Business Cycle Dynamics under Rational Inattention

Transcription

1 Business Cycle Dynamics under Rational Inattention Bartosz Maćkowiak European Central Bank and CEPR Mirko Wiederholt Northwestern University First draft: June 27 This draft: October 28 Preliminary Version Abstract This paper develops a dynamic stochastic general equilibrium model with rational inattention Decision-makers have limited attention and choose the optimal allocation of their attention We study the implications of rational inattention for business cycle dynamics For example, we study how rational inattention affects the impulse responses of prices and quantities to monetary policy, aggregate technology and micro-level shocks The impulse responses under rational inattention have several properties of empirical impulse response functions, eg, (i) prices respond slowly to monetary policy shocks, (ii) prices respond faster to aggregate technology shocks, and (iii) prices respond very fast to disaggregate shocks In addition, profit losses due to deviations of the actual price from the profit-maximizing price are an order of magnitude smaller than in the Calvo model that generates the same real effects We thank for helpful comments: Paco Buera, Larry Christiano, James Costain, Martin Eichenbaum, Christian Hellwig, Giorgio Primiceri and seminar as well as conference participants at Bank of Canada, Chicago Fed, Columbia, DePaul, European Central Bank, European University Institute, ESSIM 28, MIT, Minneapolis Fed, NBER Summer Institute 28, North American Winter Meeting of the Econometric Society 28, Riksbank, SED 28, Stony Brook and UCSD The views expressed in this paper are solely those of the authors and do not necessarily reflect the views of the European Central Bank bartoszmackowiak@ecbint and m-wiederholt@northwesternedu

2 Introduction This paper develops a dynamic stochastic general equilibrium model with rational inattention We model the idea that agents cannot attend perfectly to all available information Therefore, the mapping between economic conditions and the price and quantity decisions taken by agents is not perfect Decision-makers make mistakes However, decision-makers try to minimize these mistakes The economy consists of households, firms and a government Households supply differentiated types of labor, consume a variety of goods, and hold nominal government bonds Firms supply differentiated goods that are produced with the differentiated types of labor The central bank sets the nominal interest rate according to a Taylor rule Households take wage setting and consumption decisions Firms take price setting and factor mix decisions In the model, prices and wages are physically fully flexible The only source of slow adjustment is decision-makers limited attention We compute the impulse responses of all variables to monetary policy shocks, aggregate technology shocks and micro-level shocks under rational inattention Rational inattention means that decision-makers have limited attention and that they choose how to allocate their attention Following Sims (23), we model attention as a flow of information, and we model limited attention as a constraint on the flow of information We let agents decide how to allocate the information flow For example, agents decide how to allocate their attention across their different decision problems Furthermore, agents decide how closely they attend to the different factors that may affect an optimal decision We find that rational inattention by decision-makers in firms has the following implications First, for our parameter values, rational inattention by decision-makers in firms implies that the impulse response of the price level to monetary policy shocks resembles the impulse response in a Calvo model with an average price duration of 75 months At the same time, prices respond fairly quickly to aggregate technology shocks and almost perfectly to micro-level shocks The reason is the optimal allocation of attention Decision-makers in firms decide to pay little attention to monetary policy, about twice as much attention to aggregate technology, and a lot of attention to firm-specific conditions Therefore, prices respond slowly to monetary policy shocks, but fairly quickly to aggregate technology shocks, 2

3 and almost perfectly to micro-level shocks Second, losses in profits due to deviations of the actual price from the profit-maximizing price are an order of magnitude smaller than in the Calvo model that generates the same real effects In particular, losses in profits due to sub-optimal price responses to aggregate conditions are 23 times smaller than in the Calvo model;andlossesinprofits due to sub-optimal price responses to firm-specific conditions are 57 times smaller than in the Calvo model that generates the same real effects One reason is the optimal allocation of attention, which implies that prices respond slowly to monetary policy shocks, but fairly quickly to aggregate technology shocks, and almost perfectly to micro-level shocks In contrast, in the Calvo model prices respond slowly to all these shocks The other reason is that under rational inattention deviations of the actual price from the profit-maximizing price are less likely to be extreme than in the Calvo model This also reduces profit losses Third, rational inattention on the side of decision-makers in firms implies that firms produce with a factor mix that tends to deviate from the optimal factor mix Adding rational inattention by households yields responses of consumption to shocks that look similar to the impulse responses of consumption in a model with habit formation This paper is related to two strands of literature: (i) the literature on business cycle models with imperfect information (eg Lucas (972), Woodford (22), Mankiw and Reis (22), and Lorenzoni (28)) and (ii) the literature on rational inattention (eg Sims (23, 26), Luo (28), Maćkowiak and Wiederholt (28), Van Nieuwerburgh and Veldkamp (28), and Woodford (28)) The main difference to the existing literature on business cycle models with imperfect information is that in the model presented below agents choose the information structure The main innovation with respect to the existing literature on rational inattention is that we solve a dynamic stochastic general equilibrium model The paper is organized as follows Section 2 describes all features of the economy apart from the decision-makers problem of allocating attention Section 3 describes the steady state of the non-stochastic version of the economy In Section 4 we derive the objective that decision-makers in firms maximize when they choose the allocation of attention In Section 5 we derive the objective that households maximize when they choose the allocation of attention Section 6 describes issues related to aggregation Section 7 characterizes the 3

