.

Proposition 4 (Transitory Shocks with Sticky Prices). Suppose that prices are sticky one periodin advance. Then in response to a transitory productivity shock A0 6= A and At = A for all t ≥ 1,optimal capital controls τt are zero for t ≥ 1, and optimal capital controls τ0 are a decreasingfunction of A0 with τ0 = 0 when A = A0. In response to a transitory export demand shockΛH,0 6= 1 and ΛH,t = 1 for all t ≥ 1, in the limit of small time intervals (β→ 1), optimal capitalcontrols τt are zero for t ≥ 1, and optimal capital controls τ0 are an increasing function of ΛH,0

with τ0 = 0 when ΛH,0 = 1. In response to a transitory foreign consumption shock C∗0 6= C∗ andC∗t = C∗ for all t ≥ 1, when σ = 1, optimal capital controls τt are zero for t ≥ 1, and optimalcapital controls τ0 have the same sign as C∗0 − 1.

Proof. We start with transitory productivity shocks. The optimal allocation is constantfor t ≥ 1. Let Θ be the corresponding value of Θ, which is related to Θ0 by (Θ− 1) =1−β

β (1−Θ0). This immediately implies that τt = 0 for t ≥ 1. Then we need to solve thet = 0 problem maxΘ0 U(Θ0, A0) where

U(Θ0, A0) = C∗1−σ Θ1−σ0

1− σ− C∗1+φ

A1+φ0

[(1− α)Θ0 + α]1+φ

1 + φ+ βV

(1β

C∗1−σα (1−Θ0)

).

It is easily verified that UA0,Θ0 > 0. This implies that Θ0 is an increasing function ofA0. This in turn implies that Θ is a decreasing function of A0. The first result in theproposition follows since capital controls τ0 are given by Θ

Θ0= (1 + τ0)

1σ .

Now consider temporary export demand shocks. The optimal allocation is constantallocation for t ≥ 1, so we drop the t subscripts. This immediately implies that τt =

0 for t ≥ 1. We are interested in the limit of small time intervals (β → 1). We have1−β

β α (Λ0 −Θ0) = α(Q 1

σ−ηΘ− Sγ−1Λ)

. Using the other constraints, we can write the

19

right hand side as a function J(Θ) with J(1) = 0. We find that to a first order in 1− β wecan write

Θ = 1 +1

J′(1)(1− β)α (Λ0 −Θ0) + O(1− β)2.

We can then write welfare for t ≥ 1 as a function H(Θ). We find that up to second orderterms in 1− β, we have to solve maxΘ0 U(Θ0, Λ0) where

U(Θ0, Λ0) = C∗1−σ Θ1−σ0

1− σ− C∗1+φ

A1+φ

[(1− α)Θ0 + αΛ0]1+φ

1 + φ+

H′(1)J′(1)

α (Λ0 −Θ0) .

It is easily verified that UΛ0,Θ0 < 0. This implies that Θ0 is a decreasing function ofΛH,0. Moreover, up to first order terms in 1− β, Θ is constant. The second result in theproposition follows since capital controls τ0 are given by Θ

Θ0= (1 + τ0)

1σ .

Finally consider foreign consumption shocks. The optimal allocation is constant fort ≥ 1. Let Θ be the corresponding value of Θ, which is related to Θ0 by C∗−σ (Θ− 1) =1−β

β C∗−σ0 (1 − Θ0). This immediately implies that τt = 0 for t ≥ 1. We have to solve

maxΘ0 U(Θ0, Λ0) where

U(Θ0, C∗0 ) = C∗1−σ0

Θ1−σ0

1− σ− C∗1+φ

0A1+φ

[(1− α)Θ0 + α]1+φ

1 + φ+ βV

(1β

C∗1−σ0 α (1−Θ0)

).

We find

UC∗0 ,Θ0(Θ0, C∗0 ) = (1− σ)C∗−σ0 Θ−σ

0 − (1− α)(1 + φ)C∗φ0

A1+φ[(1− α)Θ0 + α]φ

− (1− σ)C∗−σ0 αV′

(1β

C∗1−σ0 α (1−Θ0)

)

− (1− σ)C∗1−σ0 C∗1−σ

0 (1−Θ0)1β

α2V′′(

1β

C∗1−σ0 α (1−Θ0)

).

Clearly, for σ = 1, we have UC∗0 ,Θ0(Θ0, C∗0 ) < 0. The third result follows.

To gain some intuition for these results, consider the case of a temporary negative pro-ductivity shock in the Cole-Obstfeld case. Similar intuitions can be derived for generalpreferences, for export demand shocks, foreign consumption shocks and wealth shocksbut we omit them in the interest of space. With flexible prices, the allocation would fea-ture constant labor, a temporary decrease in output, consumption, and exports (but withconstant export revenues). At t = 0, there would be a temporary improvement in theterms of trade brought about by an increase in the prices of home goods, and a constantwage. The nominal interest rate would be unchanged, but there would be a temporary

20

high real interest rate brought about by expected deflation as the prices of home goodsrevert to their original values.

Now consider what happens with one period in advance sticky prices. If the exchangerate were flexible, we would achieve the same real allocation by temporarily raising thenominal interest rate, letting the exchange rate appreciate at t = 0 and depreciate back att = 1. This can be seen as a version of the classical argument in favor of flexible exchangerate famously put forth by Milton Friedman.

If the exchange rate is not flexible, then this solution cannot be attained because theterms of trade are fixed at t = 0. Without capital controls, at t = 0, labor temporarilyincreases, consumption and output are constant. Compared with the flexible price allo-cation, output, labor and consumption are higher. Similarly, at t = 0, inflation and thereal interest rate are unchanged and are respectively higher and lower than at the flexi-ble price allocation. It is therefore intuitively desirable to impose positive capital controlsand increase the nominal interest rate to decrease consumption at t = 0. At t = 0, outputthen also decreases, but by less than consumption. Therefore the country runs positivenet exports at t = 0, which implies higher consumption and lower output for t ≥ 1.

This analysis underscores that capital controls are a second best instrument. Theyallow the country to regain some monetary autonomy and therefore some control overthe intertemporal allocation of spending. However, this reallocation is costly since it in-troduces a wedge between the intertemporal prices for home and foreign households.Moreover, capital controls cannot affect the division of spending between home and for-eign goods in the short run when prices are fixed.

Our next result deals with permanent shocks. The proof is contained in the appendix.

Proposition 5 (Permanent Shocks with Sticky Prices). Suppose that prices are sticky oneperiod in advance. Consider the Cole-Obstfeld case σ = η = γ = 1. Then in response to anyshock to the path {At} of productivity, optimal capital controls τt are zero for t ≥ 1, and optimalcapital controls τ0 are a decreasing function of A0 with τ0 = 0 when A0 = A. In response toa small enough permanent export demand shocks ΛH,t = ΛH for all t ≥ 1 where ΛH 6= 1,optimal capital controls τt are zero for t ≥ 1, and optimal capital controls τ0 are an increasingfunction of ΛH with τ0 = 0 when ΛH = 1. In response to any shock to the path {C∗t } of foreignconsumption, optimal capital controls τt are zero for t ≥ 1, and optimal capital controls τ0 arean increasing function of C∗0 with τ0 = 0 when C∗0 = C∗. In response to a small enough wealthshock W0 6= 0, optimal capital controls τt are zero for t ≥ 1, and optimal capital controls τ0 are anincreasing function of NFA0 with τ0 = 0 when NFA0 = 0. The last result is true more generallyfor any positive shock NFA0 > 0.

It is interesting to contrast the use of capital controls when prices are flexible with

21

their use when prices are sticky. First, we know that in response to permanent shocksto productivity, export demand, foreign consumption, or wealth, capital controls are notused when prices are flexible. By contrast, capital controls are used to deal with theseshocks when prices are sticky. Second, at least in the Cole-Obstfeld case, in responseto transitory shocks to productivity and export demand, capital controls are used in theopposite direction when prices are sticky than when they are flexible. Indeed, followinga negative productivity shock (A0 < A) or a positive foreign consumption shock (C∗0 >

C∗), capital controls τ0 are zero with flexible prices but positive with sticky prices, andfollowing a transitory positive export demand shock (ΛH,0 > 1), capital controls τ0 arenegative with flexible prices but positive with sticky prices.

5 Calvo Pricing

In this section, we assume that the exchange rate is fixed and focus on the case of Calvoprice setting. As most of the literature on Calvo pricing, we find it convenient to workwith a linearized model. More precisely, we log-linearize the model around the symmet-ric deterministic steady state. At t = 0, the economy is hit with a one time unanticipatedshock. We describe the flexible price allocation with no intervention (the allocation thatwould prevail if prices were flexible and capital controls were not used). We call thisallocation the natural allocation. We then summarize the behavior of the sticky priceeconomy with capital controls in log-deviations (gaps) from the natural allocation. Forboth the natural and the sticky price allocation with capital controls, the behavior of therest of the world is taken as given.

