A Model of the Asymmetric Transmission of Aggregate Shocks

Transcription

1 A Model of the Asmmetrc Transmsson of Aggregate Shocks Joshua Bernsten November 6, 208 JOB MARKET PAPER Clck here for the latest verson Abstract I stud the transmsson of aggregate shocks n a New Kenesan model n whch households ncomes are heterogeneousl exposed to changes n aggregate ncome, and borrowng frctons lmt opportuntes for aggregate rsk sharng. I analtcall show that shock transmsson s asmmetrc: output responds more to contractonar shocks than to expansonar shocks of equal magntude. Estmatng ke model parameters usng the mcro evdence on heterogeneous consumpton exposures to changes n output generates asmmetrc responses of output to monetar polc shocks that can explan at least 60% of the emprcal asmmetr. Prnceton Unverst. I thank Mkhal Golosov for hs nvaluable advce throughout ths project, and Mark Aguar and Ganluca Volante for man nsghtful conversatons. I also thank partcpants at the Prnceton and Unverst of Chcago Macro student presentaton lunches for man useful comments and suggestons. Emal: jmsb@prnceton.edu

2 Introducton A growng bod of papers studes how departng from the representatve household paradgm affects the transmsson of aggregate shocks n New Kenesan models. When households are no longer dentcal, some ma be more exposed to the effects of aggregate shocks than others. For example, Guvenen and co-authors (204, 207) use hgh qualt admnstratve ncome data for the US to document sgnfcant heterogenet n how households ncomes co-move wth busness ccle movements n GDP. However, ths heterogenet n the ncdence of aggregate shocks s somewhat understuded b the exstng lterature, whch focuses on the effects of dosncratc and unnsurable ncome rsk on the transmsson of aggregate shocks. In ths paper, I contrbute towards fllng ths gap. I stud the transmsson of aggregate shocks when households ncomes are heterogeneousl exposed to changes n aggregate ncome (output), and borrowng frctons lmt the opportuntes for rsk sharng. I couple ths model of the household sector wth a standard New Kenesan suppl-sde: frms are monopolstcall compettve and are subject to costl prce adjustments, and nomnal nterest rates are set accordng to a Talor rule. In ths settng, I explore the transmsson mechansm of aggregate shocks both theoretcall and numercall. Usng a smple verson of m model, I analtcall establsh m man novel result: output responds more to contractonar monetar polc shocks than to expansonar shocks of equal magntude. Ths result follows from the fact that households ncomes are heterogeneousl exposed to changes n aggregate ncome, and that borrowng frctons prevent households from full sharng ths aggregate ncome rsk. In the presence of bndng borrowng constrants, onl households wth the strongest ncentve to save respond to nterest rate changes on the margn. I refer to these households as savers. In equlbrum, savers are unconstraned, and ther consumpton response s governed b a standard Euler equaton. As a smple example, suppose that the elastct of ntertemporal substtuton (EIS) s one, and consder a transtor shock to the real nterest rate of +%. The Euler equaton states that, n response to ths shock, saver households reduce ther current consumpton b %. Lkewse, a shock to the real nterest rate of -% causes saver households to ncrease ther current consumpton b %. In contrast, borrowng-constraned households necessarl respond asmmetrcall to postve and negatve nterest rate changes. For a household to be borrowng-constraned n response to As an example of ths heterogenet, fgure 4 n the appendx reproduces Fgure 2 n Guvenen et al. (207), whch plots the g coeffcents from the pooled OLS regresson log,t = g + g logy t + u,t where,t s worker s ncome n perod t as reported on her W-2 form, Y t s GDP, and the groups {g} are the gender-specfc (Panel A) or age-specfc (Panel B) percentles of the permanent ncome dstrbuton computed usng ncomes n perods t 6 to t 2. In both cases, there s sgnfcant heterogenet n the estmated g coeffcents across groups. 2

3 a % nterest rate hke, her equlbrum drop n current consumpton must be greater than %. If ths were not the case, she would use the asset to save n order to acheve the unconstraned response of a % drop n consumpton. Conversel, for a household to be borrowng-constraned n response to a % nterest rate cut, her equlbrum ncrease n current consumpton must be smaller than %. If not, she would use the asset to save n order to acheve the unconstraned response of a % ncrease n consumpton. Combnng the consumpton responses of savers and borrowng-constraned households mples that on average, households decrease ther current consumpton b more than % n response to a % nterest rate hke, but ncrease t b less than % n response to a % nterest rate cut. Therefore, n equlbrum, output (equal to aggregate consumpton) must respond more to contractonar monetar polc shocks (nterest rate hkes) than to expansonar monetar polc shocks of equal magntude. I establsh analtcall that ths mechansm apples to the transmsson of two other aggregate shocks commonl used n the New Kenesan lterature: TFP shocks and cost-push shocks (drect shocks to nflaton). In each case, the transmsson of the shock that ncreases output n equlbrum s weaker than the transmsson of an equal and opposte shock that decreases output. I also stud the responses of nflaton to these shocks and show that the drecton of the asmmetr depends the tpe of shock httng the econom: nflaton nherts the output asmmetr n response to monetar polc shocks, but exhbts the opposte asmmetr pattern for TFP and cost-push shocks. I confrm that output response asmmetr s robust to the ntroducton of dosncratc rsk. I show that, n the emprcall relevant case of ver persstent dosncratc shocks, the asmmetr of the output responses to monetar polc shocks s unaffected b dosncratc rsk. Intutvel, when dosncratc shocks are ver persstent, a household does not expect her consumpton to change for dosncratc reasons, so that her ncentve to borrow or save s drven manl b her ncome senstvt to changes n output, as n the case wthout dosncratc rsk. Smlarl, I show that output response asmmetr occurs when households have heterogeneous EISs. In ths case, bndng borrowng constrants restrct the ncrease n consumpton of hgh EIS households n response to expansonar shocks, and also prevent low EIS households from achevng small consumpton declnes n response to contractonar shocks. The sze of the output response asmmetr s determned b the hghest and lowest senstvtes of household consumpton to equlbrum changes n output. Intutvel, a larger range of senstvtes mples less asset tradng n equlbrum and hence tghter borrowng constrants, whch causes a larger asmmetr. Usng mcro data on household consumpton from the Consumer Expendture Surve, I estmate a lower bound for the rato of these senstvtes of 2.5. Insertng ths rato nto the model then mples that the output response to a contractonar monetar polc shock s two and a half tmes larger than the response to an expansonar monetar polc shocks of equal sze. 3

