Fiscal policy in a currency union at the zero lower bound
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Cook, David; Devereux, Michael B. Working Paper Fiscal policy in a currency union at the zero lower bound ADBI Working Paper, No. 801 Provided in Cooperation with: Asian Development Bank Institute (ADBI), Tokyo Suggested Citation: Cook, David; Devereux, Michael B. (2018) : Fiscal policy in a currency union at the zero lower bound, ADBI Working Paper, No. 801, Asian Development Bank Institute (ADBI), Tokyo This Version is available at: https://hdl.handle.net/10419/190222 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/3.0/igo/
ADBI Working Paper Series FISCAL POLICY IN A CURRENCY UNION AT THE ZERO LOWER BOUND David Cook and Michael B. Devereux No. 801 January 2018 Asian Development Bank Institute
The Working Paper series is a continuation of the formerly named Discussion Paper series; the numbering of the papers continued without interruption or change. ADBI’s working papers reflect initial ideas on a topic and are posted online for discussion. ADBI encourages readers to post their comments on the main page for each working paper (given in the citation below). Some working papers may develop into other forms of publication. ADB recognizes “China” as the People’s Republic of China; “Hong Kong” as Hong Kong, China; and “Korea” as the Republic of Korea. Suggested citation: Cook, D. and M.B.Deveraux. 2018. Fiscal Policy in a Currency Union at the Zero Lower Bound. ADBI Working Paper 801. Tokyo: Asian Development Bank Institute. Available: https://www.adb.org/publications/fiscal-policy-currency-union-zero-lower-bound Please contact the authors for information about this paper. Email: [email protected], [email protected] David Cook is a professor of economics at the Hong Kong University of Science and Technology. Michael B Devereux is a professor at the Vancouver School of Economics, University of British Columbia. The views expressed in this paper are the views of the author and do not necessarily reflect the views or policies of ADBI, ADB, its Board of Directors, or the governments they represent. ADBI does not guarantee the accuracy of the data included in this paper and accepts no responsibility for any consequences of their use. Terminology used may not necessarily be consistent with ADB official terms. Working papers are subject to formal revision and correction before they are finalized and considered published. Asian Development Bank Institute Kasumigaseki Building, 8th Floor 3-2-5 Kasumigaseki, Chiyoda-ku Tokyo 100-6008, Japan Tel: +81-3-3593-5500 Fax: +81-3-3593-5571 URL: www.adbi.org E-mail: [email protected] © 2018 Asian Development Bank Institute
Fiscal Policy in a Currency Union at the Zero Lower Bound ∗ David Cook†and Michael B Devereux‡ Revised, November 2017 Abstract When monetary policy is constrained by the zero lower bound, fiscal policy can be used to achieve macro stabilization objectives. At the same time, fiscal policy is also a key policy variable within a single currency area that allow policy-makers to respond to regional demand asymmetries. How do these two uses of fiscal policy interact with one another? Is there an inherent conflict between the two objectives? How do the answers to these questions depend on the degree of fiscal space available to different members of the currency area? This paper constructs a two-country New Keynesian model of a currency union to address these questions. We find that the answers depend sensitively on the underlying internal structure of the currency union, notably the degree of trade openness between the members of the union. Keywords:Liquidity Trap, Monetary Policy, Fiscal Policy, International Spillovers JEL: E2, E5, E6 ∗The authors thank the Hong Kong Research Grants Council for support under CERG GRF 690513. Devereux thanks SSHRC, the Bank of Canada, and the Royal Bank of Canada for financial support as well as support from ESRC award ES/1024174/1. †Hong Kong University of Science and Technology, da[email protected] ‡CEPR, NBER, University of British Columbia, [email protected]
