International dimensions of optimal monetary policy∗ Giancarlo Corsetti University of Rome III, Yale University and CEPR Paolo Pesenti Federal Reserve Bank of New York and NBER First draft: December 2000 Revised: March 2001 ∗ We thank Pierpaolo Benigno, Caroline Betts, Luca Dedola, Gianluca Fem- minis, Marc Giannoni, C´dric Tille, and seminar participants at the 2001 ASSA e meeting, the Bank of Italy and the Universities of Milan, Tilburg, Warwick and Yale for comments. A previous version of this paper was circulated under the title “Optimal interest rate policy rules and exchange rate pass-through”. Corsetti’s work on this paper is part of a research network on ‘The Analysis of International Capital Markets: Understanding Europe’s Role in the Global Economy’, funded by the European Commission under the Research Training Network Programme (Contract No. HPRN-CT-1999-00067). The views expressed here are those of the authors, and do not necessarily reflect the position of the Federal Reserve Bank of New York, the Federal Reserve System, or any other institution with which the authors are affiliated. Abstract This paper provides a baseline general-equilibrium model of optimal mon- etary policy among interdependent economies with monopolistic firms that set prices one period in advance. Strict adherence to inward-looking policy objectives such as the stabilization of domestic output cannot be optimal when firms’ markups are exposed to currency fluctuations. Such policies in- duce excessive volatility in exchange rates and foreign sales revenue, leading exporters to set higher prices in response to higher profit risk. In general, optimal rules trade off a larger domestic output gap against lower import prices. Monetary rules in a world Nash equilibrium lead to less exchange rate volatility relative to both inward-looking rules and discretionary poli- cies, even when the latter do not suffer from any inflationary (or deflationary) bias. Gains from international monetary cooperation are related in a non- monotonic way to the degree of exchange rate pass-through. JEL classification: E31, E52, F42 Keywords: optimal cyclical monetary policy, nominal rigidities, exchange rate pass-through, international cooperation. 1 Introduction What are the basic principles underlying the design of optimal monetary policies among open and interdependent economies? From the recent policy and academic debate, it is far from obvious that monetary policy should have any ‘international’ dimension at all. In fact, there is little or no consensus on such questions as of whether domestic monetary policy should react to exchange rate fluctuations and overseas macroeconomic developments; what role is played by external factors in the choice between different rules, and between rules and discretion in policy-making; what (if anything) can be achieved through international cooperation and ‘global compacts’ with trad- ing partners. This paper presents a baseline general-equilibrium framework to assess these issues and provide some tentative answers. The paper shares the unify- ing research agenda of most recent contributions on optimal monetary policy, and adopts a common methodology based upon choice-theoretic stochastic models with nominal rigidities, imperfect competition in production, and forward-looking price-setting.1 Our main results can be summarized as fol- lows. First, inward-looking monetary policies that do not react to world cyclical conditions cannot be optimal for an open economy in which firms’ profits are exposed to exchange rate fluctuations. Recent literature has stressed that, in a closed economy with nominal rigidities, it is desirable to pursue a policy that closes the output gap completely.2 In fact, if one can disregard real rigidities and ‘cost-push’ inflation, such policy can replicate the allocation that would prevail if prices where fully flexible. In an open economy, however, a policy attempting to close the domestic output gap could induce excessive volatility in the exchange rate, affecting foreign exporters’ profits. Risk- averse foreign producers would then hedge against the risk of losses in their 1 The literature on monetary rules in open economy has boomed in recent years. A far from complete list of theoretical contributions includes Ball [1999], Benigno and Be- nigno [2000], Benigno, Benigno and Ghironi [2000], Carlstrom and Fuerst [1999], Clarida, Gali and Gertler [2001], Devereux and Engle [2000], Gali and Monacelli [1999], Ghironi and Rebucci [2000], Leitmo and S¨derstr¨m [2000], McCallum and Nelson [1999], Mona- o o celli [1999], Obstfeld and Rogoff [2000b], Parrado and Velasco [2000], Sutherland [2000], Svensson [2000], and Walsh [1999]. 2 See for instance Gali [2000], Goodfriend and King [2000] and Woodford [2000]. 1 export market by charging higher prices in domestic currency, thus reducing the purchasing power of domestic consumers. Only when markups are independent of exchange rate movements the optimal domestic policy does not react to policy shocks and cyclical con- ditions in the world economy. This may be the case, for instance, when goods are produced with domestic inputs only and are priced in the produc- ers’ currency, or when exporters are fully insured against currency volatility. Otherwise, relative to an inward-looking policy conduct, the domestic policy maker can improve welfare by trading off, at the margin, a larger domestic output gap against lower import prices. To act in the best interest of do- mestic consumers and reduce the overall magnitude of distortions affecting national welfare, policy makers must stabilize world producers’ profits in the domestic market. Second, commitment is superior to discretion even when discretionary policies do not suffer from any inflationary (or deflationary) bias. This is because, given pre-set prices by firms, a discretionary policy maker has an incentive, ex post, to ‘over-stabilize’ the domestic economy and use mone- tary policy as to tilt terms of trade in favor of domestic agents. But any systematic attempt to follow this conduct is doomed to lower domestic wel- fare on average, as foreign exporters react by pre-setting higher prices in the domestic market. Third, relative to the case of suboptimal inward-looking policies, imple- menting optimal monetary policies reduces exchange rate volatility. With optimal policies in place, the lower the degree of exchange rate pass-through and profits exposure, the lower the equilibrium volatility in the foreign ex- change market, both in nominal and real terms. There is, however, a case for exchange rate flexibility. Fixed exchange rates can be supported by optimal monetary policies only when all shocks are correlated worldwide or when local prices are fully inelastic to exchange rate fluctuations. Otherwise, the relative price adjustment associated with the implementation of the optimal policy requires exchange rate flexibility. Fourth, the magnitude of gains from cooperation are related in a non- linear way to the degree of pass-through. In our model there are no welfare gains from entering international binding agreements either when there are no deviations from the law of one price or when local prices are completely insulated from exchange rate fluctuations. That is, under the extreme as- sumptions of complete and zero pass-through, monetary policies are strate- 2 gically independent and there are no policy spillovers in equilibrium. Yet, gains from cooperation materialize in economies with intermediate levels of pass-through. Building upon our previous contribution (Corsetti and Pesenti [2001]), we setup a model that can be solved in closed form, without resorting to (loglinear) approximations. Different from our previous work, as well as from many other contributions in the ‘new open-economy’ macro literature, all our welfare results are derived without specifying a particular distribution of the stochastic disturbances underlying the economy. Also, our results are derived for intermediate degrees of exchange rate pass-through. Our analysis thus complements and generalizes recent models that explore the implications of two polar cases of nominal rigidities. In the first case (Producer Currency Pricing, or PCP) firms set prices in the cur- rency of the country where they are located. The domestic price of imports then moves one to one with the exchange rate, pass-through is 100% and profits are completely insulated from exchange rate movements. In the sec- ond case (Local Currency Pricing, or LCP) firms set the price of their goods in the currency of the market where they sell their products. The domestic price of imports does not change with the exchange rate and pass-through is zero, corresponding to a high degree of profits’ exposure to exchange rate movements.3 Both PCP and LCP assumptions are extreme in light of the empirical evidence. Pass-through to export and import prices is generally below 1 but seldom 0, and a large body of empirical studies have documented that ex- change rate pass-through is rather low at consumer price level but it is far from nul at producer price level.4 Our modelling strategy takes imperfect 3 For instance, Benigno and Benigno [2000], Corsetti and Pesenti [2001], and Obstfeld and Rogoff [1995, 2000a,b] focus on the PCP case. Bacchetta and Van Wincoop [2000], Chari, Kehoe and McGrattan [2000], and Duarte and Stockman [2001] focus on the LCP case. Corsetti, Pesenti, Roubini and Tille [2000], Devereux and Engel [2000], and Tille [2001] compare PCP and LCP allocations. Devereux, Engel and Tille [1999] consider the interdependence between a country with zero pass-through and a country with complete pass-through. A few exceptions include Betts and Devereux [2000a], who study the effect of different degrees of pass-through on exchange rate dynamics in a non-stochastic envi- ronment, and Monacelli [1999], who simulates numerically the effects of pass-through in a calibration exercise. 