Liquidity trap and optimal monetary policy in open economies

Transcription

1 Liquidity trap and optimal monetary policy in open economies Tomoyuki Nakajima Kyoto University February, 5 Abstract We consider an open-economy model with the Calvo-type sticky prices. We mainly analyze the situation in which the monetary authority in each country cooperates so as to maximize the world welfare. In the case where the zero lower bound (ZLB) on nominal interest rates never binds, the optimal inflation targeting rule in our openeconomy model has exactly the same form as in the closed-economy model. This is not the case, however, when the ZLB may bind. The optimal paths are characterized in such a situation. In contrast with what Svensson (, 3, 4) suggests, the optimal paths of the nominal exchange rate in our model typically exhibit appreciation of the currency of the country where the ZLB binds. JEL Classification Numbers: E5, F3, F4 Key words: optimal monetary policy, zero lower bound, open economies, nominal rigidities. Address: Institute of Economic Research, Kyoto University, Kyoto 66-85, Japan. Tel: nakajima@kier.kyoto-u.ac.jp. I have benefitted from discussions with Makoto Saito, and Akihisa Shibata. I also thank seminar participants at Hitotsubashi University, Osaka University and the University of Tokyo for helpful comments and criticisms.

2 Introduction How should monetary policy be conducted when the zero lower bound (ZLB) for nominal interest rates may bind? The recent experience of Japan is a well-known example of such a liquidity trap. There, the call rate has been below.5 percent per annum since October 995 and below. percent since March 999 (except for the period August -March ). Krugman (998) argues that, even when the nominal interest rate hits the zero bound, the central bank could still stimulate the current level of output by raising expectations of future inflation. This point is further elaborated in a more fully dynamic framework with staggered pricing by Eggertsson and Woodford (3). Based on an optimization-based, quadratic approximate welfare measure, they analyze the statecontingent paths of inflation, output gap, and nominal interest rates under optimal policy commitment. Furthermore, they derive a price-level targeting rule that could implement the optimal paths. In this paper, we extend the analysis of Eggertsson and Woodford (3) to a twocountry open-economy model. 3 A continuum of differentiated products are produced in each country, and each good price is adjusted at random intervals as in Calvo (983). We assume perfect exchange-rate pass-through, so that the law of one price holds. We analyze the optimal state-contingent paths of various variables, and compare our results to the proposal of Svensson (, 3, 4) that the currency of a country in a liquidity trap should depreciate. We start with a result on equilibrium shares of consumption across countries. In the literature on open-economy monetary models, it is often assumed that the equilibrium shares of consumption across countries are determined independently of the monetary policy rule adopted by each country. This assumption is a bit problematic. It is true that with complete asset markets (and isoelastic preferences), given initial financial asset holdings and policy rules, the equilibrium shares of consumption are constant at all dates and all contingencies. In general, however, even under these assumptions, given initial asset holdings, different policy rules would result in different equilibrium shares of consumption across countries. We provide sufficient conditions on preferences and initial asset holdings for the equilibrium shares of consumption across the two countries to be independent of the policy rule adopted by each country. They include unit elasticity between goods produced The call rate is an overnight interest rate, which is analogous to the federal funds rate in the US. A related problem is studied by Jung, Teranishi and Watanabe (). 3 There is growing literature on open-economy models with nominal rigidities. Examples include, among others, Corsetti and Pesenti (), Obstfeld and Rogoff (), Clarida, Galí and Gertler (), Galí and Monacelli (), Benigno and Benigno (3), Svensson (4).

3 in different countries. For simplicity, we assume that those conditions hold in this paper. As for the policy objective, we mostly focus on the case in which the two monetary authorities cooperate each other to maximize the world welfare, which is defined as the world average of expected lifetime utility of households. Based on a second-order approximation, the objective function of the monetary authorities is given by a quadratic loss function in output gaps and inflation rates. Here, the welfare-relevant inflation rates are not the CPI inflation rates, but the producer-price inflation rate in each country, as in Benigno and Benigno (3) and Clarida, Galí and Gertler (). Thus, fluctuations in the nominal exchange rate per se does not affect the welfare. We first examine the optimal policy rule when the ZLB is assumed never to bind. Surprisingly, in this case, the optimal inflation targeting rule in our open economy model takes exactly the same form with the same parameter value as the one obtained for the closed economy by Woodford (3, Section 7.5). That is, the (producer-price) inflation rate in each country must be targeted at the level given by a constant times the rate of change of its output gap. This is true even though the aggregate supply relation shows international dependence, that is, the inflation-output-gap tradeoff in each country is affected by the output gap in the other country. Thus, as long as the ZLB never binds, the monetary authority in each country may forget about international dependence and set the target rate of inflation independently, in order to maximize the world welfare. This is not the case when the ZLB may bind. In such a case, as our optimal price-level target rule shows, the price-level target in each country is not determined independently. To examine quantitative properties of the optimal state-contingent paths of key variables, we conduct a numerical experiment similar to the one in Eggertsson and Woodford (3). It shows that the optimal state-contingent paths of inflation, output gaps, and nominal interest rates of the country in a liquidity trap look very similar to those obtained for the closed economy by Eggertsson and Woodford (3). Such paths in our open economy model are, however, made possible by active policy coordination of the other country. Regarding exchange rates, our numerical experiment shows that the currency of a country in a liquidity trap must appreciate, rather than depreciate. This makes a sharp contrast to what Svensson (, 3, 4) proposes. A theoretical justification for his proposal is made in Svensson (4). While our model and Svensson s (4) differ in several ways, 4 the difference in the evolution of nominal exchange rates under optimal policy arises from his assumption that (i) the shock that generates a liquidity trap lasts 4 For instance, Svensson (4) considers prices that are set one period in advance, rather than staggered pricing. Also, in his model, a country in a liquidity trap may or may not be small; there may or may not be international coordination of monetary policy. 3