4 solution of the model under perfect information Section 8 shows numerical solutions of the model under rational inattention by decision-makers in firms Section 9 studies the implications of adding rational inattention by households Section concludes 2 Model In this section we describe all features of the economy apart from the information structure Afterwards we solve the model for alternative assumptions about the information structure: (i) perfect information and (ii) rational inattention 2 Households There are J households Households supply differentiated types of labor, consume a variety of goods, and hold nominal government bonds Each household seeks to maximize the expected sum of discounted period utility The discount factor is β (, ) The period utility function is given by with U (C jt,l jt,,l jnt )= C γ jt γ i= ϕ NX n= e χ jnt L +ψ jnt +ψ, () Ã IX! θ C jt = C θ θ θ ijt (2) Here C jt is composite consumption by household j in period t and C ijt is consumption of good i by household j in period t The household can consume I different goods The parameter θ> is the elasticity of substitution between different goods The parameter γ> is the inverse of the intertemporal elasticity of substitution We assume that households supply differentiated types of labor We also assume that each household supplies multiple types of labor and that the disutility of supplying a particular type of labor is subject to a preference shock Here L jnt denotes the household s supply of his or her nth type of labor, and χ jnt is a preference shock affecting the disutility of supplying this type of labor We introduce labor-type-specific preference shocks to generate variation in relative wage rates The assumption of a constant elasticity of substitution between different varieties is only for ease of exposition One could also use a general constant returns-to-scale aggregator 4

5 Some variation in relative wage rates is needed for an equilibrium to exist The reason is the following If there is no variation in relative wage rates, decision-makers in firms have no incentive to attend to all wage rates Then, for some type of labor, the demand for labor does not depend on the wage rate for that type of labor, implying that the household supplying this type of labor could set an infinite wage rate 2 We assume that each household supplies multiple types of labor to allow households to insure against labor-type-specific preference shocks within the household We will assume that N is large The parameter ϕ> affects the disutility of supplying labor The parameter ψ istheinverseofthe Frisch elasticity of labor supply The flow budget constraint of household j in period t reads IX NX P it C ijt + B jt = R t B jt +(+τ w ) W jnt L jnt + D t J T t J (3) i= Here P it is the price of good i in period t, B jt are bond holdings by household j between periods t and t +, R t is the nominal interest rate on bond holdings between periods t and t, W jnt is the nominal wage rate for household j s nth type of labor in period t, τ w is a wage subsidy, (D t /J) is a pro-rata share of nominal aggregate profits, and (T t /J) is a pro-rata share of nominal lump-sum taxes We assume that all J households have the same initial bond holdings, B j, > Furthermore, we assume that bond holdings cannot be negative or zero, B jt > One can think of households having an account The account holds only nominal government bonds and the balance on the account has to be positive In every period, each household chooses a consumption vector, (C jt,,c Ijt ),anda vector of nominal wage rates, (W jt,,w jnt ) Each household commits to supply any quantity of labor at the chosen nominal wage rates Bond holdings, B jt,thenfollowasa residual from the flow budget constraint (3) and the labor demand function derived below Each household takes as given: all wage rates set by other households, all prices set by n= firms, the nominal interest rate, and all aggregate variables 2 To generate variation in relative wage rates, one could also introduce labor-type-specific productivity shocks instead of labor-type-specific preference shocks 5

6 22 Firms There are I firms in the economy Firms supply differentiated goods that are produced with the different varieties of labor The technology of firm i is given by Y it = e a t e a it L α it, (4) with JX NX L it = j= n= η η L ijnt η η (5) Here Y it is output of firm i in period t, L it is composite labor input at firm i in period t and L ijnt is input of household j s nth type of labor at firm i in period t There are JN types of labor because there are J households each supplying N differentiated types of labor 3 The parameter α (, ] is the elasticity of output with respect to composite labor The parameter η> is the elasticity of substitution between different types of labor Total factor productivity, (e at e a it ), has an aggregate component, e at,andafirm-specific component, e a it Nominal profits of firm i in period t equal where τ p is a production subsidy ( + τ p ) P it Y it JX j= n= NX W jnt L ijnt, (6) ³ˆLit,,ˆL ij(n )t In every period, each firm chooses a price, P it,andafactormix, where ˆL ijnt =(L ijnt /L it ) Each firm commits to supply any quantity of the good at the chosen price The composite labor input, L it, then follows from the production function (4) and the demand function derived below Each firm takes as given: all prices set by other firms, all wage rates set by households, the nominal interest rate, and all aggregate variables 3 The assumption that all types of labor appear in the labor aggregator (5) is only for ease of exposition One could also use a labor aggregator in which a firm-specific subset of types of labor appears 6

7 23 Government There is a monetary authority and a fiscal authority Let Π µ t =(P t /P t ) denote inflation XI of a price index, P t,thatwillbedefined later, and let Y t = denote aggregate i= Y it output The central bank sets the nominal interest rate according to the rule R t R = µ Rt R ρr " µπt Π φπ µ # ρr φy Yt e εr t, (7) Y where ε R t is a monetary policy shock Here R, Π and Y are the values of the nominal interest rate, inflation and aggregate output in the non-stochastic steady state The policy parameters satisfy ρ R [, ), φ π > and φ y The government budget constraint in period t reads à JX NX IX! T t +(B t B t )=(R t ) B t + τ w W jnt L jnt + τ p P it Y it (8) j= n= i= The government has to finance interest on nominal government bonds, the wage subsidy and the production subsidy government bonds The government can collect lump-sum taxes or issue new We assume that the government sets the production subsidy, τ p, and the wage subsidy, τ w,soastocorrectthedistortionsarisingfromfirms market power in the goods market and households market power in the labor market In particular, we assume that and +τ p = ϑ ϑ, (9) +τ w = ζ ζ, () where ϑ denotes the price elasticity of demand and ζ denotes the wage elasticity of labor demand 4 24 Shocks There are four types of shocks in the economy: monetary policy shocks, aggregate technology shocks, firm-specific productivity shocks and labor-type-specific preference shocks We 4 If households have perfect information then ϑ = θ, andiffirms have perfect information then ζ = η However, under limited attention, ϑ will differ from θ and ζ will differ from η 7

