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Optimal Monetary Policy with Uncertain Fundamentals and Dispersed Information Guido Lorenzoni∗ March 2008 Abstract This paper studies monetary policy in an economy where output ﬂuctuations are driven by the private sector’s uncertainty about the economy’s fundamentals. I consider an economy where information on aggregate productivity is dispersed across agents and there are two aggregate shocks: a standard productivity shock and a “noise shock” aﬀecting public beliefs about aggregate productivity. Neither the central bank nor individual agents can distinguish the two shocks when they hit the economy. The main results are: (1) despite the lack of superior information, an appropriate monetary policy rule can change the economy’s response to the two aggregate shocks; (2) monetary policy can achieve “full aggregate stabilization,” that is, an equilibrium where aggregate activity is the same as in the case of full information; (3) under optimal monetary policy, the economy achieves a constrained eﬃcient allocation; (4) optimal monetary policy is typically diﬀerent from full aggregate stabilization. Behind these results there are two crucial ingredients. First, agents are forward looking. Second, as time passes, better information on past fundamentals becomes available. The central bank can then adopt a backward-looking policy rule, based on more precise information about past fundamentals. By announcing its response to future information, the central bank can inﬂuence the expected real interest rate faced by agents with diﬀerent beliefs and thus induce an optimal use of the information dispersed in the economy. Keywords: Optimal monetary policy, imperfect information, consumer sentiment. JEL Codes: E52, E32, D83. ∗ MIT, Federal Reserve Bank of Chicago and NBER. Email: glorenzo@mit.edu. A previous version of this paper circulated with the title “News Shocks and Optimal Monetary Policy.”I wish to thank for very useful comments and suggestions Kjetil Storesletten, three anonymous referees, Marios Angeletos, Ricardo Caballero, Marvin Goodfriend, Veronica Guerrieri, Ivan Werning and seminar participants at the SED Meetings (Budapest), UQAM (Montreal), the Kansas City Fed, MIT, UC San Diego, Chicago GSB, Northwestern, Cornell, and U. of Texas at Austin. 1 Introduction Suppose a central bank observes an unexpected expansion in economic activity. This could be due to a shift in fundamentals, say an aggregate productivity shock, or to a shift in public beliefs with no actual change in the economy’s fundamentals. If the central bank could tell apart the two shocks the optimal response would be simple: accommodate the ﬁrst shock and oﬀset the second. In reality, however, central banks can rarely tell apart these shocks when they hit the economy. What can the central bank do in this case? What is the optimal monetary policy response? In this paper, I address these questions in the context of a model with dispersed information, which allows for a micro-founded treatment of fundamental shocks and “sentiment shocks.” The US experience in the second half of the 90s has fueled a rich debate on these issues. The run up in asset prices has been taken by many as a sign of optimistic expectations about widespread technological innovations. In this context, the advice given by diﬀerent economists has been strongly inﬂuenced by the assumptions made about the ability of the central bank to identify the economy’s actual fundamentals. Some, e.g. Cecchetti et al. (2000) and Dupor (2002), attribute to the central bank some form of superior information and advocate early intervention to contain an expansion driven by incorrect beliefs. Others, e.g. Bernanke and Gertler (2001), emphasize the uncertainty associated with the central bank’s decisions and advocate sticking to a simple inﬂation-targeting rule. In this paper, I explore the idea that, even if the central bank does not have superior information, a policy rule can be designed to take into account, and partially oﬀset, aggregate mistakes by the private sector regarding the economy’s fundamentals. I consider an economy with heterogeneous agents and monopolistic competition, where aggregate productivity is subject to unobservable random shocks. Agents have access to a noisy public signal of aggregate productivity, which summarizes public news about technological advances, aggregate statistics, and information reﬂected in stock market prices and other ﬁnancial variables. The error term in this signal introduces aggregate “noise shocks,” that is, shocks to public beliefs which are uncorrelated with actual productivity shocks. In addition to the public signal, agents have access to private information regarding the realized productivity in the sector where they work. Due to cross-sectional heterogeneity, this information is not suﬃcient to identify the value of the aggregate shock. Therefore, agents combine public and private sources of information to forecast the aggregate behavior of the economy. The central 1 bank has access only to public information. In this environment, I obtain two sets of results. First, I show that the monetary authority, using a policy rule which responds to past aggregate shocks, can aﬀect the relative response of the economy to productivity and noise shocks. Actually, there exists a policy rule which achieves “full aggregate stabilization,” that is, an equilibrium where aggregate activity is the same as in the case of full information. Second, I derive the optimal policy rule and show that full aggregate stabilization is typically suboptimal. As long as the coeﬃcient of relative risk aversion is greater or equal than 1, it is optimal to let output respond less than one for one to underlying changes in aggregate fundamentals and to let it respond positively to noise shocks. At the optimal policy rule, the economy achieves a constrained eﬃcient allocation, where agents make optimal use of public and private sources of information. The fact that monetary policy can tackle the two shocks separately is due to two crucial ingredients. First, agents are forward looking. Second, productivity shocks are unobservable when they are realized, but become public knowledge in later periods. At that point, the central bank can respond to them. By choosing an appropriate policy rule the monetary authority can then alter the way in which agents respond to private and public information. In particular, the monetary authority can announce that it will increase the target for aggregate nominal spending tomorrow, following an actual increase in aggregate productivity today, so as to generate inﬂation. Under this policy, consumers observing an increase in productivity in their own sector expect higher inﬂation than consumers who only observe a positive public signal. Therefore, they expect a lower real interest rate and choose to consume more. This makes consumption more responsive to private information and less to public information and moderates the economy’s response to noise shocks. This result points to an idea which applies more generally in models with dispersed information. If future policy is set contingent on variables that are imperfectly observed today, this can change the agents’ reaction to diﬀerent sources of information, and thus aﬀect the equilibrium allocation. In the model presented, the power of policy rules to shape the economy’s response to aggregate shocks is surprisingly strong. Namely, by adopting the appropriate rule the central bank can support an equilibrium where aggregate output responds one for one to fundamentals and does not respond at all to noise in public news. However, such a policy is typically suboptimal, since it has undesirable consequences in terms of the cross-sectional allocation. In particular, full stabilization generates an ineﬃcient compression in the distribution of relative prices. 2 To deﬁne the appropriate benchmark for constrained eﬃciency, I consider a social planner who can dictate the way in which individual consumers respond to the information in their hands, but cannot change their access to information.1 My constrained eﬃciency result shows that, in a general equilibrium environment with isoelastic preferences and Gaussian shocks, a simple linear monetary policy rule, together with a non-state-contingent production subsidy, are enough to eliminate all distortions due to dispersed information and monopolistic competition. In particular, a policy rule which only depends on aggregate variables is enough to induce agents to make an optimal use of public and private information.2 Finally, I use the model to ask whether better public information regarding the economy fundamentals can have destabilizing eﬀects on the economy and whether it can lead to social welfare losses. 3 I show that increasing the precision of the public signal increases the response of aggregate output to noise shocks and this can potentially increase output gap volatility.4 However, as agents receive more precise information on average productivity, they can set relative prices that are more responsive to their idiosyncratic productivity shocks. Therefore, a more precise public signal can improve welfare by allowing a more eﬃcient allocation of consumption and labor eﬀort across sectors. What is the total welfare eﬀect of increasing the public signal’s precision? If monetary policy is kept constant, then a more precise public signal may, for some set of parameters, reduce total welfare. This provides an interesting general equilibrium counterpart to Morris and Shin’s (2002) “anti-transparency” result. However, if monetary policy is chosen optimally, then a more precise signal is always welfare improving. This follows from the fact that, as pointed out by Angeletos and Pavan (2007a), more precise information is always desirable when the equilibrium is constrained eﬃcient. In this paper, equilibrium allocations and welfare are derived in closed form. This is possible thanks to an assumption about the random selection of consumption baskets. In particular, I maintain the convenience of a continuum of goods in each basket, while, at the same time, I allow for baskets that diﬀer from consumer to consumer. This technical solution may be usefully adapted to other models of information diﬀusion with random matching, as it allows 1 See Hellwig (2005) and Angeletos and Pavan (2007a) for similar notions of constrained eﬃciency. Angeletos and Pavan (2007b) derive a similar result in the context of quadratic games with optimal policy. See Angeletos, Lorenzoni, and Pavan (2008) for an application of the same principle to a model of investment and ﬁnancial markets. 3 On the social value of public information in game-theoretic settings, see Morris and Shin (2002), Angeletos and Pavan (2007a, 2007b), Hellwig and Veldkamp (2007), Amador and Weill (2007). This literature has sparked a lively debate on the merits of transparency in monetary policy, see Amato, Morris, and Shin (2002), Svensson (2005), Hellwig (2005), Morris and Shin (2005). 4 Here the output gap is measured with respect to the equilibrium under full information. 2 3 to construct models where each agent interacts with a large number of other agents, but does not fully learn about aggregate behavior. A number of recent papers, starting with Woodford (2002) and Sims (2003), have revived the study of monetary models with imperfect common knowledge, in the tradition of Phelps (1969) and Lucas (1972).5 In particular, this paper is closely related to Hellwig (2005) and Adam (2006), who study monetary policy in economies where money supply is imperfectly observed by the public. In both papers consumers’ decisions are essentially static, as a cashin-advance constraint is present and always binding. Therefore, the forward-looking element, which is crucial in this paper, is absent in their models. In the earlier literature, King (1982) was the ﬁrst to recognize the power of policy rules in models with imperfect information. He noticed that “prospective feedback actions” responding to “disturbances that are currently imperfectly known by agents” can aﬀect real outcomes.6 However, the mechanism in King (1982) is based on the fact that diﬀerent policy rules change the informational content of prices. As I will show below, that channel is absent in this paper. Here, policy rules matter because they aﬀect agents’ incentives to respond to private and public signals. The existing literature on optimal monetary policy with uncertain fundamentals has focused on the case of common information in the private sector. This includes Aoki (2003), Svensson and Woodford (2003, 2005), and Reis (2003). A distinctive feature of the environment in this paper is that private agents have access to superior information about fundamentals in their local market but not in the aggregate. The presence of dispersed information generates a novel tension between aggregate eﬃciency and cross-sectional eﬃciency in the design of optimal policy. There is a growing literature on the eﬀect of expectations and news on the business cycle. In particular, Christiano, Motto, and Rostagno (2006) and Lorenzoni (2006) show that shocks to expectations about productivity can generate realistic aggregate demand disturbances in business cycle models with nominal rigidities.7 In Christiano, Motto, and Rostagno (2006) the monetary authority has full information regarding aggregate shocks and can adjust the nominal interest rate in such a way so as to essentially oﬀset the eﬀect of the news shock and replicate the behavior of the corresponding ﬂexible price economy. Moreover, this oﬀsetting is optimal 5 See also Moscarini (2004), Milani (2005), Nimark (2005), Bacchetta and Van Wincoop (2005), Luo (2006), Ma´kowiak and Wiederholt (2006). Mankiw and Reis (2002) and Reis (2006) explore the complementary idea c of lags in informational adjustment as a source of nominal rigidity. 