4 only for one period and (ii) the country in a liquidity trap has a productivity level which is higher than the steady-state level. Note that a country falls into a liquidity trap, say, at date, if the expected growth rate of productivity in that country from date to date is sufficiently negative. Svensson (4) creates such a situation by assuming that the productivity of a country at date is unusually high (and the expected level of productivity at date is normal). This is why he observes depreciation at date under optimal policy: Since the country produces more than normally the relative price of goods produced in that country to those produced abroad must fall. Because the price of a good is set one period in advance, this fall in the relative price is achieved by a depreciation. However, even in this scenario, if the shock lasts for more than one period so that the expected growth rate of productivity is negative, say, for dates t =,..., τ, then the currency would depreciate at all t =,..., τ under optimal policy commitment. In this sense, our claim that the currency of a country in a liquidity trap depreciates under optimal policy commitment holds more robustly. The rest of the paper is organized as follows. In Section, we describe households and give a proposition on the equilibrium share of consumption. In Section 3, the aggregatesupply relations are derived. In Section 4, a quadratic approximate world welfare measure is computed, and an optimal inflation targeting rule is derived for the case where the ZLB is assumed never to bind. In Section 5, the natural rates of interest are defined and the intertemporal IS equations are obtained. In Section 6, optimal policy in a liquidity trap is analyzed. In Section 7, the state-contingent paths of nominal exchange rates under optimal policy is discussed. Section 8 is concluding remarks. Households The model economy is an open-economy version of the sticky-price model developed by Woodford (3). In particular, goods prices are adjusted at random intervals as in Calvo (983); households supply differentiated labor; and monetary frictions are abstracted so that money is not modelled explicitly (a cashless economy in the terminology of Woodford, 3). The world economy consists of two countries, Home (H) and Foreign (F ). The size of population in country j {H, F } is n j. We normalize the world population to unity: n H + n F =. A set of differentiated products are produced in each country and they are traded between the two countries. Let N j denote the set of those products. We assume 4

5 that N H =, n H ], and N F = (n H, ]. Each country is resided by identical households, who consume differentiated commodities, supply labor, and own firms in their country. The monetary and fiscal policy is set by the government in each country. Governments do not consume, and set fiscal policy in the Ricardian way in the sense used by Benhabib, Schmitt-Grohé and Uribe (), among others. The detail of monetary policy is discussed later.. Preferences A representative household in H has preferences given by U = E { β t ũ(c t ) ṽ l t (i) ] } di, () n H N H where < β <, σ >, and l t (i), i N H, is the supply of type-i labor, which is used to produce differentiated product i. We assume that ũ and ṽ have constant elasticity: ũ(c) C σ σ, ṽ(l) + ω l+ω. The consumption index for the home household, C t, is given by C t = n ρ ρ H C ρ H,t ρ + n ρ F C ρ F,t ] ρ ρ, () where C H,t and C F,t are the consumption indexes of home and foreign goods consumed by the home household, respectively, which are defined by: C j,t = n θ j c t (i) θ θ N j di ] θ θ, j = H, F. (3) Here, θ > and c t (i) N j is the home household s consumption of good i produced in country j {H, F }. It is convenient to define the function u(c H, C F ) by ( ρ ρ ] ρ ) ρ u(c H, C F ) ũ nh C ρ H + n ρ F C ρ ρ F The lifetime utility of a representative household in F takes the same form as that of the home household: U = E β t { ũ(c t ) n F ṽ l t (i) ] } di. (4) N F 5

6 The consumption indexes for the foreign household, {Ct, CH,t, C F,t }, are defined as in () and (3): Ct = n C j,t = ρ H n θ j ρ C ρ H,t ρ + n F c t (i) θ θ N j ρ ] ρ C ρ ρ F,t, (5) di ] θ θ, j = H, F. (6) Corresponding to the consumption indexes in country H, C t, C j,t, j = H, F, the prices indexes, P t, P j,t, j = H, F, are defined as n H P ρ H,t ] + n F P ρ ρ F,t, P t = P j,t = p t (i) θ di n j N j ] θ, j = H, F, (7) where p t (i), i N j, j {H, F }, is the price of good i produced in country j quoted in the home currency. The price indexes in country F, Pt, Pj,t, j = H, F, are defined similarly by individual good prices, p t (i), i N j, j {H, F }, quoted in the foreign currency. We assume that the law of one price holds: p t (i) = E t p t (i), for all i N j, j {H, F }, where E t is the nominal exchange rate, defined as the price of foreign currency in terms of home currency. It follows that P j,t = E t Pj,t, j = H, F and P t = E t P t. Cost minimization leads to the derived demands of the home household: ( ) ρ Pj,t C j,t = n H C t, j = H, F, P t c t (i) = ( ) pt (i) θ C j,t, j = H, F. (8) n j P j,t The derived demands of the foreign household are written similarly.. Utility maximization We assume worldwide complete markets. The flow budget constraint for the home household is P t C t + E t Q t,t+ W t+ ] = W t + wt (i)l t (i) + Π t (i) ] di + T t, (9) N H where E t is the conditional expectation operator, Q t,t+ is the stochastic discount factor between dates t and t + for nominal payoffs in the home country, W t+ is the portfolio of 6

7 one-period state-contingent bonds, w t (i) is the date-t nominal wage rate for type i N H labor, Π t (i) is the date-t nominal profits from sales of good i N H, and T t is the nominal lump-sum transfers from the home government. To prevent Ponzi schemes, the home household also faces the natural debt limit (Ljungqvist and Sargent, ): W t+ s=t+ E t Q t+,s { N H ws (i)l s (i) + Π s (i) ] di + T s }], () where Q t,s is the stochastic discount factor between dates t and s, defined by Q t,s = s τ=t+ Q τ,τ. Given the initial asset holding, W, the home household maximizes the lifetime utility () subject to (9) and (). The flow budget constraint for the foreign household is analogously expressed: E t Pt C t + E t Q t,t+ E t+ Wt+] = E t Wt + E t w t (i)l t (i) + Π t (i) ] di + Tt, N F where W t+ is the portfolio of the state-contingent bonds in the foreign currency, w t (i), is the nominal wage rate of type i N F labor, Π t (i) is the nominal profit from sales of good i N F, and Tt is the nominal transfer from the foreign government. Given the initial asset holding W, the utility maximization problem for the foreign household is defined similarly to that for the home household. The first-order conditions that {C t, C t, l t (i), l t (i)} must satisfy are given by βũ c (C t+ ) ũ c (C t ) = βũ c(c t+ ) ũ c (C t ) = Q t,t+ P t+ P t, () and ṽ l l t (i)] n H ũ c (C t ) = w t(i) ṽ l l t (i)], i N H, P t n F ũ c (Ct ) = w t (i) Pt, i N F. As we shall see, what the policy makers should stabilize is not P t or Pt, but P H,t and PF,t. For this reason, the first-order conditions in terms of C j,t and Cj,t, j = N, H, turn out to be more relevant. They are given by βu H (C H,t+, C F,t+ ) u H (C H,t, C F,t ) βu F (C H,t+, C F,t+ ) u F (C H,t, C F,t ) = βu H(C H,t+, C F,t+) u H (C H,t, C F,t ) = βu F (C H,t+, C F,t+) u F (C H,t, C F,t ) = Q t,t+ P H,t+ P H,t () = Q t,t+ P F,t+ P F,t (3) ṽ l l t (i)] n H u H (C H,t, C F,t ) = w t(i) P H,t (4) ṽ l l t (i)] n F u F (CH,t, C F,t ) = w t (i) PF,t (5) u F (C H,t, C F,t ) u H (C H,t, C F,t ) = u F (CH,t, C F,t ) u H (CH,t, C F,t ) = E tpf,t P H,t (6) 7

