The fiscal multiplier and spillover in a global liquidity trap

Journal of Economic Dynamics & Control 37 (2013) 1264–1283

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The fiscal multiplier and spillover in a global liquidity trap Ippei Fujiwara a,n, Kozo Ueda b a b

Crawford School of Public Policy and CAMA, The Australian National University, Canberra ACT 0200, Australia School of Political Science and Economics, Waseda University, Japan

a r t i c l e in f o

abstract

Article history: Received 29 April 2012 Received in revised form 18 December 2012 Accepted 4 February 2013 Available online 1 March 2013

We consider the fiscal multiplier and spillover—the extent to which one country’s government expenditure increases production at home and also in another foreign country, when the two countries are caught simultaneously in a liquidity trap. Using a standard new open economy macroeconomics (NOEM) model, we show that the fiscal multiplier and spillover are contrary to textbook economics. For the country where government expenditure takes place, the fiscal multiplier exceeds one, the currency depreciates, and the terms of trade worsen. The fiscal spillover is negative if the intertemporal elasticity of substitution in consumption is less than one, and positive if it is greater than one. Incomplete stabilization of marginal costs due to the existence of the zero lower bound is critical in understanding the effects of fiscal policy in open economies. These results remain unchanged even if we incorporate incomplete markets or endogenous capital into the model, but local currency pricing yields positive fiscal spillover irrespective of the size of the intertemporal elasticity of substitution. & 2013 Elsevier B.V. All rights reserved.

JEL classifications: E52 E62 E63 F41 Keywords: Zero lower bound Two-country model Fiscal policy Beggar-thy-neighbor

1. Introduction The liquidity trap has become an issue of global concern. As shown in Fig. 1, the economic downturn following the financial turmoil that first began to emerge in 2007 has resulted in monetary policy being constrained at the zero lower bound on nominal interest rates simultaneously in a number of countries, including Japan, the United Kingdom, and the United States. Such a situation is coined by Fujiwara et al. (2011) and Jeanne (2009) ‘‘the global liquidity trap’’.1 In order to free itself from this trap, many countries currently aim to stimulate aggregate demand and production via fiscal expansion. In this paper, we investigate theoretically the effects of fiscal policy when two countries are caught simultaneously in a liquidity trap, and compare the results with that of normal circumstances. Using the standard new open economy macroeconomics (NOEM) model, a two-country sticky price model, we analyze the fiscal multiplier—the extent to which one country’s government expenditure increases production in that country—and the fiscal spillover—the extent to which this government’s expenditure boosts production in another country. We examine whether fiscal expansion yields a beggar-thy-neighbor situation. Depending on interest rate environments, fiscal expansion yields differing outcomes. According to textbook economics, fiscal expansion under flexible exchange rates is thought to be ineffective (Dornbusch et al., 2008). In the Mundell–Fleming

n

Corresponding author. E-mail addresses: [email protected] (I. Fujiwara), [email protected] (K. Ueda). 1 Jeanne (2009) illustrates the mechanism by which an adverse shock in one country pushes the world economy into the global liquidity trap. On the other hand, Fujiwara et al. (2011) investigate the optimal monetary policy under commitment under given liquidity trap. 0165-1889/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jedc.2013.02.006

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Fig. 1. Global liquidity trap.

model (Mundell, 1967), where the economy is characterized by flexible exchange rates, fixed domestic prices, and perfect capital mobility, fiscal expansion creates upward pressure on interest rates in the home country, and subsequently, the exchange rate appreciates. This offsets the increase in demand for domestically produced goods by crowding out exports. In the NOEM model with producer currency pricing, Clarida et al. (2002) and Benigno and Benigno (2003, 2006) show that the optimal monetary policy aims for price stability. When the central banks follow optimal monetary policy, as will be explained in detail below, domestic interest rates are raised after an increase in the price of domestically produced goods resulting from a positive government shock. The nominal exchange rate appreciates and the terms of trade improve for the domestic country. This induces production switching from domestic to foreign country, and consequently, the fiscal multiplier becomes less than one. The conventional wisdom of ineffective fiscal policy can be reversed in the liquidity trap. This is because in a liquidity trap, nominal interest rates are kept low despite fiscal expansion and consequently the low interest rates prevent the exchange rate from appreciating. Economic activity in the home country is thus stimulated rather than stifled.2 Using a standard two-country sticky price, namely the NOEM model, we demonstrate that the size of multipliers and the sign (positive or negative) associated with the spillover in the global liquidity trap are contrary to those predicted in the Mundell–Fleming model or in the NOEM model with optimal monetary policy. In the global liquidity trap, the fiscal multiplier exceeds one. If the intertemporal elasticity of substitution in consumption is less than one, then the fiscal spillover is negative and positive if that is greater than one. Incomplete stabilization of marginal costs due to the existence of the zero lower bound is critical in understanding the effects of fiscal policy in open economies. As a result of this failure to completely stabilize marginal costs, government spending in the home country raises the marginal costs of domestically produced goods, which in turn increases expected inflation rates and decreases real interest rates. Intertemporal optimization causes consumption to increase, so that the

2 The mechanism in the liquidity trap is somewhat similar to when monetary policy does not completely stabilize but instead accommodates the distortions resulting from a government spending shock, as examined by Obstefld and Rogoff (1995). In that case, monetary policy, namely money supply, is assumed to be constant. Relatively lowered domestic consumption due to a positive government shock yields downward pressure on money demand in the home country. To keep money supply constant, a nominal interest rate falls, resulting in the depreciation of the nominal exchange rate and the deterioration of the terms of trade for the domestic country. Consequently, the fiscal multiplier becomes larger than unity. In both cases, the key for the higher multilier is that marginal costs are not completely stabilized, which we call incomplete stabilization of marginal costs.

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fiscal multiplier exceeds one. This effect functions to increase demand for foreign-produced goods as well, but the fiscal spillover may be negative due to the following mechanism. While government spending continues, the price of domestically produced goods increases more in comparison to foreign-produced goods. Assuming that the two countries are at symmetric equilibrium when government spending ends, nominal exchange rates depreciate and the terms of trade worsen on impact in the home country when government spending begins. This in turn shifts demand for goods from foreign-produced goods to domestically produced ones. The fiscal spillover thus may become negative depending on the intertemporal elasticity of substitution in consumption. This result is widely robust to model changes. We conduct robustness analyses, incorporating incomplete markets instead of complete markets, an interest rate rule instead of optimal discretionary monetary policy, local currency pricing instead of producer currency pricing, and capital with investment adjustment cost. We obtain unchanged results, with only one exception. Under local currency pricing, the fiscal spillover becomes positive irrespective of the size of the intertemporal elasticity of substitution. Reflecting on Japan’s experience around 2000 as well as the global financial crisis that began in the summer of 2007, several studies analyze the role of fiscal policy in the liquidity trap.3 In a closed-economy model, Auerbach and Obstfeld (2005) discuss the effectiveness of large-scale open market operations as a fiscal policy tool. Regarding the fiscal multiplier, Christiano (2004) demonstrates that it exceeds one in the presence of the zero lower bound. A number of papers have also examined his results using more detailed frameworks with nonlinearity, different monetary policy, and/or various shocks and policy measures.4 Motivated by the current situation where many developed countries are trapped at virtually a zero lower bound on nominal interest rates simultaneously, in this paper, we extend their models to an open-economy model and evaluate the fiscal multiplier and spillover in a global liquidity trap. The closest study to our abovementioned research is Cook and Devereux (2011).5 Cook and Devereux (2011) also analyze the effects of fiscal policy in a global liquidity trap using a sticky price model, reporting almost the same findings. There are also several important differences. First, the two papers introduce a different assumption regarding the duration of government expenditure. In Cook and Devereux (2011), the duration of government expenditure is the same as that of the adverse shock that brings about the liquidity trap. In our model, the duration of government expenditure is up to two periods, which is shorter than the duration of the adverse shock. By using this assumption, we simplify our analysis and clarify the underlying mechanism that produces the multiplier greater than one and the spillover which can be either positive or negative. Second, as our original contributions, we highlight the importance of the size of the intertemporal elasticity of substitution in influencing the sign of spillover. Third, we incorporate some additional features to examine the robustness of our results in a wide variety of models. For example, our baseline assumption of complete markets and producer currency pricing, which is common in Cook and Devereux (2011), is perhaps overly simplified. At the other extreme assumption, we use incomplete markets and local currency pricing. Moreover, we incorporate a less strict inflation targeting rule as well as capital with investment adjustment cost. By adding these features, we discuss changes in the model dynamics, and in turn, the fiscal multiplier and spillover. One of noteworthy findings of our paper is that the fiscal spillover in a global liquidity trap becomes positive with local currency pricing even when the intertemporal elasticity of substitution is less than one. Finally, the unique contribution of Cook and Devereux (2011) is that they incorporate home bias and analyze optimal policy. Other related studies include Freedman et al. (2009), Corsetti et al. (2010), Erceg and Linde (2010), Fahri and Werning (2012), and Flotho (2012), which evaluate the fiscal multiplier and spillover in a similar context. Their analyses suggest that a low interest rate environment is the key to yielding a greater fiscal multiplier and changing the sign associated with the fiscal spillover. However, Freedman et al. (2009) and Corsetti et al. (2010) are not strictly based on a model with a zero lower bound on nominal interest rates. The latter three papers complement ours in discussing fiscal policy effects in a currency union under the liquidity trap.6 The structure of this paper is as follows. Section 2 introduces the standard two-country sticky price model (the NOEM model) used in this paper. In Section 3, we analyze the fiscal multiplier and spillover without the presence of the zero lower bound. In Section 4, we analyze the fiscal multiplier and spillover in the presence of the zero lower bound. Section 5 extends the model by incorporating incomplete financial markets, less strict interest rate rules against inflation, local currency pricing, and endogenous capital, and Section 6 concludes.

