The Intertemporal Keynesian Cross

Transcription

1 he Ineremporal Keynesan Cross Adren Aucler Mahew Rognle Ludwg Sraub January 207 Absrac We derve a mcrofounded, dynamc verson of he radonal Keynesan cross, whch we call he neremporal Keynesan cross. I characerzes he mappng from all paral equlbrum demand shocks o her general equlbrum oucomes. he aggregae demand feedbacks beween perods can be nerpreed as a nework, and he lnkages n he nework can be generalzed o reflec boh he feedback from consumpon and oher dynamc forces, such as fscal and moneary polcy responses. We explore he general equlbrum amplfcaon and propagaon of mpulses, and show how hey vary wh feaures of he economy. General equlbrum amplfcaon s especally srong when agens are consraned, face uncerany, or are unequally exposed o aggregae flucuaons, and plays a crucal role n he ransmsson of moneary polcy. Inroducon he Keynesan cross s a saple of nroducory macroeconomcs, and one of he cenral deas n he analycal radon ha began wh Keynes (936. I spells ou a smple feedback mechansm: when some shock leads o a rse n demand for aggregae oupu, par of he ncome earned from producng ha oupu goes back no consumpon demand, leadng o a furher rse n demand for aggregae oupu, and so on. Locally, he radonal Keynesan cross equaon can be wren as dy = Y + MPC dy ( where Y s he mpulse o aggregae demand, dy s he equlbrum change n oupu, and MPC s he aggregae margnal propensy o consume ou of ncome. Equaon ( can be solved o oban dy = ( MPC Y, where ( MPC = + MPC + MPC s called he mulpler and reflecs he accumulaed consumpon feedback amplfyng he orgnal mpulse. Alhough he radonal Keynesan cross embodes a useful nuon, has a number of weaknesses. I s sac, no dynamc. I does no address budge consrans: an mpulse o demand from governmen spendng comes ou of hn ar, raher han beng offse by axaon or a cu

2 n spendng a some oher dae. I does no drecly correspond o any sandard, mcrofounded model. hs paper presens a modernzed alernave, he neremporal Keynesan cross, ha addresses hese weaknesses and capures he general equlbrum feedback mechansms n a varey of dynamc macroeconomc models. I has he form dy = Y + MdY (2 whch s an neremporal generalzaon of (, wh Y beng he vecor of oupu a dfferen daes and M beng a marx of aggregae margnal propenses o consume, where each enry shows he fracon of aggregae ncome n one dae ha wll be spen, a he margn, n anoher. M can be generalzed o nclude neremporal feedbacks oher han household consumpon, such as nvesmen, fscal polcy, or moneary polcy. he soluon o (2 gves a general equlbrum mulpler ha maps mpulses Y o equlbrum oupu changes dy, remnscen of he radonal ( MPC mulpler. he mulpler, however, s now a marx, and reflecs rch hgher-order neracons: for nsance, some ncome earned n perod wll be spen n perod 2, from whch some ncome wll be spen agan n perod. hs mulpler marx characerzes smulaneously he general equlbrum consequences of all paral equlbrum shocks o he neremporal paern of demand ncludng hose comng from preference shocks, changes o fscal or moneary polcy, or changes n he dsrbuon of ncome among heerogenous agens. In dervng he general equlbrum mulpler from he neremporal Keynesan cross, we uncover sublees no presen n he radonal, sac analyss. One crucal observaon s ha n ne presen value erms, all ncome s evenually consumed. In analycal erms, hs mples ha he marx M egenvalue of one, and ha s mpossble o nver I M o solve (2 for arbrary mpulses Y. Inuvely, he problem s ha f an mpulse has posve ne presen value, hen applyng M wll preserve ha ne presen value, and he seres I + M + M of eraed consumpon feedbacks wll dverge o +. hs dffculy, however, s mgaed by anoher observaon: snce ndvdual agens mus respec budge consrans, a well-defned paral equlbrum mpulse o demand mus have zero ne presen value. For nsance, he subsuon effec from a shock o neres raes changes he paern of household consumpon over me whou changng s ne presen value; assumng asse marke clearng and Rcardan fscal polcy, he ne presen value of he combned ncome effecs mus also be zero. Due o hs zero ne presen value propery of demand shocks Y, s possble o solve he neremporal Keynesan cross (2 for general equlbrum dy. Snce M has an egenvalue of one, however, here s a leas one nonzero soluon o he equaon dy = MdY, meanng ha he general equlbrum soluon s ndeermnae usng (2 alone. hs ndeermnacy s nheren o models wh nomnal rgdes, and can be resolved by moneary polcy. Dfferen polcy rules wll lead o dfferen oucomes moneary polcy can deermne, for nsance, wheher a gven mpulse from governmen spendng wll resul n an ncrease n he 2

