ESSAYS IN INTERNATIONAL MACROECONOMICS

Transcription

1 ESSAYS IN INTERNATIONAL MACROECONOMICS by Xuan Liu Department of Economics Duke University Date: Approved: Dr. Martín Uribe, Supervisor Dr. Stephanie Schmitt-Grohé Dr. Kent Kimbrough Dr. Craig Burnside Dissertation submitted in partia fufiment of the requirements for the degree of Doctor of Phiosophy in the Department of Economics in the Graduate Schoo of Duke University 2007

2 ABSTRACT ESSAYS IN INTERNATIONAL MACROECONOMICS by Xuan Liu Department of Economics Duke University Date: Approved: Dr. Martín Uribe, Supervisor Dr. Stephanie Schmitt-Grohé Dr. Kent Kimbrough Dr. Craig Burnside An abstract of a dissertation submitted in partia fufiment of the requirements for the degree of Doctor of Phiosophy in the Department of Economics in the Graduate Schoo of Duke University 2007

3 Copyright c 2007 by Xuan Liu A rights reserved

4 Abstract This dissertation consists of two essays in internationa macroeconomics. The first essay shows that optima fisca and monetary poicy is time consistent in a standard sma open economy. Further, there exist many maturity structures of pubic debt capabe of rendering the optima poicy time consistent. This resut is in sharp contrast with that obtained in the context of cosed-economy modes. In the cosed economy, the time consistency of optima monetary and fisca poicy imposes severe restrictions on pubic debt in the form of a unique term structure of pubic debt that governments can eave to their successors at each point in time. The time consistent resut is robust: optima poicy is time consistent when both rea and nomina bonds have finite horizons. Whie in a cosed economy, governments must have both nomina and rea bonds, and have at east rea bonds over an infinite horizon to render optima poicy time consistent. The second essay uses a dynamic stochastic genera equiibrium mode to theoreticay rationaize the empirica finding that sudden stops have weaker effects on outputs when the sma open economy is more open to trade. First, wefare costs of sudden stops are decreasing in trade openness. The reason is that when the economy is more open to trade, the economy wi have ess voatie capita, which eads to ess voatie output. In terms of wefare, when the sma open economy is more open to trade, the wefare costs of sudden stops wi be smaer. Second, sudden stops may be wefare improving to the sma open economy. This is because when the representative househod is a net borrower in the internationa capita market, its consumption wi be negativey correated with country spread. Since utiity is a concave function iv

5 of consumption, it must be a convex function of country spread. That is, when the country spread is more voatie, the mean utiity is higher. The two findings are robust: they hod with one sector economy mode, and two sector economy modes with homogenous capita and heterogenous capita. In addition, this paper shows that a counter-cycica tariff rate poicy is not wefare-improving. v

6 Acknowedgements The writing of this dissertation woud never have been finished without persona and practica support of numerous peope. I sincerey thank my committee members, my friends, and my wife and famiy for their ong-term support. I woud ike to express my deepest gratitude to my advisor, Professor Martín Uribe, for his invauabe guidance and understanding. Professor invested so much time discussing and providing his wise insight into both my research and ife as a mentor and a friend. Professor Uribe is my dream advisor. He is a eader in the fied in which I want to do my research. He cares about my research; gives me detaied guidance; and pushes me to do serious research. He is a true genteman. In that same vein, I want to thank Professor Stephanie Schmitt-Grohé, Professor Kent Kimbrough, and Professor Craig Burnside for their guidance in various aspects of my dissertation writing. Many peope in the Department of Economics at Duke University assisted and encouraged me in in various ways. I am especiay gratefu to Profs. Lori Leachman, Edward Tower, Lutz Weinke, Han Hong, Michee Connoy, Stefania Abanesi, and Árpád Ábrahám for a that they have taught me. I aso thank my feow students and staff for their hep and encouragement. In particuar, I thank David Powers and Lutz Weinke for editing my paper. My famiy have a been encouraging. My dearest wife Yue Lucy Hu has unwavering faith in me. She has encouraged and supported me to go through a the peaks and troughes. My parents provided me the initia inspiration to pursue my Ph.D. study abroad. My parents, my parents in aw, and my sister give me unending support and encouragement with their ove. vi

7 Contents Abstract Acknowedgements List of Figures iv vi x 1 Optima and Time Consistent Monetary and Fisca Poicy in a Sma Open Economy Introduction The Sma Open Economy Househods Competitive Firms The Government Internationa Investors Competitive Equiibrium Characterization of Competitive Equiibrium Intertempora Budget Constraints Optima Poicy with Commitment Optima Poicy with Discretion An Economy without Nomina Bonds An Economy without Rea Bonds Perfect Substitution between Impicit Instruments and Expicit Instruments The Time Horizon of Bonds Poicy Impications vii

8 1.5 Concusion Trade Openness and the Costs of Sudden Stops Introduction One Sector Economy The Benchmark Economy Numerica Soution and Resuts Wefare Costs Working Capita Constraint Two Sector Economy with Homogenous Capita Mode Modification Caibration Updating Simuation Resuts Two Sector Economy with Heterogenous Capita Counter-Cycica Tariff Rate Poicy Concusion A Appendix for Chapter 1 94 A.1 Equivaence of constraints 1.1 and 1.2 with the singe constraint A.2 Primary form of competitive equiibrium A.3 Intertempora budget constraint of the government A.4 Impementabiity constraint of the Ramsey government Hereafter RG A.5 Simpification of Condition A.6 Time Consistency when Both Rea and Nomina Bonds Are Avaiabe viii