8 assume that, for all i and jn, the processes ε R ª t, {at }, {a it } and ª χ jnt are independent Furthermore, we assume that the firm-specific productivity processes, {a it }, are independent across firms and the labor-type-specific preference shocks, χ jnt ª, are independent across types of labor In addition, we assume that I and N are sufficiently large so that IX a it =, () I and N i= NX χ jnt = (2) n= Finally, we assume that the processes ε R ª t, {at }, {a it } and ª χ jnt are stationary Gaussian processes with mean zero In the following, we denote the period t innovation to a t, a it and χ jnt by ε A t, ε I it and εχ jnt, respectively 3 Non-stochastic steady state Before studying the economy described in Section 2 that is hit by shocks, we study the non-stochastic version of the economy In particular, we study the non-stochastic steady state defined as a solution of the non-stochastic version of the economy with the property that real quantities, relative prices, the nominal interest rate and inflation are constant over time In the following, variables without subscript t denote values in the non-stochastic steady state In this section, P t denotes the following price index P t = Ã IX i= and W t denotes the following wage index JX NX W t = j= n= P θ it W η jnt Furthermore, ˆP it denotes the relative price of good i! θ, (3) η (4) ˆP it = P it P t, (5) 8

9 Ŵ jnt denotes the relative wage rate for type jn labor Ŵ jnt = W jnt, (6) W t W jnt denotes the real wage rate for type jn labor and W t denotes the real wage index W jnt = W jnt P t, (7) W t = W t (8) P t It is straightforward to show that in the non-stochastic steady state the households first-order conditions read and R Π = β, (9) C ij C j = ˆP θ i, (2) W jn = ϕ ³Ŵ η jn L ψ C γ j, (2) where L is the aggregate composite labor input IX L = L i (22) i= Furthermore, the firms first-order conditions read L ijn L i = Ŵ η jn, (23) and ˆP i = W ³ ˆP θ α i C, (24) α where C is aggregate composite consumption C = JX C j (25) j= In the non-stochastic version of the model, all households face the same decision problem Therefore, all households choose the same composite consumption definition of aggregate composite consumption (25) that It follows from the C j C = J (26) 9

10 ³ The households wage setting equation (2), equation (26) and Ŵjn = Wjn / W imply that the wage rate is the same for all different types of labor Thus firmsdecidetohireall different types of labor in equal amounts It follows from the labor aggregator (5) and the definition of the wage index (4) that µ Lijn L i η η = Ŵ η jn = JN (27) The firms price setting equation (24) implies that all firms set the same price Thus households decide to consume the different goods in equal amounts, implying that all firms produce the same amount Since all firms also have the same technology, all firms have the same composite labor input It follows from the consumption aggregator (2), the definition of the price index (3), the definition of aggregate output and the definition of aggregate composite labor input (22) that µ Cij C j θ θ = ˆP θ i = Y i Y = L i L = I (28) We will use equations (9)-(2), equations (23)-(24) and equations (26)-(28) below In the non-stochastic steady state, only the real interest rate, (R/Π), is determined Neither the nominal interest rate, R, nor inflation, Π, is determined We will assume that Π equals some value Π Furthermore, in the model, the initial price level, P,isnot determined We will assume that P equals some value P Together Π and P pin down the price level at each point in time in the non-stochastic steady state [Finally, in the non-stochastic steady state, Ricardian equivalence holds, and we have not specified a fiscal rule Therefore, taxes are not determined We will assume that the government runs a balanced budget in the non-stochastic steady state Actually, this assumption is implied by the definition of a non-stochastic steady state] 4 Derivation of the firms objective In this section, we derive an expression for the expected sum of discounted profits We will use this expression in Section 6 when we assume that decision-makers in firms choose the allocation of their attention so as to maximize the expected sum of discounted profits

11 First, we guess that the demand function for good i has the form C it = ς µ Pit where ς> and ϑ> are undetermined coefficients satisfying P t ϑ C t, (29) ς ˆP ϑ i = ˆP θ i, (3) C t is aggregate composite consumption and P t is some price index satisfying = C t = JX C jt, (3) j= IX d i= µ Pit P t, (32) where d is a twice continuously differentiable function We will always make assumptions to ensure that the guess (29)-(32) concerning the demand function is correct When households have perfect information, ς =, ϑ = θ and the price index P t is given by equation (3) When households have limited attention, we will add assumptions concerning the exogenous processes to ensure that the demand function still has the form (29)-(32) Second, given the technology (4)-(5), the expression for nominal profits (6) and a demand function, we can derive the profit function for this demand function We first rewrite the expression for nominal profits (6) using the fact that the wage bill equals the product of the composite labor input and the wage bill per unit of composite labor input JX NX ( + τ p ) P it Y it L it W jntˆlijnt, (33) j= n= where ˆL ijnt =(L ijnt /L it ) Afterwards, we rearrange the production function (4) L it = µ Yit e a t e a it α, (34) and the labor aggregator (5) = JX NX j= n= η η ˆL ijnt