1 Introduction Over the last decade, developed economies have been operating with very low interest rates, effectively at the zero bound. This has severely constrained the use of monetary policy in these countries. Constraints on the use of monetary stabilization raises the possibility of alternative stabilization instruments, including fiscal policy. Fiscal expenditure can be used to stimulate inefficiently low demand, when aggregate demand conditions are sufficiently severe that monetary policy rates are unable to fall further (see Woodford, 2010; Christiano, Eichenbaum and Rebelo, 2011). But optimal fiscal stabilization at the zero bound may be complicated in a currency union, because different regions within the union may be subject to different degrees of severity in macro shocks. In fact, it is well recognized that this problem is not simply a facet of the zero bound environment. In a group of economies jointly using a single currency, monetary policy will always be in some way constrained from addressing regional disparities in aggregate demand. The literature has addressed these question, showing that optimal regional fiscal policy may then be targeted toward locations with low demand (see Beetsma and Jensen, 2005; and Gali and Monacelli, 2008). The different roles that fiscal policy might play in a currency union that is further constrained by the zero bound sets up the possibility of a conflict in the uses of fiscal policy. Consider the post global financial crisis European economy. The persistent disinflation experienced by the Eurozone was highly concentrated by region. More extreme downturns were observed in the peripheral countries than in the core countries such as Germany. In such a context, it is relevant to ask to what degree it is optimal to apply expansionary fiscal policy across all regions. Conversely, if fiscal demand should shift heavily enough toward the more badly affected regions, then in order to achieve a balanced degree of aggregate demand across the union, it may be potentially optimal to contract spending in the least affected regions. We will examine this question in the context of a two-region New Keynesian model of a currency union. A key element of the model is that each region has a bias in consumption toward home-produced goods. Under home bias, a disinflationary demand shock in one region will be concentrated on goods in that region and will spread only partially to the other region. But if it is sufficiently severe, even a regional shock could move the aggregate economy of the union to the effective lower bound on interest rates. Our approach to dealing with these trade-offs is to identify the optimal aggregate and 2
relative composition of fiscal policy within the model. Optimality is defined using an approximation for the social welfare function of the currency union 1. We concentrate on examining the social welfare maximizing optimal policies of the regions of the currency union, assuming that the union members can cooperate on fiscal spending policies, and where policies are chosen with discretion. Broadly speaking, we find that optimal policy in the most affected region should be expansionary. However, the optimal fiscal response of the partner region depends on a variety of factors governing the nature of the equilibrium. Crucially for our investigation, it is not always the case the the partner region should follow an expansionary fiscal policy. A key parameter governing our results is the degree of home bias in consumption. In a general sense, we can translate this parameter as that which governs the degree of internal trade openness within the union. The greater is home bias in the consumption basket, the smaller the gains from trade and the lower is the ratio of trade to GDP for each country. If domestic demand is very concentrated on domestic goods, then asymmetric shocks will result in greater regional business cycle disparities, leading to a greater need to concentrate spending in a particular region. Moreover, the spillovers of fiscal policy from one region to another are reduced accordingly, as home bias concentrates the effects of fiscal policy. Thus, fiscalpolicywillbelesseffective in addressing demand shocks in another region. A useful way to to clarify this relationship is to show the relationship between fiscal multipliers and homebias.Thisiswhatweillustratebelow. Conversely, if the union-wide economy economy is particularly vulnerable to the aggregate downturn and trade is highly integrated, then meeting aggregate optimal spending goals may require expansion across the union. Here is it not just home bias, but other structural parameters that are important. For example, if demand is sufficiently interest sensitive, then a relatively large fiscal expansion may be required. This will tend to require fiscal policy expansion in both regions. Another important factor is the duration of the underlying demand shocks which generate the downturn. If shocks are very persistent, this will also increase the likelihood of possible spillovers across regions. As noted above, we characterize social welfare as a second order approximation to the utility of the residents of the currency union. Following other literature on the zero bound in macro models, we focus on demand shocks which tend to push inflation and the output gap in the same direction. In the aggregate union economy, there would be no direct trade-off 1This follows Cook and Devereux (2011a). 3