3 Regarding monetary policy, Fujiwara et al. (2010, 2011) investigate the optimal monetary policy when two countries are simultaneously caught in a liquidity trap in the form of a commitment policy. For a closed economy model, see also Reifschneider and Williams (2000), Kato and Nishiyama (2005), Eggertsson and Woodford (2003), Jung et al. (2005), Adam and Billi (2006, 2007), and Nakov (2008). For an extension to an open economy model, see Svensson (2001, 2003), Coenen and Wieland (2003), Jeanne and Svensson (2007), and Nakajima (2008). In relation to the fiscal theory of the price level, see Benhabib et al. (2002) and Iwamura et al. (2005). 4 See Bodenstien et al. (2009), Braun (2009), Christiano et al. (2011), Eggertsson (2011), and Woodford (2011). 5 We only discovered this paper following the release of our discussion paper in March 2010. 6 There are also a number of empirical studies which investigate the international as well as domestic transmission of fiscal shocks (e.g., Perotti, 2008; Corsetti and Muller, 2008; Kim and Roubini, 2008; Feyrer and Shambaugh, 2009). On the international transmission of fiscal shocks, questions are related to whether we observe twin deficits: budget deficits and current account deficits, and their empirical results are mixed.

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2. Model 2.1. Model setup Our two-country sticky price (NOEM) model is a conventional one, similar to that used in Clarida et al. (2002), Benigno and Benigno (2003), and Fujiwara et al. (2011). The economy consists of a home country H and a foreign country F. Labor is not mobile and it is used to produce a continuum of differentiated goods on the unit intervals [0,1] in both countries. The details of the derivation are shown in Appendix. Household. A representative household in the home country maximizes ! 1þo 1 Y t s X C 1 ht t E0 bt w , 1s 1þo t ¼0t¼1 subject to the budget constraint   Pt C t þ Et Q t,t þ 1 Dt þ 1 þ Bt þ 1 ¼ W t ht þ Dt þit1 Bt þ Pt T: The consumption bundle per capita Ct is defined by     C H,t n C F,t 1n Ct ¼ : n 1n

ð1Þ

As a result, the utility based aggregate price index (CPI) Pt, is described as Pt ¼ P nH,t P 1n F,t :

ð2Þ

The consumption index represents bundles of differentiated domestically produced goods C H,t and foreign-produced goods C F,t , both spent in the home country per capita. The aggregate price index is composed of a domestic good price PH,t and a foreign good price PF,t . We denote the weight for the bundle of domestically produced goods by n ð0 on o 1Þ and for foreign-produced goods by 1n. We can also use n to interpret in terms of the relative size of the home country. The intratemporal elasticity of substitution between home and foreign goods is one. We denote hours worked by ht , nominal wage by W t , lump-sum profits by Pt , taxes by T, bond holdings by Bt and the gross nominal interest rate by it. The object Et Dt þ 1 is an Arrow security, standing for the quantity of home currency to be delivered in period t þ1 if state st þ 1 is realized at t þ1, conditional on history up to t. The associated price is Q t,t þ 1 . Household preferences are governed by factors common to the two countries: discount factor 0 o bt o 1, the inverse of the intertemporal elasticity of substitution in consumption s, the utility weight on hours worked w, and the Frisch elasticity of labor supply o. Following Eggertsson and Woodford (2003), Christiano (2004), and Christiano et al. (2011), we assume that the discount factor is time-varying and affects households in the two countries, serving as the origin of the global liquidity trap. We call this a natural rate shock. In the zero-inflation steady state, the natural rate is positive, so the zero bound does not bind. However, if a large-scale shock decreases the natural rate to negative, the zero bound starts to bind. The real interest rate remains too high, contracting the real economic activity and rendering the economy sub-optimal. The preferences are set as "Z #e=ðe1Þ 1

C H,t þ Gt ¼

C H,t ðjÞ1ð1=eÞ dj

,

ð3Þ

0

where Gt represents government expenditure in the home country and e 4 1 represents the elasticity of substitution. Similarly, we define domestically produced goods C nH,t that are purchased in the foreign country as "Z #e=ðe1Þ 1

C nH,t ¼

0

C nH,t ðjÞ1ð1=eÞ dj

,

where superscript n denotes foreign variables. Firms. Regarding firms, Yt and Y nt represent aggregate home and foreign productions, respectively. An intermediate good is produced by a continuous number of monopolists using the production technology Y t ðjÞ ¼ ht ðjÞ: Each monopolist maximizes its profits subject to its demand curve and the Calvo-type price friction where y is the probability that the monopolist cannot re-optimize its price. Under the setting, each monopolist maximizes the discounted sum of the profits s   C t þ k Pt P H,t C H,t þ k ðjÞW t þ k C H,t þ k ðjÞ s P C t þ k t t¼1 k¼0 " !e !e # s 1 k Y X   C P H,t P H,t k t þ k Pt ¼ y Et bt þ t1 s C H,t þ k þGt þ k P H,t W t þ k , PH,t þ k P H,t þ k C t Pt þ k t¼0 k¼0 1 X

yk Et

k Y

bt þ t1

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where P H,t is the reset price. The gross inflation rate of P H,t is defined by

pH,t ¼

PH,t : PH,t1

We assume producer currency pricing and therefore the law of one price. Let the aggregate price (inflation rate) of domestic goods and foreign goods be P H ðpH Þ and PnF ðpnF Þ, respectively. Also, let a nominal exchange rate as the number of home currency units per unit of foreign currency be St. Then, the price of domestic goods sold in the foreign country is described simply as P nH,t ¼

P H,t : St

Market clearing. Finally, the resource constraint is given by nY t ðjÞ ¼ nC H,t ðjÞ þ ð1nÞC nH,t ðjÞ:

ð4Þ

Fiscal and monetary policy. Regarding fiscal policy, as is illustrated in Eq. (3), we assume that the government buys only domestic goods for Gt. This assumption yields the deviation of the terms of trade from its steady-state level. Fully coordinated fiscal spending in the two countries would make our results isomorphic to those in a closed economy framework.7 Government expenditure is financed by the same amount of lump-sum tax each period. As for monetary policy, following Christiano (2004), we assume that domestic and foreign central banks conduct an optimal discretionary policy to stabilize inflation, if they are not faced with a liquidity trap. This is equivalent to the following rule: ^ıt ¼ ap^ H,t , ^ınt ¼ ap^ nF,t , where a coefficient on inflation a is infinity. Throughout the paper, we assume that the target level of inflation rate is zero. 2.2. Log-linearized system of equations n Using the above setup, we can derive the log-linearlized system of equations with respect to the four variables of Y^ , Y^ , n p^ H , and p^ F , as follows: n 1 þ nðs1Þ Et ðDY^ t þ 1 DG^ t þ 1 Þ þ ð1nÞðs1ÞEt DY^ t þ 1 ¼ ^ıt Et p^ H,t þ 1 þ b^ t , 1g_y

