What We Don t Know Doesn t Hurt Us: Rational Inattention and the. Permanent Income Hypothesis in General Equilibrium

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1 What We Don t Know Doesn t Hut Us: Rational Inattention and the Pemanent Income Hypothesis in Geneal Equilibium Yulei Luo Univesity of Hong Kong Gaowang Wang Shandong Univesity Jun Nie Fedeal Reseve Bank of Kansas City Eic R. Young Univesity of Viginia Octobe 18, 2014 Abstact This pape deives the geneal equilibium effects of ational inattention o RI; Sims 2003, 2010) in a model of incomplete income insuance Huggett 1993, Wang 2003). We show that, unde the assumption of CARA utility with Gaussian shocks, the pemanent income hypothesis PIH) aises in steady state equilibium due to a balancing of pecautionay savings and impatience. We then exploe how RI affects the equilibium joint dynamics of consumption, income and wealth, and find that elastic attention can make the model fit the data bette. We finally show that the welfae costs of incomplete infomation ae even smalle due to geneal equilibium adjustments in inteest ates. Keywods: Rational Inattention; Pemanent Income Hypothesis; Geneal Equilibium; Consumption and Income Volatility. JEL Classification Numbes: C61; E21; D83. Faculty of Business and Economics, The Univesity of Hong Kong, Hong Kong. yluo@econ.hku.hk. Reseach Depatment, Fedeal Reseve Bank of Kansas City. jun.nie@kc.fb.og. Cente fo Economic Reseach, Shandong Univesity, Jinan, China. wanggaowang@gmail.com. Depatment of Economics, Univesity of Viginia, Chalottesville, VA ey2d@viginia.edu.

2 1. Intoduction In intetempoal consumption-savings poblems, households save today fo thee easons: i) they anticipate futue declines in income, ii) they face uninsuable isk that geneates pecautionay savings, and iii) they ae elatively patient compaed to the inteest ate. When only motive i) is opeative then one obtains the pemanent income hypothesis PIH) of Fiedman 1957), whee consumption is solely detemined by pemanent income and follows a andom walk. See also Hall 1978.) The PIH has some implications that ae stongly inconsistent with the data. Two implications in paticula ae discussed in Campbell and Deaton 1989), the excess sensitivity and excess smoothness puzzles. Excess sensitivity occus if consumption esponds to pedictable changes in income; unde the PIH those changes ae pat of pemanent income and theefoe have aleady had thei effect on consumption. Excess smoothness occus if consumption esponds less than one fo one to pemanent changes in income o equivalently less than one fo one to changes in pemanent income). The two puzzles ae actually manifestations of the same undelying economic foces, as shown in Campbell and Deaton 1989), and thei absence is pofoundly ejected. Uninsuable income isk seems to be pevasive in micoeconomic data, and geneal equilibium models with uninsuable isk tend to pedict impatience of households that is, they face low inteest ates ), so that the basic PIH is violated. Wang 2003), using a simple model with CARA utility and isk fee assets in zeo net supply, shows that the PIH eemeges in geneal equilibium when decision ules ae linea, the equilibium inteest ate exactly balances the foces of pecautionay saving and dissaving due to impatience, even in the pesence of uninsuable isk. Due to the lineaity of consumption as a function of individual pemanent income, Wang 2003) is able to analytically chaacteize the foces that opeate in geneal equilibium and show they cancel out, unde some mild assumptions about the labo income pocess. Luo 2008) and Luo and Young 2010) intoduce ational inattention into the basic patial equilibium PIH envionment; RI implies that agents pocess signals slowly and theefoe appea to espond sluggishly to innovations in pemanent income. This sluggish esponse appeas to delive changes in consumption in esponse to anticipated income changes, because econometicians actually obseve moe than the agents do, and as a esult also delives smalle esponses to pemanent income changes. 1 1 The liteatue that ejects the PIH empiically is vast, suggesting that the eemegence is not something desieable. Howeve, the tests of the PIH, while obust to the possibility that agents have moe infomation than the econometician 1

3 Ou goal in this pape is to exploe the geneal equilibium implications of ational inattention in a model with pecautionay savings and ask the same question fom Wang 2003) namely, does the PIH eemege in geneal equilibium in the pesence of ational inattention. We study economies with constant absolute isk avesion CARA) pefeences, as they simultaneously geneate pecautionay savings and linea consumption ules, and chaacteize the foces that act on the geneal equilibium inteest ate. 2 We find that the PIH does descibe equilibium consumption behavio in geneal equilibium, with the appopiate substitution of actual pemanent income by peceived pemanent income. Thus, the delicate canceling of pecautionay and impatience foces found by Wang 2003) caies ove unmodified to models with incomplete infomation about the state. 3 We also compae ational inattention with habit fomation; although both models lead to slow adjustment in consumption, they have opposite effects on the equilibium inteest ate and the elative volatility of consumption to income gowth. One key esult in this pape is that thee exists geneal equilibium inteest ates cleaing the asset maket and they ae significantly affected by the degee of RI. Afte obtaining the explicit expession fo consumption dynamics, we examine how RI affects the stochastic popeties of the joint dynamics of consumption gowth to income gowth in both the fixed capacity and elastic capacity cases. Specifically, we find that the effect of RI on consumption dynamics is attenuated by geneal equilibium adjustment in the inteest ate as pocessing capacity declines the inteest ate also declines, leading to lowe consumption volatility. The implication is that the costs of incomplete infomation have likely been oveestimated in the liteatue, despite being vey tiny to stat. 4 This pape is oganized as follows. Section 2 constucts a pecautionay saving model with a continuum of inattentive consumes who have the CARA utility and face uninsuable labo income. Section 3 solves optimal consumption-saving ules unde ational inattention and establishes the geneal equilibium of this economy. Section 4 examines how RI affects the inteest ate and the joint dynamics of consumption, income, and wealth quantitatively. Section 5 consides an extension see Campbell and Deaton 1989), they ae not obust to the RI implication that the econometician may actually have moe infomation than the agents at least at the time of decisions). 2 Note that while the constant-elative-isk-avesion CRRA) utility function is moe commonly used in economics, this utility will geatly complicate ou annalysis because the intetempoal consumption model with CRRA utility and stochastic labo income has no explicit solution and leads to non-linea consumption ules. 3 Luo and Young 2014) document a obsevational equivalence between ational inattention and signal extaction in linea-quadatic-gaussian models. 4 Fo the welfae losses due to impefect infomation about cuent income o pemanent income calculated in the patial equilibium linea-quadatic LQ) pemanent income models, see Pischke 1995), Luo and Young 2010), and Luo, Nie, and Young 2014). Maćkowiak and Wiedeholt 2013) also find tiny welfae losses due to RI in a geneal equilibium business cycle model. 2