3 ne presen value of oupu or leave unchanged. hese analycal ools provde us wh a varey of resuls. In a very smple case, we show ha hey reduce o a verson of he radonal Keynesan cross. In more general cases, however, here are rch consequences for he general equlbrum amplfcaon or dampenng of varous shocks. We fnd, for nsance, ha here s greaer amplfcaon when households face gher fnancal consrans and more unceran ncomes. hs drecly ncreases he mpac of fscal polcy, whle for moneary polcy offses a declne n he paral equlbrum effec, snce consrans and uncerany make agens less drecly responsve o neres raes. As ancpaed moneary polcy shocks are pushed furher no he fuure, he nal paral equlbrum effec does less and less, whle he general equlbrum amplfcaon does more and more. Nework nerpreaon. As we wll see, one useful way o undersand he amplfcaon mechansms nheren o our model s o hnk of me perods as nodes of a nework. Each new un of ncome generaed a a gven node s spen, parly on self and parly on every oher node, accordng o relaonshps gven by he marx M. hs round of spendng generaes addonal ncome a each node, whch s agan spen accordng o he same paern, and so on. he fnal oucome for he dsrbuon of ncome across nodes s our man objec of neres. I s he general equlbrum effec on consumpon afer he neremporal Keynesan cross has run s course. If he marx M s wren n ne presen value uns as we generally do hen he fac ha all ncome s spen n ne presen value erms means ha M s a lef-sochasc marx, wh columns summng o one. I can hen be nerpreed as he ranson marx for a Markov chan, whch provdes a even more evocave vew of he nework. A number of analycal resuls can be usefully vewed n lgh of hs nerpreaon: for nsance, he ndeermnacy of GE soluons arses because M has a (generally unque Perron-Frobenus egenvecor wh egenvalue one. hs egenvecor corresponds o he saonary dsrbuon of he Markov chan. M 2,0 M 0,0 M,0 M, M 2, M 2,2 = 0 = = 2 = 0 C = = 2 M 0, M,2 C 2 (a Paral equlbrum demand shock M 0, (b General equlbrum amplfcaon Fgure : he Ineremporal Keynesan Cross 3

4 2 A deermnsc framework In hs secon we wre down a dynamc general equlbrum envronmen. Agens are heerogeneous n erms of her preferences, her ncome, he axes hey pay, and her ably o access fnancal markes. Producon resuls from an aggregae of dfferen ypes of labor. Hence, our envronmen ncorporaes many of he sources of heerogeney ha he recen leraure has consdered. In hs envronmen we derve, n Proposon, he Ineremporal Keynesan Cross: here exss a marx M such ha he general equlbrum effec on oupu dy of a shock o aggregae demand (from preferences, governmen spendng or moneary polcy sasfes dy = Y + MdY (3 where Y s he paral equlbrum effec of he shock. Furhermore, we have M = (Lemma and, for any shock, Y = 0 (Lemma 2. In he nex secons, we wll characerze he soluons o (3 and llusrae he usefulness of hs proposon n a number of mporan conexs. 2. Envronmen and flexble prce equlbrum We consder a + perod economy populaed by a fne se of I ypes of agens who face no uncerany. here are µ agens of ype =... I, wh masses normalzed such ha = I µ =. All agens whn a ype have he same preferences and face he same envronmen, so hey behave dencally. Agens. Each agen ype I has uly over consumpon U ( c 0, c,..., c ; θ (4 where θ s an aggregae preference parameers, whose effec on agen depends on he uly funcon U. Agen consumes c goods and works n uns of me n perod. Work provdes dsuly V ( n0, n,..., n. Uly n consumpon and lesure s separable, so overall uly s U V. he agen can rade n nomnal asses, and faces borrowng and savng lms whch may be - and -specfc. Specfcally, hs budge consran n perod s P c + A = ( + A + W n P, (5 A [ ] a P, a In (5, s he nomnal neres rae, and P he nomnal prce of goods, equal for all agens. W s he wage of ype labor, and are axes, whch are lump sum bu may be and specfc. As we wll see shorly, n equlbrum here are no frm profs o be dsrbued. Agens are born and de wh no wealh: we mpose A = A = 0 for all. 4