4 See the discussion in Goldberg and Knetter [1997] and Obstfeld and Rogoff [2000a]. Campa and Goldberg [2000] provide updated estimates of exchange rate pass-through 3 pass-through as a proxy for transaction and distribution costs affecting firms profitability in export markets,5 as to analyze the implications of deviating from the extreme assumptions of zero or full pass-through on the interna- tional transmission mechanism and the design of optimal monetary policy in open economy.6 The paper is organized as follows. Section 2 introduces the model. Sec- tion 3 summarizes the mechanism of international transmission of real and monetary shocks implied by the model. Section 4 analyzes optimal policy. We first establish a general principle of policy design and discuss it by de- riving three equivalent representations of the policy problem. Second, we characterize the optimal monetary rules and discuss their efficiency under different degrees of pass-through. Third, we revisit the ‘rules-vs.-discretion’ debate for an open economy. In Section 5 we assess the case for cooperation and discuss the choice of exchange rate regimes. Section 6 concludes. 2 The model 2.1 Preferences and technology We model a world economy with two countries, each specialized in one type of goods. Countries and types of goods are denoted H (Home) and F (Foreign). Within a country, the national type of goods is produced in a number of varieties or brands defined over a continuum of unit mass. Brands are indexed by h ∈ [0, 1] in the Home country and f ∈ [0, 1] in the Foreign country. All brands are traded worldwide. Each country is populated by households defined over a continuum of unit mass. Households are indexed by j ∈ [0, 1] in the Home country and j ∗ ∈ [0, 1] in the Foreign country. Households consume all brands of both types of goods. Ct (h, j) is con- sumption of Home brand h by Home agent j at time t; Ct (f, j) is consumption of Foreign brand f by Home agent j at time t. Similarly we define consump- tion of Foreign agent j ∗ as Ct∗ (h, j ∗ ) and Ct∗ (f, j ∗ ). Each Home brand is an across countries. 5 Obstfeld and Rogoff [2001] emphasize the role of international trade costs in open economy macroeconomic analysis. 6 A recent discussion of the policy implications of firms’ pass-through is provided by Taylor [2000]. 4 imperfect substitute to all other Home brands, with constant elasticity of substitution θ > 1. Similarly the elasticity of substitution among Foreign brands is θ∗ > 1. For each agent j in the Home country, the consumption indexes of Home and Foreign goods are defined as: θ θ∗ 1 θ−1 1 θ ∗ −1 θ−1 θ ∗ −1 CH,t (j) ≡ Ct (h, j) θ dh , CF,t (j) ≡ Ct (f, j) θ∗ df 0 0 (1) Similarly we define the consumption indexes of agent j ∗ in the Foreign coun- try, CH,t (j ∗ ) and CF,t (j ∗ ). ∗ ∗ Consistent with the idea that each country specializes in the production of a single type of good, the elasticity of substitution among goods produced in one country should not be lower than the elasticity of substitution across goods produced in different countries. Specifically, while both θ and θ∗ are greater than one, we assume that the elasticity of substitution between Home and Foreign types is one. Under this assumption the consumption baskets of individuals j and j ∗ can be written as a geometric average of the Home and Foreign consumption indexes: Ct (j) ≡ CH,t (j)γ CF,t (j)1−γ , Ct∗ (j ∗ ) ≡ CH,t (j ∗ )γ CF,t (j ∗ )1−γ ∗ ∗ 0<γ<1 (2) where the weights γ and 1 − γ are identical across countries. We denote pt (h) and pt (f ) the prices of brands h and f in the Home market (thus expressed in the Home currency), and p∗ (h) and p∗ (f ) the t t prices of brands h and f in the Foreign market. The utility-based price of a consumption bundle of domestically produced goods, denoted PH,t , is derived as:7 1 1 1−θ 1−θ PH,t = pt (h) dh . (3) 0 ∗ ∗ Similarly we derive the other price indexes PH,t , PF,t and PF,t , as well as the utility-based CPIs: γ 1−γ ∗ γ ∗ 1−γ PH,t PF,t PH,t PF,t Pt = , Pt∗ = (4) γW γW 7 The utility based price index PH is defined as the minimum expenditure required to buy one unit of the composite good CH , given the prices of the brands. See Appendix. 5 where γW ≡ γ γ (1 − γ)1−γ . Each brand h is produced and sold in both countries by a single Home household (that is, a ‘yeoman farmer’ as in Obstfeld and Rogoff [1996]), using labor as the only input in production.8 Technology is linear in household’s h labor, so that the resource constraint for brand h is: 1 1 t (h) ≥ Ct (h, j)dj + Ct∗ (h, j ∗ )dj ∗ (5) αt 0 0 where αt is a country-specific productivity shock. Similarly the resource constraint for brand f is: ∗ 1 1 (f ) t ∗ ≥ Ct (f, j)dj + Ct∗ (f, j ∗ )dj ∗ (6) αt 0 0 where ∗ (f ) is labor of Foreign household f and αt is a productivity shock ∗ in the Foreign country. Home agent j’s lifetime expected utility U is defined as: ∞ Mτ (j) Ut (j) ≡ Et β τ −t U Cτ (j) , , τ (j), Ωτ (7) τ =t Pτ where β < 1 is the discount rate and the instantaneous utility U is a positive function of the consumption index C(j) and real balances M (j) /P and a negative function of labor effort (j). Utility can be state-dependent, as indexed by the vector of random variables Ω. To keep algebraic complexities at a bare minimum, without sacrificing the substance of the argument, our baseline model adopts the following additively-separable parameterization: Mt (j) Mt (j) U Ct (j) , , t (j) = ln Ct (j) + χt ln − κ t (j) (8) Pt Pt In our framework money demand shocks can be introduced in the form of a stochastically varying elasticity of real balances, so that Ωt = {χt }.9 In what 8 Obstfeld [2000] considers an extension of the model to encompass trade in intermediate inputs. 9 Obstfeld and Rogoff [2000a] include κt in the set Ωt , intepreting the Home elasticity of labor disutility as a productivity shock. Their parameterization is isomorphic to our specification with the Home productivity shock indexed by αt . 6 follows, we will refer to Wt as the non-monetary component of Ut , or Wt ≡ Ut −Et ∞ β τ −t χτ ln Mτ (j) /Pτ . τ =t Foreign agents’ preferences are similarly defined as: ∞ Mτ (j ∗ ) Ut∗ (j ∗ ) ≡ Et β τ −t ln Cτ (j ∗ ) + χ∗ ln ∗ τ − κ∗ ∗ (j ∗ ) τ (9) τ =t Pt∗ where the discount rate β is the same as in the Home country. Preferences and velocity shocks can be asymmetric across countries. For instance, in the baseline model χ∗ and κ∗ in the Foreign country need not coincide with χt t and κ in the Home country. 2.2 Consumer optimization Given the specification of the consumption baskets, agent j’s demand for brand h is a function of the relative price of h and total consumption of Home goods: −θ pt (h) Ct (h, j) = CH,t (j) (10) PH,t Similar expressions can be derived for the other brands. Accounting for the demand functions above, we can rewrite the resource constraint for agents h in the Home country as: (h) θ θ t ∗ ≥ pt (h)−θ PH,t CH,t + p∗ (h)−θ PH,t ∗ CH,t (11) αt Home agents hold Home currency M , and two international bonds, BH and BF , respectively denominated in Home and Foreign currency. They receive a revenue R(j) from the sale of the good they produce — since each agent is the sole producer of a specific brand, we associate individual j with brand h. They pay non-distortionary (lump-sum) net taxes T , denominated in Home currency. Using the above specified consumption and price indexes, 7 the individual flow budget constraint10 for agent j in the Home country is: Mt (j) + BH,t+1 (j) + Et BF,t+1 (j) ≤ Mt−1 (j) + (1 + it )BH,t (j) + +(1 + i∗ )Et BF,t (j) + Rt (j) − Tt (j) − PH,t CH,t (j) − PF,t CF,t (j) (12) t In the expression above the nominal yields it and i∗ are paid at the beginning t of period t and are known at time t − 1; E is the nominal exchange rate, expressed as Home currency per unit of Foreign currency. The sales revenue of agent j, expressed in Home currency, is: 1 1 Rt (j) ≡ pt (h) Ct (h, j)dj + Et p∗ (h) t Ct (h, j ∗ )dj ∗ 0 0 ∗ θ ∗ = pt (h) 1−θ θ PH,t CH,t + Et p∗ (h)1−θ t PH,t CH,t (13) Similar expressions hold for the Foreign country, after associating household j ∗ with brand f . Taking prices as given, Home agent j maximizes (7) subject to (12), accounting for (11) and (13).11 A similar optimization problem is solved by Foreign agent j ∗ . In Section 3 we will focus on symmetric equilibria in which agents are equal within countries (though not necessarily symmetric across countries), dropping the indexes j and j ∗ and interpreting all variables in per-capita (or aggregate) terms. 