4 to the ecusive utility setting. Section 6 concludes. 2. A Caballeo-Huggett-Wang Economy with Rational Inattention 2.1. A Full-infomation Rational Expectations Model with Pecautionay Savings Following Caballeo 1990) and Wang 2003), we fomulate a full-infomation ational expectations FI-RE) model with pecautionay savings as follows: V a 0, y 0 ) = subject to the flow budget constaint max {c t,a t } t=0 { E 0 [ t=0 ) 1 t uc t )]}, 1) 1 + ρ a t+1 = 1 + ) a t + y t c t, 2) whee uc t ) = exp αc) /α is a constant-absolute-isk-avesion utility with α > 0, ρ > 0 is the agent s subjective discount ate, is a constant ate of inteest, and labo income, y t, follows a stationay AR1) pocess with Gaussian innovations y t = φ 0 + φ 1 y t 1 + w t, t 1, φ 1 < 1, 3) whee w t N 0, σ 2), φ 0 = 1 φ 1 ) y, y is the mean of y t, and the initial levels of labo income y 0 and asset a 0 ae given. 5 Solving 1) subject to 2) and 3) yields the following optimal consumption plan: { c t = a t + h t + 1 [ ) ]} 1 + ρ α 2 ln ln E t [exp αφw 1 + t+1 )]. 4) In this expession, [ h t 1 ) 1 j 1 + E t y 1 + t+j], 5) j=0 is human wealth defined as the discounted expected pesent value of cuent and futue labo income); evaluating the sum yields h t = φ y t + φ ) 0 5 Assuming that the individual income shock includes two components, one pemanent and the othe tansitoy, does not change the main esults in this pape. Hee we follow Wang 2003) and adopt specification 3), in ode to simplify the algeba. A detailed deivation of the model with the two-income shock specification is available fom the coesponding autho by equest. Fo the empiical studies on the income specification, see Attanasio and Pavoni 2011). 3

5 whee φ = 1/ 1 + φ 1 ). 6 This consumption function is the same as that obtained in Wang 2003). ) In the last two tems in 4), ln /α measues the elative impotance of impatience and the 1+ρ 1+ inteest ate in detemining cuent consumption, and ln E t [exp αφw t+1 )]) /α measues the amount of pecautionay savings detemined by the inteaction of isk avesion and income uncetainty. In ode to facilitate the intoduction of ational inattention we follow Luo 2008) and Luo and Young 2010) and educe the multivaiate model to a univaiate model with iid innovations to pemanent income. Letting pemanent income, s t = a t + h t, be defined as a new state vaiable, we can efomulate the PIH model as subject to { v s t ) = max u c t ) + 1 } c t 1 + ρ E t [v s t+1 )], s t+1 = 1 + ) s t c t + ζ t+1, 6) whee the time t + 1) innovation to pemanent income can be witten as ζ t+1 1 ) 1 j t+1) 1 + E 1 + t+1 E t ) [ ] y j, 7) j=t+1 which can be educed to ζ t+1 = φw t+1 when we use the income specification, 3), whee v s t ) is the consume s value function unde FI-RE Incopoating Rational Inattention In this section, we follow Sims 2003) and incopoate ational inattention RI) due to finite infomationpocessing capacity into the above pemanent income model with the CARA-Gaussian specification. Unde RI, consumes have only finite Shannon channel capacity available to obseve the state of the wold. Specifically, we use the concept of entopy fom infomation theoy to chaacteize the uncetainty about a andom vaiable; the eduction in entopy is thus a natual measue of infomation flow. 8 With finite capacity κ 0, ), a andom vaiable {s t } following a continuous dis- 6 See Appendix 7.1 fo the deivation. 7 See Appendix 7.1 fo the deivation. 8 Fomally, entopy is defined as the expectation of the negative of the natual) log of the density function, E [ln f s))]. Fo example, the entopy of a discete distibution with equal weight on two points is simply E [ln 2 f s))] = 0.5 ln 0.5) 0.5 ln 0.5) = 0.69, and the unit of infomation contained in this distibution is 0.69 nats. Fo altenative bases fo the logaithm, the unit of infomation diffes; with log base 2 the unit of infomation is the bit and with base 10 it is a dit o a hatley. ) In this case, an agent can emove all uncetainty about s if the capacity devoted 4

6 tibution cannot be obseved without eo and thus the infomation set at time t + 1, denoted I t+1, { } t+1 is geneated by the entie histoy of noisy signals s j. Following the liteatue, we assume the noisy signal takes the additive fom j=0 s t+1 = s t+1 + ξ t+1, whee ξ t+1 is the endogenous noise caused by finite capacity. 9 We futhe assume that ξ t+1 is an iid idiosyncatic shock and is independent of the fundamental shocks hitting the economy. The eason that the RI-induced noise is idiosyncatic is that the endogenous noise aises fom the consume s own intenal infomation-pocessing constaint. Agents with finite capacity will choose a new signal s t+1 I t+1 = { s 1, s 2,, s t+1} that educes the uncetainty about the vaiable st+1 as much as possible. Fomally, this idea can be descibed by the infomation constaint H s t+1 I t ) H s t+1 I t+1 ) κ, 8) whee κ is the investo s infomation channel capacity, H s t+1 I t ) denotes the entopy of the state pio to obseving the new signal at t + 1, and H s t+1 I t+1 ) is the entopy afte obseving the new signal. κ imposes an uppe bound on the amount of infomation flow that is, the change in the entopy that can be tansmitted in any given peiod. Finally, following the liteatue, we suppose that the pio distibution of s t+1 is Gaussian. Although we adopt the CARA-Gaussian setting in ou model, we will assume the loss function due to impefect-state-obsevation is still quadatic. Using a quadatic loss function, Sims 2003) shows that the tue state unde RI also follows a nomal distibution s t I t N E [s t I t ], Σ t ), whee Σ t = E t [s t ŝ t ) 2]. 10 In addition, given that the noisy signal takes the additive fom s t+1 = s t+1 + ξ t+1, the noise ξ t+1 will also be Gaussian. Hee we assume ex post Gaussian distibutions and Gaussian noise but adopt CARA pefeences. See Peng 2004), Mondia 2010), and to monitoing s is κ = 0.69 nats. 9 Fo othe types of impefect infomation about state vaiables, see Pischke 1995) and Wang 2004). Pischke 1995) assumes that consumes ignoe the aggegate income component, and Wang 2004) assume that consumes cannot distinguish two individual components in the income pocess. 10 Shafieepoofad and Raginsky 2013) fomally deive the ational inattention solution using ate-distotion theoy; while thei solution is not identical to the cetainty-equivalent one used hee, it is quantitatively vey simila. A compaison of the two solutions fo the standad LQ-Gaussian PIH poblem is available upon equest. 5