ð5Þ

n nðs1Þ Et ðDY^ t þ 1 DG^ t þ 1 Þ ¼ ^ınt Et p^ nF,t þ 1 þ b^ t , ½1þ ð1nÞðs1ÞEt DY^ t þ 1 þ 1g_y

ð6Þ

9 8h i 1 þ nðs1Þ ^ > ^n> ð1yÞð1ybÞ < o þ 1g_y Y t þ ð1nÞðs1ÞY t = p^ H,t ¼ bEt p^ H,t þ 1 þ > > y ; :  1 þ nðs1Þ G^ t

ð7Þ

1g_y

and

p^ nF,t ¼ bEt p^ nF,t þ 1 þ

ð1yÞð1ybÞ

y



n

½o þ1 þ ð1nÞðs1ÞY^ t þ

nðs1Þ ^ ðY t G^ t Þ : 1g_y

ð8Þ

The circumflex (\widehat) indicates the log-linearized deviation of the variable from its steady state, with the exception of G^ t ¼ ðGt GÞ=Y. A parameter g_y represents the steady state ratio of government expenditure to output. The first two equations represent IS curves with respect to domestic goods and foreign goods. Deducting G^ t from Y^ t , the IS curves show the Euler equations with respect to consumption. The last two equations represent the new Keynesian Phillips curves with respect to inflation in the home and foreign country. Government spending influences the marginal costs of domestic and foreign goods via the resource constraints and the labor market equilibrium conditions. Eqs. (5)–(8) are all forward-looking, namely expectation about future economic conditions determines current inflation rates and the output. Therefore, as will be explained below, not only current but also future government expenditure can have impacts on dynamics of endogenous variables. In this model, we assume two shocks: a shock to subjective discount factor and to government spending. The former is to create the global liquidity trap. 7 For simplicity, we neglect government expenditure in the foreign country. Given symmetry, it is straightforward to analyze this, except for the size of home country n. The effect of government expenditure in both countries can be analyzed easily, too, by summing the effect of government expenditure in the home country and that in the foreign country, as long as the first-order approximation is accurate.

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Table 1 Fiscal multiplier and spillover. Multiplier 0oso1

Spillover

s41 !

0oso1

s 41



þ

Normal

G^ t

þ , o 1 (þ )

ZLB

G^ t G^ t þ 1

1 (0)

1 (0)

0

0

þ ()

þ (þ )

þ



ZLB

þ, o1

if no nn þ if n 4nn

Note: Numbers and/or signs indicate the multiplier or spillover in an open economy. Signs inside brackets represent a comparison of the fiscal multiplier and spillover with those of a closed economy: þ , 0, and  imply that the multiplier or spillover is larger than, equal to, and smaller than in the closed economy, respectively. The first and second rows represent the multiplier and spillover under normal circumstances and in the global liquidity trap, respectively, provided contemporaneous government spending. The third row represents the fiscal multiplier and spillover in the global liquidity trap at time t, when government spending is anticipated for the next period t þ 1. nn is defined by ðo þ sÞ=2=ðs1Þ.

3. Fiscal multiplier and spillover under normal circumstances Before calculating the fiscal multiplier and spillover step by step, we summarize findings in this paper in Table 1. As we will see, incomplete stabilization of marginal costs due to the existence of the zero lower bound is critical in our results. 3.1. Fiscal multiplier To discuss Table 1 in detail, we first consider a case in which central banks in both countries are not constrained by the zero lower bound of nominal interest rates, using the above NOEM model summarized by Eqs. (5)–(8). With producer currency pricing, there are only two kinds of aggregate inflation. Therefore, two policies by two central banks are enough to stabilize inflation in both countries. Further, as shown by the NOEM models of Clarida et al. (2002) and Benigno and Benigno (2003), since price stability of producer prices is the optimal discretion as well as commitment monetary policy, central banks achieve zero inflation in both countries.8 Under this optimal monetary policy, marginal costs expressed in the first term of the right-hand side of the Phillips curves in Eqs. (7) and (8) become zero. Thus, analysis in this section can be considered as one using the simple international real business cycle model without capital, namely under the flexible price equilibrium. As a result, the fiscal multiplier associated with government spending in the home country is given by dY^ t oð1g_yÞ½1 þ o þ ðs1Þð1nÞ : ¼ 1 ½oð1g_yÞ þ1n þ sn½1 þ o þ ðs1Þð1nÞ þ ðs1Þ2 nð1nÞ dG^ t Since the second term is positive under standard calibration of parameters, the fiscal multiplier is lower than one. Compare it to the fiscal multiplier in a closed economy model as in Christiano (2004)

dY^ t

s ¼ :

oð1g_yÞ þ s dG^ t

closed

This is obtained from the Phillips curve in a closed economy, that is  ð1yÞð1ybÞ s s ^ Y^ t  p~ t ¼ oþ G t þ bEt p~ t þ 1 y 1g_y 1g_y and the zero inflation rate due to the optimal monetary policy. We can see

( ) dY^ t dY^ t

sign  ¼ ðs1Þð1nÞ½1 þ o þ ðs1Þð12nÞ:

dG^ t dG^ t

closed

The inverse of the intertemporal elasticity of substitution in consumption s as well as the relative size of the home country n influence the relative size of the fiscal multiplier to that in a closed economy. Start with the case of s 41. If n is smaller (larger) than nn  ðo þ sÞ=2=ðs1Þ, a fiscal multiplier becomes smaller (larger) than it is in a closed economy. Since nn 41=2, an equal country size of n ¼ 1=2 yields a fiscal multiplier smaller than that in a closed economy. When 0 o s o 1, the fiscal multiplier is larger than that in a closed economy, irrespective of the size of n. As shown by Clarida et al. (2002), when s ¼ 1, the two countries become insular. Irrespective of s, the two multipliers naturally become equal when n¼ 1, that is, when the relative size of the home country is infinite. 8 The optimal targeting rule of the home central bank is static and does not contain any foreign variables. This is because we do not include any distortionary shock, such as the cost-push shock examined in Clarida et al. (2002). Therefore, there is no trade-off in stabilizing between price and output.

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3.2. Fiscal spillover The fiscal spillover of the home country’s government spending in the foreign country is given by n

dY^ t onðs1Þ : ¼ ½oð1g_yÞ þ 1þ nðs1Þ½1þ o þ ðs1Þð1nÞ þ ðs1Þ2 nð1nÞ dG^ t The fiscal spillover is positive if s 4 1 and negative if 0 o s o 1. If s ¼ 1, the two countries become insular and there is no spillover. Naturally, there is no spillover in a closed economy model. The sign of the spillover is independent of the relative size of the home country n. 3.3. Mechanism The intuition behind these results and the differences from the case under the closed economy can be understood as follows. The key is the effects from the terms of trade ToTt. The terms of trade are denoted by ToT t ¼

St P nF,t Y t Gt ¼ , PH,t Y nt

ð9Þ

which is log-linearized as ^ ^ d t ¼ Y t G t Y^ n : ToT t 1g_y

ð10Þ

Government spending directly increases production and employment in the home country. That yields an upward pressure on home prices, and to prevent this, monetary policy is tightened and real interest rates rise. Monetary tightening generates home exchange rate appreciation and improves the terms of trade as shown in Eq. (9). As Clarida et al. (2002) and Fujiwara et al. (2011) point out, the terms of trade have two opposing effects on real marginal costs MCt. This is clear from the equation below, which is derived by combining the resource constraint and the labor market equilibrium condition o