8 Here, u j (C H, C F ) denotes the partial derivative of u(c H, C F ) with respect to C j, j = H, F..3 Equilibrium shares of consumption Since asset markets are complete, our formulation of the lifetime utility, () and (4), implies that, given a policy rule of each country and the initial asset holdings, W and W, the relative amount of consumption, C t/c t, becomes constant at all dates and all contingencies in equilibrium. But this amount, C t /C t, depends, in general, upon the policy rule adopted in each country. In particular, the equilibrium amount of C t /C t would depend on the process of prices, {P H,t, PF,t }, which, in turn, depends on the monetary policy conducted in the two countries. For our welfare analysis, it is convenient if the share of consumption is determined independently of policy. The next proposition shows that this is the case when the elasticity of substitution between C H and C F is unity and when the initial financial asset of the household in country j equals the initial liabilities of the country-j government. The latter assumption implies that n H W = E Q,t P H,t Y H,t n H {T t + wt (i)l t (i) + Π t (i) ] }] di N H n F E W = E Q,t E t P F,tY F,t n F {T t + w t (i)l t (i) + Π t (i) ] }] di, N F where Y j,t n H C j,t + n F Cj,t is the aggregate amount of country-j goods, j {H, F }. Proposition. Assume that (a) the elasticity of substitution between home-goods and foreign-goods is unity: ρ = ; and (b) the initial financial asset of the household in country j {H, F } equals the (percapita amount of the) initial liabilities of the country-j government. Then the relative amount of consumption between the two countries is determined independently from the policy rule adopted in each country. Indeed, the equilibrium relative consumption equals unity: C t Ct = C H,t C H,t = C F,t C F,t = c t(i) c =, i, ]. t (i) The proof is in Appendix. In what follows we assume (a)-(b) in the proposition. Thus, the consumption indexes () and (5) become ( ) nh ( ) nf ( CH CF C C =, C = H n H n F 8 n H ) nh ( C ) nf F. n F

9 It follows that u(c H, C F ) is written as u(c H, C F ) = σ and the price index, P t, become ( CH n H ) nh ( σ) ( CF n F ) nf ( σ), P t = P n H H,t P n F F,t = E tp t = E t P n H H,t P n F F,t. Let y t (i), i N H, and y t (i), i N F, be the aggregate supply of home and foreign goods, respectively: y t (i) = n H c t (i) + n F c t (i), i N H, y t (i) = n H c t (i) + n F c t (i), i N F. The corresponding production indexes of home and foreign goods are ] Y H,t n θ θ H y t (i) θ θ θ di = nh C H,t + n F CH,t, N H Y F,t Y t n θ F ( YH,t n H It follows from the proposition that yt (i) θ θ N ) F nh ( YF,t n F ] θ θ di = nh C F,t + n F C F,t ) nf = n H C t + n F C t C H,t = C H,t = Y H,t, C F,t = C F,t = Y F,t, C t = C t = Y t. (7) 3 Aggregate supply 3. Technology For simplicity, we assume that the technology to produce each good is linear in labor: y t (i) = A t n H l t (i), i N H, y t (i) = A t n F l t (i), i N F, where A t and A t represent country-specific technology shocks. For later use, it is convenient to define random variables ξ t and ξ t by ξ t ( + ω) ln A t, ξ t ( + ω) ln A t, and also functions v(y; ξ) and v (y ; ξ ) by ( ) v(y; ξ) eξ y +ω ( y = ṽ + ω n H n H A ( ) v (y ; ξ ) eξ y +ω ( y = ṽ + ω n F A n F ), ). 9

10 Thus, v(y; ξ) and v (y ; ξ ) measure the disutility of producing y and y in the home and foreign countries, respectively, when their technology shocks are ξ and ξ. Note that v y (y; ξ) = ṽl(l) n H A, v y(y ; ξ ) = ṽl(l ) n F A. Thus, first-order conditions (4) and (5) are rewritten as v y y t (i); ξ t ] u H (Y H,t, Y F,t ) = A t w t (i) P H,t, v yy t (i); ξ t ] u F (Y H,t, Y F,t ) = A t wt (i) PF,t, (8) where we have used equilibrium condition (7). 3. Natural rates of output Each producer is assumed to be a wagetaker. 5 Using (8) and (7), nominal profits of a home supplier of good i N H at date t are given by ( τ)p t (i) w ] t(i) y t (i) = ( τ)p t (i) w ] ] t(i) YH,t pt (i) θ = n H Π t (i), A t A t n H P H,t where τ is the constant tax rate on firms revenue. Monopoly profits of a foreign firm is defined similarly with τ as the tax rate on its revenue. Let us define the natural rates of output (Woodford, 3) at date t, Y n H,t and Y F,t, as the levels of home and foreign output which would prevail in the flexible-price equilibrium. Suppose, momentarily, that all prices are fully flexible. Profit maximization leads to: ( Φ) p t(i) P H,t = ( Φ ) p t (i) P F,t w t(i) = v yy t (i); ξ t ] P H,t A t u H (Y H,t, Y F,t ), i N H, = w t (i) P F,t A t = v yy t (i); ξ t ] u F (Y H,t, Y F,t ), i N F, where we have used (8) and Φ and Φ are the measures of distortions due to market power defined by Φ = θ θ ( τ), Φ = θ ( τ ). θ In the flexible-price equilibrium, p t (i) = P H,t and y t (i) = Y H,t /n H for all i N H, and p t (i) = P F,t and y t (i) = Y F,t /n F for all i N F. Thus, the natural rates of output, Y n H,t and YF,t n, are determined by v y Y n H,t /n H; ξ t ] u H (Y n H,t, Y n F,t ) = Φ, v yy n F,t /n F ; ξ t ] u F (Y n H,t, Y n F,t ) = Φ. (9) 5 See Woodford (3, Section 3.) for how to make this assumption consistent with the supposition that each producer uses a different type of labor.