7 Van Nieuwebugh and Veldkamp 2010) fo this specification. 11 In this case, 8) educes to ln Ψ t ) ln Σ t+1 ) 2κ, 9) whee Σ t+1 = va t+1 s t+1 ) and Ψ t = va t s t+1 ) = 1 + ) 2 Σ t + va t ζ t+1 ) ae the posteio and pio vaiance of the state vaiable, s t+1, espectively. In ou univaiate model, 9) fully detemines the value of the steady state conditional vaiance Σ: Σ = va t ζ t+1 ) exp 2κ) 1 + ) 2, 10) which means that Σ is entiely detemined by the vaiance of the exogenous shock va t ζ t+1 )) and finite capacity κ). 12 Kalman gain in the steady state θ as Following the steps outlined in Luo and Young 2014), we can compute the θ = 1 1/ exp 2κ) ; 11) θ measues the faction of uncetainty emoved by a new signal in each peiod. 13 The evolution of the estimated state ŝ t is govened by the Kalman filteing equation ŝ t+1 = 1 θ) 1 + ) ŝ t c t ) + θs t+1. 12) Combining 6) with 12) yields ŝ t+1 = 1 + ) ŝ t c t + ζ t+1, 13) whee ζ t+1 = θ 1 + ) s t ŝ t ) + θ ζ t+1 + ξ t+1 ) 14) is the innovation to ŝ t+1 and is independent of all the othe tems on the RHS of 13). ζ t+1 is an 11 Because both the optimality of ex post Gaussianity and the standad Kalman filte ae based on the linea-quadatic- Gaussian specification, the applications of these esults in the RI models with CARA pefeences ae only appoximately valid. 12 Note that hee we need to impose the estiction exp 2κ) 1 + ) 2 > 0. If this condition fails, the state is not stabilizable and the unconditional vaiance diveges. 13 In Section 3.3 we model RI as confonting the agent with a fixed maginal cost of acquiing channel capacity. Luo and Young 2014) show the two ae obsevationally equivalent with espect to consumption-income dynamics, but the effects on equilibium inteest ates will be diffeent. 6

8 MA ) pocess with E t [ ζ t+1 ] = 0 and ) va ζ t+1 = Γ θ, ) ωζ 2, whee Γ θ, ) = θ 1 1 θ)1+) 2 > 1 fo θ < 1, and s t ŝ t = 1 θ) ζ t 1 1 θ) 1 + ) L θξ t 1 1 θ) 1 + ) L 15) is the estimation eo with E t [s t ŝ t ] = 0 and va s t ŝ t ) = 3. Geneal Equilibium unde RI 1 θ 1 1 θ)1+) 2 ω 2 ζ Optimal Consumption and Savings functions Following the standad pocedue in the liteatue, the consumption function and the value function unde RI can be obtained by solving the following stochastic Bellman equation: { v ŝ t ) = max 1 c t α exp αc t) + 1 } 1 + ρ E t [ v ŝ t+1 )], 16) subject to 13)-15). The following poposition summaizes the main esults fom the above pecautionaysavings model with RI. Poposition 1. Fo a given Kalman gain, θ, the value function is v ŝ t ) = 1 { α exp α ŝ t 1 [ ) ρ [ )] ]}) ln 1 + ) + α 2 ln ln E t exp α ζ α 1 + t+1, 17) the consumption function is and the saving function is { c t = ŝ t + 1 [ ) 1 + ρ [ )] ]} α 2 ln ln E t exp α ζ 1 + t+1, 18) d t = 1 φ 1 ) φ y t y) + s t ŝ t ) + 1 [ )] ] [ln E t exp α ζ t+1 Ψ ), 19) α whee s t ŝ t is an MA ) estimation eo pocess given in 15) and Ψ ) = ln 7 ) 1+ρ 1+.

9 Poof. See Appendix 7.1. Compaing 4) with 18), it is clea that the two consumption functions ae identical except that we eplace s t with ŝ t and ζ t+1 φw t+1 ) with ζ t+1, espectively. Fist, given that ln E t [exp αζ t+1 )] = 1 2 α)2 ωζ 2, )] ln E t [exp α ζ t+1 = 1 2 Γ θ, ) α)2 ωζ 2, it is staightfowad to show that the pecautionay saving pemium due to limited attention is P i 1 α ln E t [ ) ] exp α ζ t+1 exp αζ t+1 ) = 1 2 Γ θ, ) 1) αω2 ζ, 20) which is clealy deceasing with the degee of attention θ), and is inceasing with the coefficient of absolute isk avesion α) and the pesistence and volatility of the income shock φ and σ) fo any given θ. In othe wods, RI can amplify the impact of the inteaction of isk avesion and income uncetainty on inceasing the amount of pecautionay savings. To futhe exploe the pecautionay savings pemium in 20), we isolate the effects of RI on individual consumption and saving by ewiting 18) as whee Ψ ) = ln c t = ŝ t + 1 α Ψ ) ln E t [exp αθ 1 + ) s t ŝ t ))] θ) Γ θ, ) ) 2 αω ζ αθωζ ) 2 +, 21) ) 1+ρ 1+ measues the elative impotance of impatience to the inteest ate in detemining optimal consumption it is geate than 0 if ρ > ), 1 α ln E t [exp αθ 1 + ) s t ŝ t ))] = αθ 1 θ) Γ θ, ) 1 + ) 2 ω 2 ζ is the pecautionay savings pemium due to the time t estimation eo, αθω ζ ) 2 /2 is the pecautionay savings pemium diven by the exogenous fundamental income shocks {w t }, and 1 θ) Γ θ) αω ζ ) 2 /2 captues the pecautionay savings pemium diven by the endogenous noise shocks, {ξ t }. 14 Note that when θ conveges to 1, the consumption function with RI, 18), educes to that of the Wang 2003) model, 4). Fom 18), fo finite capacity κ < o θ 0, 1)), the pecautionay saving pemium due to fundamental shocks is lowe than that in the full-infomation case, i.e., αθω ζ ) 2 /2 < } 14 This esult is deived by using Equation 15) and the iid popety of the pocesses { ζ t, {ζ t }, and {ξ t }. 8

