Rational Inattention and Dynamics of Consumption and Wealth in General Equilibrium

Transcription

1 Rational Inattention and Dynamics of Consumption and Wealth in Geneal Equilibium Yulei Luo, Jun Nie, Gaowang Wang, and Eic R. Young Novembe 2014; Revised May 2016 RWP 14-14

2 Rational Inattention and Dynamics of Consumption and Wealth 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 May 17, 2016 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 fist 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 exploe how the intoduction of RI can help the model fit the joint equilibium dynamics of consumption, income, and wealth. We then contast RI with habit fomation and show that the two models make vey diffeent geneal equilibium pedictions, and that RI is close to the data. We finally show that the equilibium welfae costs of incomplete infomation due to RI ae elatively low within the CARA-Gaussian setting. An ealie vesion of this pape was ciculated unde the title: What We Don t Know Doesn t Hut Us: Rational Inattention and the Pemanent Income Hypothesis. We ae gateful to Moten O. Ravn (the Edito) and two anonymous efeees fo many constuctive suggestions and comments. We also would like to thank Ken Kasa, Chis Sims, Neng Wang, Miko Wiedeholt, and Chistian Zimmemann fo helpful discussions and suggestions, and confeence and semina paticipants in the 2015 SED annual meeting, the 11th wold congess of the econometic society, the 2015 intenational confeence on computing in economics and finance, Univesity of Kansas, Kansas City Fed, Univesity of Hong Kong, Cental Univesity of Finance and Economics and Shandong Univesity fo helpful comments. Luo thanks the Geneal Reseach Fund (GRF No. HKU791913) in Hong Kong fo financial suppot. We thank Andew Palme fo excellent eseach assistance. All eos ae the esponsibility of the authos. The views expessed hee ae the opinions of the authos only and do not necessaily epesent those of the Fedeal Reseve Bank of Kansas City o the Fedeal Reseve System. All emaining eos ae ou esponsibility. Faculty of Business and Economics, The Univesity of Hong Kong, Hong Kong. yulei.luo@gmail.com. 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. 1

3 Keywods: Rational Inattention; Pemanent Income Hypothesis; Geneal Equilibium; Consumption and Wealth Volatility. JEL Classification Numbes: C61; D83; E21.

4 1. Intoduction In intetempoal consumption-savings poblems, pudent households save today fo thee easons: (i) they anticipate futue declines in income (saving fo a ainy day), (ii) they face uninsuable isks (pecautionay savings), and (iii) they ae patient elative to the inteest ate. In models whee only motive (i) is opeative, one obtains the pemanent income hypothesis" (PIH) of Fiedman (1957), in which consumption is solely detemined by pemanent income (the annuity value of total wealth) and follows a andom walk (see Hall 1978); this statement holds, fo example, if households have quadatic utility and have access to a single isk-fee bond with a constant etun. 1 In aggegate data, the PIH makes pedictions that ae inconsistent with the data. Two implications in paticula ae discussed in Campbell and Deaton (1989), the excess sensitivity" and excess smoothness" puzzles. Excess sensitivity aises if cuent consumption esponds to pedictable movements in income (o equivalently esponds too much" to tansitoy changes in income); unde the PIH these changes ae pat of pemanent income and theefoe should aleady have had thei effect on consumption. Excess smoothness occus if consumption esponds less than one fo one to pemanent changes in income (equivalent to changes in pemanent income). Unde the assumption of a single isk-fee asset, the two puzzles ae manifestations of the same undelying economic foces, as noted in Campbell and Deaton (1989), and thei joint test ejects the absence of both. 2 Regading motives (ii) and (iii), uninsuable income isk seems to be pevasive in micoeconomic data, and geneal equilibium models with uninsuable isk often imply impatience of households (that is, they face low inteest ates ). 3 Thee is a staightfowad link between uninsuable isk and excess sensitivity if agents engage in pecautionay savings and thei isk avesion is affected by wealth, then an incease in income today will lead to a lage incease in cuent consumption than justified by the incease in pemanent income; excess smoothness then aises via the intetempoal solvency condition, since futue consumption cannot ise by the equied 1 If utility is quadatic, the andom walk natue of consumption is only appoximately tue, but the PIH still holds. 2 Attanasio and Webe (1993) show that aggegation eos can geneate excess sensitivity, in paticula when households of diffeent ages ae lumped togethe. Similaly, Attanasio and Pavoni (2011) show that appaent violations of the intetempoal budget constaint made feasible via insuance makets can delive excess smoothness in aggegate data. It is substantially moe difficult to eject the PIH on individual (o cohot) data, but limitations in quality and coveage of consumption data ae paticulaly seious at low levels of aggegation. 3 In patial equilibium, low inteest ates ae needed to ensue a stationay distibution of wealth (see Caoll 2011). Howeve, Aiyagai (1994) and Huggett (1993) show they aise natually in geneal equilibium, whethe the supply of assets is elastic o inelastic in the aggegate. 1