4 I fnsh b showng that an asmmetr of ths magntude s consstent wth the macroeconometrc evdence for asmmetrc output responses to monetar polc shocks. Usng local projecton methods (Jorda, 2005), I estmate that the maxmal response of output to a % contractonar monetar polc shock s approxmatel four tmes larger than the maxmal response to a % expansonar shock. Therefore, the quanttatve mechansm s capable of explanng at least 60% of the emprcal asmmetr. Related Lterature I contrbute to a growng lterature that studes how departng from the representatve household paradgm affects the transmsson of monetar polc and other aggregate shocks n New Kenesan models. A large bod of work has replaced the representatve household wth the assumpton that households face dosncratc and unnsurable ncome rsk that causes ex-ante dentcal households to experence ex-post heterogeneous tme paths of ncome and consumpton. Ths class of Heterogeneous Agent New Kenesan (HANK) models has been used to stud the decomposton of monetar transmsson (Auclert (207) and Kaplan et al. (208)), the power of forward gudance (McKa et al. (206) and Wernng (205)), and the determnac of nterest rate rules (Achara and Dogra, 208), among other ssues. Relatve to these papers, I consder a novel dmenson of household heterogenet, and stud ts mplcatons for the transmsson of a varet of aggregate shocks, not lmted to monetar polc. An mportant excepton n the extant lterature, and the closest forebear to m paper, s Blbe (208), who analzes the transmsson of monetar polc n a tractable class of Two Agent New Kenesan (TANK) models. In ths class of models, the frst set of households do not face an frctons n the asset market, whle the second set face severe frctons that prevent them from both borrowng and savng. These households thus lve hand-to-mouth, and consume ther entre ncome n each perod wth a margnal propenst to consume of one. Usng ths set up, Blbe shows how the response of output to nterest rate changes s amplfed when changes n aggregate ncome fall manl on the hand-to-mouth households, and dampened otherwse. Whle Blbe s model also features heterogeneous ncome exposures, the responses of output reman smmetrc n hs framework. The lack of asmmetr follows from the extreme wa n whch asset markets are modeled n TANK frameworks: the frst group of households has complete access to asset markets, whle the second s completel barred from borrowng or savng an amount. Ths assumpton mples that onl the frst group of households adjust ther consumpton n response to nterest rate changes. Hence, the equlbrum output response smpl concdes wth the consumpton response of ths group, and s therefore smmetrc n the sgn of the nterest rate change. I show that relaxng ths assumpton so that all households onl face constrants to borrowng generates asmmetrc transmsson of aggregate shocks that algns well wth the macro-econometrc evdence for such asmmetr. Fnall, n contemporaneous work, Patterson (208) argues that contractonar shocks are amplfed when households who are hghl exposed to the fall n aggregate ncome also have 4

5 hgh margnal propenstes to consume (MPC), a fact that she documents n the data. M results complement ths emprcal fndng b provdng a structural theor of the amplfcaton of contractonar shocks, and b explorng ts consequences for expansonar shocks, thus establshng m ke result on asmmetrc transmsson. The paper proceeds as follows: secton 2 descrbes the economc envronment. Secton 3 establshes m man theoretcal result on asmmetrc output responses, and dscusses extensons and robustness. I estmate ke model parameters n secton 4, and compare the mpled output response asmmetr to the emprcal evdence n secton 5. Secton 6 concludes. 2 Envronment The econom s populated b a unt mass of households ndexed b 2 [0, ]. Each household has preferences over her nfnte sequence of fnal good consumpton {c,t } and labor suppl hours {n,t } gven b where E " X t= t u (c,t,n,t ) 2 (0, ) s a tme dscount factor, E t s an expectatons operator condtoned on tme t nformaton, and u s strctl ncreasng and concave n c and strctl decreasng and concave n n. If household works for n,t hours, she supples,t n,t unts of effectve labor, where,t s her labor productvt and s subject to dosncratc shocks, as descrbed below. Effectve labor earns the nomnal wage P t w t where P t s the nomnal prce of the fnal consumpton good, and w t s the real wage. In addton to wage ncome, household receves a fxed share s of dvdends from ntermedate goods frms d t measured n consumpton unts. 2 When households have heterogeneous dvdend shares and labor productvtes, ther ncomes are heterogeneousl senstve to changes n aggregate ncome. # For example, f total wage ncome ncreases more than total dvdend ncome, households wth hgher labor productvtes wll beneft more than households wth hgher dvdend shares. In order to smooth these heterogeneous exposures to aggregate ncome fluctuatons and to nsure aganst dosncratc ncome shocks, I assume that households can trade a nomnal, rsk-less one perod bond b,t, that earns the real rate r t gven b +r t = + t + t where t s the nomnal nterest rate, and t = Pt P t P t s nflaton. All households begn wth zero assets, b,0 =0for all. In ths econom, households face ncome rsk due to dosncratc and aggregate shocks. Wthout addtonal restrctons on the structure of fnancal markets, households could acheve large 2 Interpretng s as a household s equt holdngs, I assume that tradng equt s suffcentl costl so that households do not trade at busness ccle frequences n response to aggregate shocks. 5