between stabilizing the two goals, given an unconstrained monetary policy. This is not true of fiscal policy in the aggregate, since using fiscal spending for stabilization purposes changes the optimal mix of private to public goods, which will create inefficiencies in the composition of goods in the economy. When we look at relative, within union differentials in the impact of shocks, monetary policy would be ineffectual, even absent the zero bound constraint, so fiscal policy response would be necessary in order to respond to differential responses of output gaps and inflation. But again, this would create additional compositional distortions. Thus, along both aggregate and relative dimensions in response to negative demand shocks, afiscal spending policy obtains at best a second-best optimum. While it has been well recognized that fiscal policy responses may be necessary for stablization within a currency union (e.g. Beetsma and Jensen, 2005), the additional dimension brought by the zero bound constraint adds a new element. An important principle in the analysis we follow below is that while the relative responses of fiscal policy to regionally concentrated shocks is independent of the zero bound constraint, the absolute response of each region depends critically upon whether the zero bound is binding or not. In practice, this means that in some cases, depending on the structural parameters and especially on the degree of home bias, the region less affected by the shock may need to contract fiscal policy in order to achieve efficient relative price adjustment, while in other cases, it may need to expand fiscal policy, in order to attain an optimal aggregate demand expansion. We also consider optimal fiscal policy when some regions are fiscally constrained. We find that at the zero lower bound, aggregate fiscal policy will be expansionary under a wide variety of circumstances even when fiscal policy is constrained. Thus, when only one region can implement expansionary fiscal policy at the zero lower bound, that region will implement fiscal expansion regardless of whether that region is the most deeply affected. In our model (as in the continuous time model of Farhi and Werning, 2016), cross-country fiscal spillovers are positive at the zero lower bound, so a cooperative country can positively impact its demand constrained neighbor through its own fiscal expansion. This is in general not true in an environment of unconstrained monetary policy. We follow a large open economy literature which studies fiscalpolicyinstickyprice models at the zero lower bound. Fujiwara and Ueda (2013) identify fiscal multipliers in an open economy with flexible exchange rates. Cook and Devereux (2010, 2011a) examine optimal fisal policy at the zero lower bound in a two country model with flexible exchange rates, focusing on the importance of home bias. Hettig and Mueller (2015) examine the coordination of fiscal policy in a union of many small economies at the zero lower bound; 4
this paper examines only fiscal policy coordination. Blanchard, Erceg and Linde (2016) asks similar questions as this paper. They examine the welfare gains from fiscal expansion at the zero lower bound in a two country sticky price model. They find welfare gains from fiscal expansions in both countries, whether jointly or separately. We build on this by characterizing optimal policy, though in a much simpler model. 2Atwocountrymodel Consider a currency area made up of two regions. We assume that both of these regions have permanently committed to a single currency unit. In each country, households consume both private and government goods, and supply labor. Denote the countries as ‘South’ and ‘North’, with North variables denoted with an asterisk superscript. The population of each country is normalized to unity. Monopolistically competitive firmsineachregion produce differentiated goods with constant returns to scale technology. Regional governments produce government goods which are distributed uniformly across households within the region. Governments have access to lump sum taxation. Complete asset markets allow full insurance of consumption risk across countries. There is an implicit risk free interest rate which is common across the currency union. Firm’s production and supply is constrained by Calvo style sticky prices. 