5 amount. Ludvigson and Michaelides (2001) study a buffe stock model based on Bewley (1983) and find that quantitatively the model fails to epoduce the excess sensitivity obseved in the individual-level data. We ae theefoe led to look elsewhee fo explanations. Luo (2008) and Luo and Young (2010) intoduce the ational inattention hypothesis poposed by Sims (2003) 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, and as a esult also delives smalle esponses to pemanent income changes; that is, the model delives both excess sensitivity and excess smoothness. 4 Coibion and Goodnichenko (2015) and Andade and Le Bihan (2013) find pevasive evidence consistent with Sims (2003) s ational inattention theoy using the U.S. and Euopean suveys of pofessional foecastes and othe agents, espectively. Cucial fo ou puposes, the RI model delives only one new fee paamete, which maps diectly into the speed of leaning (one can think of this paamete as a filte gain). Howeve, those papes ae explicitly patial equilibium, taking as given a constant exogenous isk-fee ate. Wang (2003), using a simple model with constant absolute isk avesion (CARA) utility and isk fee assets in zeo net supply based on Cabelleo (1990), 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, and leads to consumption following a andom walk. 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. Ou main goal in this pape is to exploe the geneal equilibium implications of ational inattention in a model with pecautionay savings. We fist ask the same question fom Wang (2003) namely, whethe the PIH eemeges in geneal equilibium in the pesence of ational inattention. We study economies with CARA pefeences, as they simultaneously geneate pecautionay savings and linea consumption ules, and chaacteize the foces that act on the geneal equilibium inteest ate. 5 We find that the PIH does descibe equilibium consumption behavio in geneal 4 The tests of the PIH, while obust to the possibility that agents have moe infomation than the econometician (see Campbell and Deaton 1989), they may not be obust to the RI implication that the econometician may actually have moe infomation than the agents (at least at the time of decisions). The eason is that agents with moe infomation will still set consumption changes to be othogonal to eveything the econometician obseves; it is clea that the convese would not necessaily hold. 5 Constant-elative-isk-avesion (CRRA) utility functions ae moe common in macoeconomics, mainly due to 2