6 amounts of self-nsurance aganst these shocks usng onl the perod bond (Krusell and Smth, 998). Therefore, I follow the lterature on ncomplete markets and heterogeneous households and assume that households are subject to ad hoc borrowng constrants, b,t b,t where b,t 0 for all, t. The fact that the constrant ma depend on tme and the dentt of each household captures, n reduced form, the fact that dfferent households ma face borrowng frctons of varng severt at dfferent ponts n tme. For example, hgher ncome households are lkel to face less strngent restrctons on ther borrowng capact than lower ncome households. In sum, and takng all prces and dvdends " as gven, household solves # X t max E u (c,t,n,t ) {c,t,n,t,b,t } t subject to t= c,t + b,t = w t,t n,t + s d t +(+r t ) b,t b,t b,t b,0 =0,0 = A representatve compettve fnal good frm packages the unt mass of ntermedate goods ndexed b j 2 [0, ], usng the CES producton functon ˆ Y t = 0 t (j) t t where t > s the elastct of substtuton across ntermedate nputs, and s subject to aggregate shocks. Takng the prce of each nput and the prce of the fnal good as gven, the frm solves ˆ max P t t (j) t t dj {} 0 dj t ˆ t 0 t t p t (j) t (j) dj Optmzaton elds a demand functon for each ntermedate good and a nomnal prce ndex pt (j) t (j) = P t ˆ P t = p t (j) 0 t Y t t t dj 6

7 Each ntermedate good j s produced b a monopolstcall compettve frm emplong effectve labor E t (j) n the producton functon t (j) =A t E t (j) where A t s aggregate TFP and s also subject to aggregate shocks. Each frm faces ts own demand curve, and chooses ts path of prces to maxmze profts subject to quadratc prce adjustment costs (Rotemberg, " 982): max E X t p t (j) t (j) p(j) subject to t= P t w t E t (j) pt (j) t (j) = P t t (j) =A t E t (j) p 0 (j) =P 0 2 t Y t pt (j) p t (j)!# 2 P t Y t where, for smplct, I assume that frms dscount future profts usng the dscount factor of households. 3 I focus on the smmetrc equlbrum n whch p t (j) =P t, t (j) =Y t,and E t (j) =E t for all j 2 [0, ]. In ths case, the aggregate dvdend n perod t s gven b d t = Y t w t A t 2 2 t A monetar authort sets the nomnal nterest rate on the asset accordng to the Talor rule + t = Yt ( + t ) Y T e vt where >, 0, v t s a monetar polc shock, and Y T s some fxed target level of aggregate ncome (output). 4 In each perod, the labor, fnal good, and bond market must clear: ˆ 0 E t = ˆ 0 c,t d = Y t,t n,t d 2 2 t ˆ 0 b,t d =0 Households are subject to dosncratc shocks to ther labor productvt. Specfcall, labor 3 Ths assumpton s nnocuous for m theoretcal results snce I approxmate around equlbra n whch the true stochastc dscount factor s constant. 4 Throughout the paper, I refer to output and aggregate ncome nterchangeabl. 7

8 productvt for household follows the AR() process log,t = log,t +( ) log +,t where 2 (0, ) and,t s an..d. random varable wth mean zero and varance 2. Aggregate shocks affect aggregate TFP, the elastct of substtuton among ntermedate nputs (the source of so called cost-push shocks), and the nnovatons to monetar polc, all of whch evolve as AR() processes, log A t = a log A t +( a ) log Ā + a t log t = log t +( ) log + t v t = v v t + v t where a,, v 2 (0, ), and a t, t, v t are..d. random varables, each wth mean zero and t varance 2. Due to ther effects on nflaton, and followng the New Kenesan conventon, I refer to t < 0 as a postve cost-push shock and t > 0 as a negatve cost-push shock. 5 Compettve Equlbrum Defnton. Gven ntal condtons {b,0, o,0 },P 0, a compettve equlbrum s a sequence n{c,t,n,t,b,t },t, { t (j)} j,t,p t,d t,w t, t such that. {c,t,n,t,b,t },t solve the household problem for each. 2. { t (j)} j,t solve the fnal good frms problem. 3. {P t } t solve the ntermedate goods frms problem. 4. {d t } t satsfes the dvdend equaton. 5. { t } t satsfes the Talor rule. 6. Markets clear at ever tme t. t 3 Asmmetrc Transmsson of Aggregate Shocks In ths secton, I obtan analtcal results regardng the responses of output to aggregate shocks. I begn wth the case of monetar polc shocks, and then dscuss the extenson to cost-push and TFP shocks. I also dscuss nflaton responses, and generalzatons of the result to the ncluson of dosncratc rsk, and heterogeneous preferences. 5 Intutvel, t > 0 ncreases the elastct of demand faced b each monopolst producer, and hence causes frms to lower ther prces. 8

9 3. Asmmetrc Output Responses to Monetar Polc Shocks In order to acheve tractablt, I smplf the model econom n varous dmensons. Assumpton. Let u (c, n) = a,, v =0! 0 c =0 n + + b,t! 0 8, t The frst two condtons restrct aggregate stochastc parameters to be..d. over tme, and to have small varances so that local approxmaton technques are vald. The next two condtons restrct aspects of the household problem. The Greenwood et al. (988) (GHH) specfcaton of utlt s common n the busness ccle lterature, and s tractable snce t sets ncome effects on labor suppl to zero. The forth condton sets dosncratc rsk to zero so that,t = for all and t. I relax ths restrcton n secton 3.4. The fnal condton restrcts tradng n the asset market, and should be nterpreted as follows: as the b,t parameters approach zero, the szes of the asset postons taken b households who would borrow n response to a shock get closer to zero due to the bndng borrowng constrant. Snce the asset s n zero net suppl, n general equlbrum, the asset postons of savng households must also get closer to zero. Therefore, the household budget constrant mples that, as the b,t parameters approach zero, the equlbrum consumpton choces of a household become well approxmated b her ncome choces. I stud the equlbrum under ths approxmaton. Importantl, ths condton s not the same as mposng autark. Instead, the lmt condton mples that households can take arbtrarl small postons n the asset n equlbrum. Ths then requres that prces and quanttes adjust n equlbrum so that the asset market clears. In partcular, the adjustment must be such that households who save n equlbrum optmall choose a vanshngl small asset poston that offsets the vanshngl small postons taken b borrowng-constraned households. Therefore, ths assumpton bus tractablt wthout losng the ke transmsson mechansm from nterest rates to savngs choces. 6 Determnstc Compettve Equlbrum Under assumpton, I can defne a determnstc compettve equlbrum as a compettve equlbrum when all aggregate shocks are set to 6 Wernng (205) uses a smlar assumpton to analze how the cclcalt of dosncratc rsk affects the power of forward gudance. 9