5 Recursvely subsung n (5 and mposng our nal and ermnal condons yelds he neremporal budge consran Q c ( W = Q n =0 =0 P (6 where he real dscoun rae Q s he me-0 prce of a un of consumpon a me, Q s=0 and he gross real neres rae beween perods s and s + s defned as R s ( + s Maxmzaon of (4 subjec o (5 mples he Euler equaons R s (7 P s P s+ (8 ( ( U c c 0, c,..., c ; θ ( ( A ( Uc + c 0, c,..., c ; θ R a a P A = 0, (9 P Our preference seup s very general. I accomodaes arbrary preferences, asse marke parcpaon, and populaon srucure, and can generae a wde array of equlbrum me pahs for consumpon and labor supply, as well as margnal propenses o consume. Producon. Each perod, a perfecly compeve frm produces he unque fnal good n hs economy usng a echnology ha aggregaes labor from each ype of worker Y = F (l,... l I where, n each perod, F ( has consan reurns o scale and dmnshng reurns o each labor ype. Frm prces are perfecly flexble. Prof maxmzaon mples P = F l, W ( l,... l I, (0 where W s he wage of worker ype. Consan reurns hen mply ha frms make zero profs a all mes. Governmen. he governmen spends G n perod. I rases axes o pay for hs spendng, and adjuss he sock of nomnal publc deb B, so as o sasfy s budge consran P ( n =I µ + B = ( + B + P G ( 5

6 We mpose B = B = 0, o enforce marke asse marke clearng a he nal and ermnal dae. Flexble wage equlbrum. In a flexble wage equlbrum, frms opmze, mplyng (0. Households opmze, mplyng (5 and (9 for each ogeher wh Furher, labor markes clear, mplyng Fnally, goods markes clear: Vn ( n 0, n,..., n ( Uc c 0, c,..., c ; θ = W, (2 P l = µ n, (3 µ c + G = Y (4 { Equvalenly, asse markes clear, µ A = B,. Fx an allocaon Y n, Gn, c,n, n,n,,n, A,n sasfyng hese equaons and call he naural allocaon (n some cases, here may be mulple such allocaons, bu we wll consder perurbaons away from a parcular one. Noe ha, gven our nal and ermnal condons on deb, he flow governmen consrans ( mply ha he followng neremporal consran mus hold n equlbrum: 2.2 Scky wage equlbrum { Consder a naural allocaon ( n Q µ = =0 =I Q G (5 =0 }, R n, W,n, P n. We defne a scky wage Y n, Gn, c,n, n,n,,n, A,n equlbrum relave o ha allocaon. Followng he lead of he large New Keynesan leraure, we are hen also neresed n characerzng equlbrum oucomes relave o ha allocaon. Consder nex an arbrary pah {θ,, G } for preferences, nomnal neres raes and fscal polcy, as well as a ransfer rule = (G, R (6 specfyng ransfers a me for ndvdual as a funcon of he pah for spendng and real neres raes. he rule (6 s consraned o sasfy,n = (G n, R n as well as (5 for any pah G and R. Gven hese pahs and our nal naural allocaon, a scky wage equlbrum s defned as a se of equlbrum prces and quanes { Y, c, n, A, R, W, P } such ha wages have o reman a her a her naural level: W = W,n, (7 frms opmze, mplyng (0, households opmze her consumpon plan, mplyng (5, (6 and }, R n, W,n, P n 6