6 King (1982), p. 248. 7 See Beaudry and Portier (2006) and Jaimovich and Rebelo (2006) for ﬂexible price models of cycles driven by news about future productivity. 4 in their model.8 This leads to the question: are expectations-driven cycles merely a symptom of a suboptimal monetary regime, or is there some amount of expectations-driven volatility that survives under optimal monetary policy? This paper addresses this question in a setup with dispersed information, as in Lorenzoni (2006), and shows that optimal monetary policy does not eliminate noise-driven cycles. One may think that this result comes immediately from the assumption that the monetary authority has limited information. That is, it would seem that the central bank cannot intervene to bring output towards its “natural” level, given that this natural level is unknown. The analysis in this paper shows that the argument is subtler. The monetary authority could eliminate the aggregate eﬀect of news shocks by announcing an appropriate monetary rule. However, this rule is not optimal due to its undesirable crosssectional consequences. Finally, from a methodological point of view, this paper is related to a set of papers who exploit isoelastic preferences and Gaussian shocks to derive closed-form expressions for social welfare in heterogeneous agent economies, e.g., Benabou (2002) and Heathcote, Storesletten, and Violante (2008). The main novelty here is the presence of diﬀerentiated goods and consumer-speciﬁc consumption baskets. The model is introduced in Section 2. In Section 3, I characterize stationary, linear rational expectations equilibria. In Section 4, I show how the choice of the monetary policy rule aﬀects the equilibrium allocation. In Section 5, I derive the welfare implications of diﬀerent policies, characterize optimal monetary policy and prove constrained eﬃciency. In Section 6, I study the welfare eﬀects of public information. Section 7 concludes. All the proofs not in the text are in the appendix. 2 The Model 2.1 Setup I consider a dynamic model of monopolistic competition ` la Dixit-Stiglitz with heterogeneous a productivity shocks and imperfect information regarding aggregate shocks. Prices are set at the beginning of each period, but are, otherwise, ﬂexible. There is a continuum of inﬁnitely lived households uniformly distributed on the unit interval [0, 1]. Each household i is made of two agents: a consumer and a producer who is specialized 8 See Appendix B of Christiano, Motto, and Rostagno (2006). 5 in the production of good i. Preferences are represented by the utility function ∞ β t U (Cit , Nit ) , E t=0 with U (Cit , Nit ) = 1 1 1−γ 1+η Cit − Nit , 1−γ 1+η where Cit is a consumption index and Nit is the labor eﬀort of producer i.9 The consumption index is given by σ−1 σ Cit = Jit Cijt dj σ σ−1 , where Cijt denotes consumption of good j by consumer i in period t, and Jit ⊂ [0, 1] is a random consumption basket, which is described in detail below. The elasticity of substitution between goods, σ, is greater than 1. The production function for good i is Yit = Ait Nit . Productivity is household-speciﬁc and labor is immobile across households. The productivity parameters Ait are the fundamental source of uncertainty in the model. Let ait denote the log of individual productivity, ait = log (Ait ).10 Individual productivity has an aggregate component at and an idiosyncratic component it , ait = at + with 1 0 it di it , = 0. Aggregate productivity at follows the AR1 process at = ρat−1 + θt , with ρ ∈ [0, 1]. At the beginning of period t, all households observe the value of aggregate productivity in the previous period, at−1 . Next, the shocks 9 it and θt are realized. Agents in household If γ = 1, the per-period utility function is U (Cit , Nit ) = log Cit − 10 1 N 1+η . 1 + η it Throughout the paper, a lowercase variable will denote the natural logarithm of the corresponding uppercase variable. 6 i cannot observe it and θt separately, they only observe the sum of the two, that is, the individual productivity innovation xit = θt + it . Moreover, all agents observe a noisy public signal of the aggregate innovation st = θt + et . The random variables it , θt and et are independent, serially uncorrelated, and normally dis- 2 2 tributed with zero mean and variances σ 2 , σθ , and σe .11 I assume throughout the paper that σ 2 2 2 2 and σθ are strictly positive, and I study separately the cases σe = 0 and σe > 0, corresponding, respectively, to full information and imperfect information on θt . Summarizing, there are two aggregate shocks: the productivity shock θt and the “noise shock” et . Both are unobservable during period t, but are fully revealed at the beginning of t + 1, when at is observed. The second shock is the source of correlated mistakes in this economy, as it induces households to temporarily overstate or understate the current value of θt . The vector of past aggregate shocks is denoted by ht ≡ (θt−1 , et−1 , θt−2 , et−2 , ..., θ0 , e0 ) . Let me turn now to the random consumption baskets. Each period, nature selects a random set of goods Jit ⊂ [0, 1] with correlated productivity shocks. In this way, even though each consumer consumes a large number of goods (a continuum), the law of large numbers does not apply, and consumption baskets diﬀer across consumers. In the appendix, I give a full description of the matching process between consumers and producers. Here, I summarize the properties of the consumption baskets that arise from the process. Each consumer receives a “sampling shock” vit (unobserved by the consumer) and the goods in Jit are selected so that the distribution of the shocks jt for j ∈ Jit is normal with mean vit and variance σ 2|v . The sampling shocks vit are normally distributed across consumers, with zero mean and variance 2 σv . They are independent of all other shocks and satisfy of the matching process the variances 2 σv , σ 2|v and σ2 1 0 vit di = 0. To ensure consistency 2 have to satisfy σv + σ 2|v = σ 2 . Therefore, 2 the variance σv is restricted to be in the interval 0, σ 2 . Let me introduce here the parameter 11 2 σθ In the cases where γ = 1 and productivity is a random walk, ρ = 1, it is necessary to impose a bound on to ensure that expected utility is ﬁnite, namely 2 σθ < 2 γ+η (1 − γ) (1 + η) 7 2 (− log β) . 2 χ = σv /σ 2 which lies in [0, 1] and reﬂects the degree of heterogeneity in consumption baskets. The limit case χ = 0 corresponds to the standard case where all consumers consume the same representative sample of goods. 2.2 Trading, ﬁnancial markets and monetary policy The central bank acts as an account keeper for the agents in the economy. Each household holds an account denominated in dollars, directly with the central bank. The account is debited whenever the consumer makes a purchase and credited whenever the producer makes a sale. The balance of household i at the beginning of the period is denoted by Bit . All households begin with a zero balance at date 0. At the beginning of each period t, the bank sets the (gross) nominal interest rate Rt , which will apply to end-of-period balances. Households are allowed to hold negative balances at the end of the period and the same interest rate applies to positive and negative balances. However, there is a lower bound on nominal balances, which rules out Ponzi schemes. To describe the trading environment, it is convenient to divide each period t in three stages, (t, 0) , (t, I) , and (t, II). In stage (t, 0), everybody observes at−1 , the central bank sets Rt , and the households trade one-period state-contingent claims on a centralized ﬁnancial market. These claims will be paid in (t + 1, 0). In stages (t, I) and (t, II), the market for state-contingent securities is closed and the only assets traded are dollar balances in the central bank’s payment system and non-state-contingent bonds payable in (t + 1, 0). By arbitrage, the price of the bonds must be equal to 1/Rt at all stages. Since balances with the central bank and non-state contingent bonds are perfect substitutes and are in zero net supply, I simply assume that holdings of non-state-contingent bonds are always zero. In stage (t, I), all aggregate and individual shocks are realized, producer i observes st and xit , sets the dollar price of good i, Pit , and stands ready to deliver any quantity of good i at that price. In stage (t, II), consumer i observes the prices of the goods in his consumption basket, {Pjt }j∈Jit , chooses his consumption vector, {Cijt }j∈Jit , and buys Cijt from each producer j ∈ Jit . In this stage, consumer i and producer i are spatially separated, so the consumer does not observe the current production of good i. Figure 1 summarizes the events taking place during period t. In stages (t, I) and (t, II), households are exposed to idiosyncratic uncertainty and do not have access to state-contingent claims. Therefore, they will generally end up with diﬀerent end-of-period balances. However, households can fully insure against these shocks ex ante, by trading contingent claims in (t, 0). This implies that the nominal balances Bit will be constant 8 (t,0) (t,I) (t,II) (t+1,0) Everybody observes at-1 Household i observes Household i observes price State-contingent claims Central bank sets Rt st vector Agents trade state- xit contingent claims et t t Sets price Pit it Pjt j J it are settled and chooses consumption vector Cijt j J it Figure 1: Timeline and equal to 0 in equilibrium.12 In this way, I can eliminate the wealth distribution from the state variables of the problem, which greatly simpliﬁes the analysis.13 Let Zit+1 (ωit ) denote the state-contingent claims purchased by household i in (t, 0), where ωit ≡ ( it , vit , θt , et ). The price of these claims is denoted by Qt (ωit ). The household balances at the beginning of period t + 1 are then given by Bit+1 = Rt Bit − R4 Qt (˜ it ) Zit+1 (˜ it ) d˜ t + (1 + τ ) Pit Yit − ω ω ω Jit Pjt Cijt dj − Tt +Zit+1 (ωit ) , where τ is a proportional subsidy on sales and Tt is a lump-sum tax. Let me deﬁne aggregate indexes for nominal prices and real activity. For analytical conve12 13 The balances Bit are computed after the claims from t − 1 have been settled. The use of this type of assumption to simplify the study of monetary models goes back to Lucas (1990). 9 nience, I use simple geometric means,14 1 Pt ≡ exp log Pit di , 0 1 Ct ≡ exp 0 log Cit di . The behavior of the monetary authority is described by a policy rule. In period (t, 0), the central bank sets Rt based on the past realizations of the exogenous shocks θt and et , and on the past realizations of Pt and Ct . The monetary policy rule is described by the map R, where Rt = R (ht , Pt−1 , Ct−1 , ..., P0 , C0 ). Allowing the monetary policy to condition Rt on the current public signal st would not alter any of the results. The only other policy instrument available is the subsidy τ , which is ﬁnanced by the lump-sum tax Tt . The government runs a balanced budget so 1 Tt = τ Pit Yit di. 0 As usual in the literature, the subsidy τ will be used to eliminate the distortions due to monopolistic competition. 2.3 Equilibrium deﬁnition Household behavior is captured by three functions, Z, P and C. The ﬁrst gives the optimal holdings of state-contingent claims as a function of the initial balances Bit and of the vector of past aggregate shocks ht , that is, Zit+1 (ωit ) = Z (ωit ; Bit , ht ). The second gives the optimal price for household i, as a function of the same variables plus the current realization of individual productivity and of the public signal, Pit = P (Bit , ht , st , xit ). The third gives optimal consumption as a function of the same variables plus the observed price vector, Cit = C(Bit , ht , st , xit , {Pij }j∈Jit ). Before deﬁning an equilibrium, I need to introduce two other objects. Let D (.|ht ) denote the distribution of nominal balances Bit across households, conditional on the history of past aggregate shocks ht . The price of a ωit -contingent claim in period (t, 0), given the vector of past shocks ht , is given by Q (ωit ; ht ). 14 Alternative price and quantity indexes are ˆ Pt 1 ≡ 0 ˆ Yt ≡ 1 0 1−σ Pit di 1 1−σ , Pit Yit di . ˆ Pt ˆ ˆ All results stated for Pt and Ct hold for Pt and Yt , modulo multiplicative constants. 