2.3 Producer optimization and price setting Our baseline model allows for nominal rigidities, but abstracts from the com- plexities associated with the dynamics of price adjustment.12 To enhance analytical tractability, it is assumed that agents set the nominal price of their product one period in advance, and stand ready to meet (domestic and 10 We adopt the notation of Obstfeld and Rogoff [1996, ch.10]. Specifically, our timing convention has Mt (j) as agent j’s nominal balances accumulated during perod t and carried over into period t+1, while BH,t (j) and BF,t (j) denote agent j’s bonds accumulated during period t − 1 and carried over into period t. 11 The optimization problem is fully characterized in the Appendix. 12 The implications of Calvo-style adjustment of prices are explored in many closed or small open-economy models mentioned in the references. Benigno and Benigno [2000] consider the extension to a general equilibrium framework under full exchange rate pass- through. 8 foreign) demand at given prices for one period. The number of producers is large enough so that firms ignore the impact of their pricing decision on the aggregate price indexes. In terms of our notation, Home firms selling in the Home market choose pt (h) at time t − 1. As shown in the appendix, in a symmetric environment where all prices pt (h) are equal and pt (h) = PH,t , Home firms optimally set domestic prices such that: 1 pt (h) = PH,t = Et−1 [αt Pt Ct ] (14) Φ where we define Φ ≡ (θ − 1) /θκ. Interpreting (14), domestic firms fix prices equal to expected nominal marginal cost (that is, Et−1 [καt Pt Ct ]), augmented by the equilibrium markup θ/ (θ − 1). To account for the possibility of deviations from the law of one price and the impact of exchange rate fluctuations on firms’ markups, it is (realistically) assumed that pass-through to export prices is less than perfect. The simplest way to model imperfect pass-through is to consider the following pricing function: ∗ p∗ (h) ≡ pt (h)Et−η t ~ 0 ≤ η∗ ≤ 1 (15) The variable pt (h) is the predetermined component of the Foreign currency ~ price of good h that cannot be adjusted to variations of the exchange rate during period t. Home firms choose pt (h) one period in advance at time ~ ∗ t − 1, while the actual pt (h) depends on the realization of the exchange rate at time t. The parameter η ∗ indexes the degree of pass-through in the Foreign markets. For instance, if η ∗ = 1, pass-through in the Foreign country is complete — as in the PCP case. If η ∗ = 0, we have p∗ (h) = pt (h) which t ~ coincides with the price chosen by the Home producer in the LCP case. In a symmetric environment, optimal pricing also yields: ∗ ∗ ∗ η −η ∗ 1 1 Et−1 αt Pt Ct Et t ∗ p∗ (h) = PH,t = pt (h)Et ~ = ∗ (16) Φ Etη Et Pt∗ Ct∗ Et−1 Pt Ct Interpreting (16), domestic firms fix pt (h) so that at the margin the expected ~ disutility from an increase in labor effort is equal to the expected utility from consumption financed by additional sales revenue. 9 Analogous expressions can be derived for the pricing by Foreign firms in the Foreign and the Home market. Define the function: pt (f ) = p∗ (f )Etη , ~t 0 ≤ η ≤ 1. (17) As before, η = 1 corresponds to PCP, η = 0 to LCP. The degree of pass- through in the Home country, η, need not be equal to that in the Foreign country, η ∗ . The optimal pricing strategy is such that: ∗ 1 Et−1 αt Pt Ct ∗ 1 Etη Etη PF,t = E [αt Pt∗ Ct∗ ] , ∗ t−1 ∗ PF,t = (18) Φ Φ∗ Pt Ct Et−1 Et Pt∗ Ct∗ where Φ∗ ≡ (θ∗ − 1) /θ∗ κ∗ . In what follows, it will be useful to compare equilibrium prices with and without nominal rigidities. We denote variables under a flex-price equilib- rium with a f lex superscript. If prices were fully flexible, firms would use their monopoly power charging a fixed percentage over marginal costs. For instance, in the Home market we would have: f lex 1 PH,t = 1+ καt Pt Ct (19) θ−1 In a fix-price equilibrium instead, for given PH,t , the profit margin at time t is not constant, endogenously depending on how macroeconomic and policy shocks affect the marginal cost καt Pt Ct . Let Θ denote the ratio between the price and marginal cost of a good, that is the profit margin or markup. Clearly, firms are willing to supply goods at given prices as long as their ex-post percentage markup is non-negative: ΘH,t ≥ 1 ⇔ PH,t ≥ καt Pt Ct (20) Otherwise, agents would be better off by not accommodating shocks to de- mand. In what follows, we restrict the set of shocks so that the ‘participation constraint’ (20) and its analogs are never violated. 10 2.4 Government budget constraint and asset markets The government budget constraint in the Home country is: 1 1 [Mt (j) − Mt−1 (j)] dj + Tt (j)dj = 0 (21) 0 0 where Mt (j) are the money demand functions as determined above.13 Only Home residents hold Home money. The government affects the stock of Home liquidity by controlling the short-term rate it+1 . Similar considerations hold for the Foreign country, where the government controls the interest rate i∗ . t+1 The model is closed by posing that international bonds are in zero net- supply: 1 1 1 1 BH,t (j)dj + BH,t (j ∗ )dj ∗ = 0, ∗ BF,t (j)dj + BF,t (j ∗ )dj ∗ = 0 ∗ 0 0 0 0 (22) ∗ ∗ so that, in a symmetric environment, BH,t = −BH,t and BF,t = −BF,t . 3 The international transmission of real and monetary shocks 3.1 The solution of the model We solve the model under the assumption that, at some initial time t0 , net non-monetary wealth is zero in each country, that is, BH,t0 = BF,t0 = 0. Then, we can study an equilibrium in which non-monetary wealth is zero at any subsequent point in time: Home imports from Foreign are always equal in value to Foreign imports from Home. As trade and the current account are invariably balanced, countries consume precisely their sales revenue: Rt − Pt Ct = 0, Rt − Pt∗ Ct∗ = 0 ∗ (23) 13 The model could easily be extended to encompass government spending and public debt. Note that fiscal shocks, modeled as random fluctuations in government spending, are isomorphic to productivity shocks in our framework under the assumptions that each national government spends exclusively on domestically produced goods and utility is additively separable in private and public consumption. For a detailed treatment of fiscal interdependence and the interaction between monetary and fiscal policies see Corsetti and Pesenti [2001]. 11 This stems from the combination of three hypotheses: (i) Cobb-Douglas con- sumption indexes; (ii) logarithmic consumption preferences; (iii) zero initial non-monetary wealth. As shown in Corsetti and Pesenti [2001], under PCP the result holds also when assumption (ii) is relaxed. Before we derive the general solution of the model, it is analytically helpful to introduce an index of monetary stance µt such that: 1 1 = β(1 + it+1 )Et (24) µt µt+1 The above expression shows that, given the time path of µ (and 1/µ), there is a corresponding sequence of Home nominal interest rates: Home monetary easing at time t is associated with both a higher µt and a lower it+1 . A similar expression holds for the Foreign country.14 Table 1 presents the general solution of the model whereas all endoge- nous variables (25) through (35) are expressed in closed form as functions of real and monetary shocks (αt , αt , χt and χ∗ )15 and the Home and Foreign ∗ t ∗ monetary stances µt and µt . Interpreting Table 1: since the equilibrium current account is always bal- anced, the nominal exchange rate Et in (25) is proportional to Pt Ct /Pt∗ Ct∗ , that is, a function of the relative monetary stance. Domestic prices of do- mestic goods are predetermined according to (26) and (29), while import prices vary with the exchange rate, depending on the degree of exchange rate pass-through according to (27) and (28). Given interest rates, money is determined residually according to (30) and (31). Equilibrium consumption is determined in (32) and (33), where ΦW is defined as ΦW ≡ Φγ (Φ∗ )1−γ . Finally employment levels are determined according to (34) and (35). Note that it is always the case that: ∗ Et−1 t = Φ, Et−1 t = Φ∗ . (36) Under rational expectations, for any given distribution of the shocks and policy rule, agents optimally predetermine prices so as to stabilize expected 14 In equilibrium µt is equal to the reciprocal of the marginal utility of domestic con- sumers’ nominal wealth, that is the Lagrangean multiplier λt in the Appendix. With logarithmic utility, µt is also equal to nominal spending Pt Ct . 