7 (9 for each, and labor, goods and asse markes clear, mplyng (3 and (4. Hence he only dfference wh a flexble wage equlbrum s ha equaon (7 replaces he requremen ha househols be on her labor supply curves,.e. { equaon (2. } By consrucon, he naural allocaon under he baselne level of parameers θ n, n, Gn,,n s a scky wage equlbrum. he followng remark smplfes he analyss of equlbrum. Remark. In a scky wage equlbrum, prces reman a her naural-allocaon level a all mes: P = P n (8 Proof. Snce he producon funcon F has consan reurns o scale, s dervaves F l, are homogeneous of degree 0. Equaon (0 mples ha F l, F l j, ( l,... l I ( l,... l I = W W j = W,n W j,n ( = F l, ( and hence here exss a sequence {λ } such ha l = λ l,n, we fnd ha P = P n for all. F l j, l,n l,n,... l I,n,... l I,n for all,. Applyng (0 agan for any Gven hs remark, he analyss of moneary polcy n hs model s parcularly smple: he exogenous pah for he nomnal neres rae ranslaes no a pah for he real neres rae equal o R = ( + Pn P+ n, and we can alernavely hnk of a scky wage equlbrum as beng defned gven such a pah. Our framework s herefore approprae o sudy he hree man ypes of demand shocks consdered n he busness cycle leraure: preference shocks θ, governmen spendng shocks G, and moneary polcy shocks. owards our dervaon of he neremporal Keynesan cross (3, we now defne a number of objecs of neres. hese correspond o ceran combnaons of dervaves of polcy funcons evaluaed a he naural allocaon. 2.3 Aggregae demand and MPCs We sar by defnng margnal propenses o consume MPCs ou of ndvdual ncome Snce agens n a scky wage equlbrum are no able o choose her labor supply, we defne MPCs akng hs consran no accoun. Specfcally, we consder a modfed problem for each agen, n whch we rea ncome as an exogenous sream { y }. Agen maxmzes (4 subjec o A few smple modfcaons o hs seup would allow us o sudy producvy shocks as well. 7

8 he consrans c + a = R a + y [ ] a a, a and a = a = 0. he soluon deermnes Marshallan demand funcons ({ } c y, {R } ; θ. In parcular, he followng neremporal budge consran holds for each ({ } Q c y, {R } ; θ = Q y (9 =0 =0 For every par s,, defne Q,s Q Q s as he me-s prce of a un of consumpon a dae. Defnon. he margnal propensy o consume of ndvdual a me, for ncome receved a dae s s MPC,s c Q,s y s y = W,n P n n,n,r =R n,θ=θn I s he dervave of he Marshallan demand funcon, dscouned back o he dae of he ncome recep s, and evaluaed a he naural allocaon, where ncome s defned as y = W,n P n n,n and he real neres rae s R = R n Indvdual ncome response o aggregae macroeconomc changes In order o defne aggregae demand, we need o deermne how ndvdual ncome y = W n P s affeced by macroeconomc aggregaes Y, G and R. We have already esablshed ha W P = W,n, P,n he fscal rule (6 mples ha = (G, R. We now urn o he deermnans of n. Consder he frm problem n a scky wage equlbrum. Snce F has consan reurns and s herefore homohec, wh consan npu prces (7, labor demand for each ype scales lnearly n he level of producon l = l,n Snce he labor marke for each ype of labor clears (3, hs mples ha ndvdual hours worked scale lnearly n he amoun of aggregae oupu Y n = n,n Combnng hs wh (0, (7, and (8, he gross labor earnngs of ndvdual a me are herefore W n = W,n P P n n,n Y Y n Y Y n Y Y n (20 = γ,n Y (2 µ where γ,n = F l, l,n Y n s he share of labor ype n producon a me (whch s dencal n he 8

9 naural and he scky wage allocaon Aggregae demand Havng esablshed he way n whch ndvdual ncome responds o aggregaes Y {Y }, G {G } and R {R }, we naurally defne he consumpon demand c,d of ndvdual a me as follows ({ } c,d (Y, G, R, θ = c γ,n Y (G, R, R; θ (22 µ I s he level of consumpon ha ndvdual chooses when macroeconomc aggregaes are Y, G and R, akng no accoun he effecs hese aggregaes have on hs ncome. We use hese funcons o defne aggregae demand as Y d (Y, G, R, θ = µ c,d (Y, G, R, θ + G (23 Goods marke clearng (4 mples ha n scky wage equlbrum Y d = Y. Hence solvng for equlbrum nvolves solvng for he fxed pon n whch aggregae oupu s equal o aggregae demand: Y = Y d (Y, G, R, θ Our man proposon below characerzes hs fxed pon. We need wo furher defnons MPC marx We frs defne he marx M wh elemens Noce from dfferenang (22 ha MPC,s γ,n s M,s I γsmpc,s (24 = = µ Q,s c,d ({Y }, {R }, θ Y s (25 Hence, M,s s also he dscouned response of aggregae consumpon o an ncrease n aggregae ncome Y s a dae s. Our assumpons on producon and ransfers guaranee ha he endogenous response of consumpon a me o such an ncrease happens only va he response of ndvdual ncome n ha perod. Moreover, he aggregae ncome sensvy of ndvdual s ncome n perod s s γ s, whch s hs ype s share n producon. he ncdence-weghed MPC marx M has he followng mporan propery. Lemma. he vecor s a lef egenvecor of M wh egenvalue, ha s M = (26 9