6 equilibium, with the appopiate substitution of actual pemanent income by peceived pemanent income. Thus, the delicate cancelling of pecautionay and impatience foces found by Wang (2003) caies ove unmodified to models with incomplete infomation about the state, despite the additional pecautionay savings that RI geneates. 6 This esult is obust to the pecise way RI is modeled that is, whethe we think of agents as having fixed o flexible speeds of leaning. 7 We then poceed to use the RI model to intepet the dispesion of consumption and wealth (elative to income) that we obseve in the data. We constuct a panel fom the PSID, using the appoach fom Guvenen and Smith (2014) to impute total consumption fom the limited infomation in the PSID via a demand system estimated on the CEX (see also Blundell, Pistafei, and Peston 2008). We also use the PSID to constuct measues of wealth, and then we compae ou measues to those pedicted by the model. We find that RI substantially impoves the pedictions of the model fo these elative dispesions. The FI-RE model, fo a given income pocess estimated fom mico data, implies that these dispesions ae much lowe than in the data. With RI, as households become less attentive, elative dispesions in consumption and wealth both ise. Inteestingly, we find that the same value of the RI paamete oughly matches both moments, poviding an oveidentifying test of the model. Geneal equilibium effects tun out to be less significant when consumes ae less infomation-constained, acting to slightly educe the dispesions in the model elative to a patial equilibium execise with a fixed inteest ate. Futhemoe, we find that the welfae losses due to limited attention in geneal equilibium ae well above that in patial equilibium because of the geneal equilibium channel, but they ae still elatively small, about 0.04% of aveage annual income pe month fo highly infomation-constained consumes. We next compae ou RI model to an altenative method fo intoducing sluggish movements in consumption, namely habit fomation. We model habit fomation with one fee paamete, which govens the tansmission of cuent consumption into futue habit. As the habit paamete inceases, consumption changes become sluggish as households ty to smooth changes in consumpbalanced-gowth equiements. CRRA utility would geatly complicate ou analysis because the intetempoal consumption model with CRRA utility and stochastic labo income has no explicit solution and leads to non-linea consumption ules. Intoducing RI would then be substantially moe difficult and involve appoximations of unknown quality. 6 Luo and Young (2014) document an obsevational equivalence between ational inattention and signal extaction in linea-quadatic-gaussian models. Given that we find ou CARA model appoximately delives Gaussian uncetainty about the state, we believe that those esults would cay ove hee, meaning ou esults apply to a vey boad class of models. 7 Thee is one potential diffeence between the fixed and flexible leaning models the uniqueness of equilibium. While we can show that the fixed leaning model has a unique equilibium, the pesence of additional foces in the flexible model left us unable to show uniqueness. Nevetheless, we could not find any examples whee multiplicity aose. 3

7 tion athe than levels. Thus, thee is a sense in which the two model famewoks look simila; in fact, Luo (2008) shows that the two models ae obsevationally equivalent at the aggegate level (in tems of consumption gowth dynamics), but not at the individual level whee RI delives moe consumption volatility. We show that this similaity does not extend to the coss-sectional dispesion moments we examine hee, even though the noise shocks aggegate out; unlike RI, habit fomation moves both elative dispesions away fom thei empiical countepats. 8 In addition, we find that stonge habit fomation leads to highe, not lowe, inteest ates. 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 chaacteizes the unique 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 compaes the ational inattention model to the habit fomation model. In the appendices we povide the key poofs and deivations, and discuss an extension of the basic model to duable goods. 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+1 } t=0 { E 0 t=0 ( ) 1 t u(c t )]}, (1) 1 + ρ a t+1 = (1 + ) a t + y t c t, (2) whee u(c) = 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 8 Relative wealth dispesion is not stictly monotone in the habit paamete, but the change occus at vey high habit levels that cannot be econciled with obseved individual consumption volatility unless substantial measuement eo is assumed. 4

8 stationay AR(1) 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. 9 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, whee φ = 1/ (1 + φ 1 ). 10 (2003). In the last two tems in (4), ln This consumption function is the same as that obtained in Wang ) /α measues the elative impotance of impatience ( 1+ρ 1+ and the 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 total wealth. Letting total wealth, s t = a t + h t, be defined as a new state vaiable, we can efomulate the PIH model as { v (s t ) = max u (c t ) + 1 } c t 1 + ρ E t v (s t+1 )], subject to s t+1 = (1 + ) s t c t + ζ t+1, (6) 9 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). 10 See Appendix 7.1 fo the deivation. 5

9 whee the time (t + 1) innovation to total wealth 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. With finite capacity κ (0, ), a andom vaiable {s t } following a continuous distibution cannot be obseved without eo and thus the infomation set at time t + 1, { } t+1 denoted I t+1, is geneated by the entie histoy of noisy signals s j. Following the liteatue, we fist assume the noisy signal takes the additive fom j=0 s t+1 = s t+1 + e t+1, whee e t+1 is the endogenous noise caused by finite capacity. 12 We futhe assume that e 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 = { s1, s 2,, t+1} s 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 11 See Appendix 7.1 fo the deivation. 12 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. 6