9 A.7 Liquidity Constraint of the t = 0 Government A.8 What wi happen if househods do not cooperate? B Appendix for Chapter B.1 One Sector Mode B.1.1 Lagrange and Optimaity Conditions B.1.2 Functiona Form and Non-stochastic Steady State B.2 Two Sectors with Homogenous Capita - Tradabe Good B.2.1 Lagrange and Optimaity Conditions B.2.2 Non-Stochastic Steady State B.3 Two Sectors with Homogenous Capita - Fina Good B.3.1 Lagrange and Optimaity Conditions B.3.2 Non-Stochastic Steady State B.4 Two Sectors with Heterogenous Capita B.4.1 Lagrange and Optimaity Conditions B.4.2 Non-Stochastic Steady State B.5 Unconditiona Wefare Cost B.5.1 Notation Simpification B.5.2 Second Order Approximation B.5.3 Expressions for Derivatives B.6 Soving Dynamic Stochastic Genera Equiibrium B.6.1 Theoretica Steps B.6.2 MATLAB Programs Bibiography 158 Biography 164 ix

10 List of Figures 2.1 Steady State Potted against Tariff Rate Impuse Responses to a Positive Country Spread Shock One Sector Economy without Working Capita Constraint; Line corresponds to 0% tariff rate; Dotted Line corresponds to 10% tariff rate Unconditiona utiity potted against country spread R Impuse Responses to a Positive Country Spread Shock One Sector Economy with Working Capita Constraint; Line corresponds to 0% tariff rate; Dotted Line corresponds to 10% tariff rate Impuse Responses to a Positive Country Spread Shock Two Sector Economy with Homogenous Capita; Line corresponds to 0% tariff rate; Dotted Line corresponds to 10% tariff rate. 93 x

11 Chapter 1 Optima and Time Consistent Monetary and Fisca Poicy in a Sma Open Economy 1.1 Introduction In an important recent paper, Persson, Persson, and Svensson 2006 hereafter PPS show that optima fisca and monetary poicy is time consistent in the context of cosed-economy modes. However, the requirement of time consistency imposes an extremey strong restriction on pubic debt. Namey, there is a unique term structure of pubic debt capabe of rendering optima fisca and monetary poicy time consistent. This extremey strong restriction, pus the issue of time consistency itsef, provokes the foowing questions: a Is the optima poicy time consistent in a sma open economy? The question is important because time inconsistent optima poicy wi resut in an inferior equiibrium with ower overa wefare Chari b If so, is the term structure of pubic debt unique? c If so, must the government issue both nomina and rea pubic debt over an infinite horizon? The questions are important because they are of practica reevancy. Time consistency is defined as foows: From time 0, for any t 0, the time t + 1 government wi choose the time t government s poicy continuation. There are two types of time inconsistency: nomina and rea. Nomina time inconsistency refers to the government s action to change the outstanding 1

12 nomina debt by deviating from the promised path of price eves 1. Rea time inconsistency refers to the government s action to deviate from the promised abor income tax rate path. It is usuay characterized by changing the discounted present vaue of rea debt by deviating from the promised path of rea interest rates through changes in abor income tax rates 2. Time inconsistent poicy resuts in an inferior equiibrium, as seen through the foowing mechanism: The optima poicy requires the use of bonds to smooth out taxes and consumption over time. Whenever the poicy is not time consistent, the demand for bonds disappears, and the government cannot use bonds to smooth taxes and consumption over time. It has to rey entirey on distortionary taxes and the economy ends up with ower aggregate wefare. It is thus important for the Ramsey government to render optima poicy time consistent. Note that this argument appies to both cosed economies and open economies. The iterature suggests that one way to guarantee rea time consistent optima poicy in cosed rea economies is to choose the term structures of 1 Cavo 1978 shows that in a cosed monetary economy, when a ump-sum tax is not avaiabe, the government has an ex post incentive to use surprise infation to ower the rea vaue of the outstanding nomina pubic debt. The reason for nomina time inconsistency is that the government uses the surprise infation as a ump-sum tax. 2 Lucas and Stokey 1983 hereafter LS show that in a cosed rea economy, the government has an ex post incentive to manipuate the path of rea interest rates in order to ower the discounted present vaue of rea pubic debt. In this case, even though the government cannot find any tax instrument equivaent to ump-sum taxes, it can aways take advantage of the ineasticity of outstanding rea bonds to minimize the distortion caused by the distortionary taxes. This incentive is refected in the government s ex post action to compensate for the distortion in the bond market with counterpart distortion in other markets, usuay the abor market. These two compensating distortions are briefy mentioned in PPS Liu 2006 shows that rea time inconsistency exists even with non-zero one-period rea bonds because the government can sti change the contemporaneous abor income tax rate, even though in this case, the change in the rea interest rate path has nothing to do with the present vaue of the outstanding one-period rea bond. Further, Liu 2006 shows that the economic incentives for both nomina and rea time inconsistency are unified on the foowing two arguments: a The government wants to minimize the genera distortion eve caused by distortionary taxes. And b the ineasticity of inherited bonds gives the government a room to take time inconsistent action. 2