12 The last equation can be stated as ˆL ijnt = X jn6=jn η η ˆL ijnt η η (35) Substituting equations (34), (35) and (29) into the expression for nominal profits (33) yields the profit function for the demand function (29): ( + τ p ) P it ς ³ ς Pit e a t e a it µ Pit P t P t ϑ Ct ϑ C t α X jn6=jn W jntˆlijnt + W JNt X jn6=jn η η ˆL ijnt η η (36) Nominal profits of firm i in period t depend on variables that the decision-maker in the firm chooses (P it and ˆL it,,ˆl ij(n )t ) as well as variables that the decision-maker in the firmtakesasgiven(p t, C t, a t, a it and W t,,w JNt ) Third, we assume that in period nominal profits in period t are discounted with the following stochastic discount factor Q,t = β t λ (C t,,c Jt ) P t, (37) where λ is a twice continuously differentiable function with the property λ (C,,C J )=C γ j, (38) and P t is the price index that appears in the demand function (29) 5 Then, in period, the expected sum of discounted profits equals: X E i, β t λ (C t,,c Jt ) X jn6=jn ϑ ˆP it C t ( + τ p ) ς µ ϑ ς ˆP it e a te a it W jntˆlijnt + W JNt α C t X jn6=jn η η ˆL ijnt η η, (39) 5 For example, if the function λ is a weighted average of the marginal utilities of the different households (ie λ (C t,,c Jt )=λ C γ t + + λ J C γ Jt with λ j and J λ j =), equation (38) is satisfied j= This is because all households have the same marginal utility in the non-stochastic steady state 2

13 where E i, is the expectation operator conditioned on the information of the decisionmaker in firm i in period, ˆPit =(P it /P t ) is the relative price of good i in period t, W jnt =(W jnt /P t ) is the real wage rate for type jn labor in period t, andp t is the price index that appears in the demand function (29) and the stochastic discount factor (37) One can rewrite expression (39) in terms of log-deviations from the non-stochastic steady state In the following, small variables denote log-deviations from the non-stochastic steady state For example, c t =ln(c t /C) Using the equation for the production subsidy (9), the ³ steady state relationships (23), (24), (27) and L i = ˆP θ α i C, and the guess (3) yields the following expression for the expected sum of discounted profits: ϑ ϑ α e( ϑ)ˆp it+c t X E i, β t λ (C e c t,,c J e c Jt ) WL e ϑ α ˆp it+ α c t α (a t+a it) i JN X jn6=jn e w jnt+ˆl ijnt + e w JNt JN where WL i is the real wage bill of firm i in the non-stochastic steady state X jn6=jn e η η (4) ˆl ijnt η η, Proposition (Expected sum of discounted profits) Let E i, denote the expectation operator conditioned on the information of the decision-maker in firm i in period Letf denote the functional inside the expectation operator in expression (4) Let f denote the second-order Taylor approximation to f at the non-stochastic steady state Let x t, z t and v t denote the following vectors ³ x t = ˆp it ˆlit ˆliJ(N )t, (4) ³ z t = c t c Jt c t a t a it w t w JNt, (42) ³ v t = x t zt (43) Let v m,t denote the mth element of v t Suppose that, for all m, k and for each period t, E i, v m,t v k,t <, (44) and, for all m, k, lim t βt E i, v m,t v k,t = (45) 3

14 Then = h i E i, f (x,z,x,z,) β t E i, 2 (x t x t ) H (x t x t ) where the matrix H is given by H = C γ j WL i h i E i, f (x,z,x,z,) ϑ α + α α ϑ 2 ηjn ηjn, (46) ηjn ηjn ηjn ηjn ηjn 2 ηjn, (47) and the vector x t is given by ˆp it = α α + α α ϑc t α + α α ϑ (a t + a it )+ + α α ϑ JN JX NX j= n= w jnt, (48) and ˆl ijnt = η w jnt JN JX NX j= n= w jnt (49) Proof See Appendix A To interpret Proposition, note the following First, in expression (4) variables dated t only affect discounted profits at date t (not at other dates) Furthermore, discounted profits at date t equal β t times real profits, where real profits are defined as Q,t /β t h i times nominal profits Thus, E i, f (x,z,x,z,) is simply the expression for the expected sum of discounted profits that one obtains after a log-quadratic approximation to the real profit function around the non-stochastic steady state Second, in Appendix A the vector x t is defined as the vector that maximizes real profits in period t after the log-quadratic approximation to the real profit function For the real profit function in expression (4), the vector x t is given by equations (48)-(49) It is now easy to interpret Proposition After the log-quadratic approximation of the real profit function, the profit-maximizing price in period t is given by equation (48) and the profit-maximizing factor mix in period t isgivenbyequation(49) Furthermore,after 4

15 the log-quadratic approximation of the real profit function, the loss in profits in the case of a deviation from the profit-maximizing decisions in period t (ie x t 6= x t )isgivenbythe quadratic form in expression (46) The upper-left element of the matrix H determines the loss in profits in the case of a sub-optimal price setting decision The loss is increasing in the price elasticity of demand, ϑ, and increasing in (/α) The lower-right block of the matrix H determines the loss in profits in the case of a sub-optimal factor mix decision The loss depends on the elasticity of substitution between different types of labor, η, and the number of different types of labor, JN The diagonal elements of H determine the loss in profits in the case of a deviation in a single variable The off-diagonal elements of H determine how a deviation in one variable affects the loss in profits due to a deviation in another variable Finally, conditions (44) and (45) ensure that, after the log-quadratic approximation to the real profit function, in the expression for the expected sum of discounted profits one can change the order of integration and summation, all moments are finite and all infinite sums converge 5 Derivation of the households objective In this section, we derive an expression for the expected sum of discounted period utility We will use this expression in Section 7 when we assume that households choose the allocation of their attention so as to maximize the expected sum of discounted period utility First, we guess that the demand function for type jn labor has the form µ ζ Wjnt L jnt = ξ L t, (5) where ξ> and ζ> are undetermined coefficients satisfying W t ξŵ ζ jn = Ŵ η jn, (5) L t is the aggregate composite labor input and W t is some wage index satisfying = L t = JX j= n= IX L it, (52) i= NX h 5 µ Wjnt W t, (53)