11 3.3 New Keynesian aggregate supply relation Now suppose that goods prices are adjusted at random intervals as in Calvo (983). Let α be the probability that each good price remains unchanged in each period. We assume that this probability is identical in the two countries. Consider the price adjustment in the home country. Suppose that the price of good i N H can be adjusted at date t. The supplier of that good chooses p t (i) to maximize the expected discounted profits: T =t E t T =t { α T t Q t,t ( τ)p t (i) w ] T (i) YH,T A T n H ] } pt (i) θ The first-order condition for profit maximization is written as { ( ) E t α T t Y H,T pt (i) θ vy y T (i); ξ T ] Q t,t n H P H,T u H (Y H,T, Y F,T ) ( Φ) p ] } t(i) =. () P H,T P H,T It follows that all producers that change their prices at date t choose the same price. Let z t denote this common new price. As is shown in Appendix, log-linearization of () and the corresponding equation for the foreign country leads to the New Keynesian aggregate-supply relations: π H,t = γ H x H,t + γ HF n F x F,t + βe t π H,t+, () π F,t = γ HF n H x H,t + γ F x F,t + βe t π F,t+. () Here π H,t and πf,t are the inflation rates of goods produced in the home and foreign countries, respectively: π H,t ln P H,t ln P H,t, π F,t ln P F,t ln P F,t, x j,t is the output gap in country j = H, F : and the coefficients are given by x j,t ln Y j,t ln Y n j,t, γ H ζ + ω + (σ )n H ] >, (3) γ HF ζ(σ ), (4) γ F ζ + ω + (σ )n F ] >, (5) ζ α αβ α + ωθ. (6)

12 4 Welfare approximation In this section we follow Woodford (3) and Benigno and Woodford (4) to derive an approximate world welfare criterion and discuss optimal policy rules for the case where the zero lower bound (ZLB) on the nominal interest rates never binds. We assume that the monetary authorities in the two countries cooperate to maximize aggregate utility. The case where monetary policy is set in the noncooperative fashion is discussed in Appendix. Given that preferences of the home and foreign household are given by () and (4), respectively, the level of average expected utility between the two countries is given by n H U + n F U = E Using the conditions that y t (i) = n H Y H,t β {u(y t H,t, Y F,t ) v ] y t (i); ξ t di v } yt (i); ξ ] t di N H N F pt (i) P H,t ] ] θ θ, yt (i) = p t (i) Y F,t n F PF,t, we obtain N H v y t (i); ξ t ] di = nh v N F v y t (i); ξ t ( ) YH,t ; ξ t H,t, n H ] di = nf v ( YF,t n F ; ξ t ) F,t, where H and F are the measures of price dispersion defined by H,t n H F,t n F N H N F Then the world welfare measure is given by ( ) pt (i) θ(+ω) di, (7) ( P H,t p t (i) P F,t ) θ(+ω) di. n H U + n F U = E β {u(y t H,t, Y F,t ) (8) ( ) ( ) } YH,t n H v ; ξ t H,t n F v YF,t ; ξt F,t n H n F Now, fix a non-stochastic steady state with zero inflation. In what follows, a bar over a variable denotes its steady state value, and a hat indicates the log-deviation from the steady-state value. Following Benigno and Woodford (4), we take a second-order approximation of (8) around that steady state in terms of Ξ (ξ, ξ /, ˆ H,, / F,, ϕ),

13 where ϕ are parameters of policy rules normalized in such a way that ϕ = implies longrun output levels Y j, = Ȳj, j = H, F, and for any small enough ϕ, ln Y j, ln Ȳj = O( ϕ ), j = H, F. Then, as shown in Appendix, ( ) ( YH,t YF,t u(y H,t, Y F,t ) n H v ; ξ t H,t n F v ; ξt n H n F { = ūhȳh n H ΦŶH,t + n F Φ Ŷ F,t + n H n H + ( σ)n H n F Ŷ H,t Ŷ F,t + n F ) F,t (9) ( σ)n H ( + ω)( Φ)]Ŷ H,t ( σ)n F ( + ω)( Φ )]Ŷ F,t n H ( Φ)ŶH,tξ t n F ( Φ )ŶF,tξ t n H( Φ) + ω + t.i.p. + O( Ξ 3 ), ˆ H,t n F ( Φ ) + ω where t.i.p. denotes the terms independent of policy. Also, it is shown that β t β t n H + ω ˆ H,t = n F + ω ˆ F,t = αθ( + ωθ) ( α)( αβ) αθ( + ωθ) ( α)( αβ) 4. Small distortions at the steady state ˆ F,t } β t n H π H,t + t.i.p. + O( Ξ 3 ), (3) β t n F π F,t + t.i.p. + O( Ξ 3 ). (3) The existence of linear terms in (9), n H ΦŶH,t + n F Φ Ŷ F,t, might complicate the analysis. Here, for simplicity, we follow Woodford (3) and assume that Φ and Φ are small and treat them as expansion parameters in the Taylor series approximation. 6 j = H, F, denote the efficient levels of output in the absence of shocks, that is, v y (Y e H /n H; ) u H (Y e H, Y e F ) =, v y(y e F /n F ; ) u F (Y e H, Y e F ) =. Then, let x e j, j = H, F, denote the efficient levels of the output gaps: x e j ln Y e j ln Ȳj, j = H, F, where Ȳj, j = H, F, are the steady-state levels of output. When Φ and Φ are small, + ω + (σ )nh ] x e H + (σ )n F x e F = Φ + O( Φ ), (σ )n H x e H + + ω + (σ )n F ] x e F = Φ + O( Φ ) 6 Benigno and Woodford (4) study the case where Φ is not small. Let Y e j, 3