MC t ¼ ht C st ToT 1n t s sþo ¼ L ToT ð1sÞð1nÞ , H,t ht

where the relative price dispersion term LH,t is given by Z 1 P H,t ðjÞ e LH,t ¼ dj: P H,t 0 This equation implies that, first, by affecting output prices, the improvement in the terms of trade directly decreases real marginal costs in the home country.9 On the other hand, it increases real marginal costs in the foreign country. To completely stabilize inflation rates so that real marginal costs become constant, home employment needs to increase while foreign employment needs to decrease. Second, the improvement in the terms of trade results in production switching from domestically produced to foreign-produced goods. Given domestic output that requires a rise in output in the foreign country. Total consumption rises in both countries, and due to risk sharing, both domestic and foreign consumption rise by the same amount. The rise in domestic consumption raises real marginal costs in the home country, and to offset the rise in prices, home employment must decrease.10 Home employment decreases more with higher s. Consequently, if s 41, the second channel dominates the first channel, provided that n o nn . Home employment decreases, which makes the fiscal multiplier smaller than that in a closed economy. Foreign employment increases, which yields a positive fiscal spillover. As n rises, the effect of production switching on the home country diminishes, and over the threshold of nn , the first channel comes to dominate the second channel. If 0 o s o 1, the first channel dominates the second channel. Home employment increases, which makes the fiscal multiplier larger than that in a closed economy. Foreign employment decreases, which yields a negative fiscal spillover. If s ¼ 1, the two channels offset each other. The fiscal multiplier equals that in a closed economy and the fiscal spillover becomes zero, since the two countries are insular.11 It is worth noting that the implications for welfare are different. Higher production means higher employment. This has the effect of decreasing welfare, while because of perfect risk sharing, consumption in the two countries is the 9

This is the terms of trade effects according to Clarida et al. (2002). This is the risk sharing effects according to Clarida et al. (2002). 11 The sensitivity to s depends on the form of utility. For example, Greenwood et al. (1988) use the following utility function: 10

1þo

1 h C t w t 1s 1þo

!1s :

That function, not additively separable, abstracts the income effect on labor. Real marginal costs are independent of s.

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same.12 Recall that, if s 41 and n onn , the fiscal multiplier is lower and the fiscal spillover is higher compared with the case of a closed economy. Therefore, government spending produces superior social outcomes for the country than would be seen in the case of a closed economy, because households in the foreign country work more to produce more goods. This leads to the beggar-thy-neighbor problem. Also note that, as Tille (2001) and Benigno and Benigno (2003) point out, welfare implications are highly dependent on other parameters, such as the intratemporal elasticity of substitution between home and foreign goods, which in this paper is assumed to be one. Consequently, whether s is larger or smaller than unity is crucial to the analyses of this paper. 4. Fiscal multiplier and spillover in a global liquidity trap We next consider a case in which the zero lower bound constrains central banks—cannot achieve complete price stability any longer. Following Christiano (2004), we assume that the natural rate shock is deterministic: the adverse natural rate shock b^ t occurs at periods t ¼ 1, . . . ,T1, and returns to zero at period T.13 Central banks pursue optimal discretionary policy to stabilize inflation. For simplicity, we further assume that government spending in the home country Gt is implemented only for two periods, at t ¼1 and t ¼ 2 o T.14 Government spending is assumed to be not large enough to push the economy out of the liquidity trap. Using lag-operator L, we arrange Eqs. (5)–(8). Since we assume that government spending in the home country Gt is implemented for only two periods, at t ¼1 and t ¼ 2, first-order approximation up to L1 is sufficient. The fiscal multiplier is approximated up to the first order as  ð1yÞð1ybÞ n þ sð1nÞ Y^ t ¼ Ex CðL1 Þð^ıt þ b^ t Þ þ Ez C n ðL1 Þð^ınt þ b^ t Þ þ 1 þ ð11Þ oð1g_yÞEt L1 G^ t ,

s

y

where CðL1 Þ and C n ðL1 Þ are the first-order functions of L1 , which are of little importance for the fiscal multiplier and spillover in a global liquidity trap. A notable feature here is that contrary to the case under normal circumstances, under the global liquidity trap, price stickiness matters for the fiscal multiplier and spillover. This property is again related to incomplete stabilization of marginal costs under the liquidity trap. We will come back to this point later. 4.1. Fiscal multiplier The coefficients on G^ t and G^ t þ 1 represent the fiscal multiplier. Eq. (11) suggests that temporary government spending ^ G t has a multiplier of one. It already exceeds the multiplier in the previous section. Government spending in the next period G^ t þ 1 has a multiplier of  dY^ t ð1yÞð1ybÞ n þ sð1nÞ ¼ oð1g_yÞ , ^ y s dG t þ 1 in period t. Multi-period government spending increases the fiscal multiplier. As (1) s falls, (2) prices become more flexible (y decreases), or (3) the Frisch elasticity o becomes larger, the fiscal multiplier becomes larger.15 Note that government spending in the past period, t  1, has no effect on output at time t, because the model is purely forward-looking. According to Christiano (2004), the fiscal multiplier in a closed economy is 1 for G^ t . A one-time government expenditure has a multiplier of the same amplitude even in a global liquidity trap. The fiscal multiplier in a closed economy for G^ t þ 1 is given by  dY^ t ð1yÞð1ybÞ o ¼ ð1g_yÞ : y s dG^ t þ 1 If s 41, the fiscal multiplier in an open economy is larger than it is in a closed economy. If 0 o s o1, the fiscal multiplier is smaller than that in a closed economy. If s ¼ 1, the fiscal multiplier equals that in a closed economy, since the two countries become insular. Interestingly, the results here are quite contrary to those obtained in the previous section. 4.2. Fiscal spillover The fiscal spillover is written as n ð1yÞð1ybÞ ðs1Þno Y^ t ¼ Et Dn ðL1 Þð^ınt þ b^ t Þ þ Et DðL1 Þð^ıt þ b^ t Þ Et L1 G^ t ,

y

s

ð12Þ

12 Note that we do not assume any utility gain from government expenditure. For an analysis on the optimal level of government expenditure with a small utility gain, see Christiano et al. (2011). 13 Even if the natural rate shock is stochastic, the following results concerning the fiscal multiplier and spillover do not change as long as government spending is one-time. 14 Extending the periods of government spending does not change our main results. Also for simplicity, we focus on the fiscal multiplier and spillover only on the initial date. We can calculate those on the future dates following Christiano (2004), but the analysis becomes less intuitive. 15 As explained before, country size n has qualitatively different effects on the multiplier depending on s.

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where DðL1 Þ and Dn ðL1 Þ are the first-order functions of L1 . The coefficients on G^ t and G^ t þ 1 are zero and n  dY^ t ð1yÞð1ybÞ ðs1Þno ¼ , y s dG^ t þ 1 respectively. Temporary government spending G^ t has no spillover contemporaneously, while government spending for the next period G^ t þ 1 has negative (positive) spillover if s 4ð oÞ1. If s ¼ 1, there is no spillover. As n rises, the absolute size of the spillover tends to increase. Similarly to the case for multiplier, as (1) prices become more flexible (y decreases) or (2) the Frisch elasticity o becomes larger, the absolute sizes of the spillover become larger. Note that the signs of the spillover computed here are the opposite of those obtained under normal circumstances. 4.3. Mechanism For the fiscal multiplier, we discuss the differences from the case without a liquidity trap in the open economy and also with a liquidity trap in a closed economy. For the fiscal spillover, since there is no spillover effect in a closed economy, we only compare the results without a liquidity trap to those with a liquidity trap, using the same NOEM model. Incomplete stabilization of marginal costs due to the existence of the zero lower bound is a crucial factor in determining the size of the fiscal multiplier and spillover. 4.3.1. Fiscal multiplier The mechanism for the higher fiscal multiplier in a global liquidity trap compared to the case without zero lower bound is basically the same as those obtained in Christiano (2004), Christiano et al. (2011), and Woodford (2011).16 Government spending directly increases production and employment in the home country. This yields an upward pressure on home prices. Because of the zero lower bound, inflation rates and marginal costs are not stabilized.17 They increase, and real interest rates drop. Intertemporal optimization causes consumption to increase. This yields a greater fiscal multiplier than that of the previous section. Thus, the increase in consumption is larger as s is lower. Regarding other parameters, lower y increases inflation rates given marginal costs while higher o increases marginal costs. Consequently, as prices become more flexible or the Frisch elasticity becomes larger, the fiscal multiplier becomes larger. This mechanism also works to increase the size of the fiscal spillover. There exist open economy effects via the changes in the terms of trade through s, that yield differences from the case in a closed economy, as is explained in the previous section. The sign of the fiscal multiplier does, however, not change by s as shown in Table 1. These altogether imply that the main factor for the higher multiplier in a global liquidity trap compared to the case without zero lower bound is the inability of complete stabilization of inefficient fluctuations of marginal costs. 4.3.2. Fiscal spillover Table 1 shows that the results are reversed for the fiscal spillover depending on whether economies are caught in a global liquidity trap or not. Again, the difference stems from the incomplete stabilization of marginal costs in a global liquidity trap, but here we explain the detailed mechanism behind this result. Government spending has two opposing effects. First, it shifts demand from foreign-produced goods to domestically produced goods. When government spending ends, the terms of trade are at equilibrium: the price of domestically produced goods is as expensive as that of foreign-produced goods. While government spending continues, the price of domestically produced goods increases more in comparison to foreign-produced goods. In other words, reflecting the higher inflation of domestically produced goods in the future, nominal exchange rates jump toward deprecation immediately after the government spending shock. Therefore, when government spending begins, the terms of trade worsen: domestically produced goods are relatively cheaper than foreign-produced goods. Government spending increases demand for domestically produced goods, and decreases demand for foreign-produced goods. Second, the increase in expected inflation rates lowers real interest rates and increases demand for foreign-produced goods. The second effect becomes weaker, as the inverse of the intertemporal elasticity of substitution in consumption, s, becomes greater. If s is greater (less) than one, the first effect dominates (is dominated by) the second channel. The fiscal spillover becomes negative (positive). More precisely, the mechanism runs as follows. For temporary government spending, domestic production rises by the same amount while home consumption does not change. Real marginal costs rise, which increases prices of domestically produced goods on impact. In the foreign economy, production and consumption do not change, so prices of foreignproduced goods measured in the foreign currency do not change. The terms of trade do not change because the currency of the home country depreciates on impact. 16 As discussed in Introduction, this mechanism is somewhat similar to the situation supposed in Obstefld and Rogoff (1995) that monetary policy does not completely stabilize the inefficient fluctuations stemming from a government spending shock. 17 The importance of the fluctuations in marginal costs on the fiscal multiplier is also pointed out by Braun (2009) and implicitly by Christiano et al. (2009). If monetary policy is conducted following the Taylor type rule and induces incomplete stabilization, multipliers become larger in this case than under complete price stability.