16 where h is a twice continuously differentiable function We will always make assumptions to ensure that the guess (5)-(53) concerning the labor demand function is correct When firms have perfect information, ξ =, ζ = η and the wage index W t is given by equation (4) When firms have limited attention, we will add assumptions concerning the exogenous processes to ensure that the labor demand function still has the form (5)-(53) Second, just like we derived the profit function for firms by substituting the production function, the labor aggregator and the demand function into the expression for nominal profits, we now derive a period utility function by substituting the flow budget constraint (3), the consumption aggregator (2) and the labor demand function (5) into the expression for period utility () We first rewrite the flow budget constraint (3) using the fact that consumption expenditure equals the product of composite consumption and expenditure per unit of composite consumption C jt à IX i= P it Ĉ ijt! + B jt = R t B jt +(+τ w ) NX n= W jnt L jnt + D t J T t J, where Ĉijt =(C ijt /C jt ) is the relative consumption of good i Dividing the last equation by P t,wherep t is some price index, and rearranging yields C jt = R t Π t Bjt B jt +(+τ w ) NX n= W jnt L jnt + D t J T t J (54) IX ˆP it Ĉ ijt i= Here ˆP it =(P it /P t ) is the relative price of good i, Wjnt =(W jnt /P t ) istherealwageratefor type jn labor, B jt =(B jt /P t ) are real bond holdings by household j, D t =(D t /P t ) are real aggregate profits, Tt =(T t /P t ) are real lump-sum taxes, and Π t =(P t /P t ) is inflation Next, we rewrite the equation for the consumption aggregator (2) The last equation can be stated as = IX i= i= Ĉ θ θ ijt Ã! XI θ Ĉ Ijt = Ĉ θ θ θ ijt (55) 6

17 Substituting equations (54), (55) and (5) into the period utility function () yields a period utility function that incorporates the flow budget constraint, the consumption aggregator and the labor demand function: γ N γ ϕ X R t Π t Bjt B jt +(+τ w ) n= e χ jnt XI i= NX n= ˆP it Ĉ ijt + ˆP XI It à Ã! ζ Wjnt ξ L t +ψ W t ³ ζ Wjnt W jnt ξ Lt + D t W t J T t J i= +ψ! θ Ĉ θ θ θ ijt γ (56) Here W t =(W t /P t ) where W t is the wage index appearing in the labor demand function and P t is the price index that we divided by to obtain equation (54) Finally, we express the period utility function (56) in terms of log-deviations from the non-stochastic steady state In the following, small variables denote log-deviations from the non-stochastic steady state For example, b ³ jt =ln Bjt / j B Using the equation for the wage subsidy (), the steady state relationships (9)-(2) and (28), and the guess (5) yields the following expression for the utility of household j in period t: C γ j γ ω Bβ e r t π t+ b jt ω B e b jt + X γ C γ j ω W +ψ N I I i= ζ ω WN ζ eˆp it+ĉ ijt + I eˆp It N X e ( ζ) w jnt+ζ w t+l t + ω D e d t ω T e t t n= Ã! XI θ θ I e θ θ ĉijt i= NX e χ jnt ζ(+ψ)( w jnt w t )+(+ψ)l t, (57) n= γ where ω B, ω W, ω D and ω T denote the following steady state ratios: à ³ X N W ω B ω W ω D ω T = B jn L jn j n= C j C j D J C j T J C j! (58) Proposition 2 (Expected sum of discounted period utility) Let E j, denote the expectation operator conditioned on information of household j in period Let g denote the functional that is obtained by multiplying the expression for period utility (57) by β t and summing over 7

18 all t from zero to infinity Let g denote the second-order Taylor approximation to g at the non-stochastic steady state Let x t, z t and v t denote the following vectors x t = z t = v t = ³ bjt w jt w jnt ĉ jt ĉ I jt, (59) ³ r t π t w t l t dt t t ˆp t ˆp It χ jt χ jnt, (6) ³ x t zt (6) Let v m,t denote the mth element of v t Suppose that for all m, for all m and k, forτ =, and for each period t, and for all m and k, andforτ =,, E j, h b2 j, i <, (62) E j, bj, v m, <, (63) E j, v m,t v k,t+τ <, (64) lim t βt E j, v m,t v k,t+τ = (65) Then where = H = C γ j i h i E j, h g ³ bj,,x,z,x,z, E j, g ³ bj,,x,z,x,z, β t E j, 2 (x t x t ) H (x t x t )+(x t x t ) H xt+ x t+, (66) γω 2 B ³ + β γω B ζω W N γω B ζω W N ζω W N γω B ζω W ³ γζωw N N ++ζψ ³ 2 γ ζωw N ³ 2 γ ζωw N ζω W N γω B ζω W N ³ γζωw N ++ζψ 2 θi θi 2 θi θi (67), 8

19 H = C γ j γω B ζω W γω N B ζω W N, (68) γω 2 B and the process {x t } is given by: (i) the requirement that the vector v t with x t = x t satisfies conditions (63)-(65), and (ii) the requirement that, for each period t, " Ã c jt = E t r t π t+ γ I IX ˆp it+ + I i=! # IX ˆp it + c jt+, (69) w jnt γ = +ζψ c jt + ψ +ζψ (ζ w t + l t )+ +ζψ χ jnt + IX ˆp it, (7) +ζψ I i= Ã! ĉ ijt = θ ˆp it IX ˆp it, (7) I i= i= where the variable c jt is defined as c jt = ω B β ³ r t π t + b jt + ζ ω W ζ N +ω D dt ω T t t I NX ( ζ) w jnt + ζ w t + l t n= IX ˆp it ω B b jt (72) i= Here E t denotes the expectation operator conditioned on the complete history of the economy up to and including period t Proof See Appendix B The vector x t is the vector of variables appearing in the expression for period utility (57) that the household can affect in period t Proposition 2 states that, after the log-quadratic approximation to the sum of discounted period utility, and if information about initial bond holdings satisfies condition (62) and the processes {x t } and {x t } satisfy conditions (63)-(65), then the difference between expected utility for the process {x t } and expected utility for the process {x t } is given by equation (66) Here the process {x t } is defined as the sequence of 9