14 Then, using (9)-(3), a second-order approximation of the world welfare measure (8) is given by n H U + n F U = ūhȳh n H θ ζ β t L t + t.i.p. + O( Φ, Φ, Ξ 3 ), where ζ is as defined in (6), and the loss function L t is given by L t (x t x e ) Λ(x t x e ) + n H π H,t + n F π F,t, (3) Here, x t (x H,t, x F,t ), x e t (x e H, xe F ), and Λ is defined by ] Λ γh n H γ HF n H n F, θ γ HF n H n F γ F n F where γ H, γ HF, and γ F are defined in (3)-(5). Our welfare measure clearly shows that what must be stabilized are domestic inflation, π H,t and π F,t, rather than CPI inflation rates, π t and π t, as shown by Clarida, Galí and Gertler () and Benigno and Benigno (3) in different contexts. In particular, fluctuations in nominal exchange rate per se do not affect the world welfare. 4. Optimal targeting rule in the case where the ZLB never binds In this subsection we analyze optimal policy when the zero bound on the nominal interest rates never binds. In particular, we derive robustly optimal target criteria (Giannoni and Woodford, ; Woodford 3). Interestingly, those targeting rules have exactly the same form with the same parameter values as those obtained in the closed-economy model. 7 In order to introduce a tradeoff between inflation and output stabilization, we modify the aggregate-supply relation ()-() so that π H,t = γ H x H,t + γ HF n F x F,t + βe t π H,t+ + u H,t, (33) πf,t = γ HF n H x H,t + γ F x F,t + βe t πf,t+ + u F,t, (34) where u j,t, j = H, F, are interpreted as cost-push shocks (Woodford, 3). Suppose that the zero bounds on the nominal interest rates never binds. Then the optimal policy commitment is derived by solving min(/)e t L t subject to (33)-(34). 7 Clarida, Galí and Gertler () derive an optimal inflation targeting rule for a small open economy. 4

15 The Lagrangian is formed as { L = E β t (x t x e t) Λ(x t x e t ) + n H π H,t + n F π F,t ( ) + ψ H,t n H π H,t γ H n H x H,t γ HF n H n F x F,t βn H π H,t+ The first-order conditions are + ψ F,t (n F π F,t γ HF n H n F x H,t γ F n F x F,t βn F π F,t+ Λ(x t x e t ) = θλψ t, π j,t = ψ j,t + ψ j,t, j = H, F, where ψ t (ψ H,t, ψ F,t ). Eliminating ψ t, we obtain a robustly optimal target rule: 8 π H,t + θ (x H,t x H,t ) =, π F,t + θ (x F,t x F,t ) =. (35) Note that these targeting rules have exactly the same form with the same parameter values as those obtained for the closed economy (for example, by Woodford, 3). It follows that, under our parametric assumptions and as long as the zero bound does not bind, the world welfare is maximized when each monetary authority acts independently and adopts the policy rule which is optimal for the closed economy. Later, we shall see that this is not the case when the zero bound binds. ) } 5 Nominal interest rates and the ZLB Let i H,t and i F,t be the (one-period) nominal interest rates at date t in the home and foreign countries, respectively. By definition, + i H,t = E t Q t,t+ ], + i F,t = E t ] E t+ Q t,t+. E t Using first-order conditions ()-(3) and equilibrium condition (7), we could relate the nominal interest rates and the growth rates of output in the following way: ] ] βũc,t+ P t βuh,t+ P H,t βuf,t+ = E t = E t = E t + i H,t + i F,t = E t ũ c,t P t+ βũc,t+ Pt ũ c,t Pt+ u H,t ] = E t βu H,t+ u H,t P H,t+ P H,t P H,t+ u F,t ] = E t βu F,t+ u F,t P F,t P F,t+ P F,t P F,t+ 8 The fully date- optimal solution only satisfy (35) for t. As is well known, such a solution is not time consistent. As is discussed in Woodford (3), however, requiring (35) for t makes the solution optimal from a timeless perspective, and the policy problem time consistent. ], ], 5

16 where ũ c,s = ũ c (Y s ) and u j,s = u j (Y H,s, Y F,s ), for s = t, t + and j = H, F. Whether or not the zero bound binds depends on the policy rules that the monetary authorities adopt. A given shock may or may not result in i H,t = depending on, for example, whether the home monetary authority tries to stabilize CPI, P t, or the domestic price level, P H,t. In the previous section, we have seen that the prices which have to be stabilized are not CPI s, P t or Pt, but the producer price levels, P H,t and PF,t. In this sense, it is natural to use + i H,t = E t for our policy analysis. βuh,t+ P H,t u H,t P H,t+ ], and βu F,t+ = E t + i F,t u F,t P F,t P F,t+ As in Woodford (3), it is convenient to introduce new variables, r n H,t and rn F,t, called the natural rates of interest. They are defined as the real interest rates which would realize if the domestic price levels, P H,t and PF,t, were completely stabilized for all t. It follows that r n H,t and rn F,t satisfy + r n H,t = E t βuh (Y n H,t+, Y n F,t+ ) u H (Y n H,t, Y n F,t ) ], and + r n F,t βuf (YH,t+ n = E, Y F,t+ n ) ] t u F (YH,t n, Y F,t n ) At the non-stochastic steady state, r n j,t = r β, j = H, F. Note that even if P H,t and P F,t were completely stabilized, the nominal exchange rate, E t, may fluctuate. Thus, rh,t n rn F,t in general. Using these, log-linearizing (36) yields ] (36) (37) { ] i t,h = E t + (σ )nh (xh,t+ x H,t ) (38) } + (σ )n F (x F,t+ x F,t ) + π H,t+ + rh,t n { ] i t,f = E t + (σ )nf (xf,t+ x F,t ) (39) } + (σ )n H (x H,t+ x H,t ) + πf,t+ + rf,t n 6 Optimal policy commitment in the case where the ZLB may bind We say that country j is in a liquidity trap if i j,t hits the zero lower bound. In this section, we consider optimal policy coordination in the case where the home country may fall into a liquidity trap, that is, the case in which the inequality in (38) may bind but (39) does not. Our numerical exercise is an open-economy extension of Eggertsson and Woodford (3). 6