10 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. In a linea-quadatic-gaussian (LQG) setting, as has been shown in Sims (2003) and Shafieepoofad and Raginsky (2013), ex post Gaussian distibutions, s t I t N (E s t I t ], Σ t ), whee Σ t = E t (s t ŝ t ) 2], ae optimal. Hee we fist assume ex post Gaussian distibutions of the tue state and Gaussian noise but adopt negative exponential (CARA) pefeences. Because both the optimality of ex post Gaussianity and the standad Kalman filte ae based on the linea-quadatic- Gaussian (LQG) specification, the applications of these esults in the RI models with CARA pefeences ae only appoximately valid. 13 In the next subsection, afte solving fo the consumption function and the value function unde RI, we veify that the loss function due to RI is appoximately quadatic and consequently the optimality of the ex post Gaussianity of the state appoximately holds in the CARA model. Since both ex ante and ex post distibutions of the state ae Gaussian, (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 vaiances 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). To guaantee that the state is stabilizable and the unconditional vaiance conveges, we need to make the following assumption on the value of channel capacity: Assumption 1 κ > ln (1 + ). (11) It is woth noting that this estiction is vey weak if is small; in geneal equilibium will be smalle than ρ, so fo shot time peiods this condition is not estictive at all. 13 See Peng (2004), Mondia (2010), and Van Nieuwebugh and Veldkamp (2010) fo applications of CARA pefeences in RI models. 7

11 Following the steps outlined in Luo and Young (2014), we can compute the Kalman gain in the steady state θ as θ = 1 1/ exp (2κ) ; (12) θ measues the faction of uncetainty emoved by a new signal in each peiod, and is the only new paamete intoduced by the ational inattention famewok. 14 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, (13) whee ŝ t = E t s t ] is the conditional mean of the state, s t. Combining (6) with (13) yields ŝ t+1 = (1 + ) ŝ t c t + ζ t+1, (14) whee ζ t+1 = θ (1 + ) (s t ŝ t ) + θ (ζ t+1 + e t+1 ) (15) is the innovation to ŝ t+1 and is independent of all the othe tems on the RHS of (14). ζ t+1 is an MA( ) pocess with E t ζ t+1 ] = 0 and ) va ( ζ t+1 = Γ (θ, ) ωζ 2, whee ω 2 ζ = va t (ζ t+1 ), Γ (θ, ) = θ 1 (1 θ) (1 + ) 2 > 1 (16) fo θ < 1, and s t ŝ t = (1 θ) ζ t 1 (1 θ) (1 + ) L θe t 1 (1 θ) (1 + ) L (17) 1 θ is the estimation eo with E t s t ŝ t ] = 0 and va (s t ŝ t ) = ω 2 1 (1 θ)(1+) 2 ζ. (Hee L denotes the lag opeato.) To guaantee that the sum of these infinite seies conveges, hee we need to impose the estiction that κ > 0.5 ln (1 + ). Note that this condition always holds when we have (11). Fom (16), it is clea that Γ > 0 and Γ θ < 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 thee will be an additional foce opeating on the equilibium inteest ate in the elastic case. 8

12 3. Geneal Equilibium unde RI 3.1. 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 )], (18) subject to (14)-(17). 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, (19) 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, (20) d t = (1 φ 1 ) φ (y t y) + (s t ŝ t ) + 1 )]) ] ln (E t exp ( α ζ t+1 Ψ (), (21) α whee s t ŝ t is an MA( ) estimation eo pocess given in (17) and Ψ () = ln ( ) 1+ρ 1+. Poof. See Appendix 7.1 fo the deivations. See Appendix 7.2 fo the poof that the loss function due to RI is appoximately quadatic and the optimality of the ex post Gaussianity of the state appoximately holds in the CARA model. Compaing (4) with (20), it is clea that the two consumption functions ae identical except that 9