10 zero n all perods. In ths equlbrum, nflaton s alwas zero, aggregate prces and quanttes are constant, and all household choces of consumpton, labor suppl, and asset postons are fxed over tme snce the do not face an rsk. Defnton 2. Suppose that assumpton holds, and assume that there are no aggregate shocks. Then, gvenn ntal condtons {b,0,,0 } o,p 0, a determnstc compettve equlbrum s a sequence {c,n,b }, { (j)} j,p,d,w, such that. {c,n,b } solve the household problem for each. 2. { (j)} j solve the fnal good frms problem. 3. P = P 0 solves the ntermedate goods frms problem. 4. d satsfes the dvdend equaton. 5. Markets clear at ever tme t. I consder the dnamcs of the econom n response to aggregate shocks around ths determnstc compettve equlbrum. 7 Formall, the followng lemma condenses the econom s equlbrum dnamcs to a set of necessar and suffcent condtons expressed as log devatons around the determnstc compettve equlbrum, where I use ˆx t = log x t log x to denote such a devaton. The proofs of ths and all other results are contaned n the appendx. Lemma. Under assumpton, the econom s frst order equlbrum dnamcs n response to monetar polc shocks satsf the sstem t = + t + ŷ t + v t t = ŷ t + E t [ t+ ] n o mn E t hˆ c,t+ ˆ c,t = ( t E t [ t+ ] ) ˆ c,t = ŷt 8 where = log, { } depend onl on model prmtves, and c s consumpton net of the dsutlt of labor suppl, c = c n + +. > and 0 are suffcent to ensure that the sstem has a unque stead state, ŷ t =0, t =0, ˆ c,t =08. The frst two equatons of lemma are standard features of New Kenesan models. The frst equaton s the Talor rule for the nomnal nterest rate where the target level of output Y T s set to the level n the determnstc compettve equlbrum. The second equaton s the 7 In determnstc economes wth heterogeneous households, the wealth dstrbuton ma be ndetermnate (Sorger, 2000). However, m restrcton that b,t! 0 mposes that all households hold zero assets n all perods, thus breakng the ndetermnac, and ensurng unqueness of the determnstc compettve equlbrum. 0

11 New Kenesan Phllps Curve (NKPC) lnkng nflaton, and output. Intutvel, f output s hgher toda holdng, frms face hgher margnal costs of labor ceters parbus, and so wll optmall choose to rase ther prces, leadng to nflaton. The thrd equaton captures the effect of bndng borrowng constrants on the equlbrum. I emphasze that ths equaton does not requre the no-borrowng lmt restrcton n assumpton. It onl requres that the borrowng constrants bnd for some postve measure of households n each perod. In the presence of bndng borrowng constrants, onl households wth the strongest ncentve to save are unconstraned n equlbrum, and respond to nterest rate changes on the margn. All other households are borrowng-constraned and are unresponsve to margnal nterest rate changes. Ths logc s captured b the equaton n mn E t hˆ c,t+ ˆ c,t o = ( t E t [ t+ ] ) whch states that onl the Euler equatons of households wth the lowest consumpton growth (.e. households who save on the margn n equlbrum) hold wth equalt. 8 households as savers. 9 I refer to these In order to smplf the exposton, I nvoke the restrcton of..d. monetar polc shocks n assumpton. Ths restrcton mples that E t hˆ c,t+ =0for all, andthatshocksonl affect contemporaneous consumpton. I extend the analss to persstent shocks n secton 4. Gven ths, the Euler equaton for unconstraned saver households has the usual nterpretaton: a % ncrease n the real nterest rate translates nto a % decrease n the current consumpton of saver households, where s the elastct of ntertemporal substtuton (EIS). Note that the consumpton response of savers s smmetrc n the sgn of the nterest rate change. Borrowng-constraned households, however, necessarl respond asmmetrcall to postve and negatve nterest rate changes. In response to a % nterest rate hke, borrowng-constraned households must experence a drop n current consumpton of more than %. If not, the would not want to borrow on the margn to reduce ther drop n consumpton towards the unconstraned response of %. Conversel, n response to an nterest rate cut, borrowng-constraned households must experence an ncrease n current consumpton of less than %. If not, the would not want to borrow on the margn to ncrease ther equlbrum current consumpton towards the unconstraned response of %. Combnng the consumpton responses of savers and borrowng-constraned households mples that on average, households decrease ther current consumpton b more than % n response 8 I refer to c as consumpton nstead of consumpton net of the dsutlt of labor suppl for ease of exposton. Ths slght abuse of defntons does not affect the ntuton for an of the results. 9 Note that the soluton to the mn operator need not be unque. Multple households 2 [0, ] ma be unconstraned n perod t. Snceallofthesehouseholdswllchoosethen sameexpected consumptongrowthn equlbrum, I onl need to know the growth for a sngle 2 arg mn E t hˆ c,t+ ˆ c,t o.