17 We restrict attention to the case where there is no distortion at the steady state, Φ = Φ =, and there are no cost push shocks, u t = u t =, all t. Thus, the aggregatesupply relations are given by ()-(), and the loss function L t in (3) becomes L t x tλx t + n H π H,t + n F π F,t, (4) It follows that, without the zero bound, the optimal policy would simply be the one that attains π H,t = π F,t = x H,t = x F,t =, for all t. Assume that the world economy is in the steady state with zero inflation at date t =, namely, the one around which we have approximated the world welfare, the aggregate-supply relations, and the intertemporal Euler equations. Regarding exogenous shocks, we suppose that the productivity processes, ξ t and ξ t, are such that the natural rates of interest, rh,t n and rn F,t, follow the following stochastic processes: (i) The natural rate of interest in the foreign country equals the steady-state level at all times, rf,t n = r, all t. (ii) The natural rate of interest in the home country gets negative at date, rh, n = r, for some r <. In the following periods, it returns to the steady-state level at a constant probability, µ (, ]. That is, given that rh,t n = r, the conditional distribution of r n H,t is given by r H,t = { r, with probability µ, r, with probability µ. Once it gets back to r, it stays there from then on: if r n H,t = r, then rn H,t probability one. = r with The parameters are calibrated as follows. One period in the model corresponds to a quarter. We set α =.66, β =.99, θ = 7.88, ω =.47, which are taken from Woodford (3, Table 5.). We follow Eggertsson and Woodford (3) to set r =./4. For σ, three values are considered: σ =.5,, 5. As we shall see below, the value of σ affects how the foreign monetary authority should react to the liquidity trap that occurs in the home country. The two countries have an equal size: n H = n F = /. Finally, µ =.5, that is, the home natural rate of interest remains negative for a year on average. As discussed below, the value of µ is chosen so that zero-inflation targeting generates deflation during a liquidity trap. 6. Zero inflation targeting We first examine the consequences of strict zero inflation targeting, that is, the policy that achieves zero inflation whenever possible. It follows from the aggregate-supply relations ()-() and the Euler equations (38)-(39) that it is indeed possible to attain π H,t = πf,t = 7

18 x H,t = x F,t = after rh n returns to the steady-state level, r. When rn H,t = r, however, setting both of the two inflation rates to zero is not possible. Under our parametric assumptions, during the period when rh,t n = r, π H,t <, x H,t <, πf,t =, and x F,t as σ. Here, x F,t = when σ =, because, as the aggregate-supply relations ()-() show, when σ =, the inflation rate and the output gap in each country are completely separately determined. The state-contingent paths of π H,t, x H,t, and x F,t are drawn in Figure. 9 In each panel, the j-th (solid, dashed, dashdot) line corresponds to the sample path where rh,t n = r for t =,..., j, and rn H,t = r for t j. This example fits well with the casual statement that a bad policy during a liquidity trap results in deflation (π H,t < ) and a negative output gap (x H,t < ). It is not generally true, however. For example, Figure depicts the state-contingent paths of the same variables under strict zero inflation targeting when µ =.. In this case, there is inflation, π H,t >, and a positive output gap, x H,t >, during a liquidity trap. Indeed, we can show that there is a threshold value, µ, such that the home country experiences deflation for µ > µ during a liquidity trap, and inflation otherwise. To gain an intuition, consider the closed-economy case, and suppose that µ =, that is, the natural rate of interest remains negative for all t. It follows from the Fisher equation that the inflation rate equals the difference between the nominal and the real interest rates. Since the latter is negative and the former is zero, the inflation rate is positive. In what follows, we assume µ =.5 so that strict zero inflation targeting implies deflation in the home country during the liquidity trap. We have checked numerically that the qualitative properties of the optimal state-contingent paths of inflation and output gaps are not affected much by the choice of the value of µ. 6. Optimal policy The optimal policy problem is formulated as the minimization problem of the discounted expected value of the loss function (4) subject to the aggregate-supply relations ()-() and the inequality constraints (38)-(39). Letting ψ j,t and φ j,t, j = H, F, be the Lagrange 9 In all figures, inflation rates and interest rates show annual rates. 8

19 multipliers associated with those constraints, the Lagrangian is formed as { L = E β t x tλx t + n H π H,t + n F π F,t ( ) + ψ H,t n H π H,t γ H n H x H,t γ HF n H n F x F,t βn H π H,t+ + ψ F,t (n F π F,t γ HF n H n F x H,t γ F n F x F,t βn F π F,t+ + φ H,t (n H + (σ )n H ](x H,t x H,t+ ) + (σ )n H n F (x F,t x F,t+ ) n H π H,t+ r n H,t + φ F,t ((σ )n H n F (x H,t x H,t+ ) + n F + (σ )n F ](x F,t x F,t+ ) n F π F,t+ r n F,t Letting π t (π H,t, π F,t ), x t (x H,t, x F,t ), ψ t (ψ H,t, ψ F,t ), and φ t (φ H,t, φ F,t ), the first-order conditions are written as Λx t θλψ t + A(φ t β φ t ) =, (4) ) ) ) } π t + ψ t ψ t β φ t =, (4) where the matrix A is defined by ] nh + (σ )n H ] (σ )n H n F A (σ )n H n F n F + (σ )n F ] Eliminating ψ and φ in (4)-(4), we obtain the following optimal targeting rule for the logged price levels, p t (ln P H,t, ln P F,t ), with possibly binding ZLB. It extends the result of Eggertsson and Woodford (3) to the open-economy model. (i) In each period, there is a predetermined price-level target ˇp t = (ˇp H,t, ˇp F,t ). central banks choose the interest rates i H,t and i F,t to achieve the target relation: The p t = ˇp t if this is possible; if it is not possible because of the zero bound, then i H,t = and/or i F,t =. Here, p t = ( p H,t, p F,t ) is output-gap adjusted price indexes defined by p t p t + θ x t. In the terminology of Woodford (3), this rule is optimal from a timeless perspective, and hence time-consistent. 9

20 (ii) The target for the next period is determined as ˇp t+ = ˇp t + β (I + θa Λ)δ t β δ t (43) where δ t is the target shortfall in period t: δ t ˇp t p t. In the price-target equation (43), ] θa Λ = ζ σ + ω + (σ )ωnf (σ )ωn F, σ (σ )ωn H σ + ω + (σ )ωn H which is not a diagonal matrix. It follows that the target price levels, ˇp H,t and ˇp F,t, are not separately determined. This is in contrast to the case where the zero bound never binds, as we have seen in the previous section. Figure 3 shows the optimal state-contingent paths of inflation, output gaps, and nominal interest rates in the two countries. Figure 4 compares the paths of those variables under optimal policy and under strict zero inflation targeting in the case where rh,t n = r for t =,,..., 9 and σ = 5. The optimal state-contingent paths of π H,t, x H,t and i H,t are remarkably similar to those obtained for the closed-economy by Eggertsson and Woodford (3). When the shock hits the home country at date, the home inflation rate π H and the home output gap x H become negative, but by committing to generate inflation after the natural rate rh n turns back to normal, its impact is curbed to minimal. As Eggertsson and Woodford (3) emphasize, under optimal policy commitment, the home nominal interest rate, i H,t, typically is set to zero for a few periods after the home natural rate, rh,t n, returns to the steady-state level, r. As Figure 4 shows, the deviations of π H,t and x H,t are much smaller under optimal policy commitment than under strict zero inflation targeting. Note that such stabilization is made possible by the coordinating monetary policy in the foreign country unless σ =. During periods where rh,t n = r, the foreign nominal interest rate, i F,t, must be set lower (higher) than the steady state level when σ < (σ > ). It is true that such policy coordination generates (small) inflation/deflation in the foreign country, the output gaps in the foreign country are made much smaller than under zero inflation targeting. 7 Liquidity trap and the exchange rate In this section, we discuss the optimal state-contingent paths of the nominal exchange rate. We are particularly interested in examining the extent to which Svensson s (,