2.1 Households Utility of a representative infinitely lived home household evaluated from date 0is: 0=0 ∞ X =0 (()−()+()) (1) where felicity is the functions ,,andrepresent the utility of the composite South consumption bundle ()disutility of labor, ()andutilityofthegovernmentgood ()respectively with 1.Thevariablerepresents a demand shock to preferences, (assuming that 12 0). Define ≡− as the inverse of the elasticity of intertemporal substitution in consumption, ≡−00 0as the elasticity of the marginal disutility of hours worked and ≡−00 0as the elasticity of marginal utility of public goods. In addition, we assume that =1, consistent with empirical evidence (see, e.g. Yogo, 2004) The composite consumption consists of a geometric average of home and foreign goods. 5
=Φ2 1−2 ≥1 where Φ=¡ 2¢ 2(1−¡ 2¢) 2 is South consumption of the South produced composite good , and is South consumption of the North produced composite good2. The parameter ≥1 governs the degree of home bias in the consumption basket of each country. For =1,both countries are fully open to trade and in a symmetric equilibrium exports are %50 of GDP, while with =2, there is zero trade and the union consists of two closed economies. Consumption aggregates, and are composites, defined over a range of home and foreign differentiated goods, with elasticity of substitution 1between goods. =⎡ ⎣ 1 Z0 ()1 1−⎤ ⎦ 1 1− =⎡ ⎣ 1 Z0 ()1 1−⎤ ⎦ 1 1− The demand for good in region = is () =µ() ¶− where the price indices for home and foreign goods are: =⎡ ⎣ 1 Z0 ()1−⎤ ⎦ 1 1− =⎡ ⎣ 1 Z0 ()1−⎤ ⎦ 1 1− whiletheaggregateCPIpriceindexfortheNorthregionis=2 1−2 and for the South region is ∗ =2 1−2 . In each region, government spending has complete home bias; agents only get utility from spending on the domestic good. Government demand for each individual variety of the home good has price elasticity , the same as that for private spending. The household’s implicit labor supply at nominal wage is: ()=0()(2) Optimal risk sharing implies 2Our objective is to illustrate the main points of the analytically, Therefore, we make the simplifying assumption that the trade elasticity is unity () and the relative population sizes are equal. For more general representations, see Bhattarai and Egorov, 2016, and Erceg and Linde, 2010. 6
We can also examine the multipler if the authorities are facing negtive demand shocks which push the natural rate below the zero lower bound. e = ze + z1 =(1 −)(1 −)− (1 −)(1 −)z−z In the absence of an interest rate response the multiplier is larger than 1. 4.2 Regional Differentials Fiscal policy in one country will also create differential output responses across the currency area. Since regional disparities are invariant to monetary policy, the fiscal multiplier for the relative system e e is invariant to the zero lower bound. However, the endogenous dynamics of regional disparities also create a dynamic multiplier. In order to explore the fiscal response to demand shocks, we need to solve for the dynamics of relative magnitudes. Suppose that demand shocks follow the same Markov process as government fiscal gaps with persistence parameter .Then ∆ = 1−sothestablesolutionforthetermsof trade is given as: =−1−·∙ + ··e + 1−¸(19) where 0 ≡ z n 2z+(1+(1 −)) −o=2Ξ ½Ξ 2+1− 2+(1 −)+√[Ξ+{1+}]2−4 2¾2 We definine the parameter Ξ= 2z= 2¡ +1 ¢.Therootis characterized as 0=[Ξ+{1+}]−q[Ξ+{1+}]2−4 21 It is straightforward to show that the root is real, positive and within the unit interval. Persistence, (Ξ), is negatively related to Ξ. 0(Ξ)= 1 2[1 −(Ξ+{1+}) q[Ξ+{1+}]2−4 ]0 Intuitively, the larger is the impact of a shock on inflation, the less persistent will be changes 13
in the terms of trade. When prices are less sticky, will be larger, and hence Ξwill thus be larger, which means that and shocks will be less persistent. Prices and the terms of trade will adjust more quickly. In the limit, as prices become perfectly flexible, of course there is no persistence in the terms of trade at all, since then the exchange rate regime is irrelevant. Also, when is larger, labor supply is less elastic and the output gap has a bigger impact on inflation. Most interestingly from the perspective of this paper, the parameter Ξis negatively associated with home bias. The closer is to 2, the smaller will be and Ξand the greater the persistence in the shock. When home bias is greater, a given appreciation in the terms of trade will translate into less expenditure switching that will have less impact on inflation, prolonging the period of adjustment. 