10 A symmetric rational expectations equilibrium under the policy rule R is given by an array {Z, P, C, D, Q} that satisﬁes three conditions: optimality, market clearing, and consistency. Optimality requires that the individual rules Z, P and C are optimal for the individual household, taking as given: the exogenous law of motion for ht , the policy rule R, the prices Q, and the fact that all other households follow Z,P, and C, and that their nominal balances are distributed according to D. Market clearing requires that the goods markets and the market for state-contingent claims clear for each ht . Consistency requires that the dynamics of the distribution of nominal balances, described by D, are consistent with the individual decision rules. 3 Linear equilibria In this section, I characterize the equilibrium behavior of output and prices. Given the assumption that agents trade state-contingent claims in periods (t, 0), I can focus on equilibria where beginning-of-period nominal balances are constant and equal to zero for all households. That is, the distribution D (.|ht ) is degenerate for all ht . Moreover, thanks to the assumption of separable, isoelastic preferences and Gaussian shocks, I can analyze linear rational expectations equilibria in closed form. In particular, I will characterize stationary linear equilibria where the logs of individual prices and consumption levels take the following form pit = φa at−1 + φs st + φx xit , (1) cit = ψ0 + ψa at−1 + ψs st + ψx xit + ψx xit , (2) where φ ≡ {φa , φs , φx } and ψ ≡ {ψ0 , ψa , ψs , ψx , ψx } are vectors of constant coeﬃcients to be determined and xit is the average productivity innovation for the goods in the basket of consumer i, xit ≡ Jit xjt dj = θt + vit . I will explain in a moment why this variable enters (2). Summing (1) and (2) across agents, I obtain the aggregate price and quantity indexes pt = φa at−1 + φθ θt + φs et , (3) ct = ψ0 + ψa at−1 + ψθ θt + ψs et , (4) where φθ ≡ φs + φx and ψθ ≡ ψs + ψx + ψx . In the rest of this section, I ﬁrst characterize the optimal behavior of an individual household, assuming that the all other households follow (1) and (2). Then, I introduce a linear 11 monetary policy rule, and show that, under that rule, (1) and (2) form a rational expectations equilibrium. 3.1 Optimal consumption and prices As useful preliminary steps, let me derive the appropriate price index for household i and the demand curve for good i. Individual optimization implies that, given Cit , the consumption of good j by consumer i is Cijt = Pjt P it −σ Cit , where P it is the price index P it ≡ j∈Jit 1−σ Pjt dj 1 1−σ (5) . The assumptions made on consumption baskets imply that the productivity innovations xjt for the goods in Jit are normally distributed with mean xit and variance σ 2|v . Using my conjecture (1) for individual prices, I then obtain an exact expression for the log of the price index of consumer i,15 pit = κp + pt + φx vit . (6) ˜ Consider now the demand curve for good i. Let Jit denote the set of consumers buying good −σ i at time t. Aggregating their demand, gives Yit = Dit Pit , where Dit is the demand index Dit ≡ σ ˜ j∈Jit P jt Cjt dj. Also for the demand index Dit , I can use my assumptions on consumption baskets and conjectures (1) and (2) to obtain an exact linear expression dit = κd + ct + σpt + (ψx + σφx ) χ it . (7) Using these expressions, I can then derive the household’s ﬁrst-order conditions for Pit and Cit . The ﬁrst takes the form pit = κp + Ei,(t,I) [pit + γcit + ηnit ] − ait , (8) where Ei,(t,I) [.] denotes the expectation of household i at date (t, I). The labor eﬀort nit is determined by the technological constraint nit = yit − ait , and the output yit by the demand relation derived above, yit = dit − σpit . The expression on the right-hand side of (8) captures 15 This expression and expressions (7)-(9) below, are derived formally in the proof of Proposition 1, in the appendix, where I also derive the constant terms κp , κd , κp , and κc 12 the expected nominal marginal cost for producer i. This depends positively on the price of the consumption basket of consumer i, pit , and on the marginal rate of substitution between consumption and leisure, γcit + ηnit , and it depends negatively on the productivity ait . To compute the expectation in (8), notice that all the relevant information at (t, I) is summarized by at−1 , st and xit , so Ei,(t,II) [.] can be replaced by E [.|at−1 , st , xit ]. The optimality condition for Cit takes the form cit = κc + Ei,(t,II) [cit+1 ] − γ −1 rt − Ei,(t,II) pit+1 + pit , (9) where Ei,(t,II) [.] denotes the expectation of household i at date (t, II). Apart from the fact that expectations and price indexes are consumer-speciﬁc, this is a standard consumer’s Euler equation: current consumption depends positively on future expected consumption and negatively on the expected real interest rate. To compute the expectations in (9), notice that the consumer can observe the price vector {pjt }j∈Jit . However, if all producers follow (1), these prices are normally distributed with mean φa at−1 + φx xit + φs st and variance φ2 σ 2|v . x Given that the consumer already knows at−1 and st , he can back out φx xit from the mean of this distribution and this is a suﬃcient statistic for all the information on θt contained in the observed prices. Therefore, Ei,(t,II) [.] can be replaced by E [.|at−1 , st , xit , φx xit ]. This result is summarized in a lemma. Lemma 1 If prices are given by (1), then the information of consumer i regarding the current shock θt is summarized by the three independent signals st , xit and φx xit . This conﬁrms my initial conjecture that the individual consumption policy (2) is a linear function of at−1 , st , xit , and xit . 3.2 Policy rule and equilibrium To ﬁnd an equilibrium, I substitute (1) and (2) in the optimality conditions (8) and (9), and obtain a system of equations in φ and ψ.16 This system of equations does not determine φ and ψ uniquely. In particular, for any choice of the parameter φa in R, there is a unique pair {φ, ψ} which is compatible with individual optimality. To complete the equilibrium characterization and pin down φa , I need to deﬁne a monetary policy rule. Consider an interest rate rule which targets aggregate nominal spending. The nominal interest rate is set to rt = ξ0 + ξa at−1 + ξm (mt−1 − mt−1 ) , ˆ 16 See (32)-(39) in the appendix. 13 (10) where mt ≡ pt + ct is an index of aggregate nominal spending and mt is the central bank’s ˆ target mt = µ0 + µa at−1 + µθ θt + µe et . ˆ (11) The parameters ξ = {ξ0 , ξa , ξm } and µ = {µa , µθ , µe } are chosen by the monetary authority. The central bank’s behavior can be described as follows. At the beginning of period t, the monetary authority observes at−1 and announces its current target mt for nominal spending. ˆ The target mt has a forecastable, backward-looking component µa at−1 , and a state-contingent ˆ part which is allowed to respond to the current shocks θt and et . During trading, each agent i sets his price and consumption responding to the variables in his information set. At the beginning of period t + 1, the central bank observes the realized level of nominal spending mt and the realized shocks θt and et . If mt deviated from target in period t, in the next period the nominal interest rate is adjusted according to (10). Given this policy rule, I can complete the equilibrium characterization and prove the existence of stationary linear equilibria. In particular, the next proposition shows that the choice of µa by the monetary authority pins down φa in the system of equations described above, and thus the equilibrium coeﬃcients φ and ψ. In the proposition, I exclude one possible value for µa , denoted by µ0 , which corresponds to the pathological case where the equilibrium a construction would give φx = 0. This case is discussed in the appendix. Proposition 1 For each µa ∈ R/ µ0 a there is a pair {φ, ψ} and a vector {ξ0 , ξa , µ0 , µθ , µe } such that the prices and consumption levels in (1)-(2) form a rational expectations equilibrium under the policy rule (10)-(11), for any value of ξm ∈ R. If ξm > 1 the equilibrium is locally determinate. The value of ψa is independent of the policy rule and equal to ψa = 1+η ρ. γ+η In equilibrium, monetary policy always achieves its nominal spending target, that is, mt = mt , ˆ and the nominal interest rate is equal to rt = ξ0 − (µa + (γ − 1) ψa ) (1 − ρ) at−1 . 4 The eﬀects of monetary policy Let me turn now to the eﬀects of diﬀerent monetary policy rules on the real equilibrium allocation. By Proposition 1, the choice of the policy rule is summarized by the parameter 14 µa , so, from now on, I will simply refer to the “policy rule µa .” Proposition 1 shows that in equilibrium mt = mt and the central bank always achieves its desired target for nominal ˆ output. In particular, by choosing µa the central bank determines the response of nominal output to past realizations of aggregate productivity. This is done by inducing price setters to adjust their nominal prices in equilibrium. Since at−1 is common knowledge, price setters simply coordinate on setting prices proportionally to exp{µa at−1 }.17 The ﬁrst question raised in the introduction can now be stated in formal terms. How does the choice of µa aﬀects the equilibrium response of aggregate activity to fundamental and noise shocks, that is, the coeﬃcients ψθ and ψs in (4)? More generally, how does the choice of µa aﬀects the vectors φ and ψ, which determine the cross-sectional allocation of goods and labor eﬀort across households? The rest of this section addresses these questions. 4.1 Full information Let me begin with the case where households have full information on the aggregate shock θt . 2 This happens when st is a noiseless signal, σe = 0. In this case, households can perfectly forecast current aggregate prices and consumption, pt and ct , by observing at−1 and st . Using (6) and (7) to substitute for pit and dit in the optimal pricing condition (8), and taking the expectation E[.|at−1 , st ] on both sides, gives pt = (κp + κp + ηκd ) + pt + γct + η (ct − at ) − at . This implies that aggregate consumption is cf i = ψ0 + t 1+η at , γ+η (12) that is ψθ = (1 + η) / (γ + η). In the next proposition, I show that also ψ0 and the other coeﬃcients which determine the real equilibrium allocation, are uniquely determined and independent of µa . This is a baseline neutrality result: under full information the real equilibrium allocation is independent of the monetary policy rule.18 Proposition 2 If households have full information on θt , the allocation of consumption goods and labor eﬀort in all stationary linear equilibria is independent of the monetary policy rule µa . 17 18 The response of real output to at−1 , instead, is independent of the policy rule, as shown in Proposition 1. See McCallum (1979) for an early neutrality result in a model with pre-set prices. 15 4.2 Imperfect information: a special case 2 Let me turn now to the case of imperfect information on θt , which arises when σe is positive. In this case, the choice of µa aﬀects equilibrium prices and quantities. To understand how monetary policy operates, it is useful to start from a special case. Consider the case where the intertemporal elasticity of substitution γ is 1, the disutility of eﬀort is linear, η = 0, and productivity is a random walk, ρ = 1. In this case, the consumer’s Euler equation can be rewritten as19 pit + cit = Ei,(t,II) pit+1 + cit+1 , that is, nominal spending is a random walk. Under the nominal output target (11), the forecastable part of future nominal spending is equal to Ei,(t,II) [pt+1 + ct+1 ] = µ0 + µa Ei,(t,II) [at ] . Moreover, assume that χ is zero, so that all the consumers consume all the goods.20 Then, as I will check below, the consumers can perfectly infer the value of θt from the observed values of pt and st , so Ei,(t,II) [at ] = at . Putting together these results and using the fact that µ0 = ψ0 ,21 it follows that all consumers choose the same consumption ct = ψ0 + µa at − pt . (13) To complete the equilibrium characterization, let me turn to price setting. Households still have imperfect information when they set prices, since they only observe st and xit in (t, 0). Given that η = 0 and substituting the optimal consumption derived above, the optimality condition for prices (8) boils down to22 pit = µa E [at |st , xit ] − ait . The expectation of at can be written as E [at |st , xit ] = at−1 + βs st + βx xit , where βs and βx are positive inference coeﬃcients which satisfy βs + βx < 1.23 Aggregating across producers and 19 To obtain this expression from (9), notice that, when ρ = 1, rt is constant and equal to ξ0 , by Proposition 1. Moreover, in the proof of the same proposition I show that ξ0 = γκc , so the terms κc and γ −1 rt in (9) cancel out. 20 This shows that my basic positive results can be derived without introducing heterogeneous consumption baskets. However, Proposition 6 below shows that heterogeneous consumption baskets are necessary to obtain interesting welfare trade-oﬀs. 21 See equation (42) in the appendix. 22 Equation (32) in the appendix shows that κp + κp + ψ0 = 0 when γ = 1 and η = 0, so there is no constant term in this expression. 23 See (30) in the appendix. 