We find it convenient to work in continuous time. In continuous time, with stickyprices, no price index can jump. This greatly facilitates the derivation of the initial condi-tions for the economy. We denote the instantaneous discount rate by ρ, and the instanta-neous arrival rate for price changes by ρδ.

We use lower cases variables to denote gaps from the symmetric deterministic steadystate. We denote the natural allocation with bars, and the gaps from the natural allocationwith hats.

5.1 Summarizing the Economy and the Experiment

The natural allocation. Let ν = − log(1 + τL), µ = log M, ω = σγ + (1− α)(ση − 1),and σ = σ

1−α+αω .The natural level of output is given by

22

yt =σ−1(1 + φ)

1 + φσ−1 at −α(ω− 1)1 + φσ−1 c∗t −

αω

1 + φσ−1 θ,

the natural interest rate is given by

rt − ρ =1 + φ

1 + φσ−1 at +α(ω− 1)φ1 + φσ−1 c∗t ,

and the natural terms of trade are given by

st =1 + φ

1 + φσ−1 at −σ + φ

1 + φσ−1 c∗t −σ + φ(1− α)

1 + φσ−1 θ,

where θ is such that the country budget constraint holds:

θ =

(ωσ − 1

)σ

1 +(

ωσ − 1

)σ(1− α)

ˆ ∞

0e−ρt(yt − y∗t ).

We can also compute the natural levels of employment and consumption from the equa-tions yt = at + nt and ct = θt + c∗t +

1−ασ st.

Note that the natural interest rate rt, the natural terms of trade st, the foreign nominalinterest rate i∗t and foreign inflation π∗t must be related by

i∗t − rt = − ˙st + π∗t . (13)

Summarizing the system in gaps. The economy with sticky prices and capital controlscan be conveniently summarized in gaps from the natural allocation.

The demand block is summarized by three equations, the Euler equation

˙yt = σ−1[it − πH,t − rt]− αω ˙θt,

the UIP equationit = i∗t + σ ˙θt,

and the country budget constraint

ˆ ∞

0e−ρtθtdt =

(ωσ − 1

)σ

1 +(

ωσ − 1

)σ(1− α)

ˆ ∞

0e−ρtytdt.

23

The supply block consists of one equation, the New-Keynesian Philips Curve

πH,t = ρπH,t+1 − κyt − λσαωθt

where λ = ρδ(ρ + ρδ) and κ = λ (φ + σ).Finally, we have the initial condition

y0 = (1− α)θ0 − σ−1s0

which formalizes the requirement that the terms of trade are predetermined at t = 0 i.e.that s0 = −s0.

Note that we can back out the terms of trade gap st from the equation

yt = (1− α)θt + σ−1st.

Similarly, we can back out the employment gap nt and the consumption gap ct from theequations

yt = nt,

and

yt = −αθt + ct + αω

σst.

Finally, capital controls can be inferred from the UIP equation

τt = it − i∗t .

An experiment. Suppose we start in steady state where it = i∗t = rt = r, πH,t = π∗t = 0,yt = 0. Now suppose there is an unexpected shock that upsets the whole sequences{rt, st, i∗t , π∗t } but still satisfying the restriction i∗t − rt = − ˙st + π∗t . For simplicity, weassume that rt − i∗t = rshocke−ρrt.

This parametrization is flexible enough to accommodate pure terms of trade shocks(permanent shocks to st with no shock to rt), pure natural interest rate shock (shocks to rt

with no shocks to s0), and any combination of the two such as for example mean-revertingproductivity shocks.

Note also that it, rt and i∗t enter in the equations describing the system purely throughit − i∗t and rt − i∗t . As a result, positive world interest rate shocks to i∗t have the sameeffects as positive domestic natural interest rate shocks to rt. This is true both for the

24

allocation without capital controls, and for the optimal allocation with capital controls(the only difference is that the path of domestic interest rates is shifted by the shock to i∗t ).

We now explain how to set up the planning problem for the home economy. We focuson a special case, the Cole-Obstfeld case, where a second order approximation to thewelfare function is available.

5.2 Optimal Capital Controls in the Cole-Obstfeld Case

The Cole-Obstfeld case is attractive for two reasons. First, with flexible prices, it is op-timal not to use capital controls. Second, it is relatively easy to derive a second orderapproximation of the welfare function around the symmetric deterministic steady state.

Loss function. When σ = γ = η = 1, we can derive a simple second order approxima-tion of the welfare function. The corresponding loss function (in consumption equivalentunits) can be written as

(1− α)(1 + φ)

ˆ ∞

0e−ρt

[12

αππ2H,t +

12

y2t + αθ

12

θ2t

]dt

where απ = ελ(1+φ)

and αθ =α

1+φ

(2−α1−α + 1− α

)θ2

t .The first two terms in the loss function are familiar in New-Keynesian models. The

inflation term 12 αππ2

H,t captures the cost of inflation: with staggered price setting a laCalvo, inflation comes with spurious dispersion in markups across firms—a form of mis-allocation. The output gap term 1

2 y2t represents the average markup distortion—because

of sticky prices, the average markup can differ from the markup desired by firms.The third term in the loss function αθ

12 θ2

t is new and captures the cost of capital con-trols. In the Cole-Obstfeld case, the country budget constraint is simply

´e−ρtθtdt = 0.

Capital controls can be used to reallocate demand across time by running nonzero tradebalances −αθt. However, this reallocation is costly since it introduces a wedge betweenthe intertemporal prices for home and foreign households.

Note that απ is independent of α but that αθ goes to zero when α goes to zero. Hencein the closed economy limit (α → 0), the cost of capital controls vanishes. The reason isthat for a given path of θt, the associated trade balances−αθt vanish as α goes to zero, andso do the distortions associated with the wedge between home and foreign intertemporalprices.

25

Planning problem. The planning problem is

min{πH,t,yt,it,θt}

ˆ ∞

0e−ρt

[12

αππ2H,t +

12

y2t + αθ

12

θ2t

]dt

s.t.πH,t = ρπH,t − κyt − λαθt,

˙yt = (1− α)(it − i∗t )− πH,t + i∗t − rt,

˙θt = it − i∗t ,ˆ ∞

0e−ρtθtdt = 0,

y0 = (1− α)θ0 − s0.

This planning problem makes clear that in the Cole-Obstfeld case, welfare is necessar-ily lower with sticky prices and optimal capital controls than with flexible prices an nocapital controls. Therefore the natural allocation represents an upper bound for welfare.Moreover, by taking the limit of flexible prices, i.e. by increasing λ and simultaneouslyvarying απ = ε

λ(1+φ)and κ = λ(1+ φ), it is easily confirmed that when prices are flexible,

it is optimal not to use capital controls at all.To get a better feel for the planning problem, it is useful to imagine how it would

change if the exchange rate were flexible. The only differences are in the UIP equationand the initial condition, which become ˙θt = it − i∗t − et and y0 = (1 − α)θ0 − s0 + e0

where the initial level of the exchange rate e0 and the rate of exchange rate depreciationet are additional control variables. It is clear that the solution features πH,t = yt = θt = 0and it = i∗t +

rt−i∗t1−α . The associated path for the exchange rate is determined by et = st for

all t, which is equivalent to e0 = s0 and et =rt−i∗t1−α .

With flexible exchange rate, and for the shocks that we consider, the natural allocationcan be attained. This observation about the stabilizing role of flexible exchange ratesgoes back to Milton Friedman. But with a fixed exchange rate, perfect macroeconomicstabilization cannot be achieved. A way to understand this is to go back to Mundell’strilemma, which states that it is impossible to have at the same time free capital flows,independent monetary policy, and a fixed exchange rate.

By introducing capital controls, the home country is able to regain some monetarypolicy autonomy, even with a fixed exchange rate. But this comes at a cost since capitalcontrols distort the intertemporal allocation of consumption, for a given path of output,and also impacts inflation dynamics. Optimal policy requires optimally trading off the

26

gain in monetary autonomy against the the resulting distortions. In this section, we makethis intuition formal and characterize how to optimally use capital controls to accommo-date macroeconomic shocks.

Formulation as an optimal control problem. We can incorporate a multiplier Γ on theintertemporal budget constraint. We are left with the following optimal control problem

min{πH,t,yt,it,θt}

ˆ ∞

0e−ρt[

12

αππ2H,t +

12

y2t + αθ

12

θ2t + Γθt]dt

s.t.πH,t = ρπH,t − κyt − λαθt,

˙yt = (1− α)(it − i∗t )− πH,t + i∗t − rt,

˙θt = it − i∗t ,

y0 = (1− α)θ0 − s0.

Let µπ,t, µy,t and µθ,t be the costates. We find the following first-order conditions

µπ,t = −αππH,t + µy,t,

µy,t = −yt + µπ,tκ + ρµy,t,

µθ,t = −αθ θt − Γ + µπ,tλα + ρµθ,t,

(1− α)µy,t + µθ,t = 0,

µπ,0 = 0,

µθ,0 = −(1− α)µy,0,

together with the transversality conditions limt→∞ e−ρtµy,tyt = 0, limt→∞ e−ρtµθ,tθt = 0,and limt→∞ e−ρtµπ,tπt = 0.