15 Recall that the analysis is restricted to shocks whose size does not lower the ex-post markup below zero. 12 Table 1: The solution of the fix-price model 1 − γ µt Et = (25) γ µ∗ t 1 PH,t = Et−1 (αt µt ) (26) Φ η 1 µt 1−γ PF,t = ∗ ∗ Et−1 αt (µ∗ )η µ1−η ∗ t t (27) Φ µt γ η∗ 1 µ∗ ∗ ∗ γ ∗ PH,t = t Et−1 αt µη (µ∗ )1−η t t (28) Φ µt 1−γ 1 PF,t = ∗ Et−1 (αt µ∗ ) ∗ ∗ t (29) Φ χt µt Mt = (30) 1 − βµt Et µ−1 t+1 χ∗ µ∗ t t Mt∗ = (31) 1 − βµ∗ Et µ∗−1 t t+1 1−η(1−γ) γΦW µt (µ∗ )η(1−γ) t Ct = 1−γ (32) [Et−1 (αt µt )]γ Et−1 αt (µ∗ )η µ1−η ∗ t t ∗ ∗γ (1 − γ) ΦW (µ∗ )1−η γ µη t t Ct∗ = ∗ ∗ γ (33) [Et−1 (αt µ∗ )]1−γ Et−1 αt µη (µ∗ )1−η ∗ t t t ∗ ∗ αt µt αt µη (µ∗ )1−η t t t =Φ γ + (1 − γ) η∗ ∗ (34) Et−1 (αt µt ) Et−1 αt µt (µ∗ )1−η t ∗ ∗ αt µ∗ ∗ t αt (µ∗ )η µ1−η ∗ t t t =Φ (1 − γ) ∗ ∗ +γ ∗ ∗ η 1−η (35) Et−1 (αt µt ) Et−1 αt (µt ) µt 13 employment at the constant level Φ or Φ∗ , that we will soon show to be the flex-price equilibrium employment. 3.2 The transmission mechanism The essence of the international transmission mechanism in our model is captured by three features of the equilibrium allocation in Table 1. First, for given monetary stances µ and µ∗ , productivity shocks affect cur- rent employment but neither output nor consumption. The negative response of employment to productivity shocks is a common finding in the so-called New Keynesian or New Neoclassical synthesis model.16 Intuitively, nominal aggregate demand is controlled by the government through monetary policy, while the CPI is a function of predetermined prices and the exchange rate, in itself a function of relative monetary stances. In the presence of price rigidities, for given µ and µ∗ , an unanticipated shock to the relative supply of the Home goods cannot be matched by a fall of their relative price: all prices (26), (28), (27), and (29), are in fact invariant to the realizations of the productivity shocks. As a consequence, with a given world demand nailing down the level of output in each country, an increase in productivity simply lowers employment (34) without affecting the supply of goods available for consumption. Second, current consumption and employment in each country are non- decreasing functions of monetary policy shocks abroad. The transmission of monetary shocks works through both world aggregate demand — which always increases after a policy shock eases the global monetary stance — and movements of the terms of trade. A high degree of pass-through (η ∗ = η 1) implies that a depreciation of the Home currency is associated with a wors- ening of the Home terms of trade. The relative price of Home goods falls in each market, so that more units of Home labor are required to buy one unit of Home consumption. Correspondingly, the Foreign terms of trade improve.17 A low degree of pass-through (η ∗ = η 0) , instead, leads to the opposite result. Local prices are invariant to the exchange rate, but for each unit of goods sold abroad, Home firms’ revenue in domestic currency rises 16 Gali [1999] provides empirical evidence in support of such a view of the business cycle effects of supply shocks. 17 On the ‘beggar-thyself’ vs. ‘beggar-thy-neighbor’ effects of exchange rate depreciations see Corsetti and Pesenti [2001] and Tille [2000]. 14 in proportion to the rate of depreciation. This additional revenue finances higher consumption, thus leading to higher production and imports. For- eign exporters now have to supply more goods while suffering a fall in the domestic-currency value of their sales revenue. Third, observe that the level of the Home good price, (27), depends on Et−1 (αt µt ) and is therefore an increasing function of the covariance between α and µ. This implies that the systematic tightening of monetary policy (lower µ) in response to shocks affecting Home producers’ marginal costs (higher α) is associated, in equilibrium, with lower domestic prices PH (and higher expected utility). Similar considerations hold for the predetermined component of the Foreign good price in the domestic market, (26): as long as η > 1, PF depends on the covariance between Home monetary policy and Foreign productivity shock. Of course, in the absence of ‘fundamental’ shocks α and α∗ , monetary volatility would only have a negative effect on Home expected utility, by increasing the volatility of consumption.18 3.3 The flex-price equilibrium benchmark Before delving into policy analysis, it is useful to characterize the equilibrium allocation we would obtain if prices were flexible. In fact, we will use such allocation as a welfare benchmark in the rest of the paper. With flexible prices, the pricing equations (26) through (29) hold in any state of nature, not in expectation, so that the degree of pass-through is irrelevant, the law of one price is valid and purchasing power parity holds. It is easy to verify that the level of employment is constant and equal to: ∗f lex Ptf lex f lex ∗f lex ∗ Pt f lex t = αt f lex Ct = Φ t = αt ∗f lex Ct∗f lex = Φ∗ . (37) PH,t PF,t As opposed to the fixed-price case, shocks to productivity affect output, not employment. Note that the flex-price level of employment is determined by the magnitude of monopolistic distortions — the competitive level of Home employment corresponds in fact to 1/κ > Φ. Consumption at Home and abroad is a function of global shocks and global monopolistic distortions 18 See the considerations in Obstfeld and Rogoff [1998]. 15 ΦW : ΦW ∗f lex ΦW Ctf lex = γ γ ∗ 1−γ , Ct = (1 − γ) γ ∗ 1−γ . (38) αt (αt ) αt (αt ) Observe that, first, consumption and employment are higher in an econ- omy where global monopolistic distortions ΦW are low. Second, a produc- tivity shock anywhere in the world economy raises global consumption. To see why, consider a positive productivity innovation in the Home country. Employment being constant in equilibrium, such shock raises Home output and consumption. However, it also lowers the Home goods price, raising the real income of Foreign consumers. The terms of trade between domestic and foreign goods are in fact a function of relative output supply: ∗ f lex PF,t Et ∗ 1 − γ αt Φ = (39) PH,t γ αt Φ∗ Movements of the terms of trade ensure that the benefits (costs) from a positive (negative) productivity shock spread around the world. By the same token, large monopolistic distortions anywhere in the world economy raise the level of consumption prices worldwide, thus reducing global consumption. 4 A framework for the analysis of monetary policy in open economy 4.1 The goals of monetary policy In this section we set and discuss the policy problem faced by a benevolent policy maker whose welfare objective is the maximization of Home agents’ expected utility. In our analysis, we focus on the non-monetary components of utility — i.e. we assume that χ is arbitrarily small — adopting Et−1 Wt as a measure of national welfare. 4.1.1 The utility-based policy loss function... Using the flex-price solution for consumption (38) with the corresponding ex- pression (32), we can calculate the gap between expected utility with flexible 16 prices, indexed W f lex , and expected utility with predetermined prices: Et−1 (αt µt ) Et−1 Wtf lex − Et−1 Wt = Et−1 γ ln + αt µt Et−1 αt (µ∗ )η µ1−η ∗ t t + (1 − γ) ln ≥0 (40) αt (µ∗ )η µ1−η ∗ t t By Jensen’s inequality, the gap Et−1 Wtf lex − Et−1 Wt cannot be negative: expected utility with price rigidities is never above expected utility with flexible prices. It follows that, at best, what monetary policy rules can do is to bridge the gap between the two: a policy rule that minimizes (40) is also a policy that maximizes the expected utility of the Home representative consumer. Below we discuss three equivalent representations of the policy loss func- tion (40), each contributing to our understanding of the policy problem. 4.1.2 ... can be expressed as a function of expected markups... First, we equate the policy loss function to an index of markups charged by producers selling in the Home market. The markup charged by Home firms on domestic sales and the markup charged by Foreign firms exporting to the Home market are, respectively: PH,t θ Et−1 (αt µt ) ΘH,t = = (41) καt Pt Ct θ−1 αt µt η 1−η PF,t θ∗ Et−1 αt (µ∗ ) µt ∗ t ΘF,t = = ∗ (42) Et κ∗ αt Pt∗ Ct∗ ∗ θ −1 αt (µ∗ )η µ1−η ∗ t t Using these definitions, it is easy to show that the policy loss function (40) is equal to the expected value of the (log) average markup in the Home markets, minus a constant depending on θ and θ∗ : Et−1 Wtf lex − Et−1 Wt = Et−1 [γ ln ΘH,t + θ θ∗ +(1 − γ) ln ΘF,t − γ ln − (1 − γ) ln ∗ (43) θ−1 θ −1 17 The logic of this result is as follows. Producers optimally set their monopoly markups and prices given their expectations of future monetary policy. The higher these prices, the lower the representative consumer’s welfare. Under the constraint of rational expectations, domestic policy makers cannot sys- tematically push markups below their flex-price equilibrium levels; they can, however, design policy rules aimed at minimizing expected markups when firms set their prices. This goal is achieved by stabilizing producers’ income from the domestic market. Note that, in the case η = 1, the markup of foreign producers in the Home country ΘF,t does not depend on domestic monetary policy. In this case, Home monetary authorities are only concerned with domestic produc- ers. Conversely, if η = 0, ΘF,t does not depend on Foreign monetary policy — ruling out an important channel of policy spillovers. 