12 to a % nterest rate hke, but ncrease t b less than % n response to a % nterest rate cut. Ths suggests that, n equlbrum, output must respond more to contractonar monetar polc shocks than to expansonar monetar polc shocks. In order to pn down ths output response asmmetr, the forth equaton maps the heterogeneous senstvtes of household ncome to changes n output nto heterogeneous senstvtes of household consumpton to changes n output ˆ c,t = ŷt 8 where the senstvt coeffcents { } measure the per cent change n household s consumpton for a % change n output n equlbrum. Intutvel, the pattern of { } depends on the underlng heterogenet n ncome senstvtes, and the tghtness of the borrowng constrants that restrct asset tradng n equlbrum. For example, n the absence of an asset tradng frctons, households would full nsulate ther consumpton from ther heterogeneous ncome senstvtes, and would be fxed across. Under assumpton however, the no-borrowng lmt restrcton mples the other extreme: a household s consumpton nherts the senstvt of her ncome to changes n output. Heterogenet n consumpton senstvtes therefore reflects heterogenet n the underlng ncome senstvtes. There are two sources of heterogenet n the household ncome senstvtes, whch are most clearl seen usng the explct expresson for where = + 0 = + gven b + s s d. Frst, when output ncreases, households wth hgher labor productvtes receve more of the correspondng ncrease n wage ncome, so that s ncreasng n. Second, snce TFP s fxed n the case of monetar polc shocks, hgher wages cause the dvdend share of aggregate ncome to declne, so that s decreasng n s. It s smple to show that effect on household ncome and hence consumpton, > 0 when the labor productvt effect domnates the dvdend + > 0 () >s Antcpatng the emprcal results, whch fnd postve consumpton senstvtes for all, I assume that ths condton holds for all from now on. Gven the consumpton response of saver households = S, the equlbrum output response! 2

13 must mechancall satsf Hence, f S ŷ t = ˆ c S,t S s dfferent for contractonar and expansonar shocks, then the equlbrum path of output wll be shock-dependent, and asmmetrc, as demonstrated b the followng example. ANumercalExample Let there be two household tpes, 2{, 2}, andassumethat prces are fxed,! +, so that the central bank drectl controls the real nterest rate. Suppose that =0,and =, and assume that group households consumpton s more senstve to changes n output than group 2 households consumpton, =2,and 2 =0.5.0 Gven real nterest rate shocks of v t = ±%, the responses of output are found b solvng the sstem n o mn ˆ c,t = v t ˆ c,t = ŷt 8 where I have used E t hˆ c,t+ =0because the nterest rate shock s..d. over tme. When the real nterest rate ncreases b %, saver households reduce ther consumpton b % because =. Snce all other households are borrowng-constraned, ther equlbrum reductons n consumpton are greater than %. Hence, the equlbrum reducton n output must be greater than %. Specfcall, gven output falls n equlbrum, group 2 households must be the savers snce the experence the smallest drop n current consumpton and therefore have the lowest expected consumpton growth. Hence, ĉ 2,t = %. Invertng the senstvt equaton for group 2 households, ˆ c 2,t =0.5ŷ t, mples that ŷ t = % 0.5 = 2%, so that output falls b 2% n equlbrum. When the real nterest rate decreases b %, saver households ncrease ther consumpton b %. Snce all other households are borrowng-constraned, ther equlbrum ncrease n consumpton s less than %. Hence, the equlbrum ncrease n output must be less than %. Specfcall, gven output rses n equlbrum, group households must be the savers snce the experence the largest ncrease n current consumpton and therefore have the lowest expected consumpton growth. Hence, ĉ,t = %. Invertng the senstvt equaton for group households, ĉ,t =2ŷ t, mples that ŷ t =0.5% so that output ncreases b 0.5% n equlbrum. Therefore, output responds more to contractonar monetar polc shocks than to expansonar monetar polc shocks. Furthermore, because the saver households alwas adjust ther consumpton growth b % n equlbrum, the rato of the contractonar response to the 0 Techncall, the stead state of the sstem s no longer unque when! + and =0. However, I abstract from ths complcaton for the purposes of ths example. 3

14 expansonar response concdes wth the rato of consumpton senstvt coeffcents. I come back to ths connecton n secton 4. In general, we have the followng closed-form representaton of the asmmetrc output responses to monetar polc shocks. Proposton. Under assumpton, the frst order equlbrum dnamcs of output n response to monetar polc shocks are gven b 8 >< ŷ t = >: + v + t f v t > 0 + v + t f v t < 0 where = max { } =mn { } > mples that output responds more to postve (contractonar) monetar polc shocks than to negatve (expansonar) monetar polc shocks of equal magntude. In other words, output responds asmmetrcall to monetar polc shocks. In response to an expansonar monetar polc shock, nterest rates fall, and saver households ncrease ther current consumpton. B vrtue of beng borrowng-constraned n equlbrum, all other households must ncrease ther consumpton b less than savers. Therefore, the equlbrum ncrease n output s smaller than the consumpton response of savers alone. Equvalentl, saver households consumpton ncrease s the most senstve to the ncrease n output n equlbrum, as captured b the forth equaton of the lemma, evaluated for the saver household = S, ˆ c S,t = ŷ t, = max { } In response to a contractonar monetar polc shock, saver households decrease ther current consumpton. B vrtue of beng borrowng-constraned n equlbrum, all other households must experence a larger decrease n ther consumpton than savers do. Therefore, the equlbrum decrease n output s larger than the consumpton response of savers alone. Equvalentl, saver households consumpton s the least senstve to changes n output n equlbrum, ˆ c S,t = ŷ t, =mn { } The equlbrum dnamcs exst as long as + + > 0, whchsmpledb > 0 for all. If ths condton fals, then there does not exst an equlbrum output response to a contractonar monetar polc shock, v t > 0, of the pece-wse lnear form presented n proposton. However, gven,parameters,, and can alwas be chosen to ensure that the condton holds. 4