We can use the fourth and third of these first-order conditions to find

µπ,t

(κ + λ

α

1− α

)= yt +

αθ

1− αθt +

Γ1− α

.

We can then summarize the optimal allocation as a system of differential equations inyt, πH,t, θt and µy,t:

27

˙yt = −αθ

(1−α)2 + απ

(κ + λ α

1−α

)

1 + αθ

(1−α)2

πH,t +κ + λ α

1−α

1 + αθ

(1−α)2

µy,t −αθ

(1− α)2rt − i∗t

1 + αθ

(1−α)2

,

πH,t = ρπH,t − κyt − λαθt,

˙θt =1

1−α

(1− απ

(κ + λ α

1−α

))

1 + αθ

(1−α)2

πH,t +1

1−α

(κ + λ α

1−α

)

1 + αθ

(1−α)2

µy,t +

rt−i∗t1−α

1 + αθ

(1−α)2

,

µy,t = −λ α

1−α

κ + λ α1−α

yt + ρµy,t +κ Γ

1−α

κ + λ α1−α

+κ αθ

1−α

κ + λ α1−α

θt,

with the initial condition

y0 +αθ

1− αθ0 = − Γ

1− α,

where y0,θ0, πH,0, µy,0 and Γ must be chosen such that´ ∞

0 e−ρtθtdt = 0 and the transver-sality conditions are verified.

Rigid prices. In the case where prices are completely rigid, the optimal allocation canbe found in closed form.

Proposition 6 (Rigid Prices, Calvo). In the case where prices are completely rigid, the optimalallocation is given by

yt = −rshock

1− α + αθ1−α

[(1− α)

1ρ + ρr

+αθ

1− α

1− e−ρrt

ρr

]− s0,

it = i∗t +1

1− α + αθ1−α

(rt − i∗t ),

and capital controls are given by τt = it − i∗t .

When prices are rigid, capital controls are not used to accommodate a pure terms oftrade shock (a permanent shock in st with no shock to rt). By contrast, capital controls areused to smooth variations in output in response to a natural interest rate shock (rt), whichrequires it = i∗t +

11−α+

αθ1−α

(rt− i∗t ). Note that in this latter case, the interest must not be set

equal to the natural interest rate rt, nor must it be set to the world nominal interest rate i∗t .Note also that the smaller α, the more capital controls are used to accommodate naturalinterest rate shocks: it = i∗t +

11−α+

αθ1−α

(rt − i∗t ). When α goes to zero (closed economy

limit), the output gap is constant yt = − rshock

ρ+ρr− s0 with it = rt.

28

Closed economy limit ((α→ 0)). We can derive the optimal allocation in closed form inthe closed-economy limit (α → 0). Note that απ is independent of α and that αθ becomeszero. Let

λ− =ρ−

√ρ2 + 4α2

πκ2

2and

Γ =

[s0 +

ˆ ∞

0e−ρt(rt − i∗t )

] [1 + κ + (1− απκ)

(1

ρ2απ− 1− απκ

απ(ρ− λ−)2

)].

Proposition 7 (Closed economy limit). In the closed economy limit (α → 0), the optimalallocation is given by

yt = Γ1− απκ

−λ−(ρ− λ−)(eλ−t − 1)− Γ,

πH,t = Γ1− απκ

απ(ρ− λ−)(eλ−t − 1)− Γ

1απ

[−ρ(1− απκ) + ρ− λ−ρ(ρ− λ−)

],

it = rt + (1− απκ)πH,t −κ

ρΓ = rt + Γ

(1− απκ)2

απ(ρ− λ−)eλ−t − 1

ραπΓ,

and capital controls are given by τt = it − i∗t .

Note that if Γ 6= 0, inflation does not return to zero in the long run (and hence it −rt does not return to zero). By contrast, for α > 0, inflation always returns to zero inthe long run (and hence it − rt also returns to zero). In other words, the double limitslimα→0 limt→∞ πH,t = 0 and limt→∞ limα→0 πH,t 6= 0 are different.

Proposition 8. In the closed economy limit (α → 0), in the special case where 1− απκ = 0,the solution features constant output gaps and inflation yt = −Γ, πH,t = − 1

ραπΓ where Γ =[

s0 +´ ∞

0 e−ρt(rt − i∗t )](1 + κ). In addition the nominal interest rate is given by it = rt − κ Γ

ρ .

Note that by contrast with the case of rigid prices, capital controls are used to accom-modate pure terms of trade shocks.

5.3 Numerical exploration

In this section we explore how capital controls can be used to accommodate pure terms oftrade shocks, pure natural interest rate shocks, and mean-reverting productivity shocks.We focus on the Cole-Obstfeld case. We set φ = 3, ρ = 0.04, δ = 1− 0.754, ε = 6. We runtwo experiments, the first one with α = 0.4 and the second one with α = 0.1.

29

A terms of trade shock. Our first experiment is such that st = −0.05 for all t and rt = 0for all t. This represents an improvement in the terms of trade that causes an appreciationin the real exchange rate. This can be traced back to a permanent shock to productivityAt.

Figures 1 and 2 show the results two values of openness, α = 0.4 and α = 0.1, respec-tively. The green line is the outcome without interventions (zero capital controls). Theblue line is the outcome with optimal capital controls.

To gain intuition, consider first the outcome without intervention. On the one hand, ifprices were fully flexible there would be an immediate and permanent upward jump inthe price of home goods PH. On the other hand, with totally rigid prices the equilibriumfeatures a permanent rise in output and consumption. In the intermediate case, withCalvo pricing, the equilibrium features a smoother transition towards a higher price forhome goods. As shown in the figure, positive home inflation and positive output gaps arepositive and asymptote to zero. Due to the Cole-Obstfeld parameterization, the transitiondoes not affect trade balance. Output and consumption are higher, pushing for higherimports (foreign revenue is constant), but Home goods are also cheaper, encouraging therelative consumption of home goods over imports. These two effects cancel each otherout.

Turning to the optimal intervention, of course, with flexible prices no intervention isrequired. With fully rigid prices, our result implies that the optimum does nothing and ac-cepts the permanent change in output. Interestingly, in the intermediate case, with Calvopricing, some intervention is optimal. In the short run, the optimal policy interventionlowers home consumption and output relative to the no-intervention equilibrium. Thedrop in consumption lowers imports, which creates a trade surplus. The rest of the dy-namics are simply convergence. Note an tradeoff: the optimal policy lowers output andconsumption. This makes inflation lower, but the adjustment in home prices is what weneed. Note that when α is lower, the nominal interest rate it is higher, so that capital con-trols are used more. The paths for the output gap and inflation become much smoother(in the closed economy limit, they would be completely smooth, although at a strictlypositive level).

A (mean-reverting) natural interest rate shock (or world interest rate shock). Next welook at a shock to rt which becomes negative (with r0 = −0.05) and then mean-reverts tozero with s0 = 0. The coefficient of mean reversion is set to a relatively high value of 1,so that the half-life of the shock is 0.7 years. This shock can be traced back to a temporaryproductivity shock where at t = 0, it becomes known that productivity will be decreasing

30

in the future. This shock is isomorphic to a positive world interest rate shocks to i∗t , up toa shift (of i∗t ) in the path of the domestic interest rate it.

Figures 3 and 4 show the results two values of openness, α = 0.4 and α = 0.1, respec-tively. The green line is the outcome without interventions (zero capital controls). Theblue line is the outcome with optimal capital controls.

Consider first the outcome without intervention. If prices were flexible, there wouldbe inflation in PH with no change at impact. The inflation rate would be decreasingover time towards zero. This would reduce the real interest rate, more so in the shortrun. With sticky prices and a fixed nominal interest rate, this inflationary process issmoothed, so that inflation is at first lower, then higher and also converges to zero in thelong run. The long-run increase in PH is the same as with flexible prices. To understandthe behavior of the output gap, it is useful to keep in mind the long-run Euler equationyt = y∞ −

´ ∞t (i∗s − rs − πH,s)ds together with the fact that the long-run output gap y∞ is

zero. Because the long-run increase´ ∞

0 πH,sds in PH is the same as under flexible prices,the output gap is initially zero. Because inflation is initially lower and eventually higherthan under flexible prices, the output gap is positive throughout. Once again, due to theCole-Obstfeld parameterization, the transition does not affect trade balance.

We now turn to the optimal intervention. With fully rigid prices, capital controlswould be such that the nominal interest rate it would equal i∗t +

11−α+

αθ1−α

(rt − i∗t ). The

corresponding level of capital controls is displayed in red in the figures. This involvessetting a subsidy on inflows / tax on outflows so as to reduce the nominal interest rate.The output gap would be positive throughout but initially zero. The country would ini-tially run a trade surplus (such capital controls initially reduce consumption more thanoutput) and eventually a trade deficit. With Calvo pricing, the solution has similar fea-tures. Additionally, there is inflation. But note that both the paths for the output gap andinflation are smoother with the optimal intervention than with no intervention. Note thatwhen α is lower, the nominal interest rate is much closer to i∗t +

11−α+

αθ1−α

(rt − i∗t ) which is

turn closer to rt, so that capital controls are used more. The paths for the output gap andinflation become much smoother (in the closed economy limit, they would be completelysmooth, although at a strictly positive level).