4.1.3 ... or a function of the consumer price index under flexible and fixed prices... Second, we translate the above index of markups in terms of consumer prices, with and without nominal rigidities. Expressions for prices of domestic and foreign goods in the Home country in a flex-price equilibrium can be easily derived from (26) and (27), without the expectation operator. The relation between markups and prices can thus be written as: θ PH,t θ∗ PF,t ΘH,t = f lex , ΘF,t = f lex (44) θ − 1 PH,t θ∗ − 1 PF,t Substituting, the policy loss function becomes Pt Et−1 Wtf lex − Et−1 Wt = Et−1 ln ≥0 (45) Ptf lex The policy loss function is equal to the expected ratio between the fixed-price CPI and the flex-price CPI. For any given monetary stance, the consumer price index under sticky prices cannot be expected to be lower than the CPI under flexible prices. We therefore conclude that optimal monetary policy rules minimize the expected deviation of the (log) CPI from a target given by the (log) equilibrium CPI under flexible prices.19 19 For a recent analysis of the case for price stability in closed and small open economy see Goodfriend and King [2000]. 18 4.1.4 ... or a function of output gaps and deviations from the law of one price Third, we express (40) in terms of output gaps, in the Home and Foreign economy, and deviations from the law of one price in each market. With a linear production function, the output gap can be measured as the distance between actual and equilibrium employment levels. Using the benchmark of the flex-price allocation, we express the output gap in terms of the ratio 20 t /Φ. Observe that output gaps for the Home and the Foreign countries can be easily derived from (34) and (35). Rearranging these expressions and using the definition of markups, we obtain: t θ 1 PH,t = γ + (1 − γ) (46) Φ θ − 1 ΘH,t Et PH∗ ,t Substituting into (43) we can see that the policy loss function is equal to the expectation of a weighted average of the output gaps, domestically and abroad, and a weighted average of the deviations from the law of one price in each country: ∗ t Et−1 Wtf lex − Et−1 Wt = Et−1 γ ln + (1 − γ) ln t + Φ Φ∗ PH,t PF,t +γ ln γ + (1 − γ) + (1 − γ) ln γ + (1 − γ) (47) Et PH∗ ,t Et PF∗ ,t If the law of one price holds, i.e. if η is equal to 1, the two terms in square brackets disappear, and the foreign output gap does not respond to domestic monetary policy. As we will see below, in this case welfare maximization coincides with closing the domestic output gap and the optimal policy is fully inward-looking. In general, however, the policy loss function for an interdependent economy includes more than the domestic output gap — it also depends on the exchange rate and international cyclical conditions. 20 A different measure of output gap is /(1/κ), where the flex-price equilibrium alloca- tion is derived ruling out monopolistic distortion. 19 4.2 Stabilization and interdependence in the Nash equi- librium We are now ready to derive the optimal monetary stances in a world Nash equilibrium, whereas national policy makers are able to commit to prean- nounced rules. The policy problem faced by the Home monetary authority is to maximize the (non-monetary) indirect utility of the Home representative consumer: ∞ max Et−1 Wt = Et−1 β τ −t (ln Cτ − κΦ) (48) {µt+τ }∞ τ =0 τ =t taking {µ∗ , ατ ,ατ }∞ as given. Note that this is equivalent to minimizing τ ∗ τ =t the policy loss Et−1 Wtf lex − Et−1 Wt in (40). The Foreign authority faces a similar problem. Table 2 presents the closed-form reaction functions, respec- tively (49) for Home and (50) for Foreign, whose solution is the global Nash equilibrium. The reaction functions are expressed in three ways: as explicit functions of shocks and monetary policies; as implicit functions of output gaps and deviations from the law of one price; and as implicit functions of the ex-post profit margins. With an optimal policy in place, the Home monetary stance is eased (µt increases) in response to a positive domestic productivity shock (low αt ). Monetary policy is pro-cyclical, in the sense that it accommodates domestic productivity improvements. Yet, as seen above, in the absence of a policy re- action a productivity shock would lower employment below Φ — that is, raise Home potential output above the current level. In light of this observation, monetary policy leans against the wind and moves to close the employment and output gaps. The optimal response of Home interest rates to a Home productivity shock, however, is not necessarily aimed at closing the output gap completely: unless η = 1, will not be equal to Φ at an optimum. To gain intuition, consider two symmetric productivity shocks in the Home country, leading Home monetary authorities to tighten (if the shock is negative) or to relax (if the shock is positive) their policy stance. With a negative shock, the appreciation of the Home currency reduces the revenue of Foreign firms selling goods at Home, as PF /E falls by a percentage 1 − η with the movement in the exchange rate. Even though the Foreign firms revenue increases with a 20 Table 2: The Nash equilibrium under commitment γαt µt (1 − γ) (1 − η) αt (µ∗ )η µ1−η ∗ t t 1 − η(1 − γ) = + Et−1 (αt µt ) Et−1 αt (µ∗ )η µ1−η ∗ t t ∗ t t γ (1 − γ) (1 − η) = Φ + Φ∗ PH,t PF,t γ + (1 − γ) ∗ γ + (1 − γ) ∗ Et PH,t Et PF,t γθ 1 (1 − γ) (1 − η) θ∗ 1 = + (49) θ − 1 ΘH,t θ∗ − 1 ΘF,t ∗ ∗ ∗ (1 − γ) αt µ∗ γ (1 − η ∗ ) αt µη (µ∗ )1−η ∗ t t t 1−η γ = + η∗ ∗ 1−η ∗ Et−1 (αt µ∗ ) ∗ t Et−1 αt µt (µt ) ∗ t t (1 − γ) γ (1 − η ∗ ) = Φ∗ Φ ∗ + ∗ Et PF,t Et PH,t γ + (1 − γ) ∗ γ + (1 − γ) PF,t PH,t ∗ ∗ (1 − γ) θ 1 γ (1 − η ) θ 1 = ∗ − 1 Θ∗ + (50) θ F,t θ − 1 Θ∗ H,t 21 positive shock, the welfare costs to Foreign agents from a Home monetary contraction are larger than the gains from higher income in the event of a Home monetary expansion. Ex ante, Foreign agents will hedge against the expected loss of sales rev- enue in the Home market by pre-setting a higher price on their exports, lowering Home consumption on average. This is why the Home monetary stance required to close the domestic output gap is not optimal. Relative to such stance, domestic policy makers can improve utility by adopting a policy that equates, at the margin, the benefit from keeping domestic output close to its potential level with the loss from lower consumption, as prescribed by equation (49). The corollary of these considerations is that, in general, opti- mal monetary rules in a world Nash equilibrium imply smaller exchange rate volatility vis-`-vis inward-looking rules focused on domestic developments a regardless of global cyclical conditions. As long as η is below one, the Home monetary stance tightens when productivity worsens abroad and loosens otherwise. Rising costs abroad (an ∗ increase in αt ) lower the markup of Foreign goods sold at Home. If Home policy makers were not expected to intervene and stabilize the markup by hiking rates and appreciating the exchange rate, Foreign firms would charge higher prices onto Home consumers. Only when η = 1 Foreign firms realize that any attempt by the Home authorities to stabilize the markup is bound to fail as both PF and the exchange rate fall in the same proportion. Note that, in general, domestic monetary authorities react to Foreign policy changes: a contraction abroad is matched by an expansion at Home so that monetary stances are strategic substitutes. The only cases in which Home monetary policy does not react to Foreign policy are either when η = 0 or when η = 1. In the case η = 1 Foreign interest rate volatility increases the fluctuations of Foreign marginal costs, thus raising the markup on Foreign exports. Yet, there is nothing the Home authorities can do to reduce the volatility of Foreign firms’ profits, since PF /E does not respond to Home interest rates. In the case η = 0, instead, movements in the exchange rate and Foreign marginal costs exactly offset each other and PF does not respond to the current exchange rate. Once again, there is nothing for the Home authorities to stabilize and Home interest rates do not respond to Foreign policies. Besides these two extreme cases of PCP and LCP, however, there is always a rationale for Home policies to react negatively to Foreign policies: a Foreign 22 expansion lowers the markup on Foreign exports, requiring a Home monetary response to sustain the Foreign firms’ profits by appreciating the exchange rate. Thus, in general, there will be strategic interdependence among policy makers.21 4.3 Pass-through and Pareto-efficiency The utility gap in expression (40) does not depend on monopolistic distor- tions, whose magnitude is indexed by ΦW . As discussed by Obstfeld and Ro- goff [2000a], suppose that governments in both countries set non-monetary instruments (taxes and subsidies) so as to offset the distortions due to mo- nopolistic production. If a global monetary policy exists such that the gap (40) is bridged, such policy would implement the first-best allocation. Then, starting from the first-best allocation, removing taxes and subsidies will lower the level of both Et−1 Wtf lex and Et−1 Wt , but will not affect the difference between the two. The global monetary policy in place would still be optimal, and the corresponding allocation would be constrained-Pareto efficient (that is, efficient for the world economy subject to a given level of monopolistic distortion). Can monetary rules close the world utility gap completely? In the pres- ence of asymmetric shocks the answer is negative in all but one important case, that is PCP. With complete pass-through, the Nash equilibrium policy is: ΦPH,t Φ∗ PF,t ∗ µt = µ∗ = t ∗ (51) αt αt Monetary authorities optimally stabilize employment at its flex-price level Φ and Φ∗ .22 Domestic and global consumption optimally co-moves with productivity shocks according to (38). Thus, the Nash optimal monetary policy supports the same allocation as with flexible prices, that is constrained Pareto-efficient. 