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Given that nominal interest rates in both countries are bound at zero, the fact that the nominal exchange rates change may seem contradictory to the uncovered interest parity condition, Yet, it is not contradictory at all: the expected change in nominal exchange rates remains unchanged. Also note that because government spending is temporary, expected inflation rates in two countries do not change. Consequently, real interest rates and consumption are unchanged in both countries. Consider a case in which there is government spending at t ¼2 as well as at t¼1. Backward induction addresses the underlying mechanism. The economy at t¼ 2 is close to the economy when there is temporary government spending at t ¼1. The terms of trade are at equilibrium at t¼ 2. Prices of domestically produced goods rise by p^ H,2 while prices of foreign-produced goods do not change. The only difference is in the nominal exchange rates. From t ¼1 to t ¼ 2, nominal exchange rates do not move due to the uncovered interest parity condition. Expected appreciation (depreciation) is zero reflecting the zero nominal interest rates in both countries. It suggests that the terms of trade at t ¼1 are positive by p^ H,2 , meaning that the terms of trade worsen or nominal exchange rates depreciate at t ¼1 for the home country.18 From Eq. (10), we have

p^ H,2 ¼

Y^ 1 G^ 1 ^ n Y 1 : 1g_y

ð13Þ

The worsening of the terms of trade increases domestically produced goods but decreases foreign-produced goods. On the other hand, because of a rise in domestically produced goods at t ¼2, the aggregate inflation rate in the home country rises by np^ H,2 , which lowers the real interest rate by np^ H,2 at t¼1. Due to intertemporal optimization that increases aggregate consumption in the two countries as np^ H,2

s

¼n

n Y^ 1 G^ 1 þ ð1nÞY^ 1 : 1g_y

ð14Þ

A decrease in the real interest rate increases both domestically and foreign-produced goods. Clearly, both Eqs. (13) and n n (14) suggest that, for 0 o s o 1, Y^ 1 increases, while for s 41, Y^ 1 decreases. As s is larger, the effect of the real interest rate n n n on Y^ 1 becomes smaller. The increase in Y^ 1 is dominated by the decrease in Y^ 1 stemming from the worsening (improvement) of the terms of trade for the home (foreign) country. The fiscal spillover thus becomes negative for a large s. Regarding welfare, a lower fiscal multiplier or higher spillover is better for the country where the government spending takes place. Therefore, if s 41, government spending is socially worse for the country with government spending than in a closed economy, since households in the foreign country work less to produce less goods. It does not yield the beggar-thyneighbor problem, but instead ends up with beggar-thy-self. 5. Extension In this section, we extend the model by incorporating incomplete financial markets, a less strict interest rate rule against inflation, local currency pricing, and capital with investment adjustment cost. All these exercises aim to examine the robustness of aforementioned results in a wide variety of models. Capital with investment adjustment cost makes our model complex but more realistic, and adds an important savings channel, thereby possibly affecting the fiscal multiplier and spillover. Three other features are not necessarily realistic but help us to examine robustness, in that reality probably lies between our baseline models and these new models. 5.1. Incomplete financial markets In the main part of this paper, we assumed complete financial markets. The household has access to the Arrow security, leading to perfect international risk sharing s C nt s ¼ C  t

St Pnt s ¼ C t et : Pt

Since our model does not assume a home bias in demand for goods and the representative households of the two countries are equally wealthy in the initial period, the real exchange rate et remains at unity. Home and foreign consumption are perfectly aligned n C^ t ¼ C^ t :

When markets are incomplete, such perfect international risk sharing is not always attainable. As a particular and rather extreme type of incompleteness, let us consider financial autarky. In this instance, net nominal export must be zero, and the above risk sharing equation is replaced by ð1nÞP H,t C nH,t nP F,t C F,t ¼ 0: 18

Numerical exercise in Fig. 1 below illustrates these dynamics.

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Using demand equations and log-linearizing the above equation, we obtain n p^ H,t þ ðp^ H,t þ C^ t Þ ¼ p^ F,t þ ðp^ F,t þ C^ t Þ:

Therefore, the same international risk sharing equation holds n C^ t ¼ C^ t :

Such a result appears surprising, but is actually consistent with the finding discovered by Cole and Obstfeld (1991). They reveal that fluctuations in the terms of trade generate a similar outcome to international risk sharing, because a country’s terms of trade are negatively correlated with growth in its export sector. In our model, in the above equation of net nominal export, the log-deviation of the terms of trade p^ F,t p^ H,t has two opposing effects. The improvement of the terms of trade raises the unit price of net export, while it decreases the quantity of net export by shifting demand from domestically produced goods to import goods. Those two effects cancel each other completely, when the intratemporal elasticity of substitution between home and foreign goods is one. The terms of trade effect provide perfect insurance. Consequently, our previous results are unchanged under incomplete financial markets. When the intratemporal elasticity of substitution between home and foreign goods deviates from one, these results will change. But its degree can be minor under plausible parameter values, according to Cole and Obstfeld (1991) and Woodford (2010).

5.2. Interest rate rule To examine the effects of monetary policy on the fiscal multiplier and spillover, we simulate the model using four variants of the following interest rate rule: ^ıt ¼ r^ıt1 þ ap^ H,t : The first is our benchmark, which is equivalent to r ¼ 0 and a-1. Second, we consider the rule of r ¼ 0 and a ¼ 1:5. Third and fourth, we consider the rules with smoothing, captured by r ¼ 0:5 and a ¼ 1:5 and r ¼ 0:9 and a ¼ 1:5, respectively. Since our interest is in the fiscal multiplier and spillover in the global liquidity trap, we focus only on the case in which the two countries are faced with the zero lower bound. In the simulation, a sequence of large negative shocks to the natural rate of interest (b^ ¼ 0:05) is given for 10 quarters, t¼ 1–10. We assume fiscal spending with the size of 1% of GDP only in the domestic country. The duration of such fiscal spending is set to be two quarters, t ¼1 and 2. Table 2 describes parameter values, most of which are from Christiano et al. (2011). Fig. 2 demonstrates the time-series paths of six key variables: outputs, changes in producer prices in the two countries, the terms of trade, and changes in the nominal exchange rate. All lines illustrate their percentage deviations from those without fiscal spending. Simulation results are consistent with our prediction made in the previous section. As Eq. (11) suggests, temporary government spending G^ t has a multiplier of one in period 2. In period 1, government spending yields the multiplier larger than 1. Since we assume s 4 1 in this simulation, the fiscal spillover is negative. The terms of trade and the nominal exchange rate both increase, suggesting a worsening of the terms of trade and a currency depreciation. We cannot find any difference in both fiscal multiplier and spillover among examined policy rules. As is obvious from the two figures in the second row, in order to calculate the multiplier and spillover in the global liquidity trap, we deliberately create a situation where nominal interest rates are constrained at zero irrespective of fiscal spending. It is true that a difference among interest rate rules can alter the optimal paths of variables after the termination of the zero bound period, which also induces a jump in the nominal exchange rate in period one. This can lead to some difference in the multiplier and spillover, if negative shocks to the natural rate of interest rate are small and the duration of such shocks is short.