20 decisions that the household would take, after the log-quadratic approximation to the sum of discounted period utility, if the household had perfect information at each point in time Thus, equation (66) provides a simple way to compute the household s loss in expected utility from taking a sequence of decisions that may deviate from the household s sequence of decisions under perfect information Conditions (62)-(65) are a set of sufficient conditions for equation (66) to hold They ensure that, after the log-quadratic approximation to the sum of discounted utility, in the expression for the expected sum of discounted utility one can change the order of integration and summation, all moments are finite and all infinite sums converge 6 Aggregation In this section, we describe issues related to aggregation In the following, we will work with log-linearized equations for the aggregate variables Log-linearizing the equations for aggregate output, aggregate composite consumption (3) and aggregate composite labor input (52) yields y t = I IX y it, (73) i= c t = J and l t = I JX c jt, (74) j= IX l it (75) Log-linearizing the equations for the price index (32) and the wage index (53) yields IX = ˆp it, i= i= and JX NX = ŵ jnt The last two equations can also be stated as j= n= p t = I IX p it, (76) i= 2

21 and w t = JN JX j= n= NX w jnt (77) Furthermore, in the following, we will also work with log-linearized equations when we compute the demand for an individual good or for an individual type of labor Formally, c it = J JX c ijt, (78) j= and l jnt = I IX l ijnt (79) Finally, note that the production function (4) and the monetary policy rule (7) are already log-linear y it = a t + a it + αl it, (8) and r t = ρ R r t +( ρ R ) φ π π t + φ y y t + ε R t (8) i= 7 Case : Perfect information In this section, we study the solution of the model under perfect information This solution will serve as a benchmark We define a solution under perfect information as follows: in each period all agents know the complete history of the economy up to and including the current period; decision-makers in firms choose the profit-maximizing price and factor mix; households choose the utility-maximizing consumption vector and nominal wage rates; the government sets the nominal interest rate according to the monetary policy rule, pays subsidies so as to correct the distortions due to market power, and chooses a fiscal policy that satisfies the government budget constraint; aggregate variables are given by their respective equations; and households have rational expectations The following proposition characterizes the solution under perfect information after the log-quadratic approximation of the firms objective (see Section 4), the log-quadratic approximation of the households objective (see Section 5) and the log-linearization of the equations for the aggregate variables (see Section 6) 2

22 Proposition 3 (Solution under perfect information) A solution to the system of equations (48)-(49), (69)-(72), (73)-(8), ()-(2), (5)-(8), y it = c it, ˆl ijnt = l ijnt l it and ĉ ijt = c ijt c jt with the same initial bond holdings for each household and a non-explosive bond h sequence for each household (ie lim s E t β s+ ³ bjt+s+ jt+s i b =)satisfies: and Proof See Appendix C y t = +ψ c t = α + αγ + ψ a t, (82) l t = γ α + αγ + ψ a t, (83) w t = γ + ψ α + αγ + ψ a t, (84) r t E t [π t+ ] = +ψ γ α + αγ + ψ E t [a t+ a t ], (85) ˆp it = α + α α + α α θ a it, (86) ĉ ijt = θ α θ a it, (87) w jnt w t = +ψη χ jnt, (88) ˆlijnt = η +ψη χ jnt (89) In the case of perfect information, aggregate output, aggregate composite consumption, aggregate composite labor input, the real wage index and the real interest rate depend only on aggregate technology The relative price of good i and relative consumption of good i depend only on firm-specific productivity The relative wage rate for type jn labor and relative input of type jn labor depend only on the labor-type-specific preference shock Monetary policy has no real effects The nominal interest rate and inflation then follow from the monetary policy rule (8) and the real interest rate (85) Since ( ρ R ) φ π > and ( ρ R ) φ π + ρ R >, the equilibrium paths of the nominal interest rate and inflation are locally determinate 6 6 See Woodford (23), Chapter 2, Proposition 28 22

23 8 Case 2: Firms have limited attention, households have perfect information In this section, we solve the model under rational inattention by decision-makers in firms, ie we assume that decision-makers in firms have limited attention and that they choose how to allocate their attention For the moment, we continue to assume that households have perfect information to isolate the implications of rational inattention by decision-makers in firms 8 Firms attention problem Following Sims (23), we model attention as a flow of information, and we model limited attention as a constraint on the flow of information We let agents choose the allocation of information flow In particular, agents decide how much information flow they allocate to their different decision problems Furthermore, for each decision problem, agents decide how much information flow they allocate to the different factors that determine the optimal decision We will refer to the allocation of information flow as the allocation of attention We assume that decision-makers in firms choose the allocation of attention in period so as to maximize the expected sum of discounted profits Formally, the attention problem of the decision-maker in firm i reads: max B(L),C(L) β t E i, 2 (x t x t ) H (x t x t ), (9) where x t x t = p it ˆlit p it ˆl it, (9) ˆliJ(N )t ˆl ij(n )t subject to: the equations characterizing the profit-maximizing decisions p it = A (L) ε A t + A {z } 2 (L) ε R t {z } p A it p R it + A 3 (L) ε I it {z } p I it (92) ˆl ijnt = A 4 (L) ε χ jnt, (93) 23