21 Betts and Devereux [2000b] examine the problem of international monetary coordina- tion, in environments of differing pass-through, with discretion in policy making. 22 Note that, because of our utility parameterization, domestic monetary policy only responds to domestic shock. This will not be true, for instance, with a power utility of consumption. Yet, even with power utility the Nash allocation under PCP coincides with the flex-price allocation. 23 This result provides an extreme version of the case for flexible exchange rates made by Friedman [1953]: even without price flexibility, monetary au- thorities can engineer the right adjustment in relative prices through ex- change rate movements. In our model with PCP, expenditure switching effects makes exchange rate and price movements perfect substitutes. The Nash equilibrium will however not coincide with a flex-price equilib- rium when the pass-though is less than perfect in either market. Consider for instance the case of LCP.23 Suppose monetary authorities set interest rates so as to achieve the flex-price level of consumption in both countries: γΦW Pt ∗ (1 − γ) ΦW Pt∗ µt = , µt = (52) αW,t αW,t Note that both monetary policies would react to the same global index of pro- ductivity shocks. Then, the employment levels would be quite different from their flex-price levels, varying over time as functions of relative productivity shocks: 1−γ −γ αt ∗ αt t ∝ ∗ , t ∝ ∗ (53) αt αt ∗ Only if shocks were global (αt = αt ) would the level of employment be con- stant. In fact, with global shocks the terms of trade externality disappears, relative prices need not adjust, and the Nash equilibrium trivially supports the flex-price allocation. 4.4 Commitment vs. discretion in open economy So far we have assumed that policy-makers worldwide are able to commit to optimal policies. This section contrasts the policy problem under commit- ment with the policy problem under discretion. First, we discuss inflationary bias in open economy, identifying the determinants of its magnitude and sign. Second, we show that the optimal policy is not time consistent, even in the absence of an inflationary or deflationary bias. Under discretion, in each period t the Home policy maker solves ∞ max ∞ Wt = Et β τ −t (ln Cτ − κ τ ) (54) {µt+τ }τ =0 τ =t 23 See the discussion in Devereux and Engel [2000]. 24 taking {µ∗ , ατ , ατ }∞ as given. The first order condition is: τ ∗ τ =t ∗ ∗ 1 − η(1 − γ) γαt µt (1 − γ) η ∗ αt (µ∗ )1−η µη t t = + ∗ 1−η ∗ η ∗ (55) κΦ Et−1 (αt µt ) Et−1 αt (µt ) µt A similar expression holds in the Foreign country. 4.4.1 Inflationary vs. deflationary bias In a closed economy, the presence of monopolistic distortions are clearly asso- ciated with an inflationary bias in policy making under discretion. This need not be true in an open economy. As analyzed in Corsetti and Pesenti [2001], the bias in policy making may be deflationary rather than inflationary if the economy is sufficiently small so as to suffer from the consumption and welfare consequences of adverse terms of trade movements. Taking expectations of both sides of (55), it is clear that a discretionary policy will be subject to either inflationary or deflationary bias in the Home country if: 1 − η(1 − γ) [γ + (1 − γ) η ∗ ] κΦ. (56) Under discretion, there is an inflationary bias in Home monetary pol- icy if the economy is sufficiently ‘large’, in the sense that a large share of consumption falls on Home products and γ is sufficiently close to 1. As a depreciation of the exchange rate affects the price of a relatively small share of consumption goods, policy makers are less concerned with adverse import price movements than with the distortions associated with monopoly power in production. The reverse is true when γ is sufficiently small and η and η ∗ sufficiently large. A monetary expansion, while raising output and employment, also increases the price of a substantial proportion of consumption goods. The terms of trade movement becomes a dominant concern in discretionary policy making, leading to a deflationary bias. Reducing the degree of pass-through lowers the concern about adverse terms of trade movements associated with a monetary expansion. For given γ, Φ, and η ∗ , a lower pass-through in the domestic market (i.e. a lower η) is associated with a stronger inflationary bias.24 24 Recent literature provides evidence of a negative relationship between openness and 25 4.4.2 Time consistency issues independent of inflationary bias Assume now that condition (56) holds as an equality, or (alternatively) that the government is able to use an appropriate fiscal instrument to off- set the impact of monopolistic distortions,25 so that there is no inflation- ary/deflationary bias in policy making. One may expect that the first order conditions of the policy problems under discretion would then coincide with the first order conditions under commitment. Comparing (55) and (49), it is apparent that this is true only in the PCP case (η = η ∗ = 1), when firms profits are not exposed at all to exchange rate variability. In all other cases, discretionary policies do not coincide with policies under commitment. The reason is the following. Under discretion, for given firms prices, the Home policy maker responds to domestic productivity shocks and foreign monetary shocks depending on the degree of pass-through in the Foreign market, η ∗ , that is relevant for Home producers/exporters. Ex post, the optimal policy reaction to a shock moves exchange rates as to tilt the external terms of trade in favor of domes- tic producers. While optimal ex post, however, this rule ignores the effects of domestic monetary policy on the markup of foreign producers. The variabil- ity of this markup will be reflected in the level of domestic prices of Foreign goods, that are set one period in advance. The ability to commit to a rule allows policy makers to incorporate these effects in policy design. Under commitment, in fact, the Home policy mak- ers respond to both Home and Foreign shocks depending on the degree of pass-through at Home, η, that is relevant for Home consumers. In other words, policy makers do not attempt to manipulate terms of trade systemat- ically (as under discretion). On the contrary, they contain exchange rate and terms of trade movements as to reduce their effects on the income of Foreign producers, giving up to some extent the stabilization of Home producers’ income. This result of over-stabilization under discretion complements and extends a similar result discussed in the context of closed-economy models inflation in a large cross-section of countries (Romer [1993], Campillo and Miron [1997], Lane [1997], and Cavallari [2001]). In light of this contribution their empirical findings can be interpreted as evidence that the incentive to expand monetary policy is lower in economies that are more open, and therefore more likely to suffer from the price effects of a depreciating currency. We would expect the degree of pass-through to be a relevant variable in these empirical models. 25 See the discussion in Benigno and Benigno [2000] for the PCP case. 26 with staggered price adjustments by firms and inflation dynamics.26 Our analysis shows the applicability of the same principle to an open economy, even in the absence of persistent inflation dynamics. 5 Implications for international cooperation and exchange rate volatility 5.1 The case for cooperation Are there welfare gains from cooperating in the design and implementation of optimal monetary rules? How are these gains influenced by the degree of pass- through? To address this issue, consider a binding cooperative agreement between the two countries such that the policy makers jointly maximize the following objective function: max ∞ Et−1 [ξWt + (1 − ξ) Wt∗ ] 0≤ξ≤1 (57) { µt+τ ,µ∗ t+τ }τ =0 where ξ is a constant parameter indexing the bargaining power of the Home country. A country’s bargaining power is not necessarily related to its eco- nomic size, so that ξ need not be equal to γ. The optimal cooperative policy is the solution to the system below, ex- pressed in terms of profit margins for notational simplicity: ξγθ 1 ξ [1 − η(1 − γ)] + (1 − ξ) η ∗ = + θ − 1 ΘH,t ξ (1 − γ) (1 − η) θ∗ 1 (1 − ξ) γη ∗ θ 1 + + (58) θ∗ − 1 ΘF,t θ−1 Θ∗ ,t H (1 − ξ) (1 − γ) θ∗ 1 (1 − ξ) (1 − η ∗ γ) + ξη (1 − γ) = + θ∗ − 1 Θ∗ ,t F (1 − ξ) γ (1 − η ∗ ) θ 1 ξ (1 − γ) ηθ∗ 1 + + (59) θ−1 Θ∗ ,t H θ∗ − 1 ΘF,t 26 See for instance Gali [2000]. 