13 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, we can define the pecautionay saving pemium due to limited attention as P i 1 ( ) ]) α ln E t exp ( α ζ t+1 exp ( αζ t+1 ) = 1 2 (Γ (θ, ) 1) αω2 ζ, (22) 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 1 and σ) fo any given θ. Thus, the incomplete infomation that RI foces upon the households leads to an incease in saving. To futhe exploe the pecautionay savings pemium in (22), we isolate the effects of RI on individual consumption and saving by ewiting (20) as c t = ŝ t + 1 α Ψ () whee Ψ () = ln ln (E t exp ( αθ (1 + ) (s t ŝ t ))]) (1 θ) Γ (θ, ) ( ) 2 αω ζ ( αθωζ ) 2 +, (23) ( ) 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, {e t }. 15 Note that when θ conveges to 1, the consumption function with RI, (20), educes to that of the Wang (2003) model, (4). Fom (20), 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 < ( αωζ ) 2 /2 because of the incomplete adjustment of consumption to the fundamental shock, while we have two new positive tems that incease the total } 15 This esult is deived by using Equation (17) and the iid popety of the pocesses { ζ t, {ζ t }, and {e t }. 10

14 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 (21) can be witten as ( 1 d t = f t + 2 αγ (θ, ) ω2 ζ 1 ) α Ψ (), (24) whee the fist tem f t = (1 φ 1 ) φ (ŷ 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 and Uniqueness 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 (18). Simila to Huggett (1993), we also make the following assumption: Assumption 2 The isk-fee asset in ou model is a pue-consumption loan and is in zeo net supply. The initial coss-sectional distibution of pemanent income is a 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 (), (25) whee Π (θ, ) = 1 2 (α)2 Γ (θ, ) ωζ 2 measues the amount of pecautionay savings, and Ψ () captues the dissaving effects of impatience. Given (25), an equilibium unde RI is defined by an 11

15 inteest ate satisfying D (θ, ) = 0. (26) The following poposition shows the existence of the equilibium and the PIH holds in the RI geneal equilibium. Poposition 3. Thee exists a unique equilibium with an inteest ate (0, ρ) in the pecautionaysavings model with RI. In equilibium, each agent s consumption is descibed by the PIH, in that c t = ŝ t, (27) whee ŝ t = E s t I t ] is the peceived value of pemanent income. The evolution equations of wealth and consumption ae c t+1 = ζ t+1, (28) a t+1 = 1 φ φ 1 (y t y) + (s t ŝ t ), (29) ] ) espectively, whee ζ t+1 is specified in (15) with E t ζ t+1 = 0, va ( ζ t+1 = Γ (θ, ) ωζ 2, and Γ (θ, ) = θ 1 (1 θ)(1+ ) 2. In the geneal equilibium, the value function unde RI can be witten as v (ŝ t ) = 1 + α exp ( αŝ t). (30) 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 (20), we can obtain the individual s optimal consumption ule unde RI in geneal equilibium as c t = ŝ t. Substituting (14) and (27), we can obtain (28). (27) into (2) yields (29). The poof of uniqueness is longe and elegated to Appendix 7.3. 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 plots the equilibium in- 12

16 teest ate as a function of θ, given the paametes α = 2, φ 1 = 0.92, σ = 0.175, and ρ = Given (20) and (26), 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, because the equilibium inteest ate is pushed down to counteact this pecautionay savings incease. 17 channel capacity, the equilibium inteest ate is lowe. Fom the equilibium condition, it is staightfowad to show that d dθ = (2 + ) ( α) 2 ωζ (1 θ) (1 + ) 2] 2 ( ) whee 1 (1 θ) (1 + ) 2 > 1+ρ 0 and ln 1+ With lowe Shannon ( ) 1 2 ( α) ρ Γ (θ, ) ωζ 2 ln 1 + = 0, (31) 1 1+ ] + α 2 ωζ 2Γ (θ, 1 φ1 ) 1+ φ 1 + (1 θ)(1+ ) 1 (1 θ)(1+ ) 2 1 > 0, (32) > 0. It is clea fom this expession that is deceasing in the degee of inattention 1 θ. The fist ow of Table 2 epots the geneal equilibium inteest ates fo diffeent values of θ. 18 We can see fom the table that deceases as the degee of inattention inceases. Fo example, if θ is educed fom 1 to 0.1, is educed fom 3.27 pecent to 2.87 pecent. In addition, it is clea that d dα < 0. That is, the equilibium inteest ate deceases with the degee of isk avesion. Fom (28) and (29), we can conclude that although both the CARA model and the LQ model lead to the PIH in geneal equilibium, isk avesion plays a ole in affecting the dynamics of consumption and wealth in the CARA model via the equilibium inteest ate channel. One might ask what a easonable value of θ is, and if thee is any way to calibate it outside 16 In Section 4.1, we will povide moe details about how to estimate the income pocess using the U.S. data. The main esult hee is obust to the choices of these paamete values. 17 If we intoduced an asset with elastic supply, such as the capital stock in Aiyagai (1994), the same effects would be pesent but the stock of capital would ise (and the change in the inteest ate would be smalle as a esult). 18 Hee we also set α = 2, φ 1 = 0.92, σ = 0.175, and ρ =