A mean-reverting productivity shock. Finally we look at a temporary negative produc-tivity shock such that initially s0 = −0.05. The coefficient of mean reversion is set to 0.2so that the half-life of the shock is 3.5 years.

Figures 3 and 4 show the results two values of openness, α = 0.4 and α = 0.1, respec-tively. The green line is the outcome without interventions (zero capital controls). The

31

Capital Controls Fiscal Policyα = 0.4 α = 0.1 α = 0.4 α = 0.1

permanent terms of trade shock 0.2% 0.51% 0.13% 0.18%mean-reverting productivity shock 0.19% 0.47% 0.12% 0.16%

Table 1: Welfare Costs

blue line is the outcome with optimal capital controls. This shocks is a combination of thefirst two shocks (with a positive sign on the first one and a negative sign on the secondone). The figures can be understood accordingly.

Welfare. We can also compute the reduction in the loss function (in consumption equiv-alent units) brought about by the optimal use of capital controls. Since the numbers rep-resent a single shock, it should be kept in mind that these gains should be scaled by thesize and frequency of the shocks.

It is apparent from these numbers that the more closed the economy, the more effectivecapital controls.

5.4 Sticky wages

In this section we introduce sticky wages. We then look at the role of capital controls ina currency union. We can also look at the role of capital controls with flexible exchangerates.

Introducing sticky wages in addition to sticky prices. In this section, we incorporatesticky wages. We now assume that labor is an aggregate of different varieties of labor

Nt =

(ˆ 1

0Nt(j)

εw−1εw dj

) εwεw−1

.

We denote the corresponding wage index by

Wt =

(ˆ 1

0Wt(j)1−εw dj

) 11−εw

,

and we let Mw ≡ εwεw−1 .

Wages are set a la Calvo, with parameter δw. The corresponding first-order condition

32

is∞

∑k=0

δkwδkΠk

h=11

1 + it+hNt+k|tC

−σt+k|t(

Wrt

Pt+k−MwMRSt+k|t)

where

MRSt+k|t =Cσ

t+k

Nφ

t+k|t.

The planning problem. In order to write down the corresponding planning problem,we need to introduce a few notations. First we define λw =

ρ(ρ+ρδw )1+εwφ where ρδw is the

arrival rate of new wages.The planning problem now features additional state variables: wage inflation πw

t andthe real wage gap ωt. It can be written as

min{πH,t,πw

t ,yt,it,θt,ωt}

12

ˆ ∞

0e−ρt[αππ2

H,t + απw (πwt )

2 + y2t + αθ θ2

t ]dt

πwt = ρπw

t − λw[(1− α) + φ]yt − λwα(2− α)θt + λwωt,

πH,t = ρπH,t − λωt − λαyt + λα(1− α)θt,

˙yt = (1− α)it − (πH,t + rt) + αi∗t ,

˙θt = it − i∗t ,

˙ωt = πwt − (1− α)πH,t − απ∗t − ˙ωt,

together with the initial conditions

ω0 = −ω0,

−s0 = y0 − (1− α)θ0,

and the country budget constraint

ˆ ∞

0e−ρtθt = 0.

By putting a multiplier Γ on the country budget constraint and incorporating it in theobjective, we are left with an optimal control problem with state variables πH,t, πw

t , yt, θt,ωt, and control variable it.

33

Flexible exchange rate. As Milton Friedman forcefully argued, flexible exchange ratescan act as a substitute for flexible prices. When prices are sticky, but wages are not, aflexible exchange rate can be used to achieve the flexible price allocation. In this way,flexible exchange rates yield perfect macroeconomic stabilization. But this is no longerpossible, in general, when wages are also sticky. Perfect macroeconomic stabilization isnot possible, even with a flexible exchange rate. This raises the question we address here:are capital controls optimal when prices and wages are sticky but a flexible exchange rateis managed optimally?

In terms of the equilibrium conditions, two things change when the exchange rate isallowed to vary: first, the initial conditions and second, the relationship between ˙θt andit − i∗t since we now have ˙θt = it − i∗t − et. The planning problem is

min{πH,t,πw

t ,yt,it,θt,ωt,et}

12

ˆ ∞

0e−ρt[αππ2

H,t + απw (πwt )

2 + y2t + αθ θ2

t ]dt

πwt = ρπw

t − λw[(1− α) + φ]yt − λwα(2− α)θt + λwωt,

πH,t = ρπH,t − λωt − λαyt + λα(1− α)θt,

˙yt = (1− α)it − (πH,t + rt) + αi∗t ,

˙θt = it − i∗t − et,

˙ωt = πwt − (1− α)πH,t − α(π∗t + et)− ˙ωt,

together with the initial condition4

ω0 = −ω0 − αs0 − αy0 + α(1− α)θ0,

and the country budget constraint

ˆ ∞

0e−ρtθt = 0.

By putting a multiplier Γ on the country budget constraint and incorporating it in theobjective, we are left with an optimal control problem with state variables πH,t, πw

t , yt, θt,

4The initial condition comes from the two conditions

ω0 = −ω0 − αe0,

e0 − s0 = y0 − (1− α)θ0.

where e0 is the initial exchange rate with sticky prices and wages.

34

ωt, and control variables it and et.It seems that in general, there is a role for capital controls when prices and wages

are sticky, even when the exchange rate is flexible. The role of capital controls is quitedifferent than with a fixed exchange rate though. With a flexible exchange rate, eachcountry already has the flexibility to set its own monetary policy. However, this per se isnot enough to perfectly stabilize the economy. In this context, capital controls still emergeas a second best instrument.

However note that when prices are entirely rigid, this role for capital controls disap-pears entirely. The same is true in the closed economy limit.

Proposition 9 (Sticky Prices, Wages, Flexible Exchange Rate). With sticky prices, stickywages and a flexible exchange rate, optimal capital controls are generally nonzero. However, inthe two limit cases of perfectly rigid prices and wages on the one hand, and closed economy limit(α→ 0) on the other hand, optimal capital controls are zero.

5.5 Government Spending

For comparison, we now introduce government expenditures, building more directly Galíand Monacelli (2008). Government expenditures Gt enter households’ utility as follows

∞

∑t=0

βt

[(1− χ)

C1−σt

1− σ+ χ

G1−σt

1− σ− N1+φ

t1 + φ

]

where

Gt =

(ˆ 1

0Gt(j)

ε−1ε dj

)

where j ∈ [0, 1] denotes the good variety. Note that, following Galí and Monacelli (2008),this assumes that Gt is spent entirely on home goods.

To keep the exposition brief, we focus on the Cole-Obstfeld case, abstract from capitalcontrols. We move directly to a continuous time formulation of the planning problem, setup as an optimal control:

minˆ ∞

0[12

π2H,t +

12

λx(ct + (1− ξ)gt)2 +

12

λg gt2]dt

πH,t = ρπH,t − κ[ct + (1− ξ)gt],

˙ct = (1− Γ)[−πH,t − (rt − i∗)],

35

and the initial conditionc0 = −s0(1− Γ).

In this program, we have used the following notation. Parameter Γ = GY is the steady state

share of government spending. Let ξ =1

1−Γ1

1−Γ+φ. The parameter κ is given by κ = λ

(1−Γ)ξ =

λ(φ + 11−Γ ), and the coefficients in the loss function are given by λx = λ

ε1

(1−Γ)ξ = κε

and λg = λε

1+φΓ ξ. The variable ct denotes the log-deviation of consumption from the

optimal allocation with flexible prices. The variable gt denotes the fiscal gap Gt−GtY and the

the variable ct =Ct−Ct

Y is the consumption gap, where bar variables denote the optimalallocation with flexible prices.

A decomposition of government spending. Following Werning (2012), we decomposegovernment expenditures into two components.

gt = gopp(ct) + gstimt

wheregopp(ct) = −

λx(1− ξ)

λx(1− ξ)2 + λgct

andgstim

t =1− ξ

λx(1− ξ)2 + λgκµπ,t

The first component gopp(ct) is opportunistic spending. It is the level of expenditurethat would be chosen by a government, taking the path for private consumption ct asgiven. Its determinants are purely microeconomic. It captures the idea that it makes senseto increase government spending when the opportunity cost of labor is low. The secondcomponent, stimulus spending, gstim

t , is defined as the residual. Note that gstimt = 0

whenever the costate µπ,t is zero.

Proposition 10 (Stimulus Spending). Stimulus spending is initially zero so that gstim0 = 0. In

the special case where κ 11−Γ = λx then µπ,t = 0 for all t so that gt

stim = 0 for all t. In this case,stimulus spending is zero throughout and government spending is purely opportunistic.