27 To verify whether there are gains from cooperation, we can compare the above system (58)-(59) with the Nash equilibrium (49)-(50) in Table 2. It is straightforward to verify that the two systems are identical — and therefore there are no welfare gains from cooperation — in three special cases. The ∗ first — and well known — case is when all shocks are global, i.e. αt = αt ∗ for any t. The optimal Nash policies can be written as in (51) with αt = αt , implying a constant exchange rate. Consumption and employment coincide with the flex-price allocation. The other two cases are PCP — as discussed in Obstfeld and Rogoff [2000b] — and (perhaps surprisingly) LCP. Under the extreme assumptions of either complete or zero pass-through, the allocation is the same as whether or not monetary authorities cooperate, regardless of the value of ξ. We have established above that these are precisely the case in which monetary policies are strategically independent of each other. This new result establishes that, in either case, there are no policy spillovers in equilibrium. In the case of PCP, we have shown that the Nash equilibrium coincides with the flex-price allocation. This implies that the distortions induced by nominal rigidities and the monopoly power of a country on its terms of trade have no welfare consequences: the utility level is the same in each country, both ex-ante and ex-post. The absence of gains from cooperation in the case of LCP is less straight- forward. While there is no strategic interaction among policy makers, one may expect potentially large spillovers from monetary policy: recall that an expansion at Home increases Foreign employment without affecting Foreign demand, thus depreciating the Foreign terms of trade between labor and consumption. Key to our result is that, whether or not monetary authorities cooperate, the optimal monetary policy in each country cannot be inward- looking, and must instead respond symmetrically to shocks anywhere in the world economy. The optimal monetary policy can in fact be written as: ∗ −1 αt αt µt = γ + (1 − γ) ∗ (60) Et−1 (αt µt ) Et−1 (αt µt ) ∗ −1 αt αt µ∗ t = γ + (1 − γ) , (61) Et−1 (αt µ∗ ) t Et−1 (αt µ∗ ) ∗ t expressions that are satisfied by setting µ = µ∗ . For any given shock, con- sumption increases by the same percentage everywhere in the world economy. 28 Even if ex-post labor moves asymmetrically (so that ex-post welfare is not identical in the two countries, as is the case under PCP), ex ante the expected disutility from labor is constant. Thus, expected utility at Home is identi- cal to expected utility in the Foreign country up to a constant that does not depend on monetary policy: the Nash and the cooperative allocation coincide. Besides such extreme and unrealistic cases, a country can in general do better than simply ‘keeping its own house in order’ by engaging in binding in- ternational agreements. Whether these gains are sizable for different degrees of pass-through, however, remains an open issue left to further research. 5.2 Volatility and exchange rate regimes In our model the exchange rate remains constant during the period t as long as µt = µ∗ . It follows that optimal rules can be consistent with a t fixed exchange rate target provided that the implied monetary stances are perfectly correlated across countries. Under what conditions will this be the case? The Nash equilibrium conditions (49) and (50) are solved by µt = µ∗ t only in two cases: either when αt = αt , or when η = η ∗ = 0. In other ∗ words, fixed exchange rates can be supported by optimal monetary policies only when productivity shocks are global, or when there is no pass-through worldwide. Note that in these two cases a Nash equilibrium will coincide with a cooperative equilibrium.27 In general, however, the implementation of optimal policy is not consistent with a fixed exchange rate regime. To what extent will the exchange rate fluctuate as domestic policy makers follow optimal rules? Our analysis suggests that exchange rate volatility will be higher in a world economy close to purchasing power parity, and lower in a world economy where deviations from the law of one price are large. In fact, if the exposure of firms’ revenue to exchange rate fluctuations is lim- ited, inward-looking policy makers assign high priority to stabilizing domestic output and prices, with ‘benign neglect’ of exchange rate movements. Oth- erwise, policy makers ‘think global’, taking into account the repercussions of exchange rate volatility on import prices; hence, the monetary stances in the 27 Benigno, Benigno and Ghironi [2000] consider the feedback rules leading to exchange rate determinacy in a fixed exchange rate regime. 29 world economy come to mimic each other, reducing currency volatility. Note that in our analysis we have ignored money velocity shocks χt and ∗ χt . In fact, when utility from real holdings is additively separable, money velocity shocks do not play any role in the determination of consumption and employment. As (30) and (31) show, for given interest rate policies the amount of money held by domestic residents adjusts one-to-one to velocity shocks. In other words, money balances are determined residually in the model as functions of exogenous shocks and policy variables. An obvious corollary of this analysis is that, when velocity shocks are the only source of asymmetric disturbance hitting the two economies, optimal monetary policies imply that interest rates and exchange rate all remain constant over time. 6 Conclusion The key message of our contribution is that strict adherence to inward- looking goals in policy making — such as the stabilization of domestic output — cannot be optimal in interdependent economies in which firms’ revenues are exposed to currency fluctuations. Intuitively, producers set higher prices in response to higher profits volatility. Unless markups are insulated from exchange rate movements, inward-looking policies make the profits of foreign exporters suboptimally volatile, thus inducing higher average import prices. At an optimum, the welfare costs from higher consumer prices must equate the benefits from bringing domestic output towards its potential, flex-price level. In our study, the degree of pass-through and exchange rate exposure in domestic and foreign markets emerges as a key element determining the prop- erties of optimal monetary rules, as well as the welfare features of equilibrium allocations. Ceteris paribus, economies with higher degrees of pass-through should be characterized by a higher level of exchange rate volatility and by monetary stances more closely focused on internal conditions. A corollary of this principle is that gains from cooperation materialize only for intermediate levels of pass-through. 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Consider now the problem of allocating a given level of nominal expendi- ¯ ture Z among domestically produced goods: 1 max CH (j) s.t. ¯ p(h)C(h, j)dh = Z (A.3) C(h,j) 0 Clearly, across any pair of goods h and h , it must be true that: −θ C(h , j) p(h ) = (A.4) C(h, j) p(h) or, rearranging: θ−1 θ−1 C(h, j) θ p(h )1−θ = C(h , j) θ p(h)1−θ (A.5) Integrating both sides of the previous expression we obtain: θ θ θ−1 1 θ−1 1 θ−1 θ−1 C(h, j) θ p(h )1−θ dh = C(h , j) θ p(h)1−θ dh 0 0 (A.6) so that: θ θ θ−1 θ−1 1 1 θ−1 C(h, j) p(h )1−θ dh = p(h)−θ C(h , j) θ dh 0 0 (A.7) which can be rewritten as expression (10) in the main text. Note that: 1 p(h)C(h, j)dz = PH CH (j). (A.8) 0 i Intertemporal allocation. Consider the optimal allocation by Home agent j. The maximization problem can be written in terms of the following Lagrangian: ∞ Lt (j = h) ≡ Et β τ −t {ln Cτ (j) + χτ ln Mτ (j) /Pτ − κ τ (j) τ =t +λτ (j) [−BH,τ +1 (j) + (1 + iτ )BH,τ (j) − Eτ BF,τ +1 (j) + (1 + i∗ )Eτ BF,τ (j) τ −Mτ (j) + Mτ −1 (j) + Rτ (j) − Tτ (j) − PH,τ CH,τ (j) − PF,τ CF,τ (j)]} (A.9) where Rτ (j) and τ (j) are defined in (13) and (11). The first order conditions with respect to CH,t (j), CF,t (j), BH,t+1 (j), BF,t+1 and Mt (j), are, respectively: γ = λt (j) PH,t (A.10) CH,t (j) 1−γ = λt (j) PF,t (A.11) CF,t (j) λt (j) = βEt λt+1 (j) (1 + it+1 ) (A.12) Et λt (j) = βEt Et+1 λt+1 (j) 1 + i∗ t+1 (A.13) χ = Et [λt (j) − βλt+1 (j)] (A.14) Mt (j) First, we solve for the multiplier λt (j). Take a geometric average of (A.10) and (A.11) with weights γ and 1 − γ, respectively: γ 1−γ γ 1−γ γ γ (1 − γ)1−γ = λt (j)PH,t PF,t CH,t (j)CF,t (j) (A.15) This yields: 1 λt (j) = (A.16) Pt Ct (j) ii so that, at the optimum, the individual demand for