10 ) 2 αωζ /2 because of the incomplete adjustment of consumption to the fundamental shock, while we have two new positive tems that incease the total savings moe than the absolute value of the educed savings: i) the pemium due to the estimation eo and ii) the pemium due to the RI-induced endogenous noise. Given the time t available infomation and the fact that E t [s t ŝ t ] = 0, the conditional mean of 19) can be witten as 1 d t = f t + 2 αγ θ, ) ω2 ζ 1 ) α Ψ ), 22) whee the fist tem f t = 1 φ 1 ) φ y t y) captues the consume s demand fo savings fo a ainy day, and the second tem, 1 2 αγ θ, ) ω2 ζ, is the cetainty equivalent of the innovation to the peceived state, ŝ t Existence of Geneal Equilibium As in Wang 2003), we assume that the economy is populated by a continuum of ex ante identical, but ex post heteogeneous agents, of total mass nomalized to one, with each agent solving the optimal consumption and savings poblem with RI specified in 16). Simila to Huggett 1993), the isk-fee asset in ou model is a pue-consumption loan and is in zeo net supply. The initial cosssectional distibution of pemanent income is assumed to be the stationay distibution Φ ). By the law of lage numbes in Sun 2006), povided that the spaces of agents and the pobability space ae constucted appopiately, aggegate pemanent income and the coss-sectional distibution of pemanent income Φ ) ae constant ove time. Poposition 2. The total savings demand fo a ainy day in the pecautionay savings model with RI equals zeo fo any positive inteest ate. That is, F t ) = y t f t ) dφ y t ) = 0, fo > 0. Poof. The poof uses the LLN and is the same as that in Wang 2003). Poposition 2 states that the total savings fo a ainy day is zeo, at any positive inteest ate. Theefoe, fom 20), fo > 0, the expession fo total savings unde RI in the economy at time t is D θ, ) 1 [ ] 1 1 Π θ, ) Ψ )) = α α 2 α)2 Γ θ, ) ωζ 2 Ψ ), 23) whee Π θ, ) = 1 2 α)2 Γ θ, ) ω 2 ζ measues the amount of pecautionay savings, and Ψ ) captues the dissaving effects of impatience. Given 23), an equilibium unde RI is defined by an 9

11 inteest ate satisfying D θ, ) = 0. 24) The following poposition shows the existence of the equilibium and the PIH holds in the RI geneal equilibium: Poposition 3. Thee exists at least one equilibium with an inteest ate 0, ρ) in the pecautionaysavings model with RI. In any such equilibium, each agent s consumption is descibed by the PIH, in that c t = ŝ t, 25) whee ŝ t consumption ae = E [s t I t ] is the peceived value of pemanent income. The evolution equations of wealth and a t+1 = 1 φ φ 1 y t y) + s t ŝ t ), 26) c t+1 = ζ t+1, 27) ] ) espectively, whee ζ t+1 is specified in 14) with E t [ ζ t+1 = 0, va ζ t+1 = Γ θ, ) ωζ 2, and Γ θ, ) = θ 1 1 θ)1+ ) 2. Poof. If > ρ, the two tems, Π θ, ) and Ψ ), in the expession fo total savings D θ, ), ae positive, which contadicts the equilibium condition, D θ, ) = 0. Since Π θ, ) Ψ ) < 0 > 0) when = 0 = ρ), the continuity of the expession fo total savings implies that thee exists at least one inteest ate 0, ρ) such that D θ, ) = 0. Fom 18), we can obtain the individual s optimal consumption ule unde RI in geneal equilibium as c t = ŝ t. Substituting 25) into 2) yields 26). Using 13) and 25), we can obtain 27). The intuition behind Poposition 3 is simila to that in Wang 2003). With an individual s constant total pecautionay savings demand Π θ, ), fo any > 0, the equilibium inteest ate must be such that each individual s dissavings demand due to impatience is exactly balanced by thei total pecautionay-savings demand, Π θ, ) = Ψ ). Figue 1 shows that the aggegate saving function is inceasing with the inteest ate, and thee exists a unique inteest ate fo evey given θ such that D θ, ) = 0. Hee we choose the paamete values as follows: α = 2.5, φ 1 = 0.88, σ = 0.29, and ρ = In Section 4.1, we will povide moe details about how to estimate the income pocess using the U.S. data. The main 10

12 Given 18) and 24), it is clea that even though the individual inceases thei total pecautionay savings in esponse to infomation fictions fo a given, the level of aggegate savings equals zeo. That is, RI does not affect the aggegate wealth in the economy. In contast, RI affects the equilibium inteest ate. With lowe Shannon channel capacity, the equilibium inteest ate is lowe. Fom the equilibium condition, it is staightfowad to show that ) 1 2 α) ρ Γ θ, ) ωζ 2 ln 1 + = 0, 28) d dθ = 2 + ) θ [1 1 θ) 1 + ) 2] 21 φ 1 )[1 1 θ)1+ )]+2 1 θ)1+ ) 1+ φ 1 )[1 1 θ)1+ ) 2 ] ) ln 1+ρ 1+ ) 1 > 0, 29) ) whee 1 1 θ) 1 + ) 2 > 1+ρ 0 and ln 1+ > 0. It is clea fom this expession that is locally unique and deceasing in the degee of inattention 1 θ. Figue 2 illustates as a function of θ fo diffeent values of α, and clealy shows that the geneal equilibium inteest ate is inceasing with the degee of attention, θ. 16 The fist ow of Table 1 epots the geneal equilibium inteest ates fo diffeent values of θ. 17 We can see fom the table that deceases as the degee of inattention inceases. Fo example, if θ is educed fom 1 to 0.2, is educed fom 2.95 pecent to 2.77 pecent. One might ask what a easonable value of θ is, and whethe a dop fom 1 to 0.2 is lage in any sense. Unfotunately, it is not staightfowad to answe these questions, so we simply note that 0.2 is lage than the value needed to match potfolio holdings in Luo 2010) and is theefoe not obviously uneasonable Elastic Capacity As agued in Sims 2010), instead of using fixed channel capacity to model finite infomationpocessing ability, one could assume that the maginal cost of infomation-pocessing i.e., the shadow pice of infomation-pocessing capacity) is fixed. That is, the Lagange multiplie on 9) is constant. In the univaiate case, the objective of the agent with finite capacity in the filteing poblem is to minimize the discounted expected mean squae eo MSE), E t [ t=0 β t s t s t ) 2], subject esult hee is obust to the choices of these paamete values. 16 Hee we use the same paamete values as above. Fo diffeent values of σ and φ 1, we have the simila patten of the equilibium inteest ate as in the benchmak case. 17 Hee we also set γ = 2.5, φ 1 = 0.88, σ = 0.29, and ρ =