Government spending and openness. An important observation is that the planningproblem for fiscal policy, expressed in gaps, is independent of the openness parameter α,and hence so is its solution. The openness of the economy only introduces a proportional

36

scaling factor in the associated loss function

12(Γ + (1− Γ)(1− α))

ε

λ

ˆ ∞

0[12

π2H,t +

12

λx(ct + (1− ξ)gt)2 +

12

λg gt2]dt.

This is an important distinction with capital controls, which became more effective as theeconomy became more closed.

Rigid prices. With fixed prices, we can solve the system in closed form.

Proposition 11 (Government Spending with Rigid Prices). With rigid prices, the solution isgiven by ˙ct = (1− Γ)[−(rt − i∗)], c0 = −s0(1− Γ), gt = gopp(ct) + gstim

t , where gopp(ct) =

− λx(1−ξ)λx(1−ξ)2+λg

ct and gstimt = 0.

With rigid prices, the solution is simple: government spending is purely opportunistic,and stimulus spending is equal to zero.

6 Policy Coordination

Up to this point we have isolated the problem of a small open economy with a fixedexchange rate. The policy choices include a path for capital controls as well as a constantsubsidy on labor. This country takes as given conditions in the rest of the world, includingpolicy choices by other countries, the ensuing equilibrium interest rates and prices. Wenow consider a coordinated solution within a monetary union where countries cooperateon a constant tax and capital controls to maximize the sum of utilities. We will contrastthis solution to the uncoordinated, noncooperative equilibrium, where each country actsin isolation.

An important consideration is the level of the labor tax. At a symmetric steady statewith no coordination, the labor tax is given by M(1 + τL) =

11−α . At a symmetric steady

state with no coordination, the labor tax is instead given by M(1 + τL) = 1. The labor taxis lower in the latter case than in the former. Intuitively, coordination eliminates terms oftrade manipulation. It is interesting to compare coordination vs. no coordination of therest of policy instruments (capital controls in each country, and monetary policy at theaggregate level of the union), taking the labor tax as given.

6.1 Uncoordinated labor tax

We start with the case where there is no coordination on the labor tax at the symmetricsteady state so that M(1 + τL) = 1

1−α . We denote with a double bar the corresponding

37

flexible price allocation with no capital controls. We denote with a double hat the gap ofa variable from its flexible price counterpart. We denote with a tilde a variable minus itsmean across countries. For example yi

t = ˆyit − ˆy∗t represents the deviation of country i’s

output gap from the corresponding aggregate.

Coordination. For small α, we can then write the coordinated planning problem as5

min12

ˆ ∞

0

ˆ 1

0e−ρt[απ(π

iH,t)

2 + (yit)

2 + αθ(θit)

2

+ απ(π∗t )

2 + ( ˆy∗t )2 − α

(1− α)(1 + φ)ˆy∗t]di dt (14)

subject to

˙πiH,t = ρπi

H,t − κ ˜yit − λαθi

t,

˙yit = (1− α) ˙θi

t − πiH,t − ˙si

t,

ˆ ∞

0e−ρtθi

tdt = 0,

yi0 = (1− α)θi

0 − ¯si0,

ˆ 1

0yi

tdi = 0,ˆ 1

0πi

H,tdi = 0,

π∗t = ρπ∗t − κy∗t ,

where the minimization is over the variables πiH,t, π∗t , yi

t, ˆy∗t , θit. Note that since

´ 10

¯sitdi = 0,

the constraints imply that´ 1

0 θitdi = 0. This pins down controls through the equation

˙θit = τi

t . Intuitively, the extra linear term α1+φ

ˆy∗t can be traced back to the fact that thethat the coordinated solution abstains from terms of trade manipulation. As a result theobjective acquires a preference for higher output, leading to a classical inflationary bias.

It follows that we can break down the planning problem into two parts. First, there isan aggregate planning problem determining the average output gap and inflation ˆy∗t and

5This representation is valid only for small α, otherwise, a second order approximation of the constraintsis required.

38

π∗t

min12

ˆ ∞

0e−ρt

[απ(π

∗t )

2 + ( ˆy∗t )2 − α

(1− α)(1 + φ)ˆy∗t

]dt (15)

subject toπ∗t = ρπ∗t − κ ˆy∗t .

Second, there is a disaggregated planning problem determining deviations from the ag-gregates for output gap, home inflation and consumption smoothing, yi

t, πiH,t and θi

t

min12

ˆ ∞

0

ˆ 1

0e−ρt

[απ(π

iH,t)

2 + (yit)

2 + αθ(θit)

2]

didt

subject to

˙πiH,t = ρπi

H,t − κ ˜yit − λαθi

t,

˙yit = (1− α) ˙θi

t − πiH,t − ˙si

t,

ˆ ∞

0e−ρtθi

tdt = 0,

yi0 = (1− α)θi

0 − ¯si0,

ˆ 1

0yi

tdi = 0,ˆ 1

0πi

H,tdi = 0

Consider dropping the last two aggregation constraints´ 1

0 yitdi = 0 and

´ 10 πi

H,tdi = 0.This relaxed planning problem can be broken down into separate component planningproblems for each country i ∈ [0, 1]

min12

ˆ ∞

0e−ρt

[απ(π

iH,t)

2 + (yit)

2 + αθ(θit)

2]

dt (16)

subject to˙πi

H,t = ρπiH,t − κ ˜yi

t − λαθit,

˙yit = (1− α) ˙θi

t − πiH,t − ˙si

t,ˆ ∞

0e−ρtθi

tdt = 0,

39

yi0 = (1− α)θi

0 − ¯si0.

No coordination. With no coordination, each country i ∈ [0, 1] takes the evolution ofaggregates as given and solves the following uncoordinated component planning prob-lem:

min12

ˆ ∞

0e−ρt

[απ(π

iH,t)

2 + 2αππ∗t πiH,t + (yi

t)2 + 2 ˆy∗t yi

t + αθ(θit)

2]

dt (17)

subject to˙πi

H,t = ρπiH,t − κ ˜yi

t − λαθit,

˙yit = (1− α) ˙θi

t − πiH,t − ˙si

t,ˆ ∞

0e−ρtθi

tdt = 0,

yi0 = (1− α)θi

0 − ¯si0,

where the minimization is over the variables πiH,t, yi

t, θit, taking ˆy∗t , and π∗t as given. As

usual, capital controls in country i can be computed by τit = ˙θi

t. Note that the path foraggregates { ˆy∗t ,π∗t }t≥0 affects the solution to this problem solely through linear terms inthe objective function.

A central monetary authority, by setting monetary policy, can choose aggregates { ˆy∗t ,π∗t }subject to the following constraints. First, it must ensure that the solutions to the unco-ordinated component planning problems satisfy

´ 10 yi

tdi = 0 and´ 1

0 πiH,tdi = 0. This

amounts to verifying a fixed point, that aggregates are actually equal to their proposedpath. Second, it must ensure that the aggregate Phillips curve is verified, π∗t = ρπ∗t − κ ˆy∗t .Both requirements define a set F of feasible aggregate outcomes { ˆy∗t ,π∗t }t≥0. The set is alinear space, which we characterize in closed form below, using the following definitions:

A =

[0 κ(1−α)+λα

1−α

− λακ(1−α)+λα

ρ

], E1 =

[10

], E2 =

[01

].

Which aggregate outcome in the feasibility occurs depends on the objective of the centralmonetary authority. For example, we can examine the case where the central monetaryauthority seeks to maximize aggregate welfare, taking into account that capital controlsare set uncooperatively. For small α, this can be represented as the following planningproblem

min12

ˆ ∞

0e−ρt

[απ(π

∗t )

2 + ( ˆy∗t )2 − α

(1− α)(1 + φ)ˆy∗t

]dt,

where the minimization is over the feasible set F .

40

Proposition 12 (Coordination vs. No Coordination with Uncoordinated Labor Tax). Sup-pose that countries do not coordinate on the labor tax. At the coordinated solution, the aggregatessolve the aggregate planning problem (15): ˆy∗t = α

(1−α)(1+φ)eλ−t and π∗t = α

(1−α)(1+φ)κ

ρ−λ− eλ−t

for all t ≥ 0, where λ− =ρ−√

ρ2+4καπ

2 . At the uncoordinated solution, the aggregates { ˆy∗t ,π∗t }t≥0

are in the feasible set F if and only if

ˆy∗t = απκ(1− α) + λα

1− α

ˆ ∞

tπ∗s E′1e−A(s−t)E1ds + απ

κE′1A−1E2´ ∞

0 π∗s E′1e−AsE1ds1

1−α −κE′1 A−1E2

κ(1−α)+λα

,

π∗t = ρπ∗t − κ ˆy∗t .