15 Snce the sze of the consumpton response of savers s the same for both postve and negatve equlbrum real nterest rate changes, output must respond more to contractonar monetar polc shocks than to expansonar monetar polc shocks. For completeness, I note that n the knfe-edge case of =, there s no asmmetr. In ths case, all households ncomes are equall senstve to changes n aggregate ncome so that there s no ncentve for households to trade the asset n response to an aggregate shock. Therefore, borrowng constrants do not pla a role n determnng the equlbrum response of output to monetar polc shocks. 3.2 Other Aggregate Shocks It s smple to derve the equvalent of proposton for the cases of cost-push shocks and TFP shocks. In both cases, the same asmmetr emerges: output responds more to shocks that ncrease nterest rates and lower output, and less to shocks that decrease nterest rates and ncrease output. I brefl sketch the ntuton below. Full detals of the analss are n the appendx. The asmmetr of cost-push shocks follows from the fact that a postve cost-push shock creates nflaton whch causes nterest rates to rse va the Talor rule, whle a negatve costpush shock causes nterest rates to fall. These nterest rate movements then ntate the same mechansm as above, causng output to respond more to the contractonar movement than to the expansonar movement. Smlarl, a postve TFP shock causes deflaton and hence lower nterest rates, whle a negatve TFP shock creates nflaton and hgher nterest rates. Therefore, the same mechansm mples that output wll respond more to negatve (contractonar) TFP shocks than to postve (expansonar) TFP shocks. 3.3 Inflaton Responses Solvng the sstem of equatons n lemma elds equlbrum responses of both output and nflaton. In contrast to output, the drecton of the asmmetr of nflaton s shock-dependent. I outlne the ke economc mechansms below, and relegate the dervatons to the appendx. In the case of monetar polc shocks, nflaton nherts the asmmetr of output. Ths occurs because the response of nflaton s entrel determned b the response of output va the logc of the NKPC: hgher output mples hgher margnal costs whch causes frms to ncrease ther prces, thus rasng nflaton. Therefore, nflaton moves n the same drecton as output n response to monetar shocks. Snce output responds more to contractonar monetar shocks than to expansonar shocks, nflaton nherts ths asmmetr. In the cases of cost-push and TFP shocks, the asmmetr of the nflaton response s the opposte to that of output: nflaton responds more to shocks that ncrease output. Ths occurs 5

16 because the responses of nflaton are determned b two forces. Frst, the movement of output affects nflaton va the NKPC as n the monetar shock case. Second, both cost-push and TFP shocks drectl affect frms margnal costs and so drectl affect nflaton (postve cost-push shocks ncrease margnal costs and nflaton, whle postve TFP shocks lower margnal costs and nflaton). Crucall, ths effect pushes nflaton n the opposte drecton to the movement of output such that the frst effect offsets the second. The strength of ths offsettng force nherts the asmmetr of output s responses to both tpes of shock so that nflaton responds more overall when output responds less and the offsettng force s weaker. Therefore nflaton responds wth the opposte asmmetr to output. 3.4 Idosncratc Shocks It s straghtforward to extend the analss of ths secton to the case n whch dosncratc rsk s small, so that! 0. In ths case, the asmmetr of output responses to monetar polc shocks depends on the persstence of dosncratc shocks. In partcular, when unnsurable dosncratc shocks are ver persstence (! ), the output response asmmetr s the same as n the case wthout dosncratc rsk, so that proposton stll apples. I outlne the ke economc mechansms below, and relegate the dervatons to the appendx. In the presence of dosncratc rsk, there s an addtonal channel through whch households can have low consumpton growth, and hence be savers n equlbrum. When a household expects to experence a drop n her dosncratc labor productvt, her consumpton growth wll be low to the extent that the drop n labor productvt s unnsurable and hence transmts to her consumpton. Importantl, ths novel channel s ndependent of aggregate shocks httng the econom, and so cannot be a source of asmmetr. However, when the process for unnsurable labor productvt shocks s ver persstent (! ), households expect ther labor productvt to reman approxmatel constant across consecutve perods. Ths renders the novel savngs channel nactve. As a result, savngs choces are entrel determned b household ncome senstvtes to changes n output. Therefore, the responses of output to monetar polc shocks are dentcal to the econom wthout dosncratc rsk. The emprcal evdence suggests that the process for dosncratc shocks to labor ncome has a ver persstent component, so that! s a good approxmaton to realt (see, for example, Storesletten et al. (2004) and Guvenen et al. (206)). Furthermore, related evdence suggests that households are ver well nsured aganst shocks to the transtor component (Blundell et al. (2008) and Heathcote et al. (204)), so that these shocks do not affect consumpton growth computatons. In order to obtan analtcal nsghts, I assume that the dosncratc rsk s small so that frst order approxmatons are vald. Ths approach rules out second order phenomena, such as the cclcalt of dosncratc ncome rsk, that ma also effect the responses of output to monetar polc shocks (and other aggregate shocks). These effects are the focus of Wernng 6

17 (205) and Achara and Dogra (208), who show how the cclcalt of dosncratc ncome rsk affects the sze of the output responses to both postve and negatve monetar polc shocks. Importantl, ths effect s smmetrc n the sgn of the shock, and so does not affect the asmmetr that I am nterested n. 3.5 Heterogeneous Preferences In m benchmark model, I follow the New Kenesan lterature and assume that households have homogeneous EISs gven b. 2 However, the asset prcng lterature has suggested that heterogeneous EISs ma help to reconcle macroeconomc models and asset prcng facts (Guvenen, 2009), and have emprcal groundng (Vssng-Jorgensen (2002), Guvenen (2006)). In the case of heterogeneous EISs, the asmmetr stll emerges, but for dfferent reasons. I outlne the ke mechansm below, and relegate the dervatons to the appendx. When households have heterogeneous EISs, ther consumpton responses to changes n the real nterest rate are heterogeneous. In equlbrum, these dfferent consumpton responses occur va asset tradng. For example, n response to a contractonar monetar polc shock, households wth larger EISs wll save, lendng to households wth smaller EISs n equlbrum. However, bndng borrowng constrants lmt the asset tradng that occurs n equlbrum and causes output to respond asmmetrcall. Consder a contractonar shock. When the real nterest rate falls, households tr to decrease ther consumpton b an amount dctated b ther EIS. However, households wth small EISs become borrowng-constraned n equlbrum and so experence larger consumpton drops than the would n the absence of the constrant. Therefore, the overall equlbrum decrease n output s amplfed. When nterest rates rse after an expansonar shock, households ncrease ther consumpton. In ths case, households wth large EISs become borrowng-constraned n equlbrum, and so experence smaller consumpton ncreases then the would were the not constraned. Therefore, the overall equlbrum ncrease n output s dampened, thus creatng asmmetrc responses of output to monetar polc shocks. 4 Numercal Exercse In ths secton, I quanttatvel assess the asmmetr of output responses to monetar polc shocks. I frst use m theoretcal nsghts to hghlght the ke parameters that I need to measure to quantf the asmmetr, and then turn to parameter estmaton. M estmates of the hghest and lowest senstvtes of household consumpton to changes n output mpl that the output response to a contractonar monetar polc shock s two and a half tmes the sze of the response to an expansonar shock of the same magntude. Ths result compares favorabl to the macro-econometrc evdence for asmmetr n secton 5. 2 Techncall, s the EIS for net consumpton when preferences are of the GHH form. However, abstractng from ths complcaton does not affect the ntuton. 7