13 debt that the t = 1 government wi inherit from the t = 0 government. This is possibe because the term structure of the t = 1 government s initia debt hoding position does not enter the t = 0 government s Ramsey probem to be defined ater. This irreevance gives the t = 0 government freedom to choose the term structure of debt that it can eave to its successor government. The maturity structure of rea debt can be pinned down in the exact same way as in LS Under a setup with both rea and nomina time inconsistency, PPS 2006 incorporate direct costs to the surprise infation and show that it is possibe to construct a unique term structure of nomina debt and rea debt in such a way that the optima poicy is time consistent 3. Abanesi 2005 introduces heterogenous agents and shows that, if the weath distribution effect is strong enough, it is possibe to find time consistent optima poicy by choosing appropriate distribution of nomina and rea bonds. The present work extends the PPS 2006 framework beyond a cosed economy to a sma open economy. The main findings are as foows: 1 In a standard sma open economy, the optima poicy is generay time consistent. Specificay, there are many maturity structures of pubic debt capabe of rendering optima poicy time consistent. This is in sharp contrast 3 The iterature has a debate on whether the optima poicy can be rendered time consistent in this case. LS 1983 argue that if a monetary economy has both cash and credit consumption goods, the consumer optimization requires both nomina and rea debt be non-zero in order to smooth out consumption. However, the non zero nomina debt impies nomina time inconsistency as shown in Cavo LS 1983 concude that it is impossibe to have time consistent optima poicy since any non-zero nomina debt wi resut in nomina time inconsistency. PPS 1987 study a cosed economy with money in the utiity function. They suggest that both rea and nomina time inconsistency can be removed by choosing an appropriate maturity structure of nomina and rea bonds. In particuar, they assume that the discounted present vaue of nomina iabiities in the initia period is zero. Cavo and Obstfed 1990 hereafter CO suggest that the soution in PPS 1987 is actuay not an optimum. The CO 1990 finding puts the PPS 1987 concusion in question. Avarez, Kehoe, and Neumeyer 2004 hereafter AKN show that it is possibe to remove both nomina and rea time inconsistency when the endogenous restriction on the pubic debt, which is ignored in PPS 1987, has been taken into consideration. 3

14 with what is obtained in the cosed economy. In the cosed economy, the time consistency of optima monetary and fisca poicy imposes severe restrictions on pubic debt in the form of a unique term structure of pubic debt that governments can eave to their successors at each point in time. The main reason for the time consistent optima poicy is that the t = 0 government aways has more poicy instruments than the t = 0 government has poicy choices. The poicy choices of the t = 1 government incude: nomina interest rates and ony one abor income tax rate instead of a abor income tax rates. This is true because in the Ramsey probem, other optima abor income tax rates are pinned down by the exogenousy given rea interest rates. The poicy instruments of the t = 0 government incude: nomina pubic and externa bonds from t = 3 on; and either the present vaues of rea pubic and externa bonds or the nomina pubic and externa bonds at t = 1 4 ; and possiby Lagrange mutipiers. Since the economy is of the infinite horizon, there are far more poicy instruments than there are the poicy choices of the t = 1 government, and it is thus possibe to have time consistent optima poicy. With rea bonds, the mechanism for the time consistent optima poicy is the foowing: the t = 0 government uses rea bonds to contro the t = 1 government s choice of one abor income tax rate; and uses nomina bonds to contro the t = 1 government s choices of t 2 nomina interest rates. The mechanism of the t = 0 government using one instrument to contro for one choice of the t = 1 government is as foows. In the sma open economy, when the t = 1 househods inherit a positive 4 The nomina bonds at t = 2 are taken away because the government wi choose them in such a way that the next government wi smooth consumption the same as poicy continuation. 4

15 one-period rea bond, they fee richer so that they woud choose reativey more contemporary consumption. However, since rea interest rates are exogenous, the househods have to smooth consumption out over time. Other things being equa, the hoding position of the one-period rea bond wi determine the eve of the smoothed consumption, so does the contemporary abor income tax rate. If the t = 0 government has freedom in choosing the rea bond hoding position for the t = 1 government, it can choose the right amount. As a resut, the t = 1 government s reoptimized contemporary abor income rate wi be the same as poicy continuation 56. In this process, the t = 0 government uses rea bond hoding baance to contro the t = 1 government s optima choice of the contemporary abor income tax rate. When the t = 1 government inherits positive muti-period nomina bonds at one particuar future period, it has the incentive to increase the nomina interest rate between that period and the period before because this serves as a ump-sum tax. However, the benefit from this surprise infation wi be fuy offset by the associated decine of rea money baance. This gives the t = 1 government the option to choose positive finite nomina interest rates. If the t = 0 government has freedom in choosing hoding positions of nomina 5 Liu 2006 shows that in the cosed economy, when the t = 1 government inherits a positive one-period rea bond hoding position, it has the incentive to increase the contemporary abor income tax rate comparing to the poicy continuation, whie setting reative ow abor income tax rates for the future. This is because the ineastic rea bond hoding position provides an indirect and ess distortionary source of pubic financing. This gives arise to the economic incentive of rea time inconsistent behavior. The househods receive more utiity from their unit contemporary consumption than from unit future consumption due to the ineastic and positive one-period rea bond. They wi consume reativey more, comparing to the poicy continuation, which eads to a high contemporary abor income tax rate. The Ramsey government finds it optima to set a reativey high rea interest rate to induce househods to save. 6 In the rea sma open economy, the eve of consumption is determined by discounted vaue of pubic surpus, initia pubic debt, and the present vaue of utiity from working. It can be shown that the sum of the first two components wi be time-independent, but the third component is time-dependent if productivity is changing over time. If the government does not have poicy instruments, this time-dependent component arises time inconsistent behavior. 5

16 bonds, it wi choose the right amount so that the reoptimized positive finite nomina rates of the t = 1 government wi be same as poicy continuation. In this process, the t = 0 government uses nomina bonds to contro the t = 1 government s optima choices of nomina interest rates. Both processes are possibe at the same time because the t = 0 government wi aways have more poicy instruments than the poicy choices of the t = 1 government. More importanty, as I show ater, to find the right poicy instruments is to sove a system of inear equations. The mechanism works because I assume that given an initia asset position, there is one unique equiibrium. 2 My second finding is: The t = 0 government can render optima poicy time consistent without nomina bonds when productivity is constant. In this case, the nomina economy is effectivey reduced to a rea economy and the government obtains one Lagrange mutipier as an impicit poicy instrument. This is in sharp contrast with what is obtained in the context of cosed economy modes. In cosed economies, the government must issue nomina bonds, otherwise the t = 1 government wi choose a different path of nomina interest rates even if the productivity is assumed to be constant. 3 Under certain conditions, the t = 0 government can render optima poicy time consistent without rea bonds. In this case, the t = 0 government sti has more poicy instruments than the t = 1 government has poicy choices. However, it is required that there are no identification probems caused by the shift from cosed economy to a sma open economy. I wi return ater to these identification probems. This is aso in sharp contrast to what is obtained in cosed economies: without rea bonds, the optima poicy in cosed economies cannot be time consistent. 4 Furthermore, the optima poicy in a sma open economy may be time 6