Home and Foreign con- sumption goods are a share of total consumption expenditure: 1 1 Pt Ct (j) = PH,t CH,t (j) = PF,t CF,t (j). (A.17) γ 1−γ Using these expressions, it is easy to verify that Pt Ct (j) = PH,t CH,t (j) + PF,t CF,t (j). (A.18) Combining (A.10), (A.11) and (A.12) the intertemporal allocation of con- sumption is determined according to the Euler equation: 1 Pt 1 = β (1 + it+1 ) Et (A.19) Ct (j) Pt+1 Ct+1 (j) Finally, condition (A.14) can be written as the money demand function: Mt (j) 1 + it+1 = χt Ct (j) (A.20) Pt it+1 Define now the variable Qt,t+1 (j) as: Pt Ct (j) Qt,t+1 (j) ≡ β (A.21) Pt+1 Ct+1 (j) The previous expression can be interpreted as agent j’s stochastic discount rate. Comparing (A.21) with (A.12) and (A.13) we obtain: 1 Et+1 1 Et Qt,t+1 (j) = , Et Qt,t+1 (j) = 1 + it+1 Et 1 + i∗ t+1 (A.22) from which we obtain the risk-adjusted uncovered interest parity linking do- mestic and foreign nominal interest rates: −1 1 + it+1 Et+1 Et = Et Et 1 + i∗ t+1 Pt+1 Ct+1 (j) Pt+1 Ct+1 (j) (A.23) Note that in the absence of uncertainty the previous condition collapses to the familiar expression 1 + it+1 = 1 + i∗ Et+1 /Et . t+1 iii Using (A.22) we can write: it+1 Mt (j) Mt (j) + BH,t+1 (j) = + Et {Qt,t+1 (j) [Mt (j) + (1 + it+1 ) BH,t+1 (j)]} 1 + it+1 (A.24) and: Et BF,t+1 (j) = Et Qt,t+1 (j)(1 + i∗ )Et+1 BF,t+1 (j) t+1 (A.25) It follows that the flow budget constraint (12) can also be written as: it+1 Mt (j) + Et {Qt,t+1 (j)Wt+1 (j)} ≤ Wt (j) + Rt (j) − Tt (j) − Pt Ct (j) 1 + it+1 (A.26) where Wt+1 is wealth at the beginning of period t + 1, defined as: Wt+1 (j) ≡ Mt (j) + (1 + it+1 ) BH,t+1 (j) + (1 + i∗ )Et+1 BF,t+1 (j). t+1 (A.27) Optimization implies that households exhaust their intertemporal budget constraint: the flow budget constraint hold as equality and the transversality condition: lim Et [Qt,N (j)WN (j)] = 0 (A.28) N →∞ is satisfied, where Qt,N ≡ Ns=t+1 Qs−1,s . If an interior solution exists (as is the case given our parameterization), the resource constraint (11) holds as equality as well. Foreign constraints and optimization conditions. Similar results characterize the optimization problem of Foreign agent j ∗ . For convenience, we report here the key equations. The resource constraint for agent f is: ∗ (f ) ∗ θ∗ ∗ θ∗ ∗ ≥ pt (f )−θ PF,t CF,t + p∗ (f )−θ PF,t t ∗ ∗ CF,t (A.29) αt iv The flow budget constraint is: BH,t+1 (j ∗ ) ∗ ∗ ∗ ∗ BH,t (j ∗ ) ∗ ∗ + BF,t+1 (j ) + Mt (j ) ≤ (1 + it ) + (1 + i∗ )BF,t (j ∗ ) + t ∗ Et Et +Mt−1 (j ∗ ) + Rt (j ∗ ) − Tt∗ (j ∗ ) − PH,t CH,t (j ∗ ) − PF,t CF,t (j ∗ ) ∗ ∗ ∗ ∗ ∗ ∗ (A.30) and, associating individual j ∗ with brand f , sales revenue is: 1 ∗ θ∗ ∗ θ∗ R∗ (j ∗ ) = (pt (f ))1−θ PF,t CF,t + p∗ (f )1−θ PF,t t ∗ ∗ CF,t . Et (A.31) First order conditions yield: 1 ∗ ∗ ∗ 1 Pt∗ Ct∗ (j ∗ ) = PH,t CH,t (j ) = P ∗ C ∗ (j ∗ ) (A.32) γ 1 − γ F,t F,t 1 Pt∗ 1 ∗ ∗ = β 1 + i∗ Et t+1 ∗ ∗ , (A.33) Ct (j ) Pt+1 Ct+1 (j ∗ ) Mt∗ (j ∗ ) 1 + i∗ = χ∗ ∗ t+1 Ct∗ (j ∗ ), t (A.34) Pt∗ it+1 Pt∗ Ct∗ (j ∗ ) λ∗ (j ∗ ) Q∗ (j ∗ ) = β t,t+1 ∗ = β t+1 ∗ , (A.35) Pt+1 Ct+1 (j ∗ ) λ∗ (j ) t and: 1 + it+1 1 1 = Et . 1 + i∗ t+1 ∗ ∗ Pt+1 Ct+1 (j ∗ ) 1 Et Et ∗ ∗ (A.36) Et+1 Pt+1 Ct+1 (j ∗ ) Optimal price setting. Optimal pricing in the Home country can be derived as follows. Maximizing the Lagrangian with respect to pt (h) yields: Et−1 −κθpt (h)−θ−1 PH,t αt CH,t + (θ − 1) pt (h)−θ PH,t λt (h)CH,t = 0 θ θ (A.37) v or, in a symmetric environment: θκ Et−1 [αt CH,t ] PH,t = (A.38) θ−1 CH,t Et−1 Pt Ct and, using (A.17), we can rewrite the previous expression as (14) in the main text. Maximizing the Lagrangian with respect to pt (h) yields: ~ θ t ~ ∗ 0 = Et−1 −κθp∗ (h)−θ pt (h)−1 PH,t ∗ αt CH,t θ t ~ ∗ + (θ − 1) p∗ (h)1−θ Et pt (h)−1 PH,t ∗ λt (h)CH,t (A.39) that is: ∗ −θ ∗ ∗ θ θκ Et−1 pt (h) PH,t αt CH,t 1= ∗ (A.40) θ−1 ∗ 1−θ P ∗ θ Et CH,t Et−1 pt (h) H,t Pt Ct (h) ∗ In a symmetric environment with p∗ (h) = PH,t , we can rewrite the above as: t ∗ θκ Et−1 αt CH,t 1= ∗ (A.41) θ−1 ∗ Et CH,t Et−1 PH,t Pt Ct and, using (A.32), we obtain: Pt∗ Ct∗ Et−1 αt ∗ θκ PH,t 1= (A.42) θ−1 Et Pt∗ Ct∗ Et−1 Pt Ct or: ∗ ∗ ∗ η θκ Et−1 αt Pt Ct Et pt (h) = ~ (A.43) θ−1 Et Pt∗ Ct∗ Et−1 Pt Ct from which we obtain (16) in the main text. vi Equilibrium. Given initial wealth levels Wt0 and Wt∗ (symmetric among 0 country residents), and the processes for αt , αt , χt , χ∗ , Tt , Tt∗ , it+1 and i∗ ∗ t t+1 for all t ≥ t0 , the symmetric fix-price equilibrium is a set of processes for CH,t , ∗ ∗ CF,t , Ct , Qt,t+τ , BH,t+1 , BF,t+1 , Mt , t , Rt , Wt+1 , PH,t , PF,t , Pt , CH,t , CF,t , Q∗ , Ct∗ , BH,t+1 , BF,t+1 , Mt∗ , ∗ , Rt , Wt+1 , PH,t , PF,t , Pt∗ , and Et such that, t,t+τ ∗ ∗ t ∗ ∗ ∗ ∗ for all t ≥ t0 and τ > t0 , (i) the Home government budget constraint (21) and its Foreign analog are satisfied; (ii) Home consumer optimization conditions (A.17), (2), (A.21), (A.19), (A.23), (A.20), (11), (13), (A.27), and their For- eign analogs hold as equalities; (iii) Home firm optimization conditions (14), (16), (4), and their Foreign analogs hold; (iv) the Home transversality condi- tion (A.28) and its Foreign analog hold; (v) the markets for the international bonds clear, that is conditions (22) hold. A flex-price equilibrium is similarly defined, after imposing that for all t ≥ t0 Home conditions (14), (16), and their Foreign analogs hold in any state of nature at time t rather than in expectation at time t − 1. Equilibrium balanced current account. Aggregating the individual budget constraints and using the government budget constraint we obtain an expression for the Home current account: Et {Qt,t+1 At+1 } = At + Rt − Pt Ct (A.44) where A is defined as wealth net of money balances, or At+1 ≡ Wt+1 − Mt , (A.45) and, by the definition of sales revenue: Rt = PH,t CH,t + Et PH,t CH,t = γ (Pt Ct + Et Pt∗ Ct∗ ) . ∗ ∗ (A.46) Assume now that, at time t0 , BH,t0 = BF,t0 = 0 so that initial non- monetary wealth At0 = 0. It can then be easily showed that, for all t ≥ t0 , the equilibrium conditions above are solved by the allocation: Rt − Pt Ct = −Et (Rt − Pt∗ Ct∗ ) = 0, ∗ At = −Et A∗ = 0 t t ≥ t0 . (A.47) vii The utility gap. To derive (40) and to show its sign consider the fol- lowing steps: Et−1 Utf lex − Et−1 Ut = Et−1 ln Ctf lex − ln Ct + κΦ − κΦ 1−γ Ctf lex {Et−1 [αt µt ]}γ Et−1 αt (µ∗ )η µ1−η ∗ t t = Et−1 ln = Et−1 ln Ct γ ∗ 1−γ 1−η(1−γ) ∗ η(1−γ) αt (αt ) µt (µt ) Et−1 [αt µt ] Et−1 αt (µ∗ )η µ1−η ∗ t t = Et−1 γ ln + (1 − γ) ln ∗ ∗ η 1−η αt µt αt (µt ) µt = Et−1 γ (ln Et−1 [αt µt ] − ln αt µt ) + (1 − γ) ln Et−1 αt (µ∗ )η µ1−η − ln αt (µ∗ )η µ1−η ∗ t t ∗ t t ≥ Et−1 γ (Et−1 [ln αt µt ] − ln αt µt ) + (1 − γ) Et−1 ln αt (µ∗ )η µ1−η − ln αt (µ∗ )η µ1−η ∗ t t ∗ t t = γ (Et−1 ln αt µt − Et−1 ln αt µt ) + (1 − γ) Et−1 ln αt (µ∗ )η µ1−η − Et−1 ln αt (µ∗ )η µ1−η ∗ t t ∗ t t = 0. (A.48) Nominal equilibrium. The optimal monetary rules derived in the main text do not provide a nominal anchor to pin down nominal expec- tations. For instance, recalling the solution for PH,t , we can write the Home country reaction function (49) in order to emphasize the relation between µt and PH,t : (1 − γ) (1 − η) αt (µ∗ )η µ1−η ∗ t t ΦPH,t αt µt = 1 − η(1 − γ) − ∗ ∗ η 1−η Et−1 αt (µt ) µt γ (A.49) Now, using (26) we obtain ΦPH,t = Et−1 αt µt but taking the expectation of (A.49) we obtain Et−1 αt µt = ΦPH,t for any arbitrary choice of PH,t . Similar ∗ considerations hold for (50) and PF,t . To address this issue, following Woodford [2000], take µt and µ∗ defined t by the system (49) and (50) and define two rules µt and µ∗ such that: ^ ^t δ ∗ δ∗ PH,t PF,t µt = µt ^ ¯ µ∗ ^t = µ∗ t ¯∗ δ, δ ∗ < 0 (A.50) PH PF ¯ ¯∗ where δ and δ ∗ are two negative constants, arbitrarily small, and PH and PF are the two nominal targets for the Home and Foreign monetary authorities, viii respectively. Under the rules above, we obtain: δ PH,t Φ (PH,t )δ+1 ΦPH,t = Et−1 αt µt = Et−1 αt µt ^ ¯ =γ PH ¯ δ γ PH (A.51) ¯ solved by PH,t = PH . This implies that, in equilibrium, µt is identical to ^ µt once we replace PH,t with P¯H . Similar considerations allow to pin down ∗ ¯∗ PF,t = PF . The government sets a nominal anchor and credibly threatens to tighten monetary policy if the price of domestic goods deviates from a given target. Such threat, however, is never implemented in equilibrium. Equilibrium import prices: some special cases. Given the targets ¯ ¯∗ PH and PF , we can determine the import prices in both countries according to the rules (49) and (50). For instance, under PCP the rules (51) imply: ∗ ¯ Φ 1 − γ αt . PF,t = PH ∗ (A.52) Φ γ αt ¯ Given PH , the domestic price of Foreign goods will optimally increase with a positive productivity shock in the Home country. In the LCP case instead the rule (60) holds, and 1 1−γ ∗ PF,t = ∗ Et−1 (αt µt ) (A.53) Φ γ ∗ where Et−1 (αt µt ) is the solution to: ∗ 1 αt ¯ = Et−1 ∗ ∗¯ . (A.54) PH Φ γαt Et−1 (αt µt ) + (1 − γ) αt PH Φ If η = 1 and η ∗ = 0, we have: ∗ ∗ ¯∗ ¯ Φ 1 − γ (1 − γ) αt + γ Φ PF PF,t = PH ∗ (A.55) Φ γ αt Et−1 (αt µ∗ ) t where Et−1 (αt µ∗ ) is the solution to: t 1 αt ¯∗ = Et−1 ∗ ¯∗ . PF Φ∗ (1 − γ) αt Et−1 (αt µ∗ ) + γαt PF Φ∗ t (A.56) ix