16 rearranging gives pt = (µa − 1) at−1 + µa βs st + (µa βx − 1) θt . (14) This shows that observing pt and st fully reveals θt , except in the knife-edge case where µa = 1/βx . The following discussion disregards this case. Combining (13) and (14), I get the equilibrium value of aggregate consumption ct = ψ0 + at−1 + (1 + µa (1 − βs − βx )) θt − µa βs et , (15) that is, in this economy ψθ = 1 + µa (1 − βs − βx ) and ψs = −µa βs . The choice of the policy rule µa is no longer neutral. In particular, increasing µa increases the output response to fundamental shocks and reduces its response to noise shocks. To interpret this result, it is useful to look separately at consumers’ and price setters’ behavior. If the monetary authority increases µa , (13) shows that, for a given price level pt , the response of consumer spending to θt increases. A larger value of µa implies that, if a positive productivity shock materializes at date t, the central bank will target a higher level of nominal spending in the following period. This, given the consumers’ forward looking behavior, translates into higher nominal spending in the current period. On the other hand, the consumers’ response to a noise shock et , for given pt , is zero, irrespective of µa , given that consumers have perfect information on at and place zero weight on the signal st . Consider now the response of price setters. If the monetary authority chooses a larger value for µa , price setters tend to set higher prices following a positive productivity shock θt as they observe a positive st and, on average, a positive xit , and thus expect higher consumer spending. However, due to imperfect information, they tend to underestimate the spending increase. Therefore, their price increase is not enough to undo the direct eﬀect on consumers’ demand, and, on net, real consumption goes up. Formally, this is captured by ∂ψθ = 1 − βs − βx > 0. ∂µa On the other hand, following a positive noise shocks, price setters mistakenly expect an increase in demand, following their observation of a positive st , and tend to raise prices. Consumers’ demand, however, is unchanged. The net eﬀect is a reduction in output, that is, ∂ψs = −βs < 0. ∂µa A further result which is easily established, is that monetary policy can achieve the full information benchmark for aggregate activity, by picking the right µa . When γ = 1, (12) shows 17 that aggregate consumption under full information is cf i = ψ0 + at . Moreover, (15) shows that t the central bank can achieve the same aggregate consumption path by setting µa = 0. That is, there is a value of µa which, at the same time, achieves ψθ = 1 and ψs = 0.24 This may seem the outcome of the special assumptions made here and, in particular, of the fact that consumers have full information. In fact, it is a result that holds more generally, as I will show below. 4.3 Imperfect information: general results The following two propositions extend the results derived above to the general case. First, I extend the non-neutrality result and show that increasing µa increases the response of aggregate consumption to fundamental shocks, ψθ , and reduces its response to noise shocks, ψs . Proposition 3 If households have imperfect information on θ, the real equilibrium allocation depends on µa . The equilibrium coeﬃcients {φ, ψ} are linear functions of the policy parameter µa , with ∂ψθ > 0, ∂µa ∂ψs < 0, ∂µa ∂φx > 0. ∂µa The simple example presented above helps to build the intuition for the general result. Now, consumers no longer have perfect information on θt and form expectations based on the imperfect signals st , xit , and xit . Consider two hypothetical scenarios. In case A, there is a positive fundamental shock, both st and θt are positive, and the typical consumer receives both a positive public signal and positive private signals xit , xit > 0. In case B, there is a positive noise shock et , st is positive, θt is zero, and the typical consumer receives a positive public signal and neutral private signals xit = xit = 0. Suppose that µa increases. In both scenarios, consumers expect an increase in nominal output at t + 1 and higher future prices. The increase in Ei,(t,II) pit+1 on the right-hand side of (9) leads to an increase in consumer demand at time t, for given prices pit . Under both scenarios, the producers forecast a demand increase and tend to raise current prices. However, in case A the producers tend to underestimate the increase in Ei,(t,II) pit+1 which is driving up demand, while in case B they tend to overestimate it. The reason for this is that, in case A, the typical consumer is using both public and private information, while, in case B, he is only using public information. In the ﬁrst case, the producers can perfectly forecast the 24 This does not ensure that ψ0 will also be the same. However, the subsidy τ can be adjusted to obtain any value for ψ0 . 18 demand increase associated to a positive st , but can only partially foresee the demand increase due to the private signals. In the second case, they think that θt is positive and erroneously forecast a demand increase driven by both public and private signals, while, in the aggregate, only the public signal is operating. The underreaction of current prices in case A means that Ei,(t,II) pit+1 − pit tends to increase. The overreaction of current prices in case B leads to the opposite result. Therefore, consumers’ expected inﬂation goes up in case A and down in case B, leading to an increase in real consumption in the ﬁrst case and to a reduction in the second. There are three crucial ingredients behind this result: dispersed information, forwardlooking agents, and a backward-looking policy based on the observed realization of past shocks. The diﬀerent information sets of consumers and price setters play a central role in the mechanism described above. The presence of forward-looking agents is clearly needed so that announcements about future policy aﬀect current behavior. The backward-looking policy rule works because it is based on past shock realizations which were not observed by the agents at the time they hit. To clarify this point, notice that the results above would disappear if the central bank based its intervention at t + 1 on any variable that is common knowledge at date t, for example on st . Suppose, for example, that the backward-looking component of the nominal spending target (11) took the form µs st−1 instead of µa at−1 . Then, any adjustment in the backward-looking parameter µs would lead to identical and fully-oﬀsetting eﬀects on current prices and expected future prices, with no eﬀects on the real allocation.25 The next proposition, extends the second result obtained in 4.2. There exists a policy rule µa which achieves full aggregate stabilization, that is, an equilibrium where aggregate activity perfectly tracks the full information benchmark derived in 4.1. Proposition 4 There exists a monetary policy rule µf s which, together with the appropriate a subsidy τ f s , achieves full aggregate stabilization, that is, an equilibrium with ct = cf i . t To achieve the full information benchmark for ct , the central bank has to eliminate the eﬀect of noise shocks, setting ψs equal to zero, and ensure, at the same time, that the output response to the fundamental shocks ψθ is equal to (1 + η) / (γ + η). Given that, by Proposition 3, there is a linear relation between µa and ψs and ∂ψs /∂µa = 0, it is always possible to ﬁnd a µa such that ψs is equal to zero.26 The surprising result is that the value of µa that sets ψs 25 On the other hand, it is not crucial that the central bank can observe θt perfectly in period t + 1. In fact, it is possible to generalize the result above to the case where the central bank receives noisy information about θt at t + 1, as long as this information is not in the agents’ information sets at time t. 26 In the proof of Proposition 4, I check that µf s is diﬀerent from µ0 . a a 19 to zero does, at the same time, set ψθ equal to (1 + η) / (γ + η). This result is an immediate corollary of the following lemma. Lemma 2 In any linear equilibrium, ψθ and ψs satisfy 2 2 ψθ σθ + ψs σe = 1+η 2 σ . γ+η θ (16) Proof. Starting from the optimal pricing condition (8), take the conditional expectation E [.|at−1 , st ] on both sides and use the law of iterated expectations, to obtain E [pit |at−1 , st ] = κp + E [pit + γcit + ηdit − ησpit − (1 + η) ait |at−1 , st ] . I can then substitute for pit and dit using (6) and (7), and exploit the fact that all idiosyncratic shocks have zero mean ex ante. Then, optimal pricing implies that ct satisﬁes E ct − ψ0 − 1+η at |at−1 , st = 0. γ+η (17) Using (4) to substitute for ct and using E [at |at−1 , st ] = ρat−1 +E [θt |st ] and ψa = ρ (1 + η) / (γ + η), this equation boils down to E [ψθ θt + ψs et |st ] = 1+η E [θt |st ] . γ+η 2 2 2 2 2 2 Substituting for E [θt |st ] = (σθ /(σθ + σe ))st and E [et |st ] = (σe /(σθ + σe ))st , gives the linear restriction (16). The point of this lemma is that the output responses to the two shocks are tied together by the fact that ex ante, conditional on at−1 and st , price setters must expect their prices to be in line with their nominal marginal costs. This implies that aggregate consumption and output are expected to be, on average, at their full information level, as shown by (17). In turns, this implies that when ψθ increases ψs must decrease, otherwise the sensitivity of expected output to st would be inconsistent with optimal pricing. This also implies that, if aggregate consumption moves one for one with ((1 + η) / (γ + η)) θt , then the eﬀect of the signal st (and thus of the noise et ) must be zero. To conclude this section, let me remark that the choice of µa also aﬀect the sensitivity of individual consumption and prices to idiosyncratic shocks. That is, the policy rule has implications not only for aggregate responses, but also for the cross-sectional distribution of consumption and relative prices. This observation will turn out to be crucial in evaluating the welfare consequences of diﬀerent monetary rules. 20 5 Optimal monetary policy 5.1 Welfare I now turn to the welfare analysis and to the characterization of optimal monetary policy. In a linear equilibrium, the consumption of good j by consumer i is given by Cijt = exp {ψ0 + σκp + ψa at−1 + ψs st + ψx xit + ψx xit − σφx (xjt − xit )} , which follows substituting the equilibrium price and consumption decisions, (1) and (2), and the price index (6), in equation (5). The equilibrium labor eﬀort of household i is then given by the market clearing condition Nit = ˜ j∈Jit Cjit dj Ait . (18) Using these expressions and exploiting the normality of the shocks, it is then possible to compute the value of the expected utility of a representative household at the beginning of period 0, as shown in the following lemma. Lemma 3 Take any monetary policy µa ∈ R/µ0 and consider the associated linear equilibrium, a characterized by the coeﬃcients {φ, ψ}. Assume that the subsidy τ is chosen optimally. Then the expected utility of a representative household is given by ∞ β t U (Cit , Nit ) = E t=0 if γ = 1, and by 1+η 1 (1−γ) γ+η w(µa ) W0 e , 1−γ ∞ β t U (Cit , Nit ) = w0 + E t=0 1 w(µa ), 1−β if γ = 1. W0 and w0 are constant terms independent of µa , W0 is positive, and w(.) is a known quadratic function, which depends on the model’s parameters. The function w(µa ) can be used to evaluate the welfare eﬀects of diﬀerent policies in terms of equivalent consumption changes. Suppose I want to compare two policies µa and µa by ﬁnding the ∆ such that ∞ ∞ t β U (1 + ∆) Cit , Nit E t=0 β t U Cit , Nit =E , t=0 where Cit , Nit and Cit , Nit are the equilibrium allocations arising under the two policies. The value of ∆ represents the proportional increase in lifetime consumption which is equivalent, in 21 welfare terms, to a policy change from µa to µa . The following lemma shows that w (µa )−w (µa ) provides a ﬁrst-order approximation for ∆.27 Lemma 4 Let ∆ (µa , µa ) be the welfare change associated to the policy change from µa to µa , measured in terms of equivalent proportional change in lifetime consumption. The function ∆ (., .) satisﬁes d∆ (µa , µa + u) du 5.2 = w (µa ) . u=0 Constrained eﬃciency To characterize optimal monetary policy, I will show that it achieves an appropriately deﬁned social optimum. I consider a planner who can choose the consumption and labor eﬀort levels Cijt and Nit facing only two constraints: the resource constraint (18) and the informational constraint that Cijt be measurable with respect to at−1 , st , xit , xit and xjt . This requires that, when selecting the consumption basket of consumer i at time t, the planner can only use the information that would be available to the consumer in the market economy. Speciﬁcally, I allow the planner to use the same information available to consumers in linear equilibria with φx = 0. 