17 consistent with both pubic and private bonds issued over a finite horizon. This comes from the fact that the governments in the sma open economy aso ose their choices over nomina interest rates, on top of the fact that the governments ose their choices of abor income tax rates. However, in the cosed economy, it requires that the bonds issued be over an infinite horizon. There are severa poicy impications of my findings: a Optima poicy wi be time consistent when the government issues both rea and nomina bonds over the finite horizon; b If the government has neutra taste for rea versus nomina bonds, it wi aways issue both nomina and rea bonds; c If the government disikes rea bonds, it wi aways issue both nomina and rea bonds and set the rea debt at an optimay minimum eve; and d when some expicit poicy instruments are unavaiabe, the government may obtain some impicit poicy instruments. These poicy impications are important in the foowing sense: First, if the necessary condition for time consistent optima poicy is the avaiabiity of both rea and nomina bonds over infinite horizon, it impies that optima poicy in practice is time inconsistent. The resut that optima poicy is time consistent with either rea or nomina bonds over finite horizon is of practica importance. Secondy, one interpretation of rea bonds is nomina bonds but denominated in US$, i.e., debt doarization. From the recent financia crises, debt doarization exaggerates the adverse effects of crises. The resut that the government can have time consistent optima poicy with minimum rea bond hodings is important because the propagation effect from debt doarization is then minimized. The time consistent resuts rey on the assumption that for the given combination of monetary and fisca poicy, there is a unique competitive equiibrium. 7

18 As a resut, as ong as the monetary and fisca poicy combination uniquey determines the competitive equiibrium, the optima poicy is time consistent. Note that even though the price eve is competey determined in each period, there is sti room for the future government to seect different optima poicy as opposed to mere poicy continuation. However, when the price eve is indeterminate, the time consistency probem wi be more compicated. This paper is organized as foows: Section 2 describes the economic setup; Section 3 discusses the Ramsey probem; Section 4 discusses the time consistency of optima poicy; and Section 5 concudes. 1.2 The Sma Open Economy Househods In this mode, househods are price-takers, and they are given the price of consumption, p t, the present vaue in period 0 of goods in period t, q 0,t, the abor income tax rate, τ t, and the nomina interest rate i t+1. A representative househod chooses the time profie of consumption, c t, t 0, rea money baances, m t+1, t 1, and abor suppy, h t, t 0, to maximize ifetime utiity: β t u c t, m t, h t, t=0 subject to its period budget constraints 1.1 and the no-ponzi game condition 1.2. Rea money baances are defined as m t = M t 1 p t, where M denotes nomina money baance. The choice of beginning-of-period nomina money introduces an infation cost: when the price eve increases, rea money baances are reduced, and the househod receives ess utiity from the given eve of nomina money baances. PPS 1996 argue that high infation has arge and 8

19 we-known socia costs. Another rationa for the use of the beginning of period money baance is that infation forces househods to economize money thus bringing costs to hoding money, see Baiey 1956 and Tower The parameter β is the subjective discount factor, which weights the consumption bundes over time. The representative househod s period budget constraint is given by: [ q 0,t 1 τ t w t h t + Π t + M ] t 1 + p t q 0,t c t + M t + p t s=t+1 q 0,s t 1b P s + t 1 Bs P p s s=t q 0,s tb P s + t Bs P p s. 1.1 The variabe q 0,t can be regarded as a mutipe period discount factor. In this paper, given the exogenous rea interest rate, there is a one-to-one reationship between q 0,t and β. The same discount factor is appied to both interna and externa bonds due to the assumption of perfect capita mobiity. The variabe t 1 b P s denotes net caims by the domestic househod when entering period t on the amount of goods to be deivered in period s; t 1 B P s denotes the net caims on money to be deivered in period s. One can interpret the rea bonds as nomina bonds but denominated in US$. The bonds are rea in this sma open economy because their purchasing power wi not change when the domestic price eve changes. The sum s=t q 0,s t 1b P s + t 1 Bs P p s denotes the representative househod s initia bond hoding position. Note that both nomina and rea bonds appear in the period budget constraint. The reason for this is the foowing: In a rea economy, the government cannot use nomina assets to render optima poicy time consistency, otherwise the price eve wi go to either infinity or zero. Whie in a monetary economy, the government must use nomina assets to assure the time consistency of the 9

20 optima poicy to avoid zero or infinite price eve. When the househod in a cosed monetary economy receives utiity from both consumption and rea money baances, the government has to issue both rea and nomina assets to smooth both consumption and rea money baances. When I extend this framework to sma open economies, it at first seems natura to put both nomina and rea bonds in the domestic househod s budget constraint. However, I wi ater show that it is not necessary to issue both rea and nomina bonds in a sma open economy to render optima poicy time consistent. The no-ponzi game condition for the representative househod is given by: im j [ q t,t+j M t+j p t+j + s=t+j+1 q t,s t+jb P s + t+j Bs P p s ] 0, t The no-ponzi game condition has its usua meaning: the representative househod has to keep non-negative financia assets in the imit. This no-ponzi game condition must hod in each period. In this economy, nomina interest rates are defined as: i t+1 = q 0,t+1/p t+1 q 0,t /p t 1, t Combining the period budget constraint and the no-ponzi game condition, I write the inter-tempora budget constraint of the representative househod as equation 1.4 as: t=0 q 0,t [1 τ t w t h t + Π t ] + M 1 p 0 + q 0,t t=0 1b P t + 1 B P t p t = q 0,t c t + t=0 q 0,t m t i t