4.2.1 Dispersion Multipliers Dispersion in fiscal policy across regions will create persistent dispersion in output. To illustrate dispersion multipliers, we simplify by assuming zero natural interest rate differentials, =0. Combine (17) with (19) in the case =0to get ¡e −e ¢=¡e −1−e −1¢− 2·∙ + ··e ¸ which implies e =e −1+e −e −1− 2·∙ + ··e ¸ =£e −1−e −1¤+[1− 2· + ]·e This illustrates that the government spending shocks under a single currency area will generate persistent effects on output. From these expressions, we can deduce the following propositions Proposition 1 The impact dispersion multiplier ≡[1− 2· +]is between zero and one; i.e. 0 1. The value of captures the effect of a relative government spending shock on relative output. Although the absence or presence of the zero lower bound cannot affect this response, it can at most attain a value of unity. This depends on the degree of price stickiness, captured 14
by . For a high value of price stickiness, relative inflationisonlyslightlyaffected by the relative spending shock, and therefore there is little terms of trade appreciation, which would deflect demand away from the country receiving the relative spending shock, so the multiplier approaches unity. On the other hand, as →∞, the dispersion multiplier approaches the flexible price value of z, which must be less than unity. Proposition 2 When =1, the impact dispersion multiplier is invariant to the degree of home bias. When 1, the impact multiplier , , will be larger when home bias is more intense. When =1, relative government spending affect the terms of trade and output independently of the size of home bias, as can be seen from (15), (??), and (??). Suppose again that government spending has persistence ,sothat e =and e +=with prob or zero for all ≥0. Conditional on the government spending shock continuing we can write e +="1−Ã X =0 ! 2· +#+ We call the multiplier during the duration of the spending the expansion phase dispersion multiplier () Proposition 3 The expanison phase dispersion multiplier ()=h1−³P =0 ´ 2· +i is declining in and between 0 () (−1) 1. Theimpactofgovernmentspendingdifferentials on output dissipates over time. The region with a more concentrated level of government spending will experience a slow terms of trade appreciation. This will shift expenditure to the other region. However, the impact on local output always exceeds that on foreign output. The post expansionary multiplier depends on how long the expansion lasts. Consider if the expansion ends in period ++1. Then, we can write output as: e ++1 =£e +−e +¤=[ ()−1]0 e ++=[ ()−1]0 So, the post expansionary multiplier on output differentials is negative. After the expansion, the expanding economy will have an overvalued terms of trade. The direct effect of the 15
spending shock is eliminated, and the only remnant of the shock is a higher relative price level, reducing world demand for the country’s output. Although the dispersion multiplier is independent of monetary policy, this is not true of the multipliers for the absolute value of the response of output in each country. If the economy is at the zero lower bound when the expansion occurs, fiscal expansion in one country will increase output in the other country. Define the cross-region multiplier as the response of the North output to a fiscal expansion in South, i.e., e =− Proposition 4 The cross region multiplier is always positive at the zero lower bound, ()− ()0 A corollary of (2) and (4) is that the cross-region impact multiplier is decreasing in the level of home bias - clearly with extreme home bias the cross-region multiplier must be zero. If the expansion occurs outside the zero lower bound, the cross-region multiplier in a currency union is more complicated. A fiscal expansion concentrated in one region will lead to a real appreciation, increasing demand for the other regions goods. However, the fiscal expansion will increase aggregate inflation which will impact aggregate interest rates for all regions of the currency area. The net effect of these countervailing effects on cross-regional demand is ambiguous. The previous proposition shows that if monetary policy is sufficiently passive (i.e. zero interest response) then the positive spillovers will dominate. However, if aggregate monetary policy is sufficiently active and home bias weakens the expenditure switching effects of exchange rate appreciation, then the negative spillovers will dominate. Proposition 5 Ifthemonetarypolicyruleimplementsazeroinflation equilibrium, there exists an such that the cross-region multiplier during the expansion is negative. Clearly, in the periods after the expansion, the cross region multiplier will be positive as (+)will be negative for any ≥0. 4.2.2 Relative Demand Shocks Regional differences in the natural interest rate generate inflation and output dispersion by concentrating demand within a particular region. Here, we abstract from fiscal policies, assuming for now that government spending gaps are zero. In that case, relative output 16