Table 2 Parameters. Parameters

Values

Explanation

s

2 0.99 1 0.85 0.5 0.2

Relative risk aversion Subjective discount factor Frisch elasticity Calvo parameter Home country size Share of fiscal expenditure

b

o y n g_y

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Fig. 2. Fiscal multiplier and spillover (PCP).

5.3. Local currency pricing Under local currency pricing, the law of one price no longer holds. Instead, home firms optimize the price of goods sold in the foreign country, P nH,t , by maximizing the following profits:  1 k Y X C s P W n ykt E0 bt þ k1 tþsk t P H,t C nH,t þ k ðjÞ t þ k C nH,t þ k ðjÞ : St þ k C t Pt þ k 0 k¼0 We here assume the same price stickiness y for P nH,t as that for P H,t . Similarly, P F,t can be calculated for foreign goods sold in the home country. n ^ pH, pnF , p^ , p^ n , p^ H , and p^ nF Under those setup, we obtain nine equations for nine variables Y^ , Y^ , e, 1g_y Y^ t ¼  ½^ıt Et p^ t þ 1 þ b^ t  þ Et Y^ t þ 1 Et DG^ t þ 1

s

ð1nÞ2 ð1nÞð1g_yÞ n ð1g_yÞEt Dp^ F,t þ 1 þ Et De^ t þ 1 , n s 2 n n 1 n n n n Y^ t ¼  ½^ınt Et p^ t þ 1 þ b^ t  þ Et Y^ t þ 1 þ ð1nÞEt Dp^ F,t þ 1  Et Dp^ H,t þ 1  Et De^ t þ 1 , s 1n s ( ) ð1yÞð1ybÞ s s ^ sð1nÞ2 ^ n Y^ t  p F,t þ ð1nÞe^ t , p^ H,t ¼ bEt p^ H,t þ 1 þ oþ G t þðsn1Þp^ H,t  y 1g_y 1g_y n  2 n ð1yÞð1ybÞ sn ^ n p ne^ t , p^ nF,t ¼ bEt p^ nF,t þ 1 þ ðo þ sÞY^ t þ ½sð1nÞ1p^ F,t  y 1n H,t   1n 1n p^ nt  Dp^ nF,t ¼ bEt p^ nt þ 1  Dp^ nF,t þ 1 n n  ð1yÞð1ybÞ s s ^ ð1nÞf1sð1nÞg n p^ F,t ne^ t , Y^ t  þ oþ G t þ snp^ H,t þ n y 1g_y 1g_y

 n n p^ t  Dp^ H,t ¼ bEt p^ t þ 1  Dp^ H,t þ 1 1n 1n  n ð1yÞð1ybÞ nð1snÞ n p^ H,t þ ð1nÞe^ t , þ ðo þ sÞY^ t þ sð1nÞp^ F,t þ 1n y p^ H,t ¼ p^ t þ Dp^ H,t , þ nð1g_yÞEt Dp^ H,t þ 1 

p^ nF,t ¼ p^ nt þ Dp^ nF,t , n 1 1 ^ n 1n n Y^ t  p^  p^ F,t ¼ Y^ t : Gt þ 1g_y 1g_y 1n H,t n

ð15Þ

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Fig. 3. Fiscal multiplier and spillover (LCP). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

There are four new Keynesian Phillips curves for each price of domestic and foreign goods in both countries, while there are only two with the producer currency pricing as shown in Eqs. (5)–(8). Consequently, the real exchange rate is not unity any more and becomes time-varying. Fig. 3 illustrates the time-series paths of seven key variables: outputs, producer prices in the two countries, the terms of trade, changes in the nominal exchange rate, and the real exchange rate.19 In this simulation, we assume an interest rate rule where a ¼ 1 and r ¼ 0. The blue line represents a case when s ¼ 0:5, while the red line represents a case when s ¼ 2. Under local currency pricing, we observe a positive fiscal spillover irrespective of whether s is larger (smaller) than unity and the economy is in period 2 as well as period 1. The mechanism of positive spillover under local currency pricing is understood via the global resource constraint in the last equation of (15), which is derived first by log-linearizing Eqs. (1), (2) and (4) for each country and then combining them. Defining the terms of trade by ToT t ¼

PF,t St P nH,t

under local currency pricing, we transform the last equation of (15) into Y^ t G^ t d t ¼ Y^ n , et ToT t 1g_y

ð16Þ

when n ¼0.5. Basically, this equation states that when the terms of trade improve for the domestic country or when the domestic real exchange rate appreciates, domestic production decreases relatively more than foreign production. Fig. 3 illustrates that irrespective of s, the terms of trade decrease, indicating the improvement for the home country. This is because a domestic currency depreciation improves the terms of trade for the country under local currency pricing. In Eq. (16), the effect of the worsening of the terms of trade dominates that of the depreciation of the real exchange rate. This is due to the currency misalignments that are formally defined by Engel (2011). Under local currency pricing, changes in the nominal exchange rate do not feed into the export price. As a result, the fiscal spillover becomes positive under local currency pricing, when the zero bound of nominal interest rates is binding. On the other hand, under producer currency pricing, Eqs. (9) and (10) represent the terms of trade and the global resource constraint, respectively. As shown in Fig. 2, a relative increase in the inflation rate in the domestic country induces an initial jump in the nominal exchange rate toward depreciation. These developments worsen the terms of trade. This makes domestically produced goods relatively cheaper than foreign-produced goods and increases demand for domestically produced goods, and decreases demand for foreign-produced goods. Consequently, fiscal expenditure in the domestic country has negative spillover effects on the foreign output when s 4 1. 19

Under producer currency pricing, the real exchange rate remains unity, so was not depicted in Fig. 1.

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5.4. Capital with investment adjustment cost We embed capital with investment adjustment cost. A representative home consumer maximizes s C 1 H1 þ o t w t , 1s 1þ o

subject to the budget constraint Pt C t þ P t It þ Et ðFt,t þ 1 Dt þ 1 Þ þBt þ 1 ¼ W t Ht þ P t Rkt K t1 þDt þ ð1 þ it1 ÞBt þ Pt T and the capital transition equation K t ¼ GðIt =K t1 ÞIt þ ð1dÞK t1 1 ¼ ½1 fðIt =K t1 dÞ2 It þð1dÞK t1 , 2 where Kt and It denote capital stock at the end of period t and investment, respectively. The last equation indicates the evolvement of capital: a convex function of GðIt =K t1 Þ captures investment adjustment cost, where we assume GðI=KÞ ¼ 1 and a parameter d represents a capital depreciation rate. As for firms, we assume a production function with labor and capital as input Y t ðjÞ ¼ K t1 ðjÞa Ht ðjÞ1a : The resource constraint is modified as Y t ðjÞ ¼ n½C H,t ðjÞ þ It ðjÞ þ Gt ðjÞ þ ð1nÞC nH,t ðjÞ, where home investment goods are purchased only from domestically produced goods. Fig. 4 displays the time-series paths of eight key variables: output, changes in producer prices in the two countries, the terms of trade, changes in the nominal exchange rate, consumption and investment. In this simulation, we assume an interest rate rule where a ¼ 1 and r ¼ 0. The depreciation rate d is set at 0.02. f is calibrated following Bernanke et al. (1999) so that the elasticity of the price of capital with respect to the investment capital ratio becomes 0.25. Basically, the fiscal multiplier and spillover with capital are quite similar to those without capital as depicted in Fig. 2. As explained earlier, in a model without capital, government spending in the past period, t1, has no effect on output at time t, because the model is purely forward-looking. On the other hand, the model with capital naturally contains an endogenous state variable and is therefore forward- as well as backward-looking. Indeed, government spending at t1 has some effects on output at time t in the model with capital and therefore, contrary to Fig. 2, government spending

Fig. 4. Fiscal multiplier and spillover (PCP with capital).