21 In the Appendix, I show that the time sequences for {c t, m t+1, h t } satisfying constraints 1.1 and 1.2 are the same as those satisfying the singe constraint 1.4. Thus, the representative househod maximizes ifetime utiity subject to the singe constraint 1.4. The optimaity conditions for the domestic househod are the singe inter-tempora budget constraint 1.4 and β t u ct = λq 0,t, t τ t = w t u ht u ct, t i t+1 = u mt+1 u ct+1, t A the optimaity conditions have their usua meanings: equation 1.5 says that the margina utiity of consumption shoud equa the margina cost of consumption; equation 1.6 shows that the introduction of abor income tax distorts the margina rate of substitution between consumption and abor; and equation 1.7 states that there is cost to hoding money Competitive Firms In each period, competitive firms use decreasing returns-to-scae technoogy in production: y t = z t h η t, t 0. The technoogy is non-inear in abor input in order to rue out the possibiity of a corner soution. In the cosed economy, the choice of zero output is rued out because it is usuay assumed that consumption is positive. But in the open economy, zero output does not impy zero consumption. To rue out the case of zero output and guarantee an interior soution, I assume that at ow eves of abor input, margina product of abor is extremey high. For detais, 11

22 pease see Schmitt-Grohe and Uribe I assume that output is sod in both domestic and internationa markets so that the aw of one price for one tradabe good hods in each period, i.e., p t = S t p t, t 0, 1.8 where the variabe S t denotes the nomina exchange rate at time t and the variabe p t denotes the word price at time t. Firms maximize profit, which is given by Π t = z t h η t w t h t, t The optimaity condition for abor demand is given by: w t = z t ηh η 1 t, t The Government The government finances its expenditures by evying abor income taxes at the rate of τ t, by printing money and by trading muti-period nomina and rea bonds with both domestic househods and internationa investors. The monetary/fisca regime consists of pans for the poicy instruments: money and bonds; and for the poicy choices: nomina interest rates and abor income tax rates. Here I assume that ump-sum taxes are not avaiabe to the government. The period budget constraint of the government is given by: q 0,t g t + M t 1 + p t q 0,s t 1b G s + t 1 Bs G p s s=t q 0,t τ t w t h t + M t p t + s=t+1 q 0,s tb G s + t Bs G p s

23 The variabe t 1 b G s denotes tota net caims on the amount of goods to be deivered by the government in period s. It foows from the perfect capita mobiity assumption that the government appies the same discount factors on both interna and externa bonds. The no-ponzi game condition for the government is given by: im j [ q t,t+j M t+j p t+j + s=t+j+1 q t,s t+jb G s + t+j Bs G p s ] 0, t This condition rues out the possibiity that the government borrows infinitey to finance its expenditures. The government s intertempora budget constraint is given by: q 0,t t=0 1b G t + 1 B G t p t + M 1 p 0 = q 0,t τ t w t h t g t + t=0 q 0,t i t m1.13 t Internationa Investors Internationa investors can aways borrow and end at a nomina interest rate of i in the internationa market. Due to assumption of perfect capita mobiity, the uncovered interest rate parity condition hods: 1 + i t+1 = S t+1 S t 1 + i, t 0. Combined with the purchasing power parity condition, the foowing is obtained: 1 + i t+1 = 1 + i S t+1 S t = 1 + i 1 + π p t+1 p t, t

24 1.2.5 Competitive Equiibrium Definition A competitive equiibrium is defined as a sequence {c t, m t+1, h t, w t, Π t, q 0,t } t=0, a positive constant λ, an initia price eve p 0 > 0, and a sequence of government tax poicies {τ t, i t+1 } t=0, satisfying the conditions of 1.3, 1.4, 1.5, 1.6, 1.7, 1.9, 1.10, 1.13, 1.14, given the initia asset conditions of {M 1, 1 b P t, 1 b G t, and 1 Bt P, 1 Bt G, t 0}. Equation 1.3 describes the cearing condition in the bond markets. Equations sove the domestic househod s utiity maximization probem. Equations 1.9 and 1.10 sove the firms profit maximization probem. Equation 1.13 baances the government s budget constraint. And equation 1.14 soves the internationa investor s borrowing/ending decision. For con- 14

25 venience, I bring the various conditions together beow: q 0,t c t + t=0 q 0,t i t m t M 1 p 0 = q 0,t t=0 1b P t + 1 B P t p t + q 0,t [1 τ t w t h t + Π t ] t=0 β t u ct = λq 0,t, t 0 τ t = w t u ht u ct, t 0 Π t = z t h η t w t h t, t 0 i t+1 = u mt+1 u ct+1, t i t+1 = q 0,t+1/p t+1 q 0,t /p t 1, t 0 w t = z t ηh η 1 t, t 0 q 0,t τ t w t h t g t + t=0 1 + i t+1 = 1 + i 1 + π p t+1 p t, t 0 q 0,t i t m t = q 0,t t=0 1b G t + 1 B G t p t + M 1 p Characterization of Competitive Equiibrium To eiminate nonessentia dynamics in consumption, I make two assumptions. First, I assume that β 1+i 1+π = 1. Second, I assume that the period utiity function is separabe in goods, money and abor suppy, taking the form of uc, m, h = uc + vm + gh. These two assumptions do not change this paper s resuts with respect to time consistent optima poicy, though they simpify the anaytica computation. Given these assumptions, the discount factors wi grow at the rate of β, and consumption is constant over time, 15