follows the process e =∙e −1− −1 (1 −)¸+∙1− 2¸· (1 −) 01− 2=1− +(1 −) ©1 2Ξ+1− +(1 −)ª1 A natural rate shock following a Markov process that dissipates (ex post) at period ++1 has an impact e +="1−Ã X =0 ! 2#· (1 −) where the inequality follows from the same logic as Proposition 3. In period ++1 e ++1 =∙e +− + (1 −)¸=[−Ã X =0 ! 2] (1 −)0 e ++=[−Ã X =0 ! 2] (1 −) −[Ã X =0 ! 2]0 So in the same manner as the response to a fiscal shock a positive demand shock will have positive but diminishing effects on relative output until the shock ends, then relative output effects go in reverse, as the affected country finds itself with an appreciated real exchange rate. Relative inflation follows = −1+·∙ − −1 (1 −)¸ Again, a demand shock increases inflation differentials during the period of the shock, then reverses after the shock dissipates. Countercyclical Policy Now, consider the impact of fiscal policy rules. In the next section, we derive the optimal welfare-maximizing fiscal gaps. Here, we assume simply that the the government implements endogenous spending differentials as a function of the output gaps, so that, for Φ0, e =−Φ·e 17
Inthecaseofendogenousfiscal policy, we can redefine the parameter ←→ Ξ=Ξ·1+zΦ 1+ΦΞ The stable solution for the terms of trade has the same form as (19): =←→ −1−←→ · (1 −)(20) where the initial impact of demand shocks on the terms of trade and its persistence are a function of ←→ Ξ 0←→ ³←→ Ξ´=h←→ Ξ+{1+}i−rh←→ Ξ+{1+}i2 −4 21 ←→ ³←→ Ξ´=2←→ Ξ ⎧ ⎨ ⎩ 1 2←→ Ξ+1− 2+(1 −)+r[←→ Ξ+{1+}]2−4 2⎫ ⎬ ⎭ 2 where as shown previously←→ 0(←→ Ξ)0and as implied by the proof to proposition ( 2). Note ←→ Ξis decreasing in Φ. So, the more counter-cyclical is fiscal policy (i.e. the larger is Φ) the smaller will be the initial impact of demand shocks on the terms of trade as, as the reallocation of public sector demand across regions will offset the initial shock to private sector demand. However, at the same time demand shocks will have less impact on inflation differentials, so inflation will adjust more slowly. Hence the countercyclical public sector spending rule will slow down the adjustment to the initial demand shock. We can write the dynamics of output as µe − (1 −)−e ¶=←→ µe −1− −1 (1 −)−e −1¶−←→ 2· (1 −) ¡(1 + Φ)e ¢=←→ µ(1 + Φ)e −1− −1 (1 −)¶+"1−←→ 2#· (1 −) e =←→ ∙e −1− −1 (1 + Φ)(1−)¸+h1− ←→ 2i (1 + Φ)· (1 −) Proposition 6 Counter-cyclical fiscal policy differentials will reduce the immediate impact of a relative demand shock on the dispersion of the output gap, but at the same time prolong the period of adjustment of the real exchange rate to the shock. Inflation follows 18
=←→ +←→ ∙ − −1 (1 −)¸ Again, a natural interest rate differentials shock increases inflation differentials during the period of the shock, then reverses after the shock dissipates. 5 Optimal Fiscal Policy We now turn to the analysis of optimal fiscal policy in the monetary union. As shown in Cook and Devereux (2011a), a second order approximation to an equally weighted world social welfare can also be constructed in world averages and world differences. Welfare for any period is written as: =−(e )2· 2−(e )2 2−(e )2· 2−(e )2· 2−(e )( e )(21) −(e )( e )− 2( )2− 2( )2 where are defined in the Appendix. Thus, the social welfare function faced by the policy maker depends upon output gaps, inflation rates, fiscal gaps, and the interaction between these variables. As in the positive analysis of the response to demand shocks described above, we can separate the optimal fiscal policy problem into a choice of world average and world relative policy instruments. We focus on an optimal policy response without commitment. Hence, the Given this welfare function, cooperative optimal policy maximizes the objective function "∞ X =0 +# =+ £ −(+)e + ·e − +1¤ + £ −(+)e +e − +1¤ + £(e +1 −e )−(e +1 −e )−¡−e − +1¢¤ + ∙ +2µ£e −e ¤− ∆ ¶−2µ£e −1−e −1¤− ∆ −1¶¸ 19