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multiplier both from G^ t and G^ t þ 1 in period 2 is not unity any more. Yet, the effect of G^ t to Y^ t þ 1 turns out to be very small. n The effect of G^ t to Y^ t þ 1 becomes smaller in an absolute size, but qualitatively our benchmark results stay unchanged. 6. Conclusion In this paper, we find that the fiscal multiplier exceeds one in the global liquidity trap. On the other hand, the fiscal spillover is negative if the intertemporal elasticity of substitution in consumption is less than one and positive if the parameter is greater than one. The size of the multiplier and the sign (positive or negative) associated with the spillover are quite the reverse to those in the undergraduate textbook, those without the zero lower bound, and those under the flexible price equilibrium or the sticky price equilibrium with optimal monetary policy. Incomplete stabilization of marginal costs due to the existence of the zero lower bound is crucial in understanding the effects of fiscal policy in open economies. In addition, we also show that the fiscal spillover in a global liquidity trap becomes positive under local currency pricing, even when the intertemporal elasticity of substitution is less than unity. This result stems from the fact that a currency deprecation reflecting fiscal expansion in the home country improves the terms of trade under local currency pricing while it deteriorates the terms of trade under producer currency pricing. We have so far considered only symmetric cases. A further question could be, in a situation where only the foreign country is constrained by the zero lower bound, how does government expenditure in the home country influence the foreign country’s economy. Another issue is how the government spending is used. A part of the government spending may contribute directly to a household’s utility. Government spending may be used to purchase not only home goods but also foreign goods. The validity of uncovered interest parity becomes crucial when we consider asymmetric cases between two countries. These issues will be addressed in our future research.

Acknowledgment The authors would like to thank the editor Christopher Otrok, two anonymous referees, Toni Braun, Lawrence Christiano, Giancarlo Corsetti, Richard Dennis, Michael Devereux, Martin Eichenbaum, Akito Matsumoto, Paolo Pesenti, Bruce Preston, the participants at the IMF Seminar in July 2010 and the CAMA Workshop on ‘‘Issues in Monetary and Fiscal Policy After the Global Financial Crisis’’ held at the Australian National University in March 2012, and the staff at the Bank of Japan, for their useful comments. Views expressed in this paper are those of the authors and do not necessarily reflect the official views of the Bank of Japan. Appendix A. Model details In this appendix, we describe the details of the model. Household. A representative household in the home country maximizes ! 1þo s 1 Y t X C 1 ht t E0 bt w , 1s 1þo t ¼0t¼1 subject to the budget constraint P t C t þ Et ðQ t,t þ 1 Dt þ 1 Þ þ Bt þ 1 ¼ W t ht þ Dt þ it1 Bt þ Pt T: Its first-order conditions are given by s who t ¼ Ct

Wt , Pt

s ¼ bt ð1 þ it ÞEt C t

Q t,t þ 1 ¼ bt

Pt C s , Pt þ 1 t þ 1

s C t þ 1 Pt : s Pt þ 1 C t

The last equation combined with that in the foreign country leads to risk sharing Q t,t þ 1 ¼ bt

s C ðC n Þs St P nt t þ 1 Pt ¼ bt t þn 1s , s C t Pt þ 1 St þ 1 P nt þ 1 ðC t Þ

s C nt s ¼ C  t

St P nt : Pt

or

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As for the consumption bundle     C H,t n C F,t 1n , Ct ¼ n 1n we can derive the demand functions as   P H,t 1 Ct , C H,t ¼ n Pt   PF,t 1 C F,t ¼ ð1nÞ Ct : Pt Similarly, from C H,t þ Gt ¼

"Z

#e=ðe1Þ

1

1ð1=eÞ

C H,t ðjÞ

dj

,

0

we can derive C H,t ðjÞ ¼

 P H,t ðjÞ e ðC H,t þ Gt Þ: P H,t

Firms. Under the Calvo price frictions, each monopolist maximizes the discounted sum of the profits s C t þ k Pt ½P H,t C H,t þ k ðjÞW t þ k C H,t þ k ðjÞ s C t Pt þ k 0 k¼0 " !e !e # 1 k Y X C s P P H,t P H,t : ¼ yk Et bt þ k1 tþsk t ðC H,t þ k þ Gt þ k Þ P H,t W t þ k PH,t þ k P H,t þ k C t Pt þ k 0 k¼0 1 X

yk Et

k Y

bt þ k1

Its first-order condition with respect to P H,t becomes  e 1 k Y C s P P P H,t X yk Et bt þ k1 tþsk t ðC H,t þ k þ Gt þ k Þ H,t þ k P H,t k ¼ 0 P H,t C t Pt þ k 0    s 1 k  PH,t þ k e þ 1 W t þ k C e X k Y P  ¼ y Et bt þ k1 tþsk t C H,t þ k þ Gt þ k : PH,t P H,t þ k e1 k ¼ 0 C t Pt þ k 0 This leads to K pH,t ¼ F pH,t

P H,t , PH,t

where F pH,t 

1 X

yk Et

k¼0

k Y 0

bt þ k1

 e s C t þ k P t C H,t þ k þ Gt þ k P H,t þ k s P H,t C t Pt þ k C H,t þ Gt

and K pH,t 



1 k Y C s P C þGt þ k P H,t þ k eð1tÞ X yk Et bt þ k1 tþsk t H,t þ k e1 k ¼ 0 PH,t C t P t þ k C H,t þGt 0

e þ 1

Wt þk : PH,t þ k

In recursive forms, they are written as  e s C t þ 1 P t C H,t þ 1 þ Gt þ 1 P H,t þ 1 F pH,t þ 1 , s Pt þ 1 C H,t þ Gt PH,t C t   C s P t C H,t þ 1 þ Gt þ 1 P H,t þ 1 e þ 1 p eð1tÞ W t K pH,t ¼ þ ybEt tþs1 K H,t þ 1 : e1 PH,t PH,t C t P t þ 1 C H,t þ Gt F pH,t ¼ 1þ ybt Et

From the law of one price, the price of domestic goods sold in the foreign country is given by PnH,t ¼

P H,t : St

We define relative prices by pH,t  P H,t =P t and pF,t  P F,t =Pt and the real exchange rate by et ¼

St P nt : Pt

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Furthermore, we define Pt ¼ 1 þ pt , P t1 p P H,t ¼ ð1 þ pt Þ H,t 1 þ pH,t ¼ P H,t1 pH,t1 and 1 þ pF,t ¼

p P F,t ¼ ð1 þ pt Þ F,t : P F,t1 pF,t1

Market clearing. The resource constraint is given by nY t ðjÞ ¼ nC H,t ðjÞ þ ð1nÞC nH,t ðjÞ: Its aggregation with the production function Y t ðjÞ ¼ ht ðjÞ yields Z 1 Z 1 C H,t ðjÞ dj þ ð1nÞ C nH,t ðjÞ dj nht ¼ n 0 0 #e Z 1 Z 1" n P H,t ðjÞ P H,t ðjÞ e dj þ ð1nÞC nH,t dj ¼ nðC H,t þ Gt Þ P H,t P nH,t 0 0 ¼ nLH,t ðC H,t þ Gt Þ þ ð1nÞLnH,t C nH,t , where the relative price dispersion term LH,t is defined by Z 1 P H,t ðjÞ e LH,t ¼ dj: P H,t 0 Under the Calvo price frictions, we have !e Z 1 P P H,t1 ðjÞ e LH,t ¼ ð1yÞ H,t þy dj P H,t P H,t 0 !e   Z  P H,t PH,t1 e 1 PH,t1 ðjÞ e ¼ ð1yÞ þy dj PH,t1 PH,t P H,t 0

as well as 1e

e 1e P 1 H,t ¼ ð1yÞP H,t þ yP H,t1 :