26 which can be seen from: q 0,t = 1 + π t = β t, t 0, i u ct = λ q 0,t β t = λ 1+π 1+i t β t = λ, t A third assumption I make is that u mt 0. As a resut, it is aways the case that i t 0. This assumption aso does not change the paper s resuts with respect to time consistent optima poicy Intertempora Budget Constraints Using the optimaity conditions, I can rewrite the government s intertempora budget constraint containing ony the initia price eve, p 0, a constant, λ, and rea money and abor aocations {m t+1, h t } t=0, λ [ Q 0,t 1B t G p 0 t=0 + M 1 ] = β [ ] t ληz t h η t g t 1 b G t + u ht h t t=0 + β t u mt m t, 1.17 where Q 0,t = t i=1 1 + u mi 1. λ Simiary, combining the intertempora budget constraint of the representative domestic househod with that of the government, I obtain the intertempora budget constraint for the sma open economy: t=0 β [ ] t c t + g t + 1 b F t z t h η 1 t = p 0 Q 0,t 1 Bt F, 1.18 t=0 where 1 B F t = 1 B G t 1 B P t, and 1 b F t = 1 b G t 1 b P t. 16

27 1.3 Optima Poicy with Commitment In Appendix B, I show that sequences for {λ, m t+1, and h t } satisfying optimaity conditions 1.3, 1.4, 1.5, 1.6, 1.7, 1.9, 1.10, 1.13, and 1.14, are the same as those satisfying the optimaity conditions 1.17 and Thus, when the government can commit to poicy, it wi choose a constant λ, an initia price eve p 0, and a sequence of {m t+1, h t } t=0 to maximize the representative househod s ifetime utiity: u cλ, M 1, h 0 + p 0 β t u cλ, m t, h t, 1.19 subject to 1.17 and 1.18, given the initia money stock, M 1, the initia rea and nomina debt, 1 b t t=0 and 1 B t t=0. Let µ G 0 and µ E 0 be the Lagrange mutipiers for 1.17, the t = 0 government s intertempora budget constraint, and for 1.18, the economy s intertempora budget constraint, respectivey. Then the Lagrangian associated with the t = 0 government is given by: [ L = u cλ, M ] 1, h 0 + p 0 µ G 0 +µ G 0 { [ λ p 0 t=0 β t u [cλ, m t, h t ] Q 0,t 1B G t + M 1 ]} { β [ ] } t ληz t h η t g t 1 b G t + u ht h t + β t u mt m t t=0 +µ E 0 { β [ t z t h η t cλ g t 1 b F s t=0 ] 1 p 0 t=0 Q 0,t 1B F t }, 17

28 and the optimaity condition with respect to λ is t=0 β t u ct c λ = µe 0 t=0 β t c λ µg 0 { } β t ηz t h η t g t 1 b G t t=0 + µe 0 p 0 + µg 0 p 0 1 B F t Q 0,t λ {[ ] Q 0,t 1 Bt G + M 1 + λ t=0 1 B G t Q 0,t λ } The efthand side of 1.20 represents the margina cost in terms of utiity due to an increase of λ. Intuitivey, when it becomes more expensive to borrow to smooth consumption, the representative househod wi decrease its consumption in each period. The foregone discounted present vaue of utiity caused by the decrease in consumption is the margina cost of the change in λ. The righthand side of 1.20 represents the corresponding margina benefit in terms of utiity, which contains four components: the first represents the increased discounted present vaue utiity if the economy s intertempora resource constraint is reaxed due to the decrease of consumption; the second represents the discounted present vaue disutiity if the government s intertempora budget constraint is reaxed due to the increase of λ; the third component is the margina benefit caused by the change in the discounted present vaue of outstanding externa debt; and the ast component comes from the associated change in the discounted present vaue of outstanding pubic debt. 18

29 The optimaity condition with respect to m t is: u mt = µ G 0 u mmt m t + u mt + µg 0 λ β t p 0 + µe 0 β t p 0 s=t s=t 1 B G s Q 0,s m t 1 B F s Q 0,s m t, t The efthand side of 1.21 represents the margina cost in utiity if rea money baances decrease. The righthand side of 1.21 represents the corresponding margina benefit in utiity, which has three sources: the first source is the reaxing of the government s intertempora budget constraint; the second is the change in the discounted present vaue pubic bonds due to the change in nomina interest rates; and the ast source comes from the change in externa financing due to the change in nomina interest rates. The optimaity condition with respect to h t is: u ht = µ G 0 λη 2 z t h η 1 t + u hht h t + u ht + µ E 0 ηz t h η 1 t, t Equation 1.22 shows that abor suppy is determined by equating the margina benefit with the margina cost. This optimaity condition has the same components as the corresponding optimaity condition in the cosed economy. The ony difference is that here the Lagrange mutipier in the second component of the right-hand side of 1.22 is the mutipier for the intertempora budget constraint, whie in the cosed economy the corresponding mutipier is for the within-period resource constraint. The optimaity condition with respect to p 0 is: u m0 M 1 = µ G 0 λ [ t=0 Q 0,t 1B G t 19 ] + M 1 + µ E 0 t=0 Q 0,t 1B F t. 1.23