17 a model. Unfotunately, thee is no diect suvey evidence on the value of channel capacity of odinay households in the economics liteatue, and thus it is not staightfowad to answe these questions; estimates of leaning capacity exist, but they ae not diectly useful since we ae inteested in the capacity that will be devoted to economic activity (specifically, consumption and saving). In lieu of such evidence, we simply note that 0.1 is the value needed to match potfolio holdings in Luo (2010) and is theefoe not obviously uneasonable (a caveat can be found in Luo and Young 2016, whee a significantly lage numbe is obtained using ecusive utility). Coibion and Goodnichenko (2015) have the most model independent measue of θ, and they find θ = 0.5 povides a good fit fo a vaiety of foecast and suvey data, and a vaiety of othe papes obtain a numbe of diffeent values depending on what facts they bing to bea. We will show below that θ = 0.1 allows us to match some coss-sectional dispesion facts, but ae cognizant that this paamete s value is quite uncetain Elastic Attention Instead of using fixed channel capacity to model finite infomation-pocessing ability, one could assume that the maginal cost of infomation-pocessing (i.e., the shadow pice of infomationpocessing 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 to the infomationpocessing 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). 19 Solving this poblem yields the optimal steady state conditional vaiance: Σ = (1 + ) 2 (1 β) λ 1 + (1 + ) 2 (1 β) λ ] λ (1 + ) 2 2 (1 + ) 2 ω 2 ζ, (33) 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 ln ω 2 ζ < 1 if we 19 As in the fixed-capacity case, although we adopt the CARA-Gaussian setting, the loss function due to impefectstate-obsevation could be appoximately quadatic. See Appendix 7.2 fo the poof. 14

18 adopt the assumption that λ is fixed, while ln Σ = 1 in the fixed κ case. Compaing (33) 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 1 + (1 + ) 2 (1 β) λ ] λ (1 + ) 2 In this case, the Kalman gain is ( ) θ, λ = (1 + ) 2 (1 β) λ 1 + (1 + ) 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. 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, they have distinct implications fo the model s popagation mechanism if the economy is expeiencing egime switching (see Luo and Young 2014). The key diffeence between the elastic capacity case and the fixed capacity case in this pape 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. (36) Figue 2 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 4 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 (34) 1. (35) 15

19 fixed capacity case in which θ and move in the same diection. (See Table 2.) The main eason fo this esult is that income uncetainty affects the equilibium inteest ate via two channels: (i) the diect channel which leads to highe aggegate savings (i.e., the ωζ 2 tem in (36)) and (ii) ( ) the indiect channel which leads to lowe aggegate savings (i.e., the θ, λ tem in (36)), and the diect channel dominates. 20 Fo example, when σ inceases fom 0.15 to 0.25, the equilibium inteest ate educes fom 2.80 pecent to 2.56 pecent. (See the thid ow of Table 4.) Futhemoe, elastic attention can have significant effects on the elative volatility of consumption to income. Fo example, when σ inceases fom 0.15 to 0.25, this elative volatility declines by 35 pecent fom 0.46 to In contast, in the FI case, this elative volatility only deceases by 10 pecent fom 0.29 to (See the fouth and ninth ows of Table 4 fo the details.) 4. Empiical and Quantitative Results In this section we assess ou GE-RI model s implications fo the dynamics of consumption, income and wealth. To constuct empiical countepats that ae compaable with the theoetical moments deived in the model, we constuct a panel with individual consumption, income and wealth based on the Panel Study of Income Dynamics (PSID) and the Consume Expenditue Suvey (CEX), using the imputation appoach fom Blundell, Pistafei, and Peston (2008) and Guvenen and Smith (2014). Then, using the estimated income pocess, we show the GE-RI model significantly fits the data bette than the full infomation model egading the consumption and wealth dynamics. Ou closed-fom solutions explicitly show diffeent channels though which the RI dives the esults 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. 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 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 consump- 20 We have been unable to pove that is unique unde elastic capacity, pecisely because of the indiect channel. We can show that a necessay condition fo multiple equilibia equies the esponse of θ to changes in to be vey lage, and we did not find any examples whee this condition was satisfied. See the appendix fo details. 16