In particular, ˆy∗t = π∗t = 0 for all t ≥ 0 is feasible but ˆy∗t = α(1−α)(1+φ)

eλ−t and π∗t =α

(1−α)(1+φ)κ

ρ−λ− eλ−t for all t ≥ 0 is not. Both at the coordinated and at the uncoordinated

solutions, the disaggregated variables πiH,t, yi

t, θit solve the component planning problems (16).

Proof. We start by showing that at the coordinated solution, the disaggregated variablesπi

H,t, yit, θi

t solve the component planning problems (16). The component planning prob-lems for two different countries are identical linear quadratic problems. The only differ-ence is the path for the forcing variable si

t. We can use the fact that the solution is linear inthe path of the forcing variable { ¯si

t}t≥0 and that´ 1

0¯sitdi = 0 to conclude that the solution

of the relaxed planning problem satisfies the two aggregation constraints 0 =´ 1

0 yitdi and

0 =´ 1

0 πiH,tdi. This implies that the solution of the relaxed planning problem coincides

with that of the original planning problem.Next we show that that at the uncoordinated solution, the disaggregated variables

πiH,t, yi

t, θit also solve the component planning problems (16). The uncoordinated compo-

nent planning problem (17) is linear quadratic. Its solution is therefore linear in { ˆy∗t , π∗t , ¯sit}t≥0.

Imposing that 0 =´ 1

0 yitdi and 0 =

´ 10 πi

H,tdi then immediately implies that if { ˆy∗t ,π∗t }t≥0

is in F , then the solution of the uncoordinated component planning problem coincideswith the solution of the component planning problem (16).

The derivation of the feasible set can be found in the appendix.

6.2 Coordinated labor tax

We now consider the case where there is coordination on the labor tax at the symmetricsteady state so that M(1 + τL) = 1. With a slight abuse of notation, we keep denotingwith a double bar the corresponding flexible price allocation with no capital controls. Wedenote with a double hat the gap of a variable from its flexible price counterpart. We

41

denote with a tilde a variable minus its mean across countries. For example yit = ˆyi

t − ˆy∗trepresents the deviation of country i’s output gap from the corresponding aggregate.

Coordination. We can then write the coordinated planning problem as

min12

ˆ ∞

0

ˆ 1

0e−ρt

[απ(π

iH,t)

2 + (yit)

2 + (1− α)αθ(θit)

2 + απ(π∗t )

2 + ( ˆy∗t )2]

di dt (18)

subject to

˙πiH,t = ρπi

H,t − κ ˜yit − λαθi

t,

˙yit = (1− α) ˙θi

t − πiH,t − ˙si

t,

ˆ ∞

0e−ρtθi

tdt = 0,

yi0 = (1− α)θi

0 − ¯si0,

ˆ 1

0yi

tdi = 0,ˆ 1

0πi

H,tdi = 0,

π∗t = ρπ∗t − κy∗t ,

where the minimization is over the variables πiH,t, π∗t , yi

t, ˆy∗t , θit. Note that since

´ 10

¯sitdi = 0,

the constraints imply that´ 1

0 θitdi = 0. This pins down controls through the equation

˙θit = τi

t . There are two differences between the coordinated planning problem with acoordinated labor tax (18) and and the one with an uncoordinated labor tax 14. First, thecoefficient on the term 1

2 θ2t is (1− α)αθ instead of αθ. Second, there is no linear term in ˆy∗t .

As above we can break down the planning problem into two parts. First, there is anaggregate planning problem determining the average output gap and inflation ˆy∗t and π∗t

min12

ˆ ∞

0e−ρt

[απ(π

∗t )

2 + ( ˆy∗t )2]

dt (19)

subject toπ∗t = ρπ∗t − κ ˆy∗t .

Second, there is a disaggregated planning problem determining deviations from the ag-

42

gregates for output gap, home inflation and consumption smoothing, yit, πi

H,t and θit

min12

ˆ ∞

0

ˆ 1

0e−ρt

[απ(π

iH,t)

2 + (yit)

2 + (1− α)αθ(θit)

2]

didt

subject to

˙πiH,t = ρπi

H,t − κ ˜yit − λαθi

t,

˙yit = (1− α) ˙θi

t − πiH,t − ˙si

t,

ˆ ∞

0e−ρtθi

tdt = 0,

yi0 = (1− α)θi

0 − ¯si0,

ˆ 1

0yi

tdi = 0,ˆ 1

0πi

H,tdi = 0

Consider dropping the last two aggregation constraints´ 1

0 yitdi = 0 and

´ 10 πi

H,tdi = 0.This relaxed planning problem can be broken down into separate component planningproblems for each country i ∈ [0, 1]

min12

ˆ ∞

0e−ρt

[απ(π

iH,t)

2 + (yit)

2 + (1− α)αθ(θit)

2]

dt (20)

subject to˙πi

H,t = ρπiH,t − κ ˜yi

t − λαθit,

˙yit = (1− α) ˙θi

t − πiH,t − ˙si

t,ˆ ∞

0e−ρtθi

tdt = 0,

yi0 = (1− α)θi

0 − ¯si0.

No coordination. With no coordination, each country i ∈ [0, 1] takes the evolution ofaggregates as given and solves the following uncoordinated component planning prob-

43

lem:

min12

ˆe−ρt

[απ(π

iH,t)

2 + 2αππ∗t πiH,t + (yi

t)2 + 2 ˆy∗t yi

t +α

1 + φyi

t + (1− α)αθ(θit)

2]

dt

(21)subject to

˙πiH,t = ρπi

H,t − κ ˜yit − λαθi

t,

˙yit = (1− α) ˙θi

t − πiH,t − ˙si

t,ˆ ∞

0e−ρtθi

tdt = 0,

yi0 = (1− α)θi

0 − ¯si0,

where the minimization is over the variables πiH,t, yi

t, θit, taking ˆy∗t , and π∗t as given. As

usual, capital controls in country i can be computed by τit =

˙θit. Note that once again, the

path for aggregates { ˆy∗t ,π∗t }t≥0 affects the solution to this problem solely through linearterms in the objective function. There are two differences between the uncoordinatedplanning problems with a coordinated labor tax (21) and the one with an uncoordinatedlabor tax (17). First, the coefficient on (θi

t)2 is (1− α)αθ instead of αθ. Second, there is a

linear term in yit. Intuitively, the extra linear term α

1+φ yit can be traced back to the fact that

the that the uncoordinated solution feature terms of trade manipulation. As a result theobjective acquires a preference for lower output—a form of anti-inflationary bias.

Exactly as in the case where there is no coordination on the labor tax, a central mone-tary authority, by setting monetary policy, can choose aggregates { ˆy∗t ,π∗t }t≥0 subject to thefollowing constraints. First, it must ensure that the solutions to the uncoordinated compo-nent planning problems satisfy

´ 10 yi

tdi = 0 and´ 1

0 πiH,tdi = 0. This amounts to verifying

a fixed point, that aggregates are actually equal to their proposed path. Second, it mustensure that the aggregate Phillips curve is verified, π∗t = ρπ∗t − κ ˆy∗t . Both requirementsdefine a set F ′ of feasible aggregate outcomes { ˆy∗t ,π∗t }t≥0. Which aggregate outcome inthe feasibility occurs depends on the objective of the central monetary authority. For ex-ample, we can examine the case where the central monetary authority seeks to maximizeaggregate welfare, taking into account that capital controls are set uncooperatively. Forsmall α, this can be represented as the following planning problem

min12

ˆ ∞

0e−ρt

[απ(π

∗t )

2 + ( ˆy∗t )2]

dt,

where the minimization is over the feasible set F ′.

44

Proposition 13 (Coordination vs. No Coordination with Coordinated Labor Tax). Supposethat countries coordinate on the labor tax. At the coordinated solution, the aggregates solve theaggregate planning problem (15): ˆy∗t = 0 and π∗t = 0 for all t ≥ 0. At the uncoordinatedsolution, the aggregates { ˆy∗t ,π∗t }t≥0 are in the feasible set F ′ if and only if

ˆy∗t = − α

1 + φ+ απ

κ(1− α) + λα

1− α

ˆ ∞

tπ∗s E′1e−A(s−t)E1ds+ απ

κE′1A−1E2´ ∞

0 π∗s E′1e−AsE1ds1

1−α −κE′1 A−1E2

κ(1−α)+λα

,

π∗t = ρπ∗t − κ ˆy∗t .

In particular, ˆy∗t = π∗t = 0 for all t ≥ 0 is not feasible. Both at the coordinated and at theuncoordinated solutions, the disaggregated variables πi

H,t, yit, θi

t solve the component planningproblems (20).

Proof. The proof is almost identical to the one of Proposition 12.

Propositions 12 and 13 help understand the role of coordination. For a given labortax, the aggregates ˆy∗t and π∗t associated with the coordinated solution and the uncoordi-nated solutions differ. Indeed, for a given labor tax, the aggregates corresponding to thecoordinated solution are not feasible. By contrast, for a given labor tax, the disaggregatedvariables πi

H,t, yit, θi

t (and hence capital controls) coincide at the coordinated and unco-ordinated solutions. They depend only on the labor tax, and not on whether countriescoordinate or not for a given labor tax. In other words, the lack of coordination impactsthe solution by restricting the set of feasible aggregate outcomes, but does not impactthe disaggregated variables and the associated capital controls for any feasible aggregateoutcome.