28 An allocation that solves the planner problem is said to be “constrained eﬃcient.” The crucial assumption here is that the planner can determine how consumers respond to various sources of information, but cannot intervene to change this information. This notion of constrained eﬃciency is developed and analyzed in a broad class of quadratic games in Angeletos and Pavan (2007a, 2007b). Here, it is possible to apply it in a general equilibrium environment, even though agents extract information from prices and prices are endogenous, because the matching environment is such that the information sets are essentially exogenous. The following result shows that, with the right choice of µa and τ , the equilibrium found in Proposition 1 is constrained eﬃcient. Proposition 5 There exist a monetary policy µ∗ and a subsidy τ ∗ such that the associated a stationary linear equilibrium is constrained eﬃcient. This proposition shows that a simple backward-looking policy rule, which is only contingent on aggregate variables, is suﬃcient to induce agents to make the best possible use of the public 27 I am grateful to Kjetil Storesletten for suggesting this result. In the proof of Proposition 5, I show that φx = 0 in the best linear equilibrium. When φx = 0, the consumer can recover xit and xjt for all j ∈ Jit from {Pjt }j∈Jit . Therefore, I could allow each Cijt to be a function of the entire distribution {xjt }j∈Jit . Making it just a function of xit and xjt simpliﬁes the analysis and is without loss of generality. 28 22 and private information available to them. The resource constraint and the measurability constraint are satisﬁed by all the linear equilibrium allocations. Therefore, an immediate corollary of Proposition 5 is that µ∗ is the a optimal monetary policy which maximizes w(µa ). However, the set of feasible allocations for the planner is larger than the set of linear equilibrium allocations, since in the planner’s problem Cijt is allowed to be any function, possibly non-linear, of at−1 , st , xit , xit and xjt , and there are no restrictions on the responses of Cijt to these variables.29 5.3 Optimal accommodation of noise shocks Having obtained a general characterization of optimal monetary policy, I can turn to more speciﬁc questions: what is the economy’s response to the various shocks at the optimal monetary policy? In particular, is full aggregate stabilization optimal? That is, should monetary policy completely eliminate the aggregate disturbances due to noise shocks, setting ψs = 0? The next proposition shows that, typically, full aggregate stabilization is suboptimal. Proposition 6 Suppose there is imperfect information and η > 0, χ ∈ (0, 1). If σγ > 1 full aggregate stabilization is suboptimal and µ∗ < µf s . At the optimal policy, aggregate consumpa a tion is less responsive to fundamental shocks than under full information and noise shocks have a positive eﬀect on aggregate consumption, ∗ ψθ < 1+η , γ+η ∗ ψs > 0. If σγ < 1 full aggregate stabilization is also suboptimal, but the opposite inequalities apply. Full stabilization is optimal if one of the following conditions hold: η = 0, χ = 0, χ = 1, σγ = 1. To interpret this result, I use the following expression for the welfare index w(µa ) deﬁned in Lemma 3, 1 w = − (γ + η) E[(ct − cf i )2 ] + t 2 1 1 1 + (1 − γ) (cit − ct )2 di − (1 + η) 2 2 0 where nt is the employment index nt ≡ 1 0 nit di. 1 0 (nit − nt )2 di + (ct − at − nt ) , (19) This expression is derived in the appendix. The ﬁrst term in (19) captures the welfare eﬀects of aggregate volatility. In particular, it shows that social welfare is negatively related to the volatility of the “output gap” measure 29 In fact, it is possible to further generalize the result in Proposition 5, allowing the planner to use a general time-varying rule for Cijt , conditional on all past shocks’ realizations. 23 ct − cf i , which captures the distance between ct and the full-information benchmark analyzed t in Section 4.1. A policy of full aggregate stabilization maximizes this expression, setting it to zero. However, the remaining terms are also relevant to evaluate social welfare. Once they are taken into account, full aggregate stabilization is no longer desirable. These terms capture welfare eﬀects associated to the cross-sectional allocation of consumption goods and labor eﬀort, conditional on the aggregate shocks θt and et . Let me analyze them in order. The second and third term in (19) capture the eﬀect of the idiosyncratic variances of cit and nit . Since cit and nit are the logs of the original variables, these expressions capture both level eﬀects and volatility eﬀects. In particular, focusing on the ﬁrst one, when the distribution of cit is more dispersed, Cit is, on average, higher, given that E[Cit |at−1 , θt , et ] = exp ct + 1 2 1 0 (cit − ct )2 di , but Cit is also more volatile as V ar [Cit |at−1 , θt , et ] = exp This explains why the term 1 0 (cit 1 2 1 0 (cit − ct )2 di . − ct )2 di is multiplied by (1 − γ). When the coeﬃcient of relative risk aversion γ is greater than 1, agents care more about the volatility eﬀect than about the level eﬀect. In this case, an increase in the dispersion of cit reduces consumers’ expected utility. The opposite happens when γ is smaller than 1. A similar argument applies to the third term in (19), although there both the level and the volatility eﬀects reduce expected utility, given that the utility function is convex in labor eﬀort. The last term, ct − at − nt , reﬂects the eﬀect of monetary policy on the economy’s average productivity in consumption terms. Due to the heterogeneity in consumption baskets, a given average level of labor eﬀort, with given productivities, translates into diﬀerent levels of the average consumption index ct depending on the distribution of quantities across consumers and producers. The following expression is also derived in the appendix. 1 σ (σ − 1) ˜ ct −at −nt = − V ar[cjt +σpjt |j ∈ Jit , at−1 , θt , et ]+ V ar [pjt |j ∈ Jit , at−1 , θt , et ] . (20) 2 2 To interpret the ﬁrst term, notice that cjt + σpjt is the intercept in the log demand for good i by consumer j. A producer who faces more volatile log demand has on average to put in higher eﬀort, to achieve the same average log output. To interpret the second term, notice that consumers like price dispersion in their consumption basket, given that when prices are more variable they can reallocate their expenditure from more expensive goods to cheaper ones. 24 Therefore, a given average eﬀort by the producers translates into higher consumption indexes when relative prices are more dispersed. 30 Summing up, when the dispersion in demand is lower and the dispersion in prices is higher, a given average eﬀort nt generates a higher average consumption index ct . 5.4 A numerical example To illustrate the various welfare eﬀects just described, I will use a numerical example. The parameters for the example are in Table 1. The coeﬃcient of relative risk aversion γ is set to 1. The values for σ and η are chosen in the range of values used in the sticky-price literature. 2 2 The values for the variances σθ , σ 2 and σe are set at 1. The variance of the sampling shocks 2 2 σv must then be in [0, 1]. I pick the intermediate value σv = 1/2. γ σ 2 σθ 2 σe 1 7 1 1 η 2 σ2 2 σv 1 0.5 Table 1: Parameters for the numerical example. Figure 2 shows the relation between µa and total welfare w. Figure 3 illustrates how µa aﬀects the various terms in (19). In particular, panel (a) plots the relation between µa and the ﬁrst term in (19), capturing the negative eﬀect of aggregate volatility. Not surprisingly, the maximum of this function is reached at the full-stabilization policy µf s , where it reaches zero. a With γ = 1, the second term in (19) is always zero, so I leave it aside. Panel (b) shows the eﬀects of monetary policy on the third term, the negative eﬀect due to the dispersion in labor supply. Panels (c) and (d) show the eﬀect on the “productivity” term ct − at − nt , which is further decomposed into two eﬀects, using equation (20). In panel (c), I report the negative eﬀect of the demand dispersion faced by a given producer, in panel (d), the positive eﬀect of the price dispersion faced by a given consumer. Figure 2 shows that the optimal monetary policy is to the left of the full-stabilization policy, which is consistent with Proposition 6, given that γσ > 1. Figure 3 shows that the crucial eﬀect behind this result is the eﬀect on price dispersion in panel (d). When moving from µf s to a µ∗ , there are welfare losses both in terms of aggregate volatility and in terms of labor supply a and demand dispersion, as shown in panels (a)-(c). But the welfare gain due to increased 30 Since prices are expressed in logs, an increase in the volatility of pjt has both a level and a volatility eﬀect. Given that σ > 1, the second always dominates. 25 0.45 0.44 0.43 0.42 w 0.41 0.4 0.39 0.38 0.37 0.36 0.35 0.7 0.8 0.9 * µa 1 1.1 µ a 1.2 fs µa 1.3 1.4 1.5 Figure 2: Welfare eﬀects of monetary policy. price dispersion more than compensate for these losses. Let me provide some intuition for the mechanism behind this picture. In a neighborhood of µ∗ , increasing µa has the eﬀect of reducing price dispersion by reducing a the value of |φx |, which determines the response of individual prices to individual productivity shocks. 31 This reduction in price dispersion can be interpreted as follows. At the optimal equilibrium, producers with higher productivity must set lower prices, to induce consumers to buy more of their goods. This requires φ∗ < 0. By increasing µa , the central bank induces x household consumption to be more responsive to the private productivity signal xit .32 This implies that a more productive household faces a lower marginal utility of consumption, and, at the price-setting stage, has a weaker incentive to lower the price of its good. Through this channel an increase in µa induces relative prices to be less responsive to diﬀerences in individual productivities, leading to a more compressed price distribution, as shown in panel (d). Under the parametric assumptions made, this mechanism also leads to a reduction in labor supply dispersion and in demand dispersion, as shown in panels (b) and (c). In the economy considered, at the optimal policy, individual labor supply, nit , is increasing in individual productivity, xit . When relative prices become less responsive to individual productivity, the 31 In the proof of Proposition 6, I show that φ∗ < 0 at µ∗ . This, together with ∂φx /∂µa > 0, from Proposition x a 3, implies that |φx | is decreasing in µa , in a neighborhood of µ∗ . a 32 Expression (48) in the appendix shows that ∂ψx /∂µa > 0. 26 (a) aggregate volatility 0 −0.02 −0.04 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.3 1.4 1.5 1.3 1.4 1.5 1.3 1.4 1.5 (b) labor supply dispersion 0 −0.2 −0.4 0.7 0.8 0.9 1 1.1 1.2 (c) demand dispersion 0 −0.2 −0.4 0.7 0.8 0.9 1 1.1 1.2 (d) price dispersion 1 0.8 0.6 0.7 0.8 0.9 µ* a 1 1.1 µa 1.2 µfs a Figure 3: Welfare eﬀects of monetary policy. Decomposition. 27 relation between xit and nit becomes ﬂatter and this reduces the cross-sectional dispersion in nit . Finally, an increase in µa leads to a compression in the distribution of demand indexes cjt + σpjt faced by a given producer, because the price indexes pjt become less dispersed. Summing up, if the central bank wants to reach full stabilization it has to induce households to rely more heavily on their private productivity signals xit when making their consumption decisions. By inducing them to concentrate on private signals the central bank can mute the eﬀect of public noise. However, in doing so, the central bank reduces the sensitivity of individual prices to productivity, generating an ineﬃciently compressed distribution of relative prices. Notice that in standard new Keynesian models, relative price dispersion is typically harmful for social welfare, because producers have identical productivities. Things are diﬀerent here, because there is heterogeneity in individual productivities. This does not mean that more price dispersion is always better. An increase in price dispersion eventually leads to an excessive increase in the dispersion of labor supply (as captured by panel (b) of Figure 3). Using Lemma 4, it is possible to quantify the welfare costs associated to suboptimal policies. Panel (a) in Figure 3 shows that, focusing purely on the aggregate output gap, the planner ﬁnds that going from µ∗ to µf s leads to an approximate welfare gain of 1% in equivalent consumption. a a However, when all cross-sectional terms are considered, Figure 3 shows that, in fact, this policy change generates welfare losses of more than 2% in equivalent consumption. Although this is just an example, these numbers show that disregarding the cross-sectional implications of policy, in an environment with heterogeneity, can lead to serious welfare miscalculations. 