20 tion 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. 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 fedeal 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 Ou household sample selection closely follows that of Blundell, Pistafei, and Peston (2008) as well. 21 We exclude households in the PSID povety and Latino subsamples. We exclude households in yeas of family composition change, change in maital status, o female headship, as well as in yeas whee the head o wife is unde 30 o ove 65. Households in yeas with missing education, egion, income, and imputed consumption esponses ae also excluded. We also exclude households in yeas whee they epot a negative income o a food consumption level in the top o bottom 5 pecent of all epoted values in that yea. Income and consumption values ae then deflated by the CPI to constant dollas. Ou final panel contains 7, 111 unique households with 58, 034 yealy income esponses and 48, 990 imputed nonduable consumption values. 22 With this constucted panel of household income and consumption, we next dop households in yeas whee yea-ove-yea food consumption changes ae moe than 20 pecent o less than 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 (2011). 22 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. 17

21 pecent. To exclude exteme outlies, we then follow Floden and Lindé (2001) and nomalize both income and consumption measues as atios of the mean of each yea, and exclude household in bottom and top 1 pecent of the distibution of those atios. Figue 3 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, Pistafei, and Peston (2008) elative consumption dispesion declines in the 1980s, but this decline stops aound In ode to calculate the elative volatility of wealth to income atio, sd( a), we use wealth info- sd( y) mation included in the PSID data. Notice that the PSID only epots household wealth vaiables evey five yeas stating in 1983, and then evey othe yea stating in To be consistent with the model, we constuct household wealth in the following way. We use measue of wealth defined as the sum of the net value of liquid assets (checking, savings, money maket, etc.), vehicles, home equity, and othe assets such as bonds, insuance policies, and tusts. All epoted values ae again deflated by the CPI to constant dollas. We nomalize each epoted wealth and income value to the mean of the yea epoted, and then exclude outlies of this distibution at the top and bottom 1 pecent. We then take the standad deviations of the change in nomalized value fom the pevious epot fo both wealth and income to calculate ou atio. Ou final panel fo wealth and income has 23, 630 obsevations acoss 6232 households. This panel is somewhat smalle than ou panel of consumption and income due to the limited numbe of yeas that wealth measues ae epoted. Figue 4 epots the esults, which shows the elative volatility of wealth to income has been elatively stable in the sample peiod. When estimating the income pocess, we focus on 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 changes, we eliminate household incomes with yea-ove-yea level changes in the top and bottom 5 pecent of the distibution in each yea. Then, again following Floden and Lindé (2001), we nomalize household income measues as atios of the mean fo that yea and exclude all household values in yeas in which the income is in the top and bottom 1 pecent of the nomalized income measue fo the yea. To eliminate possible heteoskedasticity in the income measues, we follow Floden and Lindé (2001) and egess each on a seies of demogaphic vaiables in a fixed-effect panel egession 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 18