13 to the infomation-pocessing constaint, o { min β [Σ t t + λ ln {Σ t } t=0 1 + ) 2 Σ t 1 + ω 2 ζ Σ t )]}, whee Σ t is the conditional vaiance at t, λ is the Lagange multiplie coesponding to 9). Solving this poblem yields the optimal steady state conditional vaiance: Σ = [ 1 + ) 2 1 β) λ ) 2 1 β) λ ] λ 1 + ) ) 2 ω 2 ζ, 30) whee λ = λ/ωζ 2 is the nomalized shadow pice of infomation-pocessing capacity. It is staightfowad to show that as λ goes to 0, Σ = 0; and as λ goes to, Σ =. Note that ln Σ < 1 if we ln ωζ 2 adopt the assumption that λ is fixed, while ln Σ = 1 in the fixed κ case. Compaing 30) with 10), ln ωζ 2 it is clea that the two RI modeling stategies ae obsevationally equivalent in the sense that they lead to the same conditional vaiance if the following equality holds: ) κ, λ = ln 1 + ) ln [ 1 + ) 2 1 β) λ ) 2 1 β) λ ] λ 1 + ) 2 In this case, the Kalman gain is ) θ, λ = [ 1 + ) 2 1 β) λ ) 2 1 β) λ ] λ 1 + ) 2 It is obvious that κ conveges to its lowe limit κ = ln 1 + ) as λ goes to ; and it conveges to as λ goes to 0. In othe wods, using this RI modeling stategy, the consume is allowed to adjust the optimal level of capacity in such a way that the maginal cost of infomation-pocessing fo the poblem at hand emains constant. Note that this esult is consistent with the concept of elastic capacity poposed in Kahneman 1973). Given this elationship between λ and θ o κ), in the following analysis we just use the value of θ to measue the degee of attention. It is woth noting that although the above two RI modeling stategies, inelastic and elastic capacity, ae obsevationally equivalent in the static sense, 31) 1. 32) 12

14 they have distinct implications fo the model s popagation mechanism if the economy is expeiencing egime switching e.g., befoe and afte the geat modeation). With inelastic capacity, the popagation mechanism govened by the Kalman gain is fixed egadless of changes in fundamental uncetainty, while with elastic capacity the popagation mechanism will change in esponse to changes in fundamental uncetainty. The key diffeence between the elastic capacity case and the fixed capacity case is that κ and θ now depend on both the equilibium inteest ate and labo income uncetainty fo a given λ. The equilibium inteest ate is now detemined implicitly by the following function: ) D θ, λ, ) 1 [ 1 ) α 2 α) 2 Γ θ, λ, ) )] 1 + ρ ωζ 2 ln 1 + = 0. 33) Figue 3 illustates how vaies with labo income uncetainty, σ, fo fixed infomation-pocessing cost, λ. It clealy shows that the aggegate saving function is inceasing with the inteest ate and the geneal equilibium inteest ate is deceasing with labo income uncetainty. We can see fom Table 2 that when the economy becomes moe volatile i.e., lage σ), the Kalman gain θ) inceases while the equilibium inteest ate ) deceases. This esult is diffeent fom that obtained in the fixed capacity case in which θ and move in the same diection. See Table 1.) The main eason fo this esult is that income uncetainty affects the equilibium inteest ate via two channel: i) the diect channel which leads to highe aggegate savings i.e., the ωζ 2 tem in 33)) and ii) the ) indiect channel which leads to lowe aggegate savings i.e., the θ, λ tem in 33)), and the diect channel dominates Compaison with Habit Fomation Habit fomation HF) has been widely used in economic models to study the consumption and saving behavio and asset picing. It can be modelled diectly as a stuctue of pefeences in which psychological factos make consumes pefe to gadual adjustment in consumption, theeby consumption volatility is moe painful than it would be in the absence of habit pesistence. As shown in Luo 2008) within a patial equilibium PIH famewok, the key diffeence between HF and RI is that individual consumption unde RI eacts not only to income shocks but also to its own endogenous noises induced by finite capacity. In othe wods, although the two models lead to the same esponse of consumption to income shocks, consumption gowth unde RI is moe volatile than that unde habit fomation at the individual level because the noise tems due to RI also contibute to the elative volatility of consumption. In contast, the effects of HF and RI on aggegate consump- 13

15 tion could be simila because aggegating acoss all individuals would weaken o even eliminate the impacts of the endogenous noises on consumption gowth. 18 In this section, we compae the diffeent implications of HF and RI in the geneal equilibium famewok. Following Alessie and Lusadi 1997), we intoduce HF into the FI-RE model specified in Section 2.1 by assuming that the utility function takes the following fom: u c t, c t 1 ) = 1 α exp α c t γc t 1 )). 34) Using the same solution method used in Section 2.1, we can solve fo the consumption function unde HF: c t = 1 + γ) s t γ 1 + c t ln α ) [ 1 + ρ ln E t exp 1 + )]) α 1 + γ) ζ 1 + t+1 ; 35) see Appendix 7.2 fo the deivation. The coesponding saving function can thus be witten as d t = 1 φ 1 ) φ y t y) + 2 γ 1 + ζ t 1 γ L 1 ln α 1 γ) ) [ 1 + ρ ln E t exp 1 + )]) α 1 + γ) ζ 1 + t+1. Following the same definition of geneal equilibium in ou benchmak model, it is staightfowad to show that thee exists an equilibium inteest ate such that: ) 1 + ρ ln α) 2 Γ γ, ) ωζ 2 = 0, 36) whee Γ γ, ) = 1+ γ 1+ ) 2 < 1, and d dγ < 0. 37) That is, the stonge the habit pesistence, the highe the equilibium inteest ate. 19 In summay, we can conclude that although both RI and HF lead to slow adjustments in consumption, they have opposite impacts on the equilibium inteest ate. 20 RI educes the equilibium inteest ate, while 18 Reis 2006) showed that infequent adjustments due to inattentiveness also leads to gadual esponses of aggegate consumption to income shocks, which matches the empiical evidence. See Luo 2008) fo a discussion on Sims ational inattention hypothess and Reis inattentiveness hypothesis. 19 In a patial equilibium model, Alessie and Lusadi 1997) show that the stonge the habit, the less impact of income uncetainty on the pecautionay saving tem. 20 It is woth noting that the mechanisms of RI and HF that geneate slow adjustment ae distinct. Unde RI, slow adjustment is foced upon the agent due to finite infomation pocessing capacity leaning is slow). In contast, slow adjustment is optimal unde FI because consumes ae assumed to pefe to smooth not only consumption but also consumption gowth. 14