18 4. Suffcent Statstcs for Output Response Asmmetr The theoretcal analss has shown that output response asmmetr arses n equlbrum when two condtons hold: some households are borrowng-constraned, and households have heterogeneous consumpton senstvtes to changes n output n equlbrum. Whle proposton was derved under assumpton, the ntuton suggests that such strngent restrctons are not necessar to generate output response asmmetr more generall. In ths secton, I consder a sstem of equatons that s motvated b the sstem n lemma, but does not have such a restrctve structural nterpretaton attached to t. In partcular, rather than dervng the senstvtes of household consumpton to changes n output from prmtves usng strong structural assumptons, I specf the senstvtes as reduced-form parameters that have a smlar nterpretaton to ther structural counterparts, but are estmable usng mcro data on household consumpton. Let S be the sstem t = + t + ŷ t + v t t = apple ŷ t + E t [ t+ ] mn {E t [ĉ,t+ ] ĉ,t } = ( t E t [ t+ ] ) ĉ,t = c,v ŷ t 8 that descrbes the equlbrum dnamcs of output ŷ t,nflaton t, nomnal nterest rates t, and household consumpton ĉ,t, n response to monetar polc shocks v t. Even n the absence of an underlng structural model, the frst three equatons n sstem S have clear nterpretatons as a Talor rule, NKPC, and Euler equaton n the presence of bndng borrowng constrants. Furthermore, t s straghtforward to nterpret the c,v parameters as mappng heterogeneous senstvtes of household ncome to changes n output nto heterogeneous senstvtes of household consumpton to changes n output va asset market tradng. For example, f c,v s fxed for all, then households would be full sharng the ncdence of aggregate shocks, whch would ndcate that ncome senstvtes are homogeneous or that borrowng constrants are slack. As the c,v parameters become more heterogeneous, t s reasonable to nterpret ths as evdence for a combnaton of heterogeneous ncome senstvtes and restrcted asset market tradng caused b bndng borrowng constrants. In ths sense, the { c,v } coeffcents are the theoretcal analogs of the { } coeffcents when the no-borrowng restrcton s relaxed.3 In the emprcall relevant case of persstent monetar polc shocks (Gertler and Karad (205) and Chrstano et al. (2005)), the nherent non-lneart of sstem S prevents me from studng the full equlbrum dnamcs. Therefore, I nstead derve the responses of output to 3 I assume the consumpton senstvtes are fxed over tme for each household. Whle ths s a reduced-form assumpton, t s consstent wth the subsequent emprcal strateg snce the mcro data on consumpton are not rch enough to permt credble estmaton of tme-varng senstvtes. 8

19 one-tme, zero probablt ( MIT ) monetar polc shocks of the form v t = t v v, where v s the zero-probablt monetar polc shock, and v 2 (0, ) s a persstence parameter. Ths approach s tractable snce once the shock hts, the econom transtons determnstcall back to the stead state of the sstem. Proposton 2. In sstem S, theresponseofoutputtoaonetme,zeroprobabltmonetar polc shock v wth persstence v,sgvenb 8 >< ŷ t = >: ( v) c,v + apple ( v) v + ( v) c,v + apple ( v) v + t v v f v > 0 t v v f v < 0 where c,v =mn{ c,v } c,v = max { c,v } As n the case of..d shocks, output responds more to the contractonar monetar polc shock than to the expansonar shock. Appealng to the structural nterpretaton of the sstem elds the same ntuton as before: when v > 0, the decrease n savers consumpton s the smallest among all households, so that output falls a lot. When v < 0, the ncrease n savers consumpton s the largest among all households so that output ncreases a lttle. In order to assess the quanttatve magntude of the output response asmmetr, I defne the rato of the contractonar response to the expansonar response, R = ( v) c,v + ( v ) ( v ) c,v + ( v ) apple v + apple v + The ke parameters for quantfng the asmmetr are c,v and c,v, whch measure the hghest and lowest equlbrum senstvtes of household consumpton to changes n output. In partcular, when apple ( v ) v + =0, R = c,v c,v. The parameters c,v and c,v are suffcent statstcs for computng the output response asmmetr (Chett, 2009). 4 In other words, to compute R, I onl need to know the values of c,v and c,v, and do not need quanttatve nformaton on the underlng structural mechansm that generates them. In m settng, ths means that I do not need to know quanttatve detals concernng borrowng constrants or the heterogenet of ncome senstvtes. Ths s convenent because I can estmate c,v and c,v drectl usng mcro data on household consumpton, and then plug these estmates nto R to mmedatel quantf the asmmetr. 4 Suffcent statstcs approaches have recentl become popular n macroeconomcs. See, for example, Auclert and Rognle (207). 9