5.5 The role of strategic complementarity in pricing Proposition 6 identiﬁes a set of special cases where full stabilization is optimal. An especially interesting case is when η = 0, that is, when utility is linear in labor eﬀort. In this case, there is no strategic complementarity in price setting, under the nominal spending target (10)-(11). Substituting the consumer’s Euler equation (9) on the right-hand side of the pricing condition (8), and using the law of iterated expectations, after some manipulations, I obtain pit = µa − ρ γ at−1 + µa + γ−1 ρ Ei,(t,I) [θt ] − xit . γ (21) This shows that in this case prices only depend on the agents’ ﬁrst-order expectations regarding the fundamental shock θt . The analysis in Section 4.2 shows that even in this simple case an interesting form of non-neutrality is present, because of asymmetric information between price28 setters and consumers. However, in this case there is no signiﬁcant interaction among pricesetters. That is, the strategic complementarity emphasized in Woodford (2002) and Hellwig (2005) is completely muted. When η = 0, the planner can reach the constrained eﬃcient allocation by letting µa = −((γ − 1)/γ)ρ.33 This policy implies that the marginal utility of expenditure, which is proportional to exp{−pit − γcit }, is perfectly equalized across households. At the same time, by (21), relative prices are perfectly proportional to individual productivities. When η = 0 these relative prices achieve an eﬃcient cross-sectional allocation of labor eﬀort. That is, in this economy there is no tension between aggregate eﬃciency and cross-sectional eﬃciency. Actually, it is possible to prove that, under the optimal monetary policy, this economy achieves the full-information ﬁrst-best allocation.34 Once η = 0, producers must forecast their sales to set optimal prices and these sales depend on the prices set by other producers. Now the pricing decisions of the producers are fully interdependent. On the planner’s front, when η = 0, it is necessary to use individual estimates of θt when setting eﬃcient “shadow” prices. In this case, the optimal policy can no longer achieve the unconstrained ﬁrst-best. Therefore, the presence of strategic complementarity in pricing is tightly connected to the presence of an interesting trade-oﬀ between aggregate and cross-sectional eﬃciency. 6 The welfare eﬀects of public information So far, I have assumed that the source of public information, the signal st , is exogenous and outside the control of the monetary authority. Suppose now that the central bank has some control on the information received by the private sector. For example, it can decide whether or not to systematically release some aggregate statistics, which would increase the precision of public information. What are the welfare eﬀects of this decision? To address this question 2 I look at the eﬀects of changing the precision of the public signal, measured by πs ≡ 1/σe , on total welfare. This exercise connects this paper to a growing literature on the welfare eﬀects of public information.35 I consider two possible versions of this exercise. First, I assume that when πs changes the monetary policy rule µa is kept constant, while the subsidy τ is adjusted 33 This can be derived from equation (72) in the appendix. To prove this, follow the same steps as in the proof of Proposition 5.1, but allow the consumption rule to be contingent on θt . Then, it is possible to show that the optimal allocation is supported by the equilibrium described above. 35 See the references in footnote 3. 34 29 to its new optimal level. Second, I assume that for each value of πs both µa and τ are chosen optimally. Suppose the economy’s parameters are those in Table 1 and suppose that µa is ﬁxed at its optimal value for πs = 1. Figure 4 shows the eﬀect of changing πs on welfare. The solid line represents total welfare, measured by w in (19).36 The dashed line represents the relation between µa and the ﬁrst component in (19), which captures the negative welfare eﬀects of aggregate volatility. Let me begin by discussing this second relation. When the signal st is very imprecise agents disregard it and the coeﬃcient ψs goes to zero. When the signal becomes more precise, agents rely more on the public signal. So, although the volatility of et is falling, the increase in ψs can lead to an increase in aggregate volatility. In the example considered, this happens whenever log(πs ) is smaller than 1.8. In that region, more precise public information has a destabilizing eﬀect on the economy. Eventually, when the signal precision is suﬃciently large, the economy converges towards the full information equilibrium and output gap volatility goes to zero. Therefore, there is a non-monotone relation between µa and aggregate volatility. However, this only captures the ﬁrst piece of the welfare function (19). The solid line in Figure 4 shows that, when all the other pieces are taken into account, welfare is increasing everywhere in πs . To understand the relationship between the two graphs in Figure 4, notice that, when the public signal is very imprecise, agents have to use their own individual productivity to estimate aggregate productivity. This makes them underestimate the idiosyncratic component of productivity and leads to a compressed distribution of relative prices. An increase in the signal precision helps producers set relative prices that reﬂect more closely the underlying productivity diﬀerentials. The associated gain in allocative eﬃciency is always positive and more than compensates for the welfare losses due to higher aggregate volatility, in the region where log(πs ) ≤ 1.8. The notion that more precise information about aggregate variables has important crosssectional implications is also highlighted in Hellwig (2005). In that paper, agents face uncertainty about monetary policy shocks and there are no idiosyncratic productivity shocks. Therefore, the cross-sectional beneﬁts of increased transparency are reﬂected in a reduction in price dispersion. Here, instead, more precise public information tends to generate higher price dispersion. However, the underlying principle is the same: in both cases a more precise public 36 To improve readability, the values of w in the plot are normalized subtracting the value of w at πs = 0, and I use a log scale for πs . 30 0.1 total welfare aggregate volatility 0.08 0.06 w 0.04 0.02 0 −0.02 −0.04 −10 −8 −6 −4 −2 0 2 4 6 8 10 log(πs) Figure 4: Welfare eﬀects of changing the public signal precision, η = 2. signal leads to relative prices more in line with productivity diﬀerentials. Let me now consider a more intriguing example, where social welfare can be decreasing in πs . In Figure 5, I plot the relation between πs and w for an economy identical to the one above, except that the inverse Frisch elasticity of labor supply is set to a much higher value, η = 5. When η is larger, the costs of aggregate volatility are bigger, and, it is possible to have a non-monotone relationship between πs and total welfare, as shown by the solid line in Figure 5. For example, when log(π) increases from 0 to 1, social welfare falls by about one-half percent in consumption equivalent terms. This result mirrors the result obtained by Morris and Shin (2002) in a simple quadratic game. As stressed by Angeletos and Pavan (2007a), their result depends crucially on the form of the agents’ objective function and on the nature of their strategic interaction. In my model, the possibility of welfare-decreasing public information depends on the balance between aggregate and cross-sectional eﬀects. When η is large the negative welfare eﬀects of aggregate volatility become a dominant concern, and increases in public signal precision can be undesirable. This result disappears when I allow the central bank to adjust the monetary policy rule to changes in the informational environment. In this case, more precise public information is unambiguously good for social welfare. This is illustrated by the dotted line in Figure 5, which shows the relation between πs and w, when µa is chosen optimally. By Proposition 31 0.15 total welfare, fixed policy aggregate volatility, fixed policy total welfare, optimal policy aggregate volatility, optimal policy 0.1 w 0.05 0 −0.05 −10 −8 −6 −4 −2 0 2 4 6 8 10 log(πs) Figure 5: Welfare eﬀects of changing the public signal precision, η = 5. 5, the optimal µa induces agents to use information in a socially optimal way. Therefore, at the optimal policy, better information always leads to higher social welfare. This provides a general equilibrium counterpart to the results in Angeletos and Pavan (2007a), who apply the same principle to quadratic coordination games. The following proposition summarizes. Proposition 7 If µa is kept ﬁxed, an increase in πs can lead to a welfare gain or to a welfare loss, depending on the model’s parameters. If µa is chosen optimally for each πs , an increase in πs never leads to a welfare loss. 7 Conclusions In this paper, I have explored the role of monetary policy rules in an economy where information about macroeconomic fundamentals is dispersed across the economy. The emphasis has been on the ability of the policy rule to shape the economy’s response to diﬀerent shocks. In particular, the monetary authority is able to reduce the economy’s response to noise shocks by manipulating agents’ expectations about the real interest rate. The principle behind this result goes beyond the speciﬁc model used in this paper: by announcing that policy actions will respond to future information, the monetary authority can aﬀect diﬀerently agents with diﬀerent pieces of information. In this way, it can change the aggregate response to fundamental and noise shocks even if it has no informational advantage over the private sector. A second 32 general lesson that comes from the model is that, in the presence of heterogeneity and dispersed information, the policy maker will typically face a trade-oﬀ between aggregate eﬃciency and cross-sectional eﬃciency. Inducing agents to be more responsive to perfectly observed local information can lead to aggregate outcomes that are less sensitive to aggregate noise shocks, but it can also lead to a worse cross-sectional allocation. The optimal policy rule used in this paper can be implemented both under commitment and under discretion. To oﬀset an expansion driven by optimistic beliefs, the central bank announces that it will make the realized real interest rate higher if good fundamentals do not materialize. With ﬂexible prices, this eﬀect is achieved with a downward jump in the price level between t and t+1. Since at is common knowledge at time t+1, this jump only aﬀects nominal variables, but has no consequences on the real allocation in that period. Therefore, the central bank has no incentive to deviate ex post from its announced policy. In economies with sluggish price adjustment, a similar eﬀect could be obtained by a combination of a price level change and an increase in nominal interest rates. In that case, however, commitment problems are likely to arise, because both type of interventions have additional distortionary consequences ex post. The study of models where lack of commitment interferes with the central bank’s ability to deal with informational shocks is an interesting area for future research. A strong simplifying assumption in the model is that the only ﬁnancial assets traded in subperiods (t, I) and (t, II) are non-state-contingent claims on dollars at (t+1, 0). Introducing a richer set of traded ﬁnancial assets would increase the number of price signals available to both consumers and the monetary authority. In a simple environment with only two aggregate shocks, this will easily lead to a fully revealing equilibrium. Therefore, to fruitfully extend the analysis in that direction requires the introduction of a larger number of shocks, which make ﬁnancial prices noisy indicators of the economy’s fundamentals. Finally, in the model presented, the information sets of consumers, producers, and of the central bank, are independent of the monetary rule chosen. Morris and Shin (2005) have recently argued that stabilization policies may have adverse eﬀects, if they reduce the informational content of prices. Here this concern does not arise, as the information in the price indexes observed by consumers is essentially independent of monetary policy. A natural extension of the model in this paper would be to introduce additional sources of noise in prices, so as to make their informational content endogenous and sensitive to policy. 33 8 Appendix 8.1 Random consumption baskets At the beginning of each period, household i is assigned two random variables, it and vit , independently 2 drawn from normal distributions with mean zero and variances, respectively, σ 2 and σv . These variables are not observed by the household. The ﬁrst random variable represents the idiosyncratic productivity shock, the second is the sampling shock that will determine the sample of ﬁrms visited by consumer i. Consumers and producers are then randomly matched so that the probability that a producer with shock jt is matched to a consumer with shock vit is given by the bivariate normal density φ ( it , vjt ), with covariance matrix √ σ2 χσv σ , 2 . σv where χ is a parameter in [0, 1]. Since the variable vit has no direct eﬀect on payoﬀs, I can normalize 2 ˜ its variance and set σv = χσ 2 . Let Jit denote the set of producers met by consumer i and Jit the set of consumers met by producer i. Given the matching process above the following properties follow. The ˜ distribution { jt : j ∈ Jit } is a normal N (vit , σ 2|v ) with σ 2|v = (1 − χ) σ 2 . The distribution {vjt : j ∈ Jit } 2 2 2 is a normal N (χ it , σv| ), with σv| = χ(1 − χ)σ . 