10 Proof. Dfferenang he budge consran (6 for each s, we see ha Applyng (25 and (27, we herefore fnd M,s = =0 ( I =0 = MPC,s = s, (27 =0 γs,n MPC,s = I γs,n = ( MPC,s = =0 I γs,n = = In words, Lemma comes from he fac ha our economy s closed, n he sense ha any addonal money earned by he agens n some perod s wll be spen Paral equlbrum effecs We nex defne he paral equlbrum response Y X o any shock X {θ, G 0 G, R 0 R } as dscouned value of he paral dervave of he aggregae demand funcon (23. Y X Y d Q dx (28 X he followng lemma s a fundamenal propery of paral equlbrum shocks. Lemma 2. Vecors of paral equlbrum responses have zero presen value, e Y X = 0 (29 for any shock X {θ, G 0 G, R 0 R }. he proof, wren separaely for each shock n appendx A., s a consequence of he agen s and he governmen s budge consrans holdng wh equaly n paral equlbrum. Snce hese are pure demand shocks ha leave oal earnngs unchanged, all he paral equlbrum responses have o ne ou o have a zero ne presen value. For example, a preference shock alers he paern of neremporal spendng of each agen bu, wh labor supply consraned o be consan, does no generae more ncome. Hence, he presen value of consumpon s consan, mplyng (29. Smlarly, he governmen s neremporal budge consran (5 mples ha an exra un of spendng mus be pad for by an equvalen ncrease n he presen value of axes. Snce each agen responds o a un-sze ncrease n he presen value of axes by reducng he presen value of her spendng by a un, he overall reducon n he presen value of aggregae consumpon mus exacly offse he ncrease n ha of governmen spendng, so ha (29 holds. Fnally, a moneary polcy shock can generae presenvalue redsrbuon beween varous ndvduals or he governmen dependng on her paerns 0

11 of neremporal rade n he naural allocaon, bu when summed up, hese ndvdual responses ne ou o a zero aggregae presen value effec. 2.4 he Ineremporal Keynesan Cross We are now ready for our man proposon of hs secon. Proposon. Consder any varable X {θ, G 0 G, R 0 R }. he change n oupu dy X ha resuls from a change dx s characerzed o frs order by dy X = Y X + MdY X (30 where M s defned n (24, Y X s defned n (28, and dy X n oupu a me relave o he naural allocaon. Q dy s he dscouned value of he change Proof. Sar by oally dfferenang he ndvdual consumpon demand funcon (22 aggregang, we herefore have dc,d = c X dx + c dy s s=0 Y s Bu from (24 and (25, dc = µ dc,d c = µ X dx + c µ Q dy s s=0 Y s Y Q dy = Q d (C + G = d Q X dx + M,s Q s dy s s=0 whch, by (28 and our defnon dy X Q dy, resuls n dy X = Y X + MdY as we se ou o prove. Proposon derves our neremporal analogue of he radonal Keynesan Cross, descrbng any general equlbrum response dy X as sum of he paral equlbrum response and he feedback of dy X hrough he MPC marx. As urns ou, equaon (30 descrbes he deermnaon of aggregaed demand n even more more general sengs han he one we have nroduced n hs secon. One could herefore hnk of as a fundamenal law ha underles many modern macroeconomc models. he nex secons explore he mplcaons of Proposon.

12 3 Solvng he Ineremporal Keynesan Cross We now nvesgae he neremporal Keynesan cross (30 n deal. Our man resul s a characerzaon of he soluon(s of (30. As wll urn ou, snce our MPC nework s closed ha s, here s no demand flowng n or ou of he sysem an equaon lke