22 specified by equation (3) by unning panel egessions on lagged income. As the last ow of Table 1 epots, the estimated values of φ 0, φ 1, and σ ae , 0.919, and 0.175, 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, it will have an additional effect on consumption dynamics. Using (29) and (28), 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 µ cy sd ( c t ) sd ( y t ) = and the elative volatility of financial wealth to income is (1 + φ1 ) Γ (θ, ) 1 +, (37) φ 1 2 µ ay sd ( ) ( a t ) sd ( y t ) = 1 1 φ 2 ( (1 θ) (1 + φ 1 ) φ 1 ) 1 (1 θ) (1 + ) (1 θ) 1 φ φ 1 (1 θ) (1 + ). (38) Poof. See Appendix 7.1. Expession (37) 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 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, σ = 0.175, and φ 1 = The second to fouth ows of Table 2 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 2 that RI can significantly affect the equilibium inteest ate. Fo example, 19

23 when θ deceases fom 1 to 0.10, deceases fom 3.27 pecent to 2.87 pecent, which is vey close to 2.97 pecent, the aveage annual equilibium eal inteest ate fom 1980 to 1996 estimated in Laubach and Williams (2015) using U.S. quately data. (Note that when θ = 0.12, the equilibium inteest ate obtained in ou model is exactly the same as its empiical countepat.) Hee we focus on the peiod because we use it to estimate the income pocess and the elative volatility of consumption to income. Fom the thid ow of Table 2, the elative volatility of consumption gowth to income gowth inceases with the degee of inattention. Fo example, when θ deceases fom 1 to 0.1, µ cy inceases fom to 0.375, which is the same as the empiical countepat. It is clea fom these esults that the diect effect of inattention via the Γ (θ, ) tem in (37) 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 2, 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. 23 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 2, we can see that when θ is educed fom fom 1 to 0.1, µ ay inceases fom to 2.63, which is much close to the empiical countepat. (Fo example, µ ay is 3.11 in 1993 and is 2.59 in 1998.) Fom (29), 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. 24 To biefly summaize the key discussions above, Table 3 compaes the pefomances of the FI- RE model, the geneal equilibium ational inattention model (RI-GE), and the patial equilibium ational inattention model (RI-PE) with the data. Oveall, it shows unde the estimated income 23 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. 24 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. 20

24 pocess and at a single value of ational inattention paamete (θ), the GE-RI model can do a significantly bette job than the FI-RE model in geneating a lowe inteest ate, a highe consumption volatility, and a highe wealth volatility, bing all of them much close to the data. In tems of welfae loss, as the last ow in Table 3 shows and will be discussed in detail in the next subsection, the patial equilibium model significantly undeestimates the welfae loss, though the welfae loss is geneally small. Table 4 epots how elastic Kalman gain, the geneal equilibium inteest ate, and the elative volatility of consumption and wealth to income vay with diffeent values of income uncetainty measued by σ (and σ y ). We have eached thee key findings. Fist, it is clea fom the second ow of Table 4 that the Kalman gain inceases with income volatility. Fo example, if λ = 400 (the value calibated to the data as explained in the next paagaph), θ is doubled when σ inceases fom 0.15 to This means agents optimally allocate moe attention to the state vaiable when income uncetainty inceases. Second, RI has significant effects on the equilibium inteest ate. Fo example, declines fom pecent to 2.23 pecent when σ inceases fom 0.15 to It is woth noting that in the elastic capacity case an incease in income volatility affects the equilibium inteest ate via two channels: (i) the diect channel (the ωζ 2 tem in (36)) and (ii) the indiect channel (the elastic capacity θ tem in (36)). The second panel of Table 4 epots the esults when we shut down the indiect channel and assume that θ = 1. Compaing the thid and sixth ows of Table 4, we can see that the indiect channel is moe impotant when σ is elatively low. Fo example, given that σ = 0.15, deceases fom 3.39 pecent to pecent when we switch fom the FI economy to the RI economy, wheeas only deceases fom 2.43 pecent to 2.23 pecent when σ = Thid, 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 0.46 to 0.25 when σ inceases fom 0.15 to The last finding highlighted above might povide a potential explanation fo the empiical evidence documented in Blundell, Pistafei, and Peston (2008) that income and consumption inequality diveged ove the sampling peiod they study. 26 To exploe this issue in ou model, we do the following execise. Fist, we divide the full sample into two sub-samples ( 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 Othe mechanisms have been poposed fo this decline; see Kuege and Pei (2006) and Atheya, Tam, and Young (2009) fo examples. 21