8.2 Proof of Proposition 1 The proof is split in steps. First, I derive price and demand indexes that apply in the linear equilibrium conjectured. Second, I use them to setup the individual optimization problem and derive necessary conditions for individual optimality. Third, I use these conditions to characterize a linear equilibrium. Fourth, I show how choosing µa uniquely pins down the coeﬃcients {φ, ψ} and derive the remaining coeﬃcients of the monetary policy rule that implements {φ, ψ}. The proof of local determinacy is in the supplementary material. 8.2.1 Price and demand indexes Lemma 5 If individual prices and quantities are given by (1) and (2) then the price index for consumer i and the demand index for producer i are equal to (6) and (7) where κp and κd are constant terms equal to κp = κd = 1−σ 2 2 φx σ |v , 2 1 2 2 1 1−σ 2 2 2 2 ψx σ + (ψx + σφx ) σv| + σ φx σ |v . 2 2 2 (22) (23) Proof. Recall that the shocks jt for j ∈ Jit have a normal distribution N (vit , σ 2|v ). Then, given (1), the prices observed by consumer i are log-normally distributed, with mean pt + φx vit and variance φ2 σ 2|v , therefore, x j∈Jit 1−σ Pjt dj = e(1−σ)(pt +φx vit )+ (1−σ)2 2 φ2 σ 2|v x . Taking both sides to the power 1/ (1 − σ) gives the desired expression for P it , from which (6) and (22) follow immediately. Using this result and expression (2), the demand index for producer i can be written as σ Dit = Cjt P jt dj = ect +σpt eσκp eψx jt +ψx vjt eσφx vjt dj. ˜ j∈Jit ˜ j∈Jit ˜ Recall that the distribution {vjt : j ∈ Jit } is a normal N (χ It follows that ˜ j∈Jit e ψx jt +ψx vjt +σφx vjt 1 2 dj = e 2 ψx σ 34 2 2 it , σv| +(ψx +σφx )χ ), and jt and vjt are independent. 2 2 1 it + 2 (ψx +σφx ) σv| . Substituting in the previous expression gives (7) and (23). 8.2.2 Individual optimization Consider an individual household, who expects all other households to follow (1)-(2) and the central bank to follow (10)-(11). In period (t, 0), before all current shocks are realized, the household’s expectations about the current and future path of prices, quantities and interest rates depend only on at−1 and Rt . Moreover, the only relevant individual state variable is given by the household nominal balances Bit . Therefore, I can analyze the household’s problem using the Bellman equation V (Bit , at−1 , Rt ) = max {Zit+1 (.)},{Bit+1 (.)}, {P (.,.)},{C(.,.,.)} Et [U (Cit , Nit ) + βV (Bit+1 , at , Rt+1 )] subject to the constraints Bit+1 (ωit ) = Rt Bit − −σ Yit = Dit Pit , q (˜ it ) Zit (˜ it ) d˜ it + (1 + τ ) Pit Yit − P jt Cit − Tt + Zit (ωit ) , ω ω ω Yit = Ait Nit , Pit = P (st , xit ) , Cit = C (st , xit , xit ) , and the law of motions for at and Rt+1 . Et [.] represents expectations formed at (t, 0) and, in the equilibrium conjectured, it can be replaced by E [.|at−1 ]. This problem gives the following optimality conditions for prices and consumption −1 −γ Ei,(t,I) (1 + τ ) P it Cit Yit − σ A−1 N η P −1 Yit σ − 1 it it it −1 −1 −γ −γ Ei,(t,II) P it Cit − βRt P it+1 Cit+1 = 0, (24) = 0, (25) where Ei,(t,I) [.] and Ei,(t,II) [.] denote the expectations of agent i at (t, I) and (t, II). Given the conjectured equilibrium and, given Lemma 1, they are equal to E [.|at−1 , st , xit ] and E [.|at−1 , st , xit , φx xit ]. By Lemma 5 all the random variables in the expressions above are log-normal, including the output −σ and labor supply of producer i which are equal to Yit = Dit Pit and Nit = A−1 Yit . Rearranging and it substituting in (24) and (25) gives (8) and (9) in the text, which I report here for completeness, pit pit + γcit = κp + η Ei,(t,I) [dit ] − σpit − ait + Ei,(t,I) [pit + γcit ] − ait , (26) = γκc − rt + Ei,(t,II) pit+1 + γcit+1 . (27) The constant terms κp and κc are equal to κp κc = H (ψs , ψx , ψx , φx ) − log (1 + τ ) , = G (ψs , ψx , ψx , φx ) , (28) (29) and H and G are known quadratic functions of ψs , ψx , ψx and φx . 8.2.3 Equilibrium characterization To check for individual optimality, I will substitute the conjectures made for individual behavior, (1) and (2), in the optimality conditions (26) and (27) and obtain a set of restrictions on {φ, ψ}. Notice that all the shocks are i.i.d. so the expected value of all future shocks is zero. Let me assume for now that φx = 0, so that E [.|st , xit , xit ] can replace E [.|st , xit , φx xit ]. Let βs , βx and δs , δx , δx be coeﬃcients such that E [θt |st , xit ] = βs st + βx xit and E [θt |st , xit , xit ] = δs st + δx xit + δx xit . Deﬁning the precision −1 2 −1 2 −1 2 −1 parameters πθ ≡ σθ , πs ≡ σe , πx ≡ σ 2 , and πx ≡ σv , the coeﬃcients βs , βx and δs , δx , δx are πx πs , βx = , (30) βs = πθ + πs + πx πθ + πs + πx πs πx πx δs = , δx = , δx = . (31) πθ + πs + πx + πx πθ + πs + πx + πx πθ + πs + πx + πx 35 I use (6) and (7) to substitute for pit and dit in the optimality conditions (26) and (27), and then I use (1)-(4) to substitute for pit , cit , pt and ct . Finally, I use it = xit −θit and vit = xit −θit , and I substitute for E [θt |st , xit ] and E [θt |st , xit , xit ]. After these substitutions, (26) and (27) give two linear equations in at−1 , st , xit , xit . Matching the coeﬃcients term by term and rearranging gives me the following set of equations. (γ + η)ψ0 = −(κp + κp + ηκd ), (γ + η)ψa = (1 + η)ρ, rt = γκc − (φa + γψa ) (1 − ρ) at−1 , (1 + ησ) φs (1 + ησ) φx = = η (ψs + σφs + (ψx + ψx + σφx ) βs − (ψx + σφx ) χβs ) + (φa + γψa ) βs , η ((ψx + ψx + σφx ) βx + (ψx + σφx ) χ (1 − βx ) − 1) + (φa + γψa ) βx − 1, φs + γψs γψx φx + γψx = (φa + γψa ) δs , = (φa + γψa ) δx , = (φa + γψa ) δx . (32) (33) (34) (35) (36) (37) (38) (39) Notice that ψa is given immediately by (33) and is independent of all other parameters. Next, notice that to ensure that (34) always holds in equilibrium, for any choice of ξm ∈ R, the following conditions need to be satisﬁed ξ0 ξa µ0 µa µθ µe 8.2.4 = = = = = γκc , − (φa + γψa ) (1 − ρ) , ψ0 , φ a + ψa , φs + φx + ψs + ψx + ψx , = φ s + ψs . (40) (41) (42) (43) (44) (45) Constructing the linear equilibrium for given µa Given µa , I immediately get φa = µa − ψa from (43). To ﬁnd the values of the remaining parameters in {φ, ψ} as a function of µa , I use (35)-(39) together with the following condition, which follows from (43) φa + γψa = µa + (γ − 1) ψa . (46) To simplify the notation, the right-hand side of this expression is denoted by µ ≡ µa + (γ − 1) ψa . ˜ First, using (36), (38) and (39), I solve for φx , ψx and ψx , φx = βx + ηγ −1 (δx βx + δx (βx + χ (1 − βx ))) µ − 1 − η ˜ , −1 ) (β + χ (1 − β )) 1 + ησ − η (σ − γ x x ψx ψx = = γ −1 δx µ, ˜ γ −1 (δx µ − φx ) . ˜ (47) (48) (49) Note that a solution for φx always exists since 1+ησ−η σ − γ −1 (βx + χ (1 − βx )) > 0. This inequality follows from βx ∈ [0, 1] and χ ∈ [0, 1]. Next, combining (35) and (37), and using the expressions above for φx , ψx and ψx , I ﬁnd φs and ψs , φs = βs + ηγ −1 δs + ηγ −1 (δx + δx (1 − χ)) βs µ + η σ − γ −1 (1 − χ) φx βs ˜ , −1 1 + ηγ 36 (50) ψs = γ −1 (δs µ − φs ) . ˜ (51) Substituting the values of ψs , ψx , ψx and φx thus obtained in (22), (23), (28), I obtain values for κp , κd and κp . Substituting these values in (32), shows that ψ0 takes the form ψ0 = J (ψs , ψx , ψx , φx ) + log (1 + τ ) , γ+η (52) where J is a known quadratic function of ψs , ψx , ψx and φx . To ﬁnd the remaining parameters of the monetary policy rule ξ0 , µ0 , µa , µs , µθ , use (40)-(42) and (44)-(45). To show that the prices and quantities above form an equilibrium, I need to check that the market for state-contingent claims clears and Bit is constant and equal to 0. Let f ( it , vit , θt , et ) denote the joint density of the shocks it , vit , θt , and et . Recall that ωit ≡ ( it , vit , θt , et ) and let the prices of state-contingent claims at (t, 0) be −1 Q(ωit ) = Rt f ( it , vit , θt , et ) g (θt ) , (53) where g (.) is a function to be determined. Suppose Bit = 0. Let the portfolio of state-contingent claims be the same for each household and equal to Zit+1 (ωit ) = Rt P it Cit − (1 + τ ) Yit Pit + Tt . For each realization of the aggregate shocks θt and et , goods markets clearing and the government budget balance condition imply that ∞ ∞ 1 Pit Yit − P it Cit di = 0. Zit+1 ({ , v, θt , et }) f ( , v, θt , et ) d dv = Rt −∞ −∞ 0 This implies that the market for state-contingent claims clears for each aggregate state θt . It also implies that the portfolio {Zit+1 (ωit )} has zero value at date (t, 0) given that ∞ R4 ∞ ∞ ∞ −∞ −∞ −∞ −∞ Q (ωit ) Zit+1 (ωit ) dωit = Zit+1 ({ , v, θ, e}) f ( , v, θ, e) g (θ) d dv dθ de = 0. Substituting in the household budget constraint shows that Bit+1 = 0. Let me check that the portfolio just described is optimal. The ﬁrst-order conditions for Zit+1 (ωit ) and Bit+1 (ωit ) are, respectively, λ (ωit ) = Rt Q (ωit ) λ (ωit ) = ∂V (0, at , Rt+1 ) f (ωit ) , ∂Bit+1 R3 λ (˜ it ) d˜ it , ω ω where λ (ωit ) is the Lagrange multiplier on the budget constraint. Combining them and substituting for ∂V /∂Bit+1 , using the envelope condition ∂V (0, at , Rt+1 ) −1 −γ = E P it+1 Cit+1 |at , ∂Bit+1 I then obtain ∞ −1 −γ E P it+1 Cit+1 |at f (ωit ) = Rt Q (ωit ) −∞ −1 −γ ˜ ˜ ˜ E P it+1 Cit+1 |ρat−1 + θ f (θ)dθ. Substituting (1), (2), and (53), and eliminating the constant factors on both sides, this becomes e−(φa +γψa )(ρat−1 +θt ) = g (θt ) ∞ −∞ 37 ˜ ˜ ˜ e−(φa +γψa )(ρat−1 +θ) f (θ)dθ, which is satisﬁed as long as the function g (.) is given by g (θt ) ≡ exp − (φa + γψa ) θt − 1 2 2 (φa + γψa ) σθ . 2 Finally, to complete the equilibrium construction, I need to check that φx = 0. From (47), this requires µa = µ0 where a µ0 ≡ a 1+η 1+η − ρ (γ − 1) . βx + ηγ −1 (δx βx + δx (βx + χ (1 − βx ))) γ+η Notice that when µa = µ0 a stationary linear equilibrium fails to exists. A stationary equilibrium with a φx = 0 can arise, but under a policy µa which is typically diﬀerent from µ0 . If φx = 0 all the derivations a above go through, except that E [θt |st , xit , φx xit ] = βs st + βx xit . Therefore, it is possible to derive the analogous of condition (47) and show that φx = 0 iﬀ 1 + ηγ −1 βx βx µ − 1 − η ˜ = 0. −1 ) (β + χ (1 − β )) 1 + ησ − η (σ − γ x x This shows that an equilibrium with φx = 0 arises when µa = µa , where ˆ µa ≡ ˆ 1+η 1+η − ρ (γ − 1) . −1 β ) βx (1 + ηγ γ+η x However, µa is also consistent with an equilibrium with φx = 0. Summing up, if µa = µ0 there is no ˆ a stationary linear equilibrium; if µa = µa there are two stationary linear equilibria, one with φx = 0 and a one with φx = 0; if µa ∈ R/ µ0 , µa , there is a unique stationary linear equilibrium. a ˆ 8.3 Proof of Proposition 2 2 If σe = 0 then βs = δs = 1 and βx = δx = δx = 0. Substituting in (47)-(51) gives φx = φs = 1+η , 1 + ησ − η (σ − γ −1 ) χ η σ − γ −1 (1 − χ) µ+ ˜ φx , 1 + ηγ −1 − (54) and ψx = −γ −1 φx , ψx = 0, ψs = γ −1 (˜ − φs ) = −γ −1 µ η σ − γ −1 (1 − χ) φx , 1 + ηγ −1 and ψ0 can be determined from (52). Notice φs is the only coeﬃcient which depends on µa . However, the real equilibrium allocation only depends on the consumption levels cit and on the relative prices pit − pt , and, given (1) and (2), these are independent of φs . 8.4 Proof of Proposition 3 For the following derivations recall that under imperfect information all the coeﬃcients βs , βx , δs , δx , δx are in (0, 1) and χ ∈ [0, 1]. Diﬀerentiating (47) with respect to µa (recalling that µ = µa + (γ − 1) ψa ˜ and ψa is constant), gives ∂φx βx + ηγ −1 (δx βx + δx (βx + χ (1 − βx ))) = > 0, ∂µa 1 + ησ − η (σ − γ −1 ) (βx + χ (1 − βx )) 38 where the denominator is positive since βx + χ (1 − βx ) < 1 and η σ − γ −1 < ησ. Diﬀerentiating (50) gives ηβs σ − γ −1 (1 − χ) ∂φx ∂φs ηγ −1 (δx + δx (1 − χ)) βs βs + ηγ −1 δs + + = . −1 −1 ∂µa 1 + ηγ 1 + ηγ 1 + ηγ −1 ∂µa Some lengthy algebra, in the supplementary material, shows that the term in square brackets is positive, so ∂φs /∂µa > 0. Next, diﬀerentiating (51) gives ∂ψs ∂φs = γ −1 δs − ∂µa ∂µa . To prove that this quantity is negative notice that ∂φs βs + ηγ −1 δs > > δs , ∂µa 1 + ηγ −1 where the last inequality follows from βs > δs , which follows immediately from (30) and (31). To prove that ∂ψθ /∂µa > 0 it is suﬃcient to use of the last result together with Lemma 2, which immediately 2 2 implies that ∂ψθ /∂µa = − σe /σθ ∂ψs /∂µa . 8.5 Proof of Proposition 4 The argument in the text shows that there is a µa that gives coeﬃcients ψs = 0 and ψθ = (1+η)/(γ +η), if one assumes that consumers form expectations based on at−1 , st , xit , and xit . It remains to check that this value of µa is not equal to µ0 , so that φx = 0 and observed prices reveal xit . The algebra is a presented in the supplementary material. 