30 There is a margina benefit in utiity due to an increase in the price eve since infation reduces the outstanding nomina pubic debt and the externa debt. This margina benefit is given by the righthand side of There is aso an associated margina cost in utiity due to an increase in the price eve since infation erodes rea money baances. This margina cost exacty offsets the margina benefit in equiibrium. A Ramsey equiibrium is defined as a choice of λ, p 0 {h t } t=0, {m t+1 } t=0 satisfying the optimaity conditions 1.17, 1.18, 1.20, 1.21, 1.22, and 1.23, given the initia asset positions, { 1 Bt G, 1 Bt F, 1 b G t, 1 b F t } t=0 and M 1. In going from a sma cosed economy to a sma open economy, there arise two potentia identification probems. First, there is an identification probem in recovering the Lagrange mutipiers. To see this, note that the Lagrange mutipiers are the ony choice variabes that matter in the optimaity condition with respect to abor suppy and in the optimaity condition with respect to price eve. The first is true because the period resource constraint does not necessariy hod with equaity. The second is true because the discounted vaues of nomina iabiities are predetermined from equation 1.17 and equation Thus, equations 1.22 at t 1 may be enough in recovering the two Lagrange mutipiers, and equation 1.23 becomes an extra restriction on the choice of Lagrange mutipiers. To sove this identification probem associated with the Lagrange mutipiers, we need one poicy instrument that enters 1.22 and equation 1.23 asymmetricay, and this is where rea bonds have an important roe. Second, there may be another potentia identification probem. Consider that the optimaity conditions with respect to consumption and with respect to t = 2 rea money baances are functions of the same set of nomina bonds 20

31 this fact wi become cear in the next section, Section 4. It is thus possibe that there is again an identification probem: the predetermined t = 2 nomina interest rate is inconsistent with the other poicy continuation. This impies that it may be impossibe to recover the term structure of the rea and nomina bonds. This second identification probem is not of major concern because it is very unikey that the two optimaity conditions are homogeneous in nomina bonds. We wi return to this issue in detai when we discuss time consistent optima poicy without rea bonds in Section 4 beow. 1.4 Optima Poicy with Discretion To prove the time consistency of optima poicy, I foow the procedure shown in PPS 2006 and in other papers in the iterature. I prove time consistency by showing that the poicy continuation of the t = 0 government satisfies the optimaity conditions of the t = 1 government. The poicy continuation of the t = 0 government refers to the t = 0 government s optima choice of λ, h t, m t+1, p 1. The optimaity conditions of the t = 1 government, , are the one-period updated version of the optimaity conditions of the t = 0 government, 1.17, 1.18, 1.20, 1.21, 1.22, and In particuar, in ine with the iterature, the time consistency probem becomes whether the t = 0 government can find a profie 0 Bt G, 0 Bt F, 0 b G t, 0 b F t such that the poicy continuation of the t=0 government λ, h t, m t+1, p 1 and the predetermined M 0 satisfy the optimaity conditions If so, the optima poicy is time consistent. Otherwise, the optima poicy is not time consistent. To faciitate the discussion, the optimaity conditions of the t = 1 govern- 21

32 ment are rewritten as foows: Q 1,t 0B t G + p1 β t 0b G t = D Q 1,t 0B t F + p1 β t 0b F t = D µ E 1 A 1.26,t 0 Bt F + t=2 µ G 1 λa 1.26,t 0B t G t=2 = D µ E 1 Q 1,s 0 Bs F + s=t µ G 1 λq 1,s 0 Bs G = D 1.27,t, t s=t µ G 1 λη 2 z t h η 1 t + u hht h t + u ht + µ E 1 ηz t h η 1 t = u ht, t µ E 1 Q 1,t 0Bt F + µ G 1 λ Q 1,t 0B t G = um1 M 0 µ G 1 λm 0,1.29 where D 1.24 = p 1 β t 1 [ ηz t h η t g t + u ht λ h t D 1.25 = p 1 β t 1 [z t h η t cλ g t ] A 1.26,t = Q 1,t [ t i=2 u mi λλ + u mi D 1.26 = p 1 λ µ E 1 1 β p 1µ G 1 ] [ β t 1 u ht λ h t + ] + p 1 t=2 t=2 β t 1 u mt λ m t M 0 β t 1 u mt λ m t D 1.27,t = µ G 1 m t β t 1 p 1 λ + u mt + u mt 1 + µ G 1 β t 1 p 1 λ + u mt. u mmt If the t = 1 government foows the t = 0 government s poicy function, a the A s and D s are functions of the poicy continuation. D 1.26 and D 1.27,t are aso 22 ]

33 functions of the Lagrange mutipiers, µ G 1 and µ F 1. D 1.26 and D 1.27,t appear on the righthand sides of equations 1.26 and 1.27 because the Lagrange mutipiers, µ G 1 and µ F 1, are determined through equations Proposition 1 states that in a sma open economy the optima poicy is time consistent. Proposition 1. In a sma open economy with perfect capita mobiity and with Svensson timing, if the government issues both nomina and rea bonds and if the government is free to choose the term structure of externa bonds, the optima monetary and fisca poicy is time consistent. This resut is independent of the productivity process, the government expenditure process, and the initia asset position of the government. Further, there are many maturity structures of bonds that are capabe of rendering optima monetary and fisca poicy time consistent. Proof. of Proposition 1 consists of two parts, P1 and P2. The first part P1 shows by construction that there exist many possibe maturity structures or profies of 0 Bt G, 0 Bt F, 0 b G t, 0 b F t. The second part P2 shows that the constructed term structure of bonds satisfies the sovency conditions of the t = 0 government and of the economy. P1: I show by construction in six steps in the text beow, S# denotes Step # that the optima poicy is time consistent. I then show that there exist many term structures of nomina and rea bonds that can make the optima poicy time consistent by changing the arbitrary choices assumed in the construction. S1: From equations 1.28, it is impied that the t = 1 government chooses the Lagrange mutipiers as: µ G 1 = µ G 0 = µ G ; µ E 1 = µ E 0 = µ E