20 4.2 Parameter Estmaton In prncple, estmates for c,v and c,v can be recovered usng data on household consumpton and output. Formall, let {c,t },t and {Y t } t be data on household consumpton and output respectvel, and consder the lnear regresson log c,t = + log Y t + u,t where {u,t } are dosncratc shocks uncorrelated wth changes n output, and runnng the regresson n growth rates rather than n levels controls for long run growth trends n household consumpton and output. As t stands, ths specfcaton has two ssues: shock condtonng, and data feasblt. Shock Condtonng In order to measure the senstvtes of household consumpton to changes n output drven b monetar polc shocks, the varaton n log Y t must be due to monetar polc shocks onl. However, the varaton n raw output data s drven b multple aggregate shocks httng the econom smultaneousl n each perod. Runnng the above regresson would therefore result n estmates of { } that measure the senstvt of household consumpton to changes n output drven b multple shocks, and would not correspond to the theoretcal parameters { c,v }. In order to allevate ths ssue, I frst project the output data onto a set of dentfed, lagged monetar polc shocks (descrbed n more detal below), Z t = v t,..., v t L, n so that the ftted values shocks onl. 5 log Y t = + Z 0 t + e t o log ˆ Y t capture the varaton n log Y t drven b monetar polc Then, usng these ftted values, I estmate the regresson log c,t = + ˆ log Yt + u,t whch correctl dentfes as the senstvt of household consumpton n to changes n output drven b monetar polc shocks onl. Gven these estmates ˆo, the ke asmmetr n o n o parameters are estmated as c,v = max ˆ and c,v =mn ˆ. Intutvel, ths process amounts to Two-Stage-Least-Squares (2SLS) estmaton, where the frst stage extracts the varaton n log Y t due to monetar polc shocks onl, and the second stage estmates the household senstvt parameters usng ths varaton alone. 5 M theoretcal results suggest that logy t should depend non-lnearl on the hstor of monetar polc shocks. However, for the purposes of extractng the varaton n logy t drven b monetar polc shocks, I abstract from ths complcaton. I nvestgate non-lnear responses n secton 5. 20

21 Household Consumpton Data There do not exst data that contan precse measures of consumpton at the household level and busness ccle frequenc. The Consumer Expendture Surve (CEX) s the closest substtute, but s known to have measurement error problems (Aguar and Bls, 205), and onl features the same household for four consecutve quarters. In order to allevate these ssues, I group households together wthn the CEX data and estmate pooled OLS regressons nstead. Ths approach helps to mtgate the effects of measurement error n the cross-secton, and creates longer snthetc panels n the tme seres dmenson. Smlar methods are common among analses that use CEX data to analze trends and fluctuatons n household consumpton (see, for example, Parker and Vssng-Jorgensen (2009), Prmcer and van Rens (2009), and De Gorg and Gambett (207)). The choce of groupng naturall affects the estmates obtaned from runnng regressons at the group level. Formall, let G be a surjectve functon that maps household n perod t,.e. the household-perod tuple (, t), nto a fnte set of groups {, 2,...,G}. G represents an arbtrar group formaton process, and nests fxed group assgnment as a specal case, G (, t) fxed for all t. Gven a choce of G functon, consder the pooled OLS regresson for a group g 2{, 2,...,G}, log c,t = g + g ˆ log Yt + e,t where the poolng occurs over the set {(, t) :G (, t) =g} of household-perods assgned to group g. Estmatng ths regresson for each group mples n that o the ke parameters n ofor quantfng the asmmetr can be estmated as c,v = max g ˆg and c,v =mn g ˆg. When the G functon assgns each household to a fxed group over tme, the mpled asmmetr parameters wll alwas be weakl bounded b the true asmmetr parameters max { } and mn { }. Therefore, pooled OLS usng fxed group assgnments wll alwas weakl underestmate the true asmmetr. Proposton 3. Suppose the model for household consumpton growth s gven b log c,t = + ˆ log Yt + u,t If G does not depend on t for all, thentheasmmetrparametersmpledbthepooledols regressons log c,t = g + g ˆ log Yt + e,t are weakl bounded b max { } and mn { },.e. n o max plm g T! ˆg n o mn plm g T! ˆg apple max { } mn { } 2

22 Intutvel, when group assgnments are fxed over tme, the estmated consumpton exposure of a group g s a convex combnaton of the consumpton exposures of each household n that group. Therefore, each group s consumpton exposure s weakl smaller than the largest household exposure, and weakl larger than the smallest household exposure. Ths mmedatel sas that the asmmetr mpled b the estmates must be bounded b the true asmmetr at the household level. When G assgns households to dfferent groups over tme, t s dffcult to sa whether the mpled asmmetr from pooled OLS over- or underestmates the true asmmetr. As an extreme example, suppose that =for all (so that the true asmmetr s nl) and consder the followng assgnment process for a fxed group g. When log ˆ Yt > 0, assgn households wth the hghest consumpton growths to group g. When log ˆ Yt < 0, assgn households wth the lowest consumpton growths to group g. Such a process wll result n an estmate of ˆg much larger than, due to the selecton bas created b the assgnment mechansm s dependence on dosncratc shocks, and wll therefore overestmate the true asmmetr. Furthermore, the opposte assgnment process wll clearl result n an underestmate of the true asmmetr. In lght of ths dscusson, I choose as a benchmark, an assgnment mechansm that s fxed over tme, so that the estmated asmmetr s known to be a lower bound on the true asmmetr (n the lmt T! ). In practce, ths amount to defnng groups based on household characterstcs that are fxed n the sample of households that I observe. 4.3 Data Monetar Polc Shocks In order to extract the varaton n log Y t drven b monetar polc shocks, I follow Cobon et al. (207), who use the methods ntroduced b Romer and Romer (2004) to dentf nnovatons to monetar polc that are orthogonal to economc condtons. Formall, the authors run the regresson FFR t = x 0 t + v t where FFR t s the change n the federal funds rate from perod t to t, andx t s a vector of controls that contans forecasts of GDP growth, nflaton, and the unemploment rate taken from the Greenbooks at each Federal Open Market Commttee meetng. The resduals from ths regresson, {ˆ v t }, are then taken as the seres of monetar polc shocks, wth the nterpretaton that ˆ v t > 0 s a contractonar shock, and ˆ v t < 0 s an expansonar shock. Usng ths method, Cobon et al. (207) generate a seres of monetar polc shocks at monthl and quarterl frequences from 969 to 2008, whch I plot n fgure. The shocks are evenl spread over postve and negatve values, and are ver volatle durng the Volcker dsnflaton perod n the earl 980s. 22