8.6 Proof of Lemma 3 Let me consider the case γ = 1, the proof for the case γ = 1 follows similar steps and is presented in 1−γ the supplementary material. First, I derive expressions for the conditional expectations E[Cit |at−1 ] 1+η and E[Nit |at−1 ]. Substituting for cit in the ﬁrst, using (2), I obtain 1−γ E Cit |at−1 = 2 2 2 2 2 2 2 2 2 1 e(1−γ)(ψ0 +ψa at−1 )+ 2 (1−γ) (ψθ σθ +ψs σe +ψx σ +ψx σv ) . Using (7) to substitute for dit , and the fact that ait = at + equilibrium labor supply Nit = it and pit − pt = φx it , I derive the −σ Dit Pit = eκd +ψ0 +ψa at−1 +ψθ θt +ψs et −at −(1+σφx −(ψx +σφx )χ) it . Ait (55) From this expression, I obtain 2 2 2 2 2 2 2 1 1+η E Nit |at−1 = e(1+η)(κd +ψ0 +(ψa −ρ)at−1 )+ 2 (1+η) ((ψθ −1) σθ +ψs σe +(1+σφx −(ψx +σφx )χ) σ ) . Using the fact that ψa = ρ (1 + η) / (γ + η) to group the terms in at−1 , the instantaneous conditional expected utility takes the form E [U (Cit , Nit ) |at−1 ] = 1 (1+η)(k2 +ψ0 ) (1−γ)ψa at−1 1 (1−γ)(k1 +ψ0 ) e − e e . 1−γ 1+η (56) where k1 = k2 = 1 2 2 2 2 2 2 2 (1 − γ) ψθ σθ + ψs σe + ψx σ 2 + ψx σv , 2 1 2 2 2 2 2 κd + (1 + η) (ψθ − 1) σθ + ψs σe + (1 + σφx − (ψx + σφx ) χ) σ 2 . 2 39 (57) (58) The equilibrium equation (52) shows that, for each value of µa , there is a one-to-one correspondence between τ and ψ0 , and ψ0 is the only equilibrium coeﬃcient aﬀected by τ . Therefore, if τ is set optimally, ψ0 must maximize the term in square brackets on the right-hand side of (56). Solving this problem shows that the term in square brackets must be equal to exp{(((1 − γ) (1 + η))/(γ + η))(k1 − k2 )} and ψ0 = (1 − γ) k1 − (1 + η) k2 . γ+η (59) Then, I can take the unconditional expectation of (56) and sum across periods to obtain ∞ ∞ β t E [U (Cit , Nit )] = t=0 (1−γ)(1+η) (1−γ)(1+η) γ+η e γ+η (k1 −k2 ) β t E e γ+η ρat−1 . (1 + η) (1 − γ) t=0 Letting w ≡ k1 − k2 , and W0 ≡ γ+η 1+η ∞ βtE e (60) (1−γ)(1+η) ρat−1 γ+η , t=0 I then obtain the expression in the text. Combining (57), (58), and (60), shows that w can be expressed as follows w = − 1 (1 − γ)(1 + η) 2 1 σθ − (γ + η) 2 γ+η 2 ψθ − 1+η γ+η 2 2 2 2 σθ + ψs σe + 1 1 2 2 2 2 + (1 − γ) ψx σ 2 + ψx σv − (1 + η) (1 + σφx − (ψx + σφx ) χ) σ 2 , 2 2 1 1 2 2 2 − ψx σ 2 + (ψx + σφx ) σv| + σ (σ − 1) φ2 σ 2|v . (61) x 2 2 This shows that w is a quadratic functions of the equilibrium coeﬃcients φ and ψ. Moreover, Proposition 3 shows that φ and ψ are linear functions of µa . Therefore, (61) implicitly deﬁnes w as a quadratic function of µa . The discounted sum in W0 is always ﬁnite if ρ < 1, because each term is bounded by 2 2 exp{((1 − γ)ψa ρa−1 + (1/2)((1 − γ)2 /(1 − ρ2 ))ψa σθ } . If, instead, ρ = 1, to ensure that the sum is 2 2 ﬁnite it is necessary to assume that β exp (1/2)(1 − γ)2 ψa σθ < 1, which is equivalent to the inequality in footnote 11. 8.7 Derivation of equations (19) and (20) I will ﬁrst show that (19) corresponds to (61) in the proof of Lemma 3. For ease of exposition, the 2 expression in the text omits the constant term −(1/2)((1 − γ)(1 + η)/(γ + η))σθ . The ﬁrst two terms in (19) can be derived from the two terms after the constant in (61), simply using the deﬁnitions of ct 1 2 and cit . Equation (55), can be used to derive 0 (nit − nt ) di, and check that the third term in (19) equals the third term after the constant in (61). Finally, the last line of (61) corresponds to −κd , by (23), while equation (55) implies that nt = κd + ct − at . This shows that the last terms in (19) and (61) are equal. To derive (20) notice that, as just argued, the last line of (61) is equal to −κd . The derivations in Lemma 5 can then be used to obtain the expression in the text. 8.8 Proof of Lemma 4 I concentrate on the case γ = 1, the case γ = 1 is proved along similar lines. Given two monetary policies µa = µa and µa = µa + u, let Cit , Nit and Cit , Nit denote the associated equilibrium allocations, and deﬁne the function −1 ∞ βtE e f (δ, u) ≡ t=0 (1−γ)(1+η) ρat−1 γ+η ∞ ∞ β t E U eδ Cit , Nit t=0 40 β t E [U (Cit , Nit )] . − t=0 Proceeding in as in the proof of Lemma 3, it is possible to show that f (δ, u) = (1+η)(1−γ) 1 (1−γ)(δ+k1 +ψ0 ) 1 (1+η)(k2 +ψ0 ) γ+η e − e − e γ+η w(µa +u) , 1−γ 1+η (1 + η) (1 − γ) where k1 and k2 are deﬁned in (57) and (58), for the coeﬃcients {φ, ψ} associated to the policy µa , and the function w (.) is deﬁned by (61). Let the function δ (u) be deﬁned implicitly by f (δ (u) , u) = 0. It is immediate that δ (0) = 0. Moreover, ∂f (δ, u) ∂δ ∂f (δ, u) ∂u = e(1−γ)(k1 +ψ0 ) , δ=u=0 = e (1+η)(1−γ) w(µa ) γ+η w (µa ) , δ=u=0 and (59) implies that e(1−γ)(k1 +ψ0 ) = e (1+η)(1−γ) w(µa ) γ+η . It follows that δ (u) = w (µa ) . Since ∆ (µa , µa + u) ≡ exp {δ (u)} − 1, the result follows from diﬀerentiating this expression at u = 0. 8.9 Proof of Proposition 5 Let me begin by setting up and characterizing the planner’s problem. Then, I will show that there is a monetary policy that reaches the constrained optimal allocation. Let a− be a given scalar representing productivity in the previous period. Let θ be a normally distributed random variable with mean zero and 2 variance σθ and let s be a random variable given by s = θ + e, where e is also a normal random variable 2 with mean zero and variance σe . Let x, x and x be random variables given by x = θ + , x = θ + v, and ˜ 2 x = x + ˜, where , v and ˜ are independent random variables with zero mean and variances σ 2 , σv , σ 2|v . ˜ ˜ The planner’s problem is to choose functions C (s, x, x, x), C (s, x, x), and N (s, x, θ) that maximize ˜ E [U (C (s, x, x) , N (s, x, θ))] subject to C (s, x, x) = x eρa− +˜ N (s, x, θ) = ˜ E ˜ C (s, x, x, x) ˜ σ−1 σ ˜ E C (s, x, x, x) |s, x, θ ˜ ˜ σ σ−1 |s, x, x for all s, x, x, (62) for all s, x, θ. ˜ (63) Let Λ (s, x, θ) denote the Lagrange multiplier on constraint (63). Substituting (62) in the objective ˜ ˜ function, one obtains the following ﬁrst-order conditions with respect to C (s, x, x, x) and N (s, x, θ): ˜ ˜ ˜ C (s, x, x, x) ˜ 1 −σ 1 (C (s, x, x)) σ −γ η (N (s, x, θ)) ˜ ˜ = E [Λ (s, x, θ) |s, x, x, x] , ˜ (64) x = eρa− +˜ Λ (s, x, θ) . ˜ (65) The planner’s problem is concave, so (64) and (65) are both necessary and suﬃcient for an optimum. To prove the proposition, I take the equilibrium allocation associated to a generic pair (µa , τ ), and I derive conditions on µa and τ which ensure that it satisﬁes (64) and (65). An equilibrium allocation immediately satisﬁes the constraints (62) and (63), the ﬁrst by construction, the second by market 41 ˜ clearing. Take a linear equilibrium allocation characterized by ϕ and ψ. Let C (., ., ., .) and N (., ., .) take the form ˜ C (s, x, x, x) = exp {σκp + ψ0 + ψa a− + ψs s + ψx x + ψx x − σφx (˜ − x)} , ˜ x (66) N (s, x, θ) = exp {κd + ψ0 + ψa a− + ψs s + (ψx + ψx ) θ − ρa− − x − (σφx − (ψx + σφx ) χ) (˜ − θ)} ˜ ˜ x I conjecture that the Lagrange multiplier Λ (s, x, θ) takes the log-linear form ˜ Λ (s, x, θ) = exp {λ0 + λs s + λx x + λθ θ} . ˜ ˜ (67) Let me ﬁrst check the ﬁrst-order condition for consumption, (64). Substituting (66) in (62) and using the deﬁnition of κp , I get C (s, x, x) = exp {ψ0 + ψa a− + ψs s + ψx x + ψx x} . After some simpliﬁcations, the right-hand side of (64) then becomes 1 1 − −γ ˜ ˜ = exp {−κp + φx (˜ − x)} exp {−γ (ψ0 + ψa a− + ψs s + ψx x + ψx x)} . x C (s, x, x, x) σ C (s, x, x) σ The left-hand side of (64), using (67), is equal to 1 E [Λ (s, x, θ) |s, x, x, x] = exp λ0 + λs s + λx x + λθ E [θ|s, x, x] + λ2 σθ , ˜ ˜ ˜ ˆ2 2 θ −1 where σθ is the residual variance of θ, equal to (πθ + πs + πx + πx ) ˆ2 holds for all s, x, x, x, the following conditions must hold, ˜ 1 λ0 + λ2 σθ ˆ2 2 θ λs + λθ δ s λx λθ δx λθ δx . Therefore, to ensure that (64) = −κp − γ (ψ0 + ψa a− ) , = −γψs , = φx , = −γψx , = −γψx − φx . 2 Set λθ = − (φa + γψa ) , λs = φs , λx = φx and λ0 = −κp − γ (ψ0 + ψa a− ) − (1/2)λ2 σθ . Then, the θˆ ﬁrst and the third of these conditions hold immediately. The other three follow from the equilibrium relations (37)-(39). Let me now check the ﬁrst order condition for labor eﬀort, (65). Substituting (67) and matching the coeﬃcients on both sides, gives λ0 + ρa− λs λx + 1 λθ = = = = η (κd + ψ0 + (ψa − ρ) a− ) , ηψs , −η (1 + σφx − (ψx + σφx ) χ) , η (ψx + ψx + σφx − (ψx + σφx ) χ) . Substituting, the λ’s derived above, using ψa = ρ (1 + η) / (γ + η) and rearranging, gives 2 ˆ2 (γ + η) ψ0 + ηκd + κp + (1/2) (φa + γψa ) σθ = 0, φs − ηψs = 0, (1 + ησ) φx + 1 + η − η (ψx + σφx ) χ = 0, φa + γψa + η (ψx + ψx + σφx − (ψx + σφx ) χ) = 0. (68) (69) (70) (71) To complete the proof, I need to ﬁnd µa and τ such that the corresponding equilibrium coeﬃcients ϕ and ψ satisfy (68)-(71). Setting µ = µ∗ where ˜ ˜ µ∗ ≡ ˜ η σ − γ −1 (1 − χ) (1 + η) , (1 + ηγ −1 (δx + δx (1 − χ))) (1 + ησ − η (σ − γ −1 ) χ) + η 2 (σ − γ −1 ) (1 − χ) γ −1 δx χ 42 (72) ensures that (69)-(71) are satisﬁed. To see why these three conditions can be jointly satisﬁed, notice that the equilibrium conditions (35) and (36) can be rewritten as φs − ηψs = (1 + ησ) φx + 1 + η − η (ψx + σφx ) χ = [η ((ψx + ψx + σφx ) − (ψx + σφx ) χ) + φa + γψa ] βs , [η ((ψx + ψx + σφx ) − (ψx + σφx ) χ) + φa + γψa ] βx , so that, in equilibrium, (71) implies the other two. Finally, the subsidy τ can be set so as to ensure that (68) is satisﬁed. The value of φx at the optimal monetary policy is φ∗ = x −1 − η + ηγ −1 δx χ˜∗ µ . −1 ) χ 1 + ησ − η (σ − γ (73) Substituting (72) in the expression ηγ −1 δx χ˜∗ , shows that this expression is strictly smaller than 1 + η, µ which implies that φ∗ < 0. This conﬁrms that µ∗ = µ0 , so that, by Proposition 1, the associated a a x coeﬃcients ϕ∗ and ψ ∗ form a linear equilibrium. 8.10 Proof of Proposition 6 Let me derive the value of ψs at the constrained eﬃcient allocation. From condition (69) and the equilibrium condition φs + γψs = µδs , I get ˜ ∗ ψs = 1 δs µ ∗ . ˜ γ+η ∗ If χ = 0, the consumer extracts perfect information from xit = θt and δs = 0, which implies that ψs = 0. ∗ ∗ If, instead χ > 0, ψs inherits the sign of µ . Inspecting (72) shows that if η > 0, χ < 1 and σγ = 1, ˜ µ∗ is not zero and has the sign of σγ − 1. In all other cases, µ∗ = 0. Therefore, if η > 0, χ ∈ (0, 1) ˜ ˜ ∗ ∗ and σγ = 1, ψs is not zero and has the sign of σγ − 1. In all remaining cases ψs = 0. The inequalities ∗ ∗ for ψθ follow from Lemma 2. To prove the inequalities for µa , notice that, by Proposition 3 there is a decreasing relation between µa and ψs , and ψs = 0 at µa = µf s . a 8.11 Proof of Proposition 7 The ﬁrst part of the Proposition is proved by the two examples discussed in the text. Let me prove the second part. By Proposition 5, social welfare under the optimal monetary policy is the value of a single decision maker’s optimization problem (the planner’s). 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Taking expectations at time (T, 0) and integrating across agents (in that order) gives ET [(1 + ησ) pt − η (˜t + σ pt ) + pt + γ˜t ] = 0, ˜ c ˜ ˜ c ET [˜t + γ˜t + rt − pt+1 − γ˜t+1 ] = 0, p c ˜ ˜ c (74) for all t ≥ T . The ﬁrst condition implies that ET [˜t ] = 0 for all t ≥ T. c (75) Moreover, notice that mt = mt − mt . Therefore, under the policy rule (10) ˜ ˆ rt = ξm mt−1 for all t ≥ T. ˜ ˜ Rewrite (74) as ET [mt + (γ − 1) ct + rt − mt+1 − (γ − 1) ct+1 ] = 0. ˜ ˜ ˜ ˜ ˜ Using (75) and deﬁning ht ≡ ET [mt ], this gives a diﬀerence equation for ht , ˜ ht+1 − ht − ξht−1 = 0 for all t ≥ T with initial condition hT −1 = mT −1 . The assumption ξm > 1 ensures that any hT −1 = 0 gives an ˜ explosive solution. This shows that any equilibrium in a neighborhood of the original equilibrium must display mt = mt for all realizations of the aggregate shocks. Using this result one can show that the ˆ individual prices and consumption are the same as under the original equilibrium. 9.2 Algebra for the proof of Proposition 3 I need to prove that γ −1 (δx + δx (1 − χ)) + σ − γ −1 (1 − χ) ∂φx > 0, ∂µa which is equivalent to proving (δx + δx (1 − χ)) 1 + ησ − η σ − γ −1 (βx + χ (1 − βx )) + (γσ − 1) (1 − χ) βx + ηγ −1 (δx βx + δx (βx + χ (1 − βx ))) > 0. Let me show separately that (δx + δx (1 − χ)) (1 + ησ − ησ (βx + χ (1 − βx ))) + (γσ − 1) (1 − χ) βx > 0, and (δx + δx (1 − χ)) (βx + χ (1 − βx )) + − (1 − χ) (δx βx + δx (βx + χ (1 − βx ))) > 0. 48 For the ﬁrst, it is suﬃcient to observe that δx + δx (1 − χ) − (1 − χ) βx > 0, follows from δx + δx > βx . For the second, it is enough to see that δx ((1 − χ) βx + χ) − (1 − χ) δx βx > 0. 9.3 Algebra for the proof of Proposition 4 To prove that µa = µ0 , I suppose the contrary and use the necessary conditions for an equilibrium to a obtain a contradiction. By deﬁnition, µa = µ0 implies φx = 0. Summing (38) and (39) and using a ψx + ψx = shows that φa + γψa = Substituting in (39) then gives ψx = 1+η , γ+η (76) 1 γ (1 + η) . δx + δx γ + η δx 1 + η . δx + δx γ + η Substituting the last three expression in (36) then gives 0=η 1+η δx 1 + η βx γ (1 + η) βx + χ (1 − βx ) − 1 + − 1. γ+η δx + δx γ + η δx + δx γ + η This leads to a contradiction because η δx η βx γ βx + χ (1 − βx ) + < 1, γ+η δx + δx γ + η δx + δx γ + η where the inequality follows because βx + and δx χ (1 − βx ) < 1 δx + δx βx < 1. δx + δx The last inequality follows from (30) and (31). 9.4 Proof of Lemma 3: case γ = 1 When γ = 1 steps analogous to the ones used in case γ = 1 lead to E [U (Cit , Nit ) |at−1 ] = ψ0 + ψa at−1 − 1 (1+η)(k2 +ψ0 ) e , 1+η and the optimal choice of τ (and ψ0 ) gives the ﬁrst-order condition 1 = e(1+η)(k2 +ψ0 ) , which implies that ψ0 = k1 − k2 = w, 49 as γ = 1 implies that k1 = 0. The unconditional expected utility is then ∞ ∞ 1 1 1 β E [U (Cit , Nit )] = − + β t E [ρat−1 ] + w, 1 + η 1 − β t=0 1−β t=0 t which gives the expression in the text, with ∞ 1 1 w0 ≡ − + β t E [ρat−1 ] . 1 + η 1 − β t=0 50