21 3, 4) proposal that emphasizes currency depreciation to escape optimally from a liquidity trap holds true in our setting. First-order condition (6) for utility maximization implies that the nominal exchange rate, E t, must satisfy E t = n H P H,t Y H,t n F PF,t. Y F,t Note that E t is the price of the foreign currency quoted in the home currency. To determine dynamic paths of the nominal exchange rate, it is not enough to specify the stochastic process of the natural rates of interest, rh,t n and rn F,t. We need to identify the process of the (growth rates of the) natural rates of output, YH,t n and Y F,t n. By definition of the natural rates of interest, (37), we obtain ) Y n ( σ)nh ( ) β( + rh,t n ) = E t ( H,t+ Y n ( σ)nf F,t+ YH,t n YF,t n ) Y n ( σ)nh ( ) β( + rf,t n ) = E t ( H,t+ Y n ( σ)nf F,t+ YH,t n YF,t n Log-linearization of these equations yields lnβ( + r n H,t)] = ( σ)n H ]E t g n H,t+ + ( σ)n F E t g n F,t+ lnβ( + r n F,t)] = ( σ)n H E t g n H,t+ + ( σ)n F ]E t g n F,t+ where gj,t+ n, j = H, F, are the growth rates of the natural rates of output in country j: g n j,t+ ln(y n j,t+/y n j,t), j = H, F. Since β( + rf,t n ) =, all t, we obtain ] ] Et gh,t+ n = ( nf + σn F ) lnβ( + r E t gf,t+ n σ H,t)] n ( σ)n H This equation relates the natural rate of interest rh,t n at date t and the expected growth rates of the natural rates of output in the next period, E t gj,t+ n, j = H, F. Thus, given a process of rh,t n, there are many processes of gn j,t+, j = H, F, which are consistent of it. To be specific, we assume that, given rh,t n at date t, the growth rate of the natural rates of output in the next period, gj,t+ n, j = H, F, are perfectly forecastable. That is, we assume that gj,t+ n, j = H, F, t, follow the process given by ] ] g n H,t+ = ( nf + σn F ) lnβ( + r σ H,t)] n ( σ)n H g n F,t+

22 For normalization, we assume that E = and gj, n =, for j = H, F. Note that the value of g j, is irrelevant for rh,t n, t. We discuss the normalization regarding gn j, later, when we compare our result to Svensson s proposal. Let ɛ t ln E t. Then our parametric assumptions imply that ɛ t = ɛ t + lnβ( + r n H,t )] + π H,t π F,t + x H,t x H,t (x F,t x F,t ). Figure 5 draws the optimal state-contingent paths of the log nominal exchange rate, ɛ t. The figure shows that, regardless of the value of σ, the optimal nominal exchange rate of the home currency appreciates during periods when rh,t n <. The optimal evolution of the nominal exchange rate does not exhibit a depreciation as opposed to the proposal of Svensson. Where does the difference between our results and Svensson s come from? His theoretical analysis is given in Svensson (4). His model is different from ours in several ways. For instance, in his model, all prices are set one period in advance; the home country may be small or large; in the large-country case, there may or may not be international policy coordination. However, the difference in the results is not due to those differences in the models considered. Instead, it owes to the difference in the normalization of g n H,. To illustrate it, following Svensson (4), consider the case in which the home natural rate of interest becomes negative only at date : r n H, = r, r n H,t = r, for t. This implies that the natural rate of home output, Y n H,t, evolves as Y n H, > Y n H, = Y n H, =, or equivalently, gh, n < and gn H,t =, for t. The process of rn H,t does not pin down gh, n. What we have assumed above is that gn H, =, i.e., Y H, n = Y H, n. In contrast, what Svensson (4) assumes is that YH, n > Y H, n = Y H,t n, t, and so, g n H, >, g n H, <, g n H,t =, for t. Thus, he considers the case in which r n H, < at date because the home country experiences an unusually high level of productivity. Figure 6 shows the processes of gh,t n assumed here (panel a) and in Svensson (4) (panel b). Figure 7 plots the optimal paths of the nominal exchange rate, ɛ t, (a) when gh, n = and (b) when gh, n >, for the three values of σ. Case (a), corresponding to our results in Figure 5, shows that the home currency appreciates during a liquidity trap. In contrast, corresponding to Svensson (4), case (b) shows depreciation of the home currency at date. The home currency depreciates in case (b), because the home country produces goods more than usually so that the relative price of home goods must fall.

23 Note, however, that our difference lies only in the choice of gh, n, and hence, in ɛ. It follows that in the case where the shock lasts for more than one period, the evolution of nominal exchange rates for t would be independent of the assumption on gh, n, and thus, the home currency would appreciate for t even in the model of Svensson (4). Hence, our previous remark that the home currency appreciates as long as rh,t n = r along the optimal path holds true for t, regardless of the choice of gh, n. Finally, Figure 8 compares the optimal path of the nominal exchange rate and its path under strict zero inflation targeting in the case where rh,t n = r for t =,,..., 9. The nominal exchange rate under strict zero inflation targeting shows much larger appreciation of the home currency than under optimal policy commitment. This is because, under our parametric assumptions, strict zero inflation targeting leads to very large deflation as shown in Figure. 8 Conclusions We have analyzed optimal policy in an open economy. First, when the ZLB for nominal interest rate is assumed never to bind, the optimal inflation targeting rule for each country is exactly the same with the same parameter value as the one for the closed-economy model (Woodford, 3). Indeed, in such a case, each monetary authority can forget about international dependence in setting its inflation target, in order to maximize the world welfare. Second, this is not the case when the ZLB may bind. The optimal pricelevel target rule shows significant international dependence. The optimal state-contingent paths of inflation, output gaps, and nominal interest rates for the country in a liquidity trap look very similar to those obtained for the closed-economy model by Eggertsson and Woodford (3). However, such paths are made possible by policy coordination by the other country. Third, in spite of the proposal of Svensson (, 3, 4), the nominal exchange rate of the country in a liquidity trap appreciates under optimal policy (except possibly at the initial date). We have restricted our attention to the case where the steady-state distortions due to market power is (close to) zero. Benigno and Woodford (4) have developed the approach to deal with the case of a distorted steady state. Applying their approach to open economies is left for future research. 3