16 HF inceases it. 4. Quantitative Results In this section we assess ou GE-RI model s implications fo consumption and income dynamics. We fist constuct time seies fo the coss-sectional dispesons of consumption and income, using the imputation appoach fom Blundell, Pistafei, and Peston 2008) and Guvenen and Smith 2014). Ou cental finding hee is that consumption inequality declined duing the 1980s but then flattened out elative to income. We then use ou model to assess how changes in attention might have played a ole in contibuting to this decline, given the movements in the volatility of income Empiical Evidence In ode to measue the elative consumption dispesion in the data, sd c), we constuct a panel sd y) data set which contains both consumption and income at the household level. We also use this panel to estimate the household income pocess. As panel data fo consumption geneally ae not available, we fist descibe how we constuct a consumption panel based on the Panel Study of Income Dynamics PSID) and the Consume Expenditue Suvey CEX). Following Blundell, Pistafei, and Peston 2008), we define the household income as total household income including wage, financial, and tansfe income of head, wife, and all othes in household) minus financial income defined as the sum of annual dividend income, inteest income, ental income, tust fund income, and income fom oyalties fo the head of the household only) minus the tax liability of non-financial income. This tax liability is defined as the total tax liability multiplied by the non-financial shae of total income. Tax liabilities afte 1992 ae not epoted in the PSID and so we estimate them using the TAXSIM pogam fom the National Bueau of Economic Reseach. Ou final household income measue can be expessed as: income measue = total HH income financial income) taxes total HH income financial income. total HH income The PSID does not include enough consumption expenditue data to ceate full pictue of household nonduable consumption. Such detailed expenditues ae found, though, in the CEX fom the Bueau of Labo Statistics. But households in this study ae only inteviewed fo fou consecutive 21 Othe mechanisms have been poposed fo this decline; see Kuege and Pei 2006) and Atheya, Tam, and Young 2009) fo examples. 15

17 quates and thus do not fom a panel. To ceate a panel of consumption to match the PSID income measues, we use an estimated demand function fo imputing nonduable consumption ceated by Guvenen and Smith 2014). Using an IV egession, they estimate a demand function fo nonduable consumption that fits the detailed data in the CEX. The demand function uses demogaphic infomation and food consumption which can be found in both the CEX and PSID. Thus, we use this demand function of food consumption and demogaphic infomation including age, family size, inflation measues, ace, and education) to estimate nonduable consumption fo PSID households, ceating a consumption panel. Ou household sample selection closely follows that of Blundell, Pistafei, and Peston 2008) as well. 22 We exclude households in the PSID low-income and Latino samples. We exclude household incomes in yeas of family composition change, divoce o emaiage, and female headship. We also exclude incomes in yeas whee the head o wife is unde 30 o ove 65, o is missing education, egion, o income esponses. We also exclude household incomes whee non-financial income is less than $1000, whee yea-ove-yea income change is geate than $90, 000, and whee yea-ove-yea consumption change is geate than $50, 000. Ou final panel contains 7, 220 unique households with 54, 901 yealy income esponses and 50, 422 imputed nonduable consumption values. 23 With this constucted panel of household income and consumption, Figue 4 shows the elative dispesion of consumption, defined as the atio of the standad deviation of the consumption change to the standad deviation of the income change between 1980 and The basic patten confims but extends the findings in Blundell et al. 2008) elative consumption dispesion declines in the 1980s, but this decline stops aound Based on ou theoetic model, we ae able to quantify the contibution of ational inattention to this decline. In ode to estimate the income pocess, we naow the sample peiod to the yeas , due to the PSID suvey changing to a biennial schedule afte To futhe estict the sample to exclude households with damatic yea-ove-yea income and consumption changes, we eliminate household obsevations in yeas whee eithe income o consumption has inceased moe than 200 pecent o deceased moe than 80 pecent fom the pevious yea. Following Floden and Lindé 22 They ceate a new panel seies of consumption that combines infomation fom PSID and CEX, focusing on the peiod when some of the lagest changes in income inequality occued. Fo othe explanations fo obseved consumption and income inequality, see Kuege and Pei 2006) and Attanasio and Pavoni 2012). 23 Thee ae moe household incomes than imputed consumption values because food consumption - the main input vaiable in Guvenen and Smith s nonduable demand function - is not epoted in the PSID fo the yeas 1987 and Dividing the total income esponses by unique households yields an aveage of 7 8 yeas of esponses pe household. These yeas ae not necessaily consecutive as ou sample selection pocedue allows households to be excluded in cetain yeas but etun to the sample if they late meet the citeia once again. 16