References

Aoki, Kosuke, Gianluca Benigno, and Nobuhiro Kiyotaki, “Adjusting to Capital Ac-count Liberalization,” Oct 2010. C.E.P.R. Discussion Papers.

Caballero, Ricardo J. and Guido Lorenzoni, “Persistent Appreciations and Overshoot-ing: A Normative Analysis,” 2007. NBER Working Papers.

Calvo, Guillermo A. and Enrique G. Mendoza, “Rational contagion and the globaliza-tion of securities markets,” Journal of International Economics, 2000, 51 (1), 79–113.

45

Costinot, Arnaud, Guido Lorenzoni, and Ivan Werning, “A Theory of Capital Controlsas Dynamic Terms-of-Trade Manipulation,” NBER Working Papers 17680, National Bu-reau of Economic Research, Inc December 2011.

Galí, Jordi and Tommaso Monacelli, “Monetary Policy and Exchange Rate Volatility ina Small Open Economy,” Review of Economic Studies, 07 2005, 72 (3), 707–734.

and , “Optimal monetary and fiscal policy in a currency union,” Journal of Interna-tional Economics, September 2008, 76 (1), 116–132.

Jeanne, Olivier and Anton Korinek, “Excessive Volatility in Capital Flows: A PigouvianTaxation Approach,” American Economic Review, May 2010, 100 (2), 403–407.

Keynes, J.M., “The German transfer problem,” The Economic Journal, 1929, 39 (153), 1–7.

Martin, Alberto and Filippo Taddei, “International Capital Flows and Credit MarketImperfections: a Tale of Two Frictions,” 2010. Collegio Carlo Alberto.

McKinnon, Ronald I. and William E. Oates, “The Implications of International EconomicIntegration for Monetary, Fiscal, and Exchange-Rate Policy,” Princeton Studies in Inter-national Finance, 1966.

Obstfeld, Maurice and Kenneth Rogoff, “Exchange Rate Dynamics Redux,” Journal ofPolitical Economy, June 1995, 103 (3), 624–60.

Ohlin, B., “The reparation problem: a discussion,” Economic Journal, 1929, 39 (154), 172–182.

Ostry, Jonathan D., Atish R. Ghosh, Karl Habermeier, Marcos Chamon, Mahvash S.Qureshi, and Dennis B.S. Reinhardt, “Capital Inflows: The Role of Controls,” 2010.IMF Staff Position Note.

Werning, Ivan, “Managing a Liquidity Trap: Monetary and Fiscal Policy,” 2012. NBERWorking Papers.

A Appendix

A.1 Proof of Proposition 1

It is clear that it is optimal to pick constant values for {Ct, Yt, Nt, Θt,Qt, St}. This imme-diately implies that τt = 0 and that τL is constant.

46

Dropping the t subscripts and substituting some of the constraints, we can rewrite theplanning problem as

maxS

[(1− α) (S)η−1 + α

] (1−σ)ηη−1 S(1−σ)(γ−1)C∗1−σ

1− σ− 1

1 + φ

C∗1+φ

A1+φ

[(1− α) (S)η−1 + α

]1+φS(1+φ)γ

This yields a function S(C∗). We then need to solve for S(C∗) = 1. We can then back outthe corresponding τL from the labor-leisure condition.

We find

0 =

(1− α)ηSη−1

S[(1− α) (S)η−1 + α

] + (γ− 1)S

[(1− α) (S)η−1 + α

] (1−σ)ηη−1 S(1−σ)(γ−1)C∗1−σ

− (η − 1)Sη−1

S[(1− α) (S)η−1 + α

] + γ

S

C∗1+φ

A1+φ

[(1− α) (S)η−1 + α

]1+φS(1+φ)γ

We now impose S = 1 and solve for C∗

0 = [(1− α)η + (γ− 1)]C∗1−σ − [(η − 1) + γ]C∗1+φ

A1+φ

i.e.C∗φ+σ

A1+φ=

(1− α)η + γ− 1η + γ− 1

We can now plug this back in the labor-leisure condition

1 = M(1 + τL)C∗φ+σ

A1+φ

to obtain the proposition.

A.2 Proof of Proposition 5

In the Cole-Obstfeld case, we can write V(NFA1) as the maximization over {Θt}t≥1 of

(1− α) log(At) + α log C∗t +1− α

1 + φlog(1− α) +

φ + α

1 + φlog(Θt)

− (1− α)φ

1 + φlog [αΛt + (1− α)Θt]−

(1− α) αΛt+(1−α)ΘtΘt

1 + φ

47

subject to

NFA1 = −∞

∑t=0

βtC∗−σt α(Λt −Θt).

Consider first productivity shocks. For all t ≥ 1 productivity At, the solution of the plan-ning problem for t ≥ 1 features a constant allocation, for which we drop the t subscripts.This immediately implies that optimal capital controls are zero for t ≥ 1. It is immediatethat the optimal Θ is independent of the path At for t ≥ 1. The result for productivityshocks then simply follows from Proposition 4.

Consider next foreign consumption shocks. For all t ≥ 1 productivity C∗t , the solutionof the planning problem for t ≥ 1 features a constant allocation, for which we drop thet subscripts. This immediately implies that optimal capital controls are zero for t ≥ 1. Itis immediate that the optimal Θ is independent of the path C∗t for t ≥ 1. The result forproductivity shocks then simply follows from Proposition 4.

Consider next permanent export demand shocks ΛH,t = ΛH for all t ≥ 0. For allt ≥ 1 the solution of the planning problem for t ≥ 1 features a constant allocation, forwhich we drop the t subscripts. This immediately implies that optimal capital controlsare zero for t ≥ 1. We also find it convenient to define Θ0 = Θ0 −ΛH and Θ = Θ−ΛH.The country budget constraint establishes the following relationship between Θ and Θ0:Θ = −1−β

β Θ0. We are left with the following maximization problem

max

[log[Λ0 + Θ0

]+ log C∗ − 1

1 + φ

(C∗

A

)1+φ (ΛH + (1− α)Θ0

)1+φ+ βV(−α

βΘ0)

]

where

(1− β)V(Θ) = (1− α) log(A) + α log C∗ +1− α

1 + φlog(1− α) +

φ + α

1 + φlog[Θ + ΛH

]

− (1− α)φ

1 + φlog[ΛH + (1− α)Θ

]−

(1− α)ΛH+(1−α)ΘΛ+Θ

1 + φ

Let U(Θ0, ΛH) be the objective function. We can compute

UΘ0,ΛH=

φ + α

1 + φ

1(Θ + ΛH)2

+α(1− α)

1 + φ

Θ−ΛH

(ΛH + Θ)3− 1

(ΛH + Θ0)2

− (1− α)φ

(C∗

A

)1+φ (ΛH + (1− α)Θ0

)φ−1 − (1− α)2φ

1 + φ

1[ΛH + (1− α)Θ]2

For Θ0 = 0 (and hence Θ = 0) we always have UΘ0,ΛH(0, ΛH) < 0, so that at least

48

for small shocks, Θ0 is decreasing in ΛH and Θ is increasing in ΛH with Θ = Θ0 = 0for ΛH = 1. The result in the proposition follows since capital controls τ0 are given byΛH+ΘΛH+Θ0

= 1 + τ0.Finally, let’s us consider wealth shocks W0 6= 0. For all t ≥ 1 the solution of the plan-

ning problem for t ≥ 1 features a constant allocation, for which we drop the t subscripts.This immediately implies that optimal capital controls are zero for t ≥ 1. We also findit convenient to define Θ0 = Θ0 − 1− (1− β) 1

α NFA0 and Θ = Θ− 1− (1− β) 1α NFA0.

The country budget constraint establishes the following relationship between Θ and Θ0:Θ = −1−β

β Θ0. We are left with the following maximization problem over Θ0 :

log(

Θ0 + 1 + (1− β)1α

NFA0

)+ log C∗

− 11 + φ

(C∗

A

)1+φ ((1− α)Θ0 + (1− β)

1− α

αNFA0 + 1

)1+φ

+ βV(Θ)

where

(1− β)V(Θ) = (1− α) log(A) + α log C∗ +1− α

1 + φlog(1− α)

+φ + α

1 + φlog[

Θ + 1 + (1− β)1α

NFA0

]− (1− α)φ

1 + φlog[(1− α)Θ + (1− β)

1− α

αNFA0 + 1

]

−(1− α)

(1−α)Θ+(1−β) 1−αα NFA0+1

Θ+1+(1−β) 1α NFA0

1 + φ.