where the constraints are the equation describing the dynamics of the economy and =Max(0+)(22) This implies that optimal fiscal policy 5.1 First Order Conditions 5.1.1 Aggregate Economy The first order conditions describing optimal policy for the aggregate economy are −e −e =(+) + (23) e +e = + (24) = (25) = Theconstraintonat the zero lower bound implies that either the shadow value, ,is zero or the zero lower bound binds, . When is zero and policy rates are unconstrained by the zero lower bound, =e , and price stability is the cooperative optimal policy under both commitment and discretion, (see Benigno and Benigno, 2003); the aggregate fiscal gap will be zero along with output gap and inflation. Intuitively, if policy is determined relative to an initial steady state without monopoly distortions, and there are no mark-up shocks, optimal cooperative policy will close all gaps, whether under discretion or commitment. A policy of price stability can be implemented by setting nominal interest rates equal to natural interest rates as defined in (10) and (11). Note that in order to implement this policy, it is necessary that there be an interest rate feedback rule on inflation or other endogenous variables, in order to avoid indeterminacy (see e.g. Gali, 2008, and Benigno and Benigno, 2008). When the zero bound on interest policy binds, there is a role for aggregate discretionary government spending in response to natural interest rate shocks. When natural interest rate shocks follow a Markov process in which the current natural interest rate is below the zero lower bound and has a probability of (1 −)of returning permanently to a positive range, 20
we can solve for the optimal fiscal policy in closed form. =−[(1 −) +(+)] (1 −)·e [(1 −)(+· )+(+{ −1})] [(1 −)(1 −)−(+)] (26) In a liquidity trap, the fiscal multiplier 1, so it is clear that optimal fiscal gaps are negatively associated with demand shocks. 5.1.2 Regional Differentials We can separately solve for the optimal relative responses of fiscal policy. The first order conditions governing regional differentials are: −e −(e )= (+)−2[ − +1)(27) e +(e )= −2[ − +1)(28) + = (29) As noted by, Corsetti, Keester and Muller (2012) and Cook and Devereux (2015), a currency union contains a commitment to maintaining stationary price level differentials. Thus, the optimal rules are dynamic in nature, even when no fiscal commitment is available. The first order conditions can be solved forward: "∞ X =0 ∗∗¡ +−e +−e +¢#=·"∞ X =0 ∗¡(+) ++(e +)+e +¢# ∞ X =0 ©∗∗+∗ªe +=− ∞ X =0 ©∗∗+∗ªe +− ∞ X =0 ©∗·(+)−∗∗ª where =(2+) (2+)1,∗≡1 1+Ξwhere ∗∗ =1 1+zΞ. Proposition 7 As →0, optimal relative consumption is a negative function of inflation and output gap differentials. e +=−{+} {+}e +− {+} 2 (2+)() + 21
The optimal choice of fiscal policy implies a trade-offbetween the various distortions. If relative demand shocks are creating distortions in the relative output gap and relative inflation, then optimal government spending gaps cannot be zero. Since autoregressive relative demand shocks shift the output gap and inflation in the same direction, the fiscal gap should move in a counter-cyclical direction. 5.2 Numerical Experiments The above results establish that the aggregate government spending gap should respond to demand shocks only if monetary policy is constrained by the zero lower bound. However, in a currency union, there is a role for regional differentials in monetary policy to mitigate against regional differentials in demand, regardless of whether the zero lower bound binds or not. We show this in a numerical solution to the optimal fiscal policy. In the numerical solution we assume there is a preference shock to the South economy which shifts the aggregate natural interest rate to =−02 (-8% on annualized basis) for a fixed number of periods7. Following this, the aggregate natural interest rate shifts to the rate of home preference, ++1 =forever. Rather than assume optimal monetary policy, we simplify by assuming a policy rule =Max(0+)(30) We search for a solution to the set of equations −{−−1} =(+)©e −ze ª+ +1 −{−−1}=(+)©e −ze ª−[+1 −] where optimal fiscal policy accords with ( 27), (28), and (34). We search for solutions for the dynamics of the real exchange rate. =−1−· 5.2.1 Benchmark Parameterization The benchmark parameterization of preference parameters are taken from Cook and Devereux (2011a). The subjective discount factor is =099; the inverse of the Frisch elasticity 7Note that we do not consider infra-marginal shifts in fiscal policy that might endogenously change the duration of the liquidity trap (see Erceg and Linde, 2014). 22
can write [1 − 2]=(Ξ)=1−2Ξ {Ξ++(Ξ)} =+(Ξ)−Ξ {Ξ++(Ξ)} So that 0(Ξ)=(0(Ξ)−1) {Ξ++(Ξ)}−(0(Ξ)+1){+(Ξ)−Ξ} {Ξ++(Ξ)}2 =(0(Ξ)) [{Ξ++(Ξ)}−{+(Ξ)−Ξ}]+[{Ξ−−(Ξ)}−{Ξ++(Ξ)}] {Ξ++(Ξ)}2 =(0(Ξ)2Ξ)−2{+(Ξ)} {Ξ++(Ξ)}2=2 (Ξ) (Ξ) (0(Ξ)−{+(Ξ)} {Ξ++(Ξ)}2 =2 ((Ξ)0(Ξ)Ξ−(Ξ){+(Ξ)} (Ξ){Ξ++(Ξ)}2=2 [(Ξ)0(Ξ)Ξ−(Ξ)2]−(Ξ) (Ξ){Ξ++(Ξ)}2 Note (Ξ)0(Ξ)Ξ=Ξ2+{1+}Ξ,and(Ξ)2=[Ξ2+2{1+}Ξ]+(1−)2]which implies (Ξ)0(Ξ)Ξ−(Ξ)2=−{1+}Ξ−(1 −)2 0(Ξ)=2 −{1+}Ξ−(1 −)2−(Ξ) (Ξ){Ξ++(Ξ)}20 So is declining in Ξ= 2z= 2 + = 2 + , and thus declining in .Sinceis declining in when 1≤2,then so will be larger when home bias is larger. Proof for Proposition 3 Proof. We can write ()= (−1) −· 2· (−1).Forany, () (∞)=h1−1 1− 2· +i.Wecanwrite 1−=2 2−[Ξ+{1+}]−q[Ξ+{1+}]2−4 2 =µq[Ξ+{1−}]2+4Ξ−[Ξ+{1−}]¶1 2 29