18 2001), we nomalize household income measues as atios of the mean fo that yea. We then exclude all household values in yeas in which the income is less than 10 pecent of the mean fo that yea o moe than ten times the mean. To eliminate possible heteoskedasticity in the income measues, we follow Floden and Lindé 2001) and egess each on a seies of demogaphic vaiables to emove vaiation caused by diffeences in age and education. We next subtact these fitted values fom each measue to ceate a panel of income esiduals. We then use this panel to estimate the household income pocess as specified by equation 3) by unning panel egessions on lagged income. The estimated values of φ 0, φ 1, and σ ae , 0.88, and 0.29, espectively Empiical Implications fo the Dynamics of Consumption, Wealth, and Income Luo 2008) examines how RI affects consumption volatility in a patial equilibium vesion of the PIH model pesented above. In geneal equilibium, since RI affects the equilibium inteest ate via inteacting with the coefficient of absolute isk avesion and income uncetainty, it will have an additional effect on consumption dynamics. Using 26) and 27), we can obtain the key stochastic popeties of the joint dynamics of individual consumption, income, and saving. The following poposition summaizes the implications of RI fo the elative volatility of consumption to income as well as the elative volatility of financial wealth to income: Poposition 4. Unde RI, the elative volatility of individual consumption gowth to income gowth is and the elative volatility of financial wealth to income is µ cy sd c t ) 1 + sd y t ) = φ1 ) Γ θ, ) φ, 38) 2 µ ay sd a t ) sd y t ) = 1 1 φ1 + ) 2 ) 1 θ) 1 + φ 1 ) φ 1 ) ) 2 1 θ) θ) 1 φ φ 1 1 θ) 1 + ). 39) Poof. See Online Appendix. Expession 38) shows that RI has two opposing effects on consumption volatility. The fist effect is diect though its pesence in the expession of Γ θ, ), wheeas the second effect is though the geneal equilibium inteest ate ) and is thus indiect. Using the expession of Γ θ, ), it is staightfowad to show that the diect effect of RI is to incease consumption volatility. The intuition is vey simple: the pesence of the RI-induced noise dominates the slow adjustment of consumption in detemining consumption volatility at the individual level. In contast, the indiect 17

19 effect of RI will educe consumption volatility because it educes the geneal equilibium inteest ate and Γ θ, ) / > 0. Following the liteatue of pecautionay savings and the estimated income pocess in the peceding subsection, we set ρ = 0.04, α = 2.5, σ = 0.29, and φ 1 = The second to fouth ows of Table 1 epots how the inteest ate and the elative volatility of consumption and wealth to income vay with θ in geneal equilibium. It is clea fom the second ow of Table 1 that RI can significantly affect the equilibium inteest ate. Fo example, when θ deceases fom 1 to 0.1, deceases fom 2.95% to 2.51%. Fom the thid ow of Table 1, the elative volatility of consumption gowth to income gowth inceases with the degee of inattention. Fo example, when θ deceases fom 100% to 10%, µ cy inceases fom to That is, the diect effect of inattention via the Γ θ, ) tem dominates its indiect geneal equilibium effect via. We can get the same conclusion by shutting down the geneal equilibium GE) channel, see the coesponding patial equilibium PE) esults epoted in the same table. Compaing the GE and PE esults in Table 1, we can see the values of µ cy ae slightly lowe in the GE case if the inteest ate is fixed as θ deceases. In othe wods, the geneal equilibium effect of RI tends to educe the volatility of individual consumption in this case. 24 Anothe impotant implication of RI in geneal equilibium is that RI leads to moe skewed wealth inequality measued by µ ay, the elative volatility of financial wealth to labo income. Fom the fouth ow of Table 1, we can see that when θ is educed fom 1 to 0.1, µ ay inceases fom to Fom 26), it is clea that the main diving foce behind this esult is the pesence of the estimation eo, s t ŝ t because va s t ŝ t ) / θ < 0. Note that although / θ > 0, the estimation eo channel dominates the geneal equilibium channel and aises the wealth inequality. Theefoe, RI can incease wealth inequality, which makes the model fit the data bette. 25 Table 2 epots how elastic Kalman gain, the geneal equilibium inteest ate, and the elative volatility of consumption and wealth to income fo diffeent values of σ. It is clea fom the second ow of Table 2 that the Kalman gain inceases with income volatility. Fo example, if λ = 50, θ is almost doubled when σ inceases fom 0.2 to 0.4. Fom the thid ow of Table 2, we can see that RI has significant effects on the equilibium inteest ate. Fo example, deceases fom 3.184% to 2.402% when σ inceases fom 0.2 to 0.4. It is woth noting that in the elastic capacity case an 24 We cannot examine the stochastic popeties of aggegate consumption dynamics because all idiosyncatic shocks income shocks and RI-induced noise shocks) cancel out afte aggegating aoss consumes. 25 The liteatue has typically found that simple models based on standad CRRA pefeences and on uninsuable shocks to labo income cannot account fo the obseved U.S. wealth distibution. Fo example, Aiyagai 1994) finds consideably less wealth concentation in a model with only idiosyncatic labo eanings uncetainty. Given the CARA- Gaussian setting, the model hee is not suitable to addess the issue like why the top 1 pecent o 5 pecent ichest families hold a lage faction of financial wealth in the U.S. economy. 18

20 incease in income volatility affects the equilibium inteest ate via two channels: i) the diect channel the ωζ 2 tem in 33)) and ii) the indiect channel the elastic capacity θ tem in 33)). The second panel of Table 2 epots the esults when we shut down the indiect channel and assume that θ = 1. Compaing the thid and sixth ows of Table 2, we can see that the indiect channel is moe impotant when σ is elatively low. Fo example, given that σ = 0.1, deceases fom 3.811% to 3.528% when we switch fom the FI economy to the RI economy. Futhemoe, the elative volatility of consumption gowth to income gowth deceases with the value of σ in geneal equilibium. That is, consumption becomes smoothe when income becomes moe volatile. Fo example, in the equilibium RI economy, µ cy deceases fom to when σ is doubled fom 0.2 to This esult is consistent with ou obsevations on consumption inequality. To futhe exploe this issue in ou model, we do the following execise. Fist, we divide the full sample into two sub-samples and ; o and ) and apply the same estimation pocedue to e-estimate σ and φ 2 see Table 3 fo the estimation esults). Household income is moe volatile in late sub-peiods than ealie ones. Fo example, the standad deviation of y is 0.47 in the sub-sample ), while it is 0.59 in the sub-sample ). The aveage values of µ cy ae and in the fist and second sub-samples, espectively. In the elastic capacity case, using the estimated income pocesses in the fist sub-sample, we fist use µ cy = to calibate that λ = 365. Using this calibated value of λ, we find that µ cy = 0.196, which matches the empiical countepat almost pefectly Welfae Losses due to RI in Equilibium We now tun to the welfae cost of RI how much utility does a consume lose if the actual consumption path he chooses unde RI deviates fom the fist-best FI-RE path in which θ = 1? To answe this question, we follow Bao 2007) and Luo and Young 2010) by computing the maginal welfae cost due to RI. The following poposition summaizes the main esult/ Poposition 5. Given the initial value of the state, ŝ 0, the maginal welfae cost mwc) due to RI is given by mwc θ) v ŝ 0) / θ) θ v ŝ 0 ) / ŝ 0 ) ŝ 0 = θ [ 2 α + α ) ŝ 0 ] d dθ, 40) whee d /dθ is given in 29) and v ŝ 0 ) = exp αŝ 0 + ln 1 + )) / α). The monthly dolla loss 26 It is not supising that µ cy is geate in the equilibium RI economy than in the equilibium FI economy because the value of θ is less than 1 in the RI case. This esult is the same as that we obtained in the fixed capacity case and epoted in Table Choosing anothe sub-samples and ) does not change ou main esult. 19