24 the equations specifying the admissible processes for the actual decisions p it = B (L) ε A t + C (L) ν A it + B {z } 2 (L) ε R t + C 2 (L) ν R it + B {z } 3 (L) ε I it + C 3 (L) ν I it {z } p A it p R it p I it ˆlijnt = B 4 (L) ε χ jnt + C 4 (L) ν χ ijnt, (95) (94) and the information flow constraint I ³n p A it,p R it,p I it, ˆl o it,,ˆl ij(n )t ; np A it,p R it,p I it, ˆl o it,,ˆl ij(n )t κ (96) The noise terms ν A it, νr it, νi it and νχ ijnt are assumed to follow unit-variance Gaussian white noise processes that are (i) firm-specific, (ii) independent of all other processes in the economy and (iii) independent of each other Here A (L) to A 4 (L), B (L) to B 4 (L) and C (L) to C 4 (L) are infinite-order lag polynomials The operator I measures the information flow between stochastic processes and will be defined below Finally, κ is the decision-maker s overall attention devoted to setting the price and choosing the factor mix in every period After the log-quadratic approximation to the real profit function, expression (9) equals the expected discounted sum of profit losses due to sub-optimal decisions See Proposition The matrix H is given by equation (47) and the vector x t x t isgivenbyequation (9) The last equation follows from the definition of x t in Proposition and the equality ˆp it ˆp it = p it p it Equations (92)-(93) characterize the profit-maximizing decisions in period t After the log-quadratic approximation to the real profit function, the profitmaximizing decisions in period t are given by equations (48)-(49), p it =ˆp it + p t and the variables p t, c t, a t, a it and w jnt Here we guess that, after substituting in the equilibrium processes for p t, c t, a t, a it and w jnt, the profit-maximizing decisions in period t have the representation (92)-(93) We will verify this guess Equations (94)-(95) specify the actual decisions Here we assume that actual decisions follow a Gaussian process and that actual decisions in period t cannot depend on the realization of shocks drawn in the future The decision-maker chooses the lag polynomials B (L) to B 4 (L) and C (L) to C 4 (L) The lag polynomials are a way of specifying the joint distribution of actual decisions and profitmaximizing decisions For example, if B (L) =A (L) and C (L) =,thenp A it = pa it, which simply means that the price set by the decision-maker responds perfectly to aggregate technology shocks The decision-maker has to respect the constraint on information flow 24

25 (96) In the next two paragraphs, we discuss the information flow constraint (96) in detail, and we discuss the assumptions about the noise terms The operator I measures the information flow between stochastic processes The operator is defined as follows Let H (X,,X T ) denote the entropy of the sequence of random variables X,,X T given information in period LetH (X,,X T Y,,Y T ) denote the conditional entropy of the sequence of random variables X,,X T and information in period The operator I is then defined as given Y,,Y T I ({X t } ; {Y t })= lim T T [H (X,,X T ) H (X,,X T Y,,Y T )] (97) Since entropy is a measure of uncertainty, the operator I is a measure of how much knowledge of Y,,Y T reduces uncertainty concerning X,,X T We have assumed that the actual decisions follow a Gaussian process One can show that a Gaussian process for the actual decisions is optimal because the objective (9) is quadratic and the profit-maximizing decisions (92)-(93) follow a Gaussian process 7 Furthermore, we have assumed that the noise in the actual decisions is firm-specific This assumption accords well with the idea that the friction is the limited attention of an individual decision-maker rather than the limited availability of information Finally, we have assumed that the noise terms ν A it, νr it, νi it and νχ ijnt are independent of each other This assumption captures the idea that paying attention to the state of aggregate technology, to monetary policy shocks, to firm-specific productivity and to relative wage rates are independent activities In the future, we also plan to study the case where the noise terms can be correlated 82 Computing the equilibrium We use an iterative procedure to solve for the equilibrium of the model First, we make a guess concerning the process for the profit-maximizing price (92) and a guess concerning the process for the profit-maximizing factor mix (93) Second, we solve the firms attention problem (9)-(96) Third, we aggregate the individual prices to obtain the aggregate price level: p t = I IX p it (98) i= 7 See Sims (26) or Maćkowiak and Wiederholt (28) 25

26 Fourth, we compute the aggregate dynamics implied by those price level dynamics The following equations have to be satisfied in equilibrium: r t = ρ R r t +( ρ R ) φ π (p t p t )+φ y y t + ε R t, (99) c t = E t γ (r t p t+ + p t )+c t+, () The first equation is the monetary policy rule y t = c t, () y t = a t + αl t, (2) a t = ρ A a t + ε A t, (3) w t = γc t + ψl t (4) The second equation follows from the assumption that households have perfect information and equations (69), (74) and (76) The third equation follows from the requirement that output has to equal demand The fourth equation follows from the production function The fifth equation is the assumed process for aggregate technology The last equation follows from the assumption that households have perfect information and equations (7), (74), (76), (77) and (2) We employ a standard solution method for linear rational expectations models to solve the system of equations containing the price level dynamics and those six equations We obtain the law of motion for (r t,c t,y t,l t,a t, w t ) implied by the price level dynamics Fifth, we compute the law of motion for the profit-maximizing price from equations (48) and (77): p it = p t + α α + α α ϑc t α + α α ϑ (a t + a it )+ + α α ϑ w t (5) In the last equation, we set ϑ = θ because the assumption that households have perfect information and equations (7) and (78) imply that the price elasticity of demand equals θ If the process for the profit-maximizing price differs from our guess, we update our guess until we reach a fixed point Finally, we compute the equilibrium relative wage rates and the equilibrium factor mix This is explained in Appendix D 26