and 2=Ξ ½Ξ 2+1− 2+(1 −)+√[Ξ+{1+}]2−4 2¾ =2Ξ ½[Ξ+{1−}]+2(1 −)+q[Ξ+{1−}]2+4Ξ¾ Define =[Ξ+{1−}]=p2+4Ξ=2(1 −) 1 1−=2 − 2=2Ξ ++ 1 1− 2=4Ξ (−)(++)=4Ξ (−)(+)+(−) =4Ξ (2−2)+(−)=4Ξ 4Ξ+(−)=4Ξ 4Ξ+2(1−) (−) =4Ξ 4Ξ+4(1−)(1 −)=Ξ Ξ+(1−)(1 −)1 So 1 1− 2[1 −z]1 Proof for Proposition 4 Proof. The zero bound multiplier exceeds one for ≤and is zero for . The differentials multiplier ()1for ≤and is zero for .So () () for all Proof for Proposition 5 Proof. If monetary policy implements zero inflaitonineveryperiod(i.e. →∞), the aggregage multiplier =z. The long run multiplier is (∞)=(1−1 1− 2· +)= £1−1 1− 2·(1 −z)¤=1 1− 2·+(1−1 1− 2). Calculate1− 2 1 1− 2=Ξ Ξ+(1−)(1 −) 1−1 1− 2=(1 −)(1 −) Ξ+(1−)(1 −) and (∞)= 1 1− 2z+(1−1 1− 2) (∞)= 1 1− 2z+(1−1 1− 2)=Ξz+(1−)(1 −) Ξ+(1−)(1 −) 30
Rewrite =z =z=Ξz+z(1 −)(1 −) Ξ+(1−)(1 −) Calculate −(∞)=Ξ(z−z)+(z−1)(1 −)(1 −) Ξ+(1−)(1 −) At =2=1and z=zSince q[Ξ+{1+}]2−4Ξand (z−1) 0,if=2 then −(∞)0. By continuity, there exists 2where −(∞)0. Since(∞) ()for any finite then −()0. Proof for Proposition 6 Proof. The impact effect is a function of Φ ³←→ Ξ´ (1 + Φ)=h1− ←→ 2i (1 + Φ) where ()is defined as in the proof of Proposition 2,0()0The impact multiplier is a function of Φwhosederivativeiswrittenas 0³←→ Ξ´z−1 (1+Φ)2Ξ(1 + Φ)−³←→ Ξ´ (1 + Φ)2=−0³←→ Ξ´←→ Ξ1−z (1+zΦ)−³←→ Ξ´ (1 + Φ)2−0³←→ Ξ´←→ Ξ−³←→ Ξ´ (1 + Φ)2 where the last inequality follows since −0³←→ Ξ´←→ Ξ−0³←→ Ξ´←→ Ξ1−z (1+zΦ)0.Fromthe proofofProposition2 [1 − 2]=(Ξ)= +(Ξ)−Ξ {Ξ++(Ξ)}=(Ξ)2−Ξ2+2+2(Ξ) {Ξ++(Ξ)}2 where ≡1−+2(1 −)and (Ξ)≡q[Ξ+{1−}]2+4Ξso (Ξ)2=[Ξ+{1−}]2+4Ξ=Ξ2+[2{1−}+4]Ξ+{1−}2 (Ξ)2−Ξ2+2+2(Ξ)=[2{1−}+4]Ξ+£{1−}2+2¤+2(Ξ)= [2+4]Ξ+£{1−}2+2¤+2(Ξ) From the proof of Proposition2,0(Ξ)=2 [0(Ξ)Ξ−(Ξ)2]− {Ξ++(Ξ)}2 −0(Ξ)Ξ=2Ξ+2(Ξ)Ξ−20(Ξ)Ξ2 {Ξ++(Ξ)}2 31
So we can write −0(Ξ)Ξ−(Ξ)=2Ξ+2(Ξ)Ξ−20(Ξ)Ξ2 {Ξ++(Ξ)}2− [2+4]Ξ+[{1−}2+2]+2(Ξ)2 {Ξ++(Ξ)}2 Calculating =2(Ξ)(Ξ−)−20(Ξ)Ξ2−4Ξ−[{1−}2+2] {Ξ++(Ξ)}2 2(Ξ)(Ξ−)−20(Ξ)Ξ2−4Ξ {Ξ++(Ξ)}2 =2(Ξ)2(Ξ−)−2(Ξ)0(Ξ)Ξ2−4Ξ(Ξ) (Ξ){Ξ++(Ξ)}2 Note (Ξ)0(Ξ)Ξ=Ξ2+{1+}Ξ,and(Ξ)2=[Ξ2+2{1+}Ξ]+(1−)2]which implies (Ξ)2(Ξ−)−(Ξ)0(Ξ)Ξ2 =£Ξ3+2{1+}Ξ2¤+(1−)2Ξ]−Ξ3−{1+}Ξ2 −£Ξ2+2{1+}Ξ¤+(1−)2] ={1+}Ξ2+(1−)2Ξ−£Ξ2+2{1+}Ξ¤+(1−)2] =[{1+}−]Ξ2+£(1 −)2−2{1+}¤Ξ−(1 −)2 We know [{1+}−]=[{1+}−1−−2(1 −)] = 2 and (1 −)2−2{1+}= (1 −)[(1 −)−2(1 + )−2(1 −)] = −(1 −)[(1 + 3+2(1 −)] 0,so 2(Ξ)2(Ξ−)−2(Ξ)0(Ξ)Ξ2−4Ξ(Ξ) (Ξ){Ξ++(Ξ)}2 =4[Ξ2−Ξ(Ξ)] −(1 −)[(1 + 3+2(1 −)]Ξ−(1 −)2 (Ξ){Ξ++(Ξ)}2 −0(Ξ)Ξ−(Ξ) Since(Ξ)Ξ,then[Ξ2−Ξ(Ξ)] 0so all of the elements are less than zero. Thus, the impact effect of demand shocks is a declining function of ΦwhenΦ0. Proof for Proposition 7 32
Proof. The limit can be written as: {+}e =−{+}e −{·−(1 −)} Write the parameter Proof. 1−=(1−) (2+)=() (+) (2+) Solving for the coefficient on inflation: (1 −)=() (+) (2+)= (+) (2+)() = (2+)() {·−(1 −)}=(2+) (2+)()− (2+)()= 2 (2+)() 7.2 Parameter Derivations Here we define the parameters used in the loss function, which is taken from that used in Cook and Devereux (2011). ≡½(1 + ) 2 +(−) (1 + (1 −2 ) 2 )¾=( +) ≡(−1)(1 −2 ) +() (1+2 (−1) ) ≡(+) 2 =(+) ≡1 (1 −) 2 =1 (1 −) ≡− 2 =− ≡∙−1 2 −(−) 2 2(1 + (−1)(−1)2 )¸ ≡((1 −)+) (1 −)2 +(−) 2 2(1 + (−1)(−1)2 ) Given this, cooperative optimal policy maximizes the objective function 33
"∞ X =0 +# =+ £ −(+)e + ·e − +1¤ + £ −(+)e +e − +1¤ + £(e +1 −e )−(e +1 −e )−¡−e − +1¢¤ + ∙ +2µ£e −e ¤− (1 −)¶−2µ£e −1−e −1¤− −1 (1 −)¶¸ + =−(e )2· 2−(e )2 2−(e)2·+ 2+(e )( e)(31) −(e )( e)− 2( )2− 2( )2 {+}e−(e )+e = + + −2[ − +1)(32) −e −e=(+) + = = −e +(e )= (+)−2[ − +1)(33) + = (34) References [27] Adam, Klaus, and Roberto M. Billi (2006). "Optimal Monetary Policy under Commitment with a Zero Bound on Nominal Interest Rates." Journal of Money, Credit and Banking,38(7), 1877-1905. [27] Adam, Klaus, and Roberto M. Billi (2007). "Discretionary monetary policy and the zero lower bound on nominal interest rates." Journal of Monetary Economics, 54(3), 728 -752. 34
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Figure1showstheresponseoftheaggregateeconomytoashocktodemandshockconcentratedinthe Southregionthatlastsfor5periods.Theshockisconstructedtobringtheaggregatenaturalinterest rateto‐8percentonanannualizedrate.Thefigureshowstheresponsewhen:Optimalbothregions implementoptimalcooperativefiscalpolicy;Countercyclicaleachregionsetsthefiscalpolicygapasa constantnegativeratiooftheoutputgap;andZeroGapwhenthefiscalgapineachregionissetto zero.Panel(B)showsthenominalinterestratewhichgoestothezerolowerboundduringtheperiods oftheshock.