Consequently, we obtain 2   3e=ðe1Þ PH,t1 1e   1 y 6 7 PH,t1 e P H,t 7 LH,t ¼ ð1yÞ6 þy LH,t1 : 6 7 4 5 1y PH,t n

Summary. The model is summed up by 31 equations for 31 endogenous variables Y, Y n , h, h , C, C n , CH, C nH , C F , C nF , pH, pnH , p pF, pnF , w  W=P, wn  W n =P n , p, pn , pH , pnH , pF , pnF , DH , DnH , DF , DnF , K pH , K pF n , FH, F pF n and e, except for nominal interest rates s who t ¼ C t wt ,

s ¼ bt ð1 þ it ÞEt C t

1 C s , 1þ pt þ 1 t þ 1

C H,t ¼ np1 H,t C t , C F,t ¼ ð1nÞp1 F,t C t , 1 ¼ pnH,t p1n F,t ,

wðhnt Þo ¼ ðC nt Þs wnt , ðC nt Þs ¼ bt Et

1 þint ðC n Þs , 1þ pnt þ 1 t þ 1

1 n C nH,t ¼ npnH,t Ct , n1 n C nF,t ¼ ð1nÞpF,t Ct ,

1 ¼ ðpnH,t Þn ðpnF,t Þ1n , s C nt s ¼ C  t et ,

I. Fujiwara, K. Ueda / Journal of Economic Dynamics & Control 37 (2013) 1264–1283

nLH,t ðC H,t þGt Þ þð1nÞLnH,t C nH,t ¼ nY t , Y t ¼ ht , 2

 1e 3e=ðe1Þ 1  e 61y 1 þ p 7 1 7 H,t LH,t ¼ ð1yÞ6 þy LH,t1 , 6 7 4 5 1 þ pH,t 1y 2

1 61y 6 1þ pnH,t LnH,t ¼ ð1yÞ6 6 6 1y 4

!1e 3e=ðe1Þ 7 7 7 7 7 5

þy

1 1þ pnH,t

!e

LnH,t1 ,

2

 1e 31=ð1eÞ 1 61y 1 þ p 7 6 7 H,t , K pH,t ¼ F pH,t 6 7 4 5 1y F pH,t ¼ 1 þ ybt Et K pH,t ¼

e wt e1 pH,t

s e C t þ 1 ð1 þ pH,t þ 1 Þ C H,t þ 1 þ Gt þ 1 p F H,t þ 1 , s 1 þ p C H,t þ Gt Ct tþ1

þ ybt Et

s eþ1 C C H,t þ 1 þ Gt þ 1 p t þ 1 ð1 þ pH,t þ 1 Þ K H,t þ 1 , s 1 þ pt þ 1 C H,t þ Gt Ct

pH,t ¼ et pnH,t , nLF,t C F,t þð1nÞLnF,t C nF,t ¼ ð1nÞY nt , n

Y nt ¼ ht ,

2

 1e 3e=ðe1Þ 1  e 61y 1 þ p 7 1 7 F,t LF,t ¼ ð1yÞ6 þy LF,t1 , 6 7 4 5 1þ pF,t 1y 2

1 61y 6 1 þ pnF,t 6 n LF,t ¼ ð1yÞ6 6 1y 4 2

1 61y 6 1þ pnF,t 6 K nF,tp ¼ F nF,tp 6 6 1y 4 np ¼ 1 þ ybt Et F F,t

K nF,tp ¼

e wnt e1 pF,t n

!1e 3e=ðe1Þ 7 7 7 7 7 5

þy

1 1 þ pnF,t

!e

LnF,t1 ,

!1e 31=ð1eÞ 7 7 7 7 7 5

,

s ð1 þ pnF,t þ 1 Þe C nF,t þ 1 np C nt þ 1 F , n s 1 þ pnt þ 1 Ct C nF,t H,t þ 1

þ ybt Et

s ð1þ pnF,t þ 1 Þe þ 1 C nF,t þ 1 np C nt þ 1 K F,t þ 1 , ns 1 þ pnt þ 1 Ct C nF,t

pF,t ¼ et pnF,t , 1 þ pH,t ¼ ð1 þ pt Þ

pH,t , pH,t1

1 þ pnH,t ¼ ð1 þ pnt Þ

pnH,t , pnH,t1

1 þ pnF,t ¼ ð1 þ pnt Þ

pnF,t pnF,t1

and 1 þ pF,t ¼ ð1 þ pt Þ

pF,t : pF,t1

n n By log-linearizing the above equations, we can derive Eqs. (5)–(8) with respect to Y^ , Y^ , p^ H , and p^ F .

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Appendix B. Derivation of Eqs. (11) and (12) In the appendix, we explain very briefly how to derive Eqs. (11) and (12). We express Eqs. (5)–(8) by lag operator L n 1 þ nðs1Þ ð1L1 ÞðY^ t G^ t Þ þ ð1nÞðs1Þð1L1 ÞY^ t ¼ ð~ıt L1 p~ H,t þ b^ t Þ, 1g_y n nðs1Þ ð1L1 ÞðY^ t G^ t Þ ¼ ð~ınt L1 p~ nF,t þ b^ t Þ, f1þ ð1nÞðs1Þgð1L1 ÞY^ t þ 1g_y  n 1n þ sn ^ 1 þnðs1Þ ^ Y t þ ð1nÞðs1ÞY^ t  ð1bL1 Þp~ H,t ¼ k o þ Gt , 1g_y 1g_y  n nð s 1Þ nð b t  s1Þ G^ t , ð1bL1 Þp~ nF,t ¼ k ½o þ 1þ ð1nÞðs1ÞY^ t þ Y 1g_y 1g_y

where we define

k

ð1yÞð1ybÞ

y

:

Eliminating p~ H,t and p~ nF,t , we obtain n 1 þ nðs1Þ ð1L1 ÞðY^ t G^ t Þ þ ð1nÞðs1Þð1L1 ÞY^ t 1g_y  k 1n þ sn ^ ^ n  1 þ nðs1Þ G^ t , Y ¼ ð~ıt þ b^ t Þ þ L1 o þ þ ð1nÞð s 1Þ Y t t 1g_y 1g_y 1bL1 n nðs1Þ ð1L1 ÞðY^ t G^ t Þ f1 þð1nÞðs1Þgð1L1 ÞY^ t þ 1g_y  k b t  nðs1Þ G^ t : ^ n þ nðs1Þ Y ½  ¼ ð~ınt þ b^ t Þ þL1 o þ1 þ ð1nÞð s 1Þ Y t 1g_y 1g_y 1bL1 n Approximating the two equations up to the first order of L1 and separating terms of Y^ t from those of Y^ t , we obtain 9 8 1 1 þ sn > > < ð1 þ nðs1ÞÞð1L Þð1g_yÞkL ½o þ 1n 1g_y  = h n oi Y^ t 2 s1Þ > >  nð1nÞð 1L1 1k n þ skð1nÞ o ; : n þ sð1nÞ

¼ ð1g_yÞð~ıt þ b^ t Þ 2 3 ð1g_yÞðs1Þð1nÞðkL1 þ 1L1 Þ n þ s1ð1nÞ 6 n o7 4 5ð~ınt þ b^ t Þ  1 þ L1 þ n þ s1ð1nÞ kL1 ½1 þ o þ ðs1Þð1nÞ 8 9 1 1 þ sn > > < ð1 þnðs1ÞÞð1L Þð1g_yÞkL 1n 1g_y = h n oi G^ t , þ nð1nÞðs1Þ2 1 > > 1k n þ skð1nÞ o :  n þ sð1nÞ 1L ;

ð17Þ

which is summarized as Eq. (11) by the first-order approximation  n þ sð1nÞ oð1g_yÞL1 G^ t : Y^ t ¼ CðL1 Þð~ıt þ b^ t Þ þ C n ðL1 Þð~ınt þ b^ t Þ þ 1þ k

s

The forms of CðL1 Þ and C n ðL1 Þ, which are unimportant for the multiplier, can be defined from Eq. (17). Similarly, we can n obtain Y^ t as in Eq. (12) n ðs1Þno 1 ^ Y^ t ¼ Dn ðL1 Þð~ınt þ b^ t Þ þ DðL1 Þf~ıt þ b^ t gk L Gt:

s

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