27 83 Benchmark parameter values and solution In this section, we report the numerical solution of the model for the following parameter values We set β =99, γ =, ψ =, θ =4, α =2/3 and η =4 To set the parameters governing the process for aggregate technology, equation (3), we consider quarterly US data from 96 Q to 26 Q4 We first compute a time series for aggregate technology, a t, using equation (2) and measures of y t and l t Weusethelog of real output per person, detrended with a linear trend, as a measure of y t Weusethelog of hours worked per person, demeaned, as a measure of l t 8 We then fit equation (3) to thetimeseriesfora t obtaining ρ A =96 and a standard deviation of the innovation equal to 85 In the benchmark economy, we set ρ A =95 and we set the standard deviation of ε A t equal to 85 To set the parameters of the Taylor rule, we consider quarterly US data on the Federal Funds rate, inflation and real GDP from 96 Q to 26 Q4 9 We fit thetaylorrule (99) to the data obtaining ρ R =89, φ π =53, φ y =33, and a standard deviation of the innovation equal to 2 In the benchmark economy, we set ρ R =9, φ π =5, φ y =33, and the standard deviation of ε R t equal to 2 We assume that firm-specific productivity follows a first-order autoregressive process Recent papers calibrate the autocorrelation of firm-specific productivity to be about twothirds in monthly data, eg Klenow and Willis (27) use 68, Midrigan (26) uses 5, and Nakamura and Steinsson (27) use 66 Since (2/3) 3 equals about 3, we set the autocorrelation of firm-specific productivity in our quarterly model equal to 3 We then choose the standard deviation of the innovation to firm-specific productivity such that the average absolute size of price changes in our model equals 97 percent under perfect information The value 97 percent is the average absolute size of price changes excluding sales reported in Klenow and Kryvtsov (27) This yields a standard deviation of the 8 We use data for the non-farm business sector The source of the data is the website of the Federal Reserve Bank of StLouis 9 We compute a time series for four-quarter inflation rate from the price index for personal consumption expenditures excluding food and energy We compute a time series for percentage deviations of real GDP from potential real GDP The sources of the data are the websites of the Federal Reserve Bank of StLouis and the Congressional Budget Office 27

28 innovation to firm-specific productivity equal to 22 We assume that labor-type-specific preference shocks follow a white noise process This simplifying assumption implies that solving for the equilibrium relative wage rates and the equilibrium factor mix is straightforward See Appendix D We compute the solution of the model by fixing the marginal value of information flow instead of κ The overall information flow, κ, is then determined within the model The idea is the following When the marginal value of information flow is high, decision-makers in firms have a high incentive to increase information flow in order to take better decisions In contrast, when the marginal value of information flow is low, decision-makers in firms have little incentive to increase information flow We set the marginal value of information flowequaltopercentofafirm s steady state output We first report the optimal allocation of attention at the rational inattention fixed point The decision-maker in a firm allocates 3 bits of information flow to tracking firm-specific productivity, bit of information flow to tracking aggregate technology, and 35 bits of information flow to tracking monetary policy The expected per-period loss in profits due to imperfect tracking of firm-specific productivity equals 7 percent of the firm s steady state output; the expected per-period loss in profits due to imperfect tracking of aggregate technology equals 5 percent of the firm s steady state output; and the expected perperiod loss in profits due to imperfect tracking of monetary policy equals 3 percent of the firm s steady state output Together these numbers imply that the expected per-period loss in profits due to deviations of the actual price from the profit-maximizing price equals 5 percent of the firm s steady state output We think this is a reasonable number Figures and 2 show impulse responses of the price level, inflation, output, and the nominal interest rate at the rational inattention fixed point (green lines with circles) For comparison, the figures also include impulse responses of the same variables at the equilibrium under perfect information derived in Section 6 (blue lines with points) All impulse responses are to shocks of one standard deviation All impulse responses are drawn such that an impulse response equal to one means a one percent deviation from the non-stochastic steady state Time is measured in quarters along horizontal axes Consider Figure The price level shows a dampened and delayed response to a mone- 28

29 tary policy shock compared with the case of perfect information The response of inflation to a monetary policy shock is persistent Output falls after a positive innovation in the Taylor rule and the decline in output is persistent The nominal interest rate increases on impact and then converges slowly to zero The impulse responses to a monetary policy shock under rational inattention differ markedly from the impulse responses to a monetary policy shock under perfect information Under perfect information, the price level adjusts fully on impact to a monetary policy shock, there are no real effects, and the nominal interest rate fails to change Consider Figure 2 The price level and inflation show a dampened response to an aggregate technology shock compared with the case of perfect information The output gap is negative for a few quarters after the shock Output and the nominal interest rate show hump-shaped impulse responses to an aggregate technology shock Note that under rational inattention the response of the price level to an aggregate technology shock is less dampened and less delayed than the response of the price level to a monetary policy shock The reason is the optimal allocation of attention Decision-makers in firms allocate about three times as much attention to aggregate technology than to monetary policy Therefore, prices respond faster to aggregate technology shocks than to monetary policy shocks As a result, the output gap is negative for only 5 quarters after an aggregate technology shock, while the output gap is negative for quarters after a monetary policy shock Figure 3 shows the impulse response of an individual price to a firm-specific productivity shock Prices respond almost perfectly to firm-specific productivity shocks The reason is the optimal allocation of attention 84 Comparison to the Calvo model For comparison, we solved the Calvo model for the same parameter values and assuming that prices change every 25 quarters on average Figures 4 and 5 show the impulse responses See also Paciello (27) Paciello solves the white noise case of a similar model analytically, where white noise case means that: (i) all exogenous processes are white noise processes, (ii) there is no lagged interest rate in the Taylor rule, and (iii) the price level instead of inflation appears in the Taylor rule The analytical solution in the white noise case helps to understand in more detail the differential response of prices to aggregate technology shocks and to monetary policy shocks 29