34 This choice of Lagrange mutipiers comes from the fact that the poicy continuation of h t satisfies the optimaity conditions of the t = 0 government, as shown beow: µ G 0 λη 2 z t h η 1 t + u hht h t + u ht + µ E 0 ηz t h η 1 t = u ht, t 0. It is cear from equations 1.28 that the optima choice of abor suppies by the t = 1 government is sensitive to the Lagrange mutipiers. By choosing the same Lagrange mutipiers, we can guarantee that the continuation of abor suppy wi satisfy equations However, constant Lagrange mutipiers do not rue out the possibiity that under certain circumstances, it is possibe for the t = 1 government to choose different Lagrange mutipiers from those of the t = 0 government. In that case, the choice of constant Lagrange mutipiers is among the t = 1 government s choice set. For exampe, if the productivity is constant, equations 1.28 reduce to a singe equation and the t = 1 government can have many choices. In another words, the t = 0 government gains an impicit poicy instrument. The choice of constant Lagrange mutipiers in equation 1.30 is a very strong resut derived from this framework. It says that governments want to keep the margina financing costs constant over time in order to have time consistent optima poicy. In this mode, constant Lagrange mutipiers impy constant margina financing costs ony if optima poicy is time consistent. This comes from the fact that the product of λµ G denotes the margina pubic financing cost; whie the Lagrange mutipier µ E denotes the margina externa financing cost. 24

35 In the context of this sma open economy, the constant margina financing costs do not impy constant abor income tax rates, however, because abor suppies can change over time. The constant margina financing costs aso do not impy constant nomina interest rates because rea money baances can change over time. This choice of constant Lagrange mutipiers refects the fact that the poicy continuation of abor suppy puts a restriction on the t = 1 government s choice of Lagrange mutipiers and that governments want to rue out the effect of endogenous responses of abor suppy on the time consistency of optima poicy. In the cosed economy, it does not have to consider the possibe restriction due to the poicy continuation of abor suppy because there is a Lagrange mutipier for every period resource constraint, and these Lagrange mutipiers can change over time. The changing Lagrange mutipiers in the period resource constraint wi absorb the effect of endogenous responses in abor suppy. S2: To revea the term structure of nomina and rea bonds, I make the foowing arbitrary assumptions, 0 b F t = 0 bf t, t 2; 0 b G t = 0 bg t, t 1; 0 Bt G = 0 BG t, t where the variabes 0 bf t, 0 bg t, and 0 BG t denote the vaues arbitrariy chosen for 0 b F t, 0 b G t, and 0 B G t, respectivey. There are two ayers of arbitrary assumptions here. First, the vaues for these bonds are arbitrary. Second, the format of assumption 1.31 itsef is arbitrary in the sense that I can interchange the superscript of G by F and vice versa across time. Note that the arbitrary vaues for 0 B G t start at t = 3 instead of 25

36 t = 2. This comes from the fact that there is no period resource constraint, rather an intertempora resource constraint in this sma open economy. Due to that fact, when Lagrange mutipiers and { 0 Bt G, 0 Bt F } t=3 are found, equation 1.26 and equation 1.27 at t = 2 wi become two equations in two unknowns, 0 B2 G and 0 B2 F. So, there is no extra degree of freedom in arbitrariy choosing 0 B2 G for the t = 0 government, given Lagrange mutipiers and { 0 Bt G, 0 Bt F } t=3. The fact that the arbitrary vaues for 0 b F t and 0 b G t incude a the vaues after t = 2 refects that there is no rea time inconsistency probem and that the term structure of rea bonds is indeterminate. Further, the arbitrary vaues for 0 b G t actuay start at t = 1. This is due to the equivaence between 0 b G 1 and 0 B1 G that occurs if the t = 1 government foows the poicy continuation. This point wi become more evident in the foowing steps. Assumption 1.31 is of interest because it appies to the case in which the t = 0 government has the maximum number of degrees of freedom in choosing the term structure of nomina bonds in order to render time consistent optima poicy. In fact, this assumption says that the t = 0 government shoud pay attention ony to one particuar mutipe-period pubic bond, and the government is free in choosing a other nomina pubic bonds. S3: Given the choice of constant Lagrange mutipiers, subtracting equation 1.27 hed at t = S from equation 1.27 hed at t = S + 1 produces the foowing equation invoving 0 BS F and 0BS G: µ E Q 1,S 0 BS F + µ G λq 1,S 0 BS G = D 1.27,S D 1.27,S+1, S

37 Equation 1.32 is the ony generic restriction on the term structure of nomina bonds. It shows that if the government wants to decrease government financing by one unit, it must increase externa financing by µ G λ/µ E units. The economic interpretation of equation 1.32 is that once the t = 1 government foows the poicy continuation, it does not care about the particuar source of financing. Equation 1.32 and assumption 1.31 are sufficient to pin down 0 BS F and 0BS G for any S 3. Equation 1.26 and equation 1.27 hed at t = 2 are two equations in two unknowns, 0 B2 F and 0 B2 G, and I can consequenty sove for the nomina bond hoding positions at t = 2. S4: After 0 BS F and 0BS G for a S 2 are soved, equation 1.24 is an equation in one unknown 0 B1 G given assumption 1.31, and I can sove for it. This construction of 0 B1 G shows that there is oneto-one reation between 0 B1 G and 0 b G 1. The t = 1 government has a degree of freedom to choose one of these two, and once the vaue for 0 b G 1 is chosen, a corresponding vaue for 0 B1 G is determined. S5: I pug the soutions of 0 BS F, S 2 and 0BS G, S 1, into equation 1.29, and I can then sove for 0 B1 F. The t = 0 government has to choose 0 B1 F in such a way that, under assumption 1.31, the benefit of surprise infation is competey neutraized by the cost of surprise infation. S6: Finay, I pug a the soved optima bond hoding positions into equation 1.25, and can sove then for 0 b F 1. This shows that under assumption 1.31, the choice of 0 b F 1 has to satisfy the intertem- 27