21 due to deviating fom the FI-RE path θ = 1) can be witten as $ loss θ < 1) 1 12 mwc 1) 1 θ) ŝ 0. 41) Poof. See Appendix 7.3. Since we ae inteested in the deviation fom the FI-RE path, θ = 1 is consideed as the stating point. When the household deviates fom 1 to θ, the pecentage change in just θ 1). ŝ 0 is initial total wealth. Finally, we need to convet the change in the ŝ 0 tem to monthly ates by multiplying by /12. Expession 40) gives the popotionate eduction in the initial level of the peceived state ŝ 0 ) that compensates, at the magin, fo a pecentage decease in θ i.e., stonge degee of RI) in the sense of peseving the same effect on welfae fo a given ŝ 0. To do quantitative welfae analysis we need to know the value of ŝ 0. Fist, we set ŷ 0 E [y t ] = 1, φ 1 = 0.88, and the atio of the initial level of financial wealth â 0 ) to mean income ŷ 0 ) equal to Second, given that ŝ 0 = â 0 + ŷ 0 / 1 + φ 1 ) + ŷ 0 / = [5 + 1/ 1 + φ 1 ) + 1/ ] ŷ 0, we can calculate the values of the monthly dolla loss $ loss) fo diffeent values of θ and the coesponding values of the geneal equilibium inteest ate. The fifth ow of Table 1 epots the welfae losses fo diffeent degees of inattention. Fo example, when θ = 0.4, i.e., when the household deviate fom the FI-RE path by 60%, the monthly dolla loss is ŷ 0, wheeas it deceases to ŷ 0 when θ = Anothe implication of the welfae losses due to RI epoted in Table 1 is that thee is a geneal equilibium effect of RI on the welfae loss. Fo example, when is set to be 2.95% in the PE case, the monthly dolla loss is lage than that obtained in the GE case fo low values of θ. Fo example, when θ = 0.1, the welfae loss in the GE case is 3.57 pecent lowe than that in the PE case. Thus, the patial equilibium esults might ovestate the welfae losses. These two esults thus povide some evidence that it is easonable fo consumes to lean the tue state slowly due to finite capacity because the welfae loss fom deviating fom the FI-RE case is tivial. In othe wods, although consumes could devote much moe capacity to pocessing economic infomation and thus impove thei consumption decisions, it may be ational fo them not to do so even when infomation costs ae negligible. 28 This numbe vaies lagely fo diffeent individuals, fom 2 to is the aveage wealth/income atio in the Suvey of Consume Finances We find that changing the value of this atio only has mino effects on the welfae implication. 29 The main conclusion hee is obust to changes in the values of α, σ, and φ 1. 20

22 5. Sepaation of Risk Avesion and Intetempoal Substitution In this section, we conside an extension in which we assume that consumes have ecusive utility and thus isk avesion and intetempoal substitution ae contolled by diffeent paametes. In ou benchmak model, we discussed how the inteaction of isk avesion and intetempoal substitution affects the equilibium inteest ate and the elative volatility of consumption. Howeve, given the time-sepaable utility setting, we cannot examine how intetempoal substitution affects the equilibium outcomes. In this section, we conside a ecusive utility RU) model with iso-elastic intetempoal substitution and exponential isk avesion poposed in Weil 1993): U t = η 1 η 1 β) ct + β 1 α ln E t [exp αu t+1 )] ) η 1 η η η 1, whee β > 0 is the agent s subjective discount facto, α > 0 is the coefficient of absolute isk avesion, and η > 0 is the elasticity of intetempoal substitution. Following the same state-space-eduction method, the Bellman equation fo the optimization poblem in the Weil model can be witten as: η 1 v s t ) = max {c t } 1 β) c η t + β 1 α ln E t [exp αv s t+1 ))] ) η 1 η η η 1, whee s t a t + φy t + φ 0 φ/, φ = 1/ 1 + φ 1 ), and s t follows the same state tansition equation, 6). As shown in Weil 1993), the consumption function unde FI-RE can be expessed as: whee A = c t = 1 + δ) s t 1 ) αa ln E t [exp αϕaσw t+1 )]), 42) ) 1 + δ η 1 η η δ) 1 η 1 and δ = β 1 + )) η Consumption and Saving Rules unde RI As in the benchmak model, we now intoduce RI. The coesponding Bellman equation unde RI can be witten as η 1 v ŝ t ) = max {c t } 1 β) c η t + β 1 α ln E t [exp αv ŝ t+1 ))]) ) η 1 η η η 1, 21

23 subject to the tansition equation of ŝ t, 13). The following poposition chaacteizes the consumption function and value function unde RI. Poposition 6. Given finite capacity κ, the consumption function unde RI is c t = 1 + δ) ŝ t 1 ) ε 43) whee ε = 1 )]) αa ln E t [exp αa ζ t+1, 44) ] ) A and δ ae defined above, and ζ t+1 is specified in 14) with E t [ ζ t+1 = 0, va ζ t+1 = Γ θ, ) ωζ 2, and Γ θ, ) = θ 1 1 θ)1+) 2. Poof. See Appendix 7.4. When β = 1/ 1 + ), we have δ = 1, A =, and the RU model is educed to ou benchmak model. In contast, when β = 1/ 1 + ), A is inceasing with η fo given see Figue 5). Compaing 42) with 43), it is clea that 43) can be obtained by eplacing s t and ζ t+1 in 42) with ŝ t and ζ t+1, espectively. Fist, given that ln E t [exp Aαζ t+1 )]) = 1 2 αa)2 ωζ 2, [ )]) ln E t exp Aα ζ t+1 = 1 2 Γ θ, ) αa)2 ωζ 2, the pecautionay saving pemium due to RI can be witten as P i = 1 2 Γ θ, ) 1) αaω2 ζ, 45) which clealy shows that the pemium is deceasing with the degee of attention θ, and is inceasing with the degee of isk avesion α, the elasticity of intetempoal substitution η, and the pesistence and volatility of the income shock φ and σ) fo any value of θ. Using 43), we can expess the saving function as follows: d t = f t Θ t Ψ t + Π, 46) 22