Let U(Θ0, NFA0) be the objective function. We have

1(1− β) 1

α

UΘ0,NFA0=

φ + α

1 + φ

1[Θ + 1 + (1− β) 1

α NFA0]2− 1

[Θ0 + 1 + (1− β) 1α NFA0]2

− (1− α)2φ

(C∗

A

)1+φ ((1− α)Θ0 + (1− β)

1− α

αNFA0 + 1

)φ−1

− (1− α)3φ

1 + φ

1[(1− α)Θ + (1− β)1−α

α NFA0 + 1]2− 2

α(1− α)

1 + φ

1[Θ + 1 + (1− β) 1

α NFA0]3

For Θ0 = 0 (and hence Θ = 0) we always have UΘ0,NFA0(0, NFA0) < 0, so that at least

for small shocks, so that at least for small shocks, Θ0 is decreasing in W0 and Θ is in-creasing in W0 with Θ = Θ0 = 0 for ΛH = 1. The result in the proposition follows since

capital controls τ0 are given by 1+(1−β) 1α NFA0+Θ

1+(1−β) 1α NFA0+Θ0

= 1 + τ0. . More generally we have

UΘ0,NFA0(Θ0, NFA0) < 0 for all Θ0 < 0 which implies, together with the concavity of U

49

in Θ0, that for positive shocks to NFA0, Θ0 is a decreasing function of NFA0 so that theresult generalizes to any positive shock to initial wealth NFA0 > 0.

A.3 Derivation of the feasible sets F and F ′ in Propositions 12 and 13

We deal with the case where the labor tax is set at its uncoordinated level. The other caseis similar. Because the uncoordinated component planning problem is linear quadratic,we know that ({ ˆy∗t }t≥0, {π∗t }t≥0) is such that 0 =

´ 10 yi

tdi and 0 =´ 1

0 πiH,tdi if and only if

the solution of the following minimization problem is πH,t = yt = θt = 0:

min12

ˆ ∞

0e−ρt

[απ(πH,t)

2 + 2αππ∗t πH,t + (yt)2 + 2 ˆy∗t yt + αθ(θt)

2]

dt

subject to˙πH,t = ρπH,t − κ ˜yt − λαθt,

˙yt = (1− α) ˙θt − πH,t,ˆ ∞

0e−ρtθtdt = 0,

y0 = (1− α)θ0,

where the minimization is over the variables πH,t, yt, θt, taking ˆy∗t , and π∗t as given. Weput a multiplier Γ on the constraint

´e−ρtθtdt = 0 and incorporate it in the objective to

obtain an optimal control problem. The first-order conditions are

−µπ,t = αππH,t + αππ∗t − µy,t,

ρµy,t − µy,t = yt + ˆy∗t − κµπ,t,

ρµθ,t − µθ,t = αθ θt + Γ− λαµπ,t,

(1− α)µy,t + µθ,t = 0,

µπ,0 = 0.

We can combine these equations to get

µπ,t =1− α

κ(1− α) + λαyt +

αθ

κ(1− α) + λαθt +

Γκ(1− α) + λα

+1− α

κ(1− α) + λαˆy∗t ,

50

leading to the following reduced system

1− α

κ(1− α) + λα˙yt +

αθ

κ(1− α) + λα˙θt +

1− α

κ(1− α) + λα˙y∗t = −αππH,t + µy,t − αππ∗t ,

ρµy,t − µy,t =λα

κ(1− α) + λαyt −

καθ

κ(1− α) + λαθt −

κΓκ(1− α) + λα

+λα

κ(1− α) + λαˆy∗t ,

˙πH,t = ρπH,t − κ ˜yt − λαθt,

˙yt = (1− α) ˙θt − πH,t,

y0 = (1− α)θ0,

1− α

κ(1− α) + λαy0 +

αθ

κ(1− α) + λαθ0 +

Γκ(1− α) + λα

+1− α

κ(1− α) + λαˆy∗0 = 0.

In order for πH,t = yt = θt = 0 to be the solution, we must have

1− α

κ(1− α) + λα˙y∗t = µy,t − αππ∗t ,

ρµy,t − µy,t = −κΓ

κ(1− α) + λα+

λα

κ(1− α) + λαˆy∗t ,

Γκ(1− α) + λα

+1− α

κ(1− α) + λαˆy∗0 = 0.

Let Xt = [ ˆy∗t , µy,t]′ and Bt = [− κ(1−α)+λα1−α αππ∗t , κ

κ(1−α)+λαΓ]′. We can write this system as

Xt = AXt + Bt

together with the initial condition

E′1X0 = − 11− α

Γ.

The characteristic polynomial of A is x2 − ρx + λα1−α . This implies that two eigenvalues of

A are either positive, or complex with positive real parts. The solution is hence given by

Xt = −ˆ ∞

te−A(s−t)Bsds.

51

We can solve for Γ:

Γ = −κ(1−α)+λα

1−α

11−α −

κE′1 A−1E2κ(1−α)+λα

απ

ˆ ∞

0π∗s E′1e−AsE1ds.

We conclude that { ˆy∗t ,π∗t }t≥0 is in the feasibility set F if and only if

ˆy∗t = απκ(1− α) + λα

1− α

ˆ ∞

tπ∗s E′1e−A(s−t)E1ds + απ

κE′1A−1E2´ ∞

0 π∗s E′1e−AsE1ds1

1−α −κE′1 A−1E2

κ(1−α)+λα

,

π∗t = ρπ∗t − κ ˆy∗t .

52

0 1 2 3 4 5−0.04−0.02

00.020.04

θ

0 1 2 3 4 5−0.04−0.02

00.020.04

y

0 1 2 3 4 50

0.050.1

0.150.2

πH

0 1 2 3 4 5−0.01

0

0.01

0.02nx

0 1 2 3 4 5−0.1−0.05

00.050.1

τ = i − i∗

Figure 1: Permanent terms of trade shock, α = 0.4.

53

0 1 2 3 4 5−0.06

−0.04

−0.02

0θ

0 1 2 3 4 50

0.02

0.04

0.06y

0 1 2 3 4 50

0.050.1

0.150.2

πH

0 1 2 3 4 50

0.002

0.004

0.006nx

0 1 2 3 4 5−0.01

0

0.01

0.02τ = i − i∗

Figure 2: Permanent terms of trade shock, α = 0.1.

54

0 1 2 3 4 5−0.02

−0.01

0

0.01θ

0 1 2 3 4 50

0.0010.0020.0030.004

y

0 1 2 3 4 50

0.010.020.030.04

πH

0 1 2 3 4 5−0.004−0.002

00.0020.004

nx

0 1 2 3 4 5−0.04

−0.02

0

0.02τ = i − i∗

Figure 3: Mean-reverting natural interest rate shock, α = 0.4.

55

0 1 2 3 4 5−0.04

−0.02

0

0.02θ

0 1 2 3 4 50

0.0010.0020.0030.004

y

0 1 2 3 4 50

0.010.020.030.04

πH

0 1 2 3 4 5−0.002

0

0.002

0.004nx

0 1 2 3 4 5−0.04−0.02

00.020.04

τ = i − i∗

Figure 4: Mean-reverting natural interest rate shock, α = 0.1.

56

0 1 2 3 4 5−0.04−0.02

00.020.04

θ

0 1 2 3 4 5−0.04−0.02

00.020.04

y

0 1 2 3 4 5−0.2−0.1

00.10.2

πH

0 1 2 3 4 5−0.01

0

0.01

0.02nx

0 1 2 3 4 5−0.1−0.05

00.050.1

τ = i − i∗

Figure 5: Mean-reverting productivity shock, α = 0.4.

57

0 1 2 3 4 5−0.06

−0.04

−0.02

0θ

0 1 2 3 4 5−0.04−0.02

00.020.04

y

0 1 2 3 4 5−0.2−0.1

00.10.2

πH

0 1 2 3 4 50

0.002

0.004

0.006nx

0 1 2 3 4 50

0.010.010.020.02

τ = i − i∗

Figure 6: Mean-reverting productivity shock, α = 0.1.

58

0 1 2 3 4 50

0.01

0.02

0.03

0.04y

0 1 2 3 4 50

0.05

0.1

0.15

0.2π

0 1 2 3 4 50

0.01

0.02

0.03

0.04c

0 1 2 3 4 5

−0.02

−0.01

0

{gopp, gstim, g

}

Figure 7: Government spending for a permanent terms of trade shock, α = 0.4.

59

0 1 2 3 4 5

0

0.002

0.004y

0 1 2 3 4 50

0.01

0.02

0.03

0.04π

0 1 2 3 4 50

0.002

0.004

0.006

0.008c

0 1 2 3 4 5−0.008

−0.006

−0.004

−0.002

0

{gopp, gstim, g

}

Figure 8: Government spending for a mean-reverting natural interest rate shock, α = 0.4.

60

0 1 2 3 4 5

0

0.02

0.04y

0 1 2 3 4 5−0.05

0

0.05

0.1

0.15π

0 1 2 3 4 5

0

0.02

0.04c

0 1 2 3 4 5−0.03

−0.02

−0.01

0

0.01

{gopp, gstim, g

}

Figure 9: Government spending for a mean-reverting productivity shock, α = 0.4.

61