24 References ] Benhabib, J., S. Schmitt-Grohé and M. Uribe (), Monetary policy and multiple equilibria, American Economic Review, 9, ] Benigno, G. and P. Benigno (3), Price stability in open economies, Review of Economic Studies, 7, ] Benigno, P. and M. Woodford (4), Inflation stabilization and welfare: The case of a distorted steady state, mimeo. 4] Calvo, G. (983), Staggered prices in a utility-maximizzing framework, Journal of Monetary Economics,, ] Clarida, R., J. Galí and M. Gertler (999), The science of monetary policy: A new Keynesian perspective, Journal of Economic Literature, 37, ] Clarida, R., J. Gali and M. Gertler (), Optimal monetary policy in open versus closed economies: An integrated approach, American Economic Review, 9, ] Corsetti, G. and P. Pesenti (), Welfare and macroeconomic interdependence, Quarterly Journal of Economics, 6, ] Eggertsson, G. and M. Woodford (3), The zero bound on interest rates and optimal monetary policy, Brookings Papers on Economic Activity, 3:, ] Galí, J. and T. Monacelli (), Monetary policy and exchange rate volatility in a small open economy, NBER Working Paper 895. ] Giannoni, M. P. and M. Woodford (), Optimal interest-rate rules: I. General theory, NBER Working Paper No ] Jung, T., Y. Teranishi and T. Watanabe (), Zero bound on nominal interest rates and optimal monetary policy, Discussion Paper no. 55, Kyoto Institute of Economic Research. ] Krugman, P. (998), It s baaack! Japan s slump and the return of the liquidity trap, Brookings Papers on Economic Activity, 998:, ] Ljungqvist, L. and T. J. Sargent (), Recursive macroeconomic theory, Cambridge, MA: MIT Press. 4

25 4] Obstfeld, M. and K. Rogoff (), Global implications of self-oriented national monetary rules, Quarterly Journal of Economics, 7, ] Svenssion, L. E. O. (), The zero bound in an open-economy: A foolproof way of escaping from a liquidity trap, Monetary and Economic Studies, 9, pp ] Svensson, L. E. O. (3), Escaping from a liquidity trap and deflation: The foolproof way and others, Journal of Economic Perspective, 7:4, ] Svensson, L. E. O. (4), The maginc of the exchange rate: Optimal escape from a liquidity trap in small and large open economies, mimeo (Version., July 4). 8] Woodford, M. (3), Interest and prices, Princeton, NJ: Princeton University Press. 5

26 A Appendix A. Proof of the proposition Here, we prove the proposition in Section.3. Then the consumption indexes become ( ) nh ( ) nf ( CH CF C C =, C ) nh ( = H C ) nf F. n H n F n H n F Under the natural debt limit, the household s utility maximization problem is reexpressed by using the lifetime budget constraints: and ( ) E Q,t PH,t C H,t + P F,t C F,t E W + E Q,t wt (i)l t (i) + Π t (i) { N ] } di + T t, H ( Q,t E t P H,t CH,t + PF,tC F,t ) E W + E Q,t E t w t (i)l { N t (i) + Π t (i) ] di + Tt F }. (44) (45) Let λ and λ be the Lagrange multipliers associated with the lifetime budget constraint for the home household and for the foreign household, respectively. The first-order conditions for utility maximization of the home household with respect to C j,t are given by β t ( CH,t n H β t ( CH,t n H ) nh ( σ) ( CF,t ) nh ( σ) ( CF,t n F ) nf ( σ) n F ) nf ( σ) = λq,t P H,t, = λq,t P F,t. Analogous conditions hold for the foreign household. Those conditions imply that P H,t C H,t P F,t C F,t C H,t C H,t = P H,tCH,t P F,t CF,t = n H n F (46) ( ) λ σ = λ (47) = C F,t C F,t The net revenue of the home government at date t is P H,t Y H,t n H {T t + wt (i)l t (i) + Π t (i) ] } di, N H 6

27 where Y H,t n H C H,t + n F C F,t. Since W equals the initial liabilities of the home government, the lifetime budget constraint for the home government implies that n H W = E Q,t P H,t Y H,t n H {T t + wt (i)l t (i) + Π t (i) ] }] di. (48) N H Similarly, the lifetime budget constraint for the foreign government is given by n F E W = E Q,t E t PF,tY F,t n F {T t + w t (i)l t (i) + Π t (i) ] }] di. (49) N F Note that the lifetime budget constraint for each household holds with equality in equilibrium. Then, combining the governments and households lifetime budget constraints, we obtain the equilibrium condition: E Q,t (n H P F,t C F,t n F P H,t CH,t) =. (5) Using (46)-(47), this equation is rewritten as { ( ) } λ σ λ E Q,t P H,t C H,t =. Thus, λ = λ in equilibrium, and C t Ct = C H,t C H,t = C F,t C F,t =. A. Steady-state equilibrium It is convenient to define the real marginal cost of producing good i N j in country j {H, F } as s y t (i), Y H,t, Y F,t ; ξ t ] v y y t (i); ξ t ] u H (Y H,t, Y F,t ) = A t w t (i) P H,t (5) s yt (i), Y H,t, Y F,t ; ξt ] vyy t (i); ξt ] u F (Y H,t, Y F,t ) = A t w t (i) P F,t Consider a non-stochastic steady state with zero inflation: ξ t = ξ t =, and p t (i) = P H,t = P H, i N H, p t (i) = P F,t = P F, i N F, E t = E. In such a steady state, aggregate output levels, ȲH and ȲF, are determined by ) ) (ȲH (ȲF s, n ȲH; = Φ, and s, H n ȲF ; = Φ. F Output of individual products are y t (i) = ȲH n H, i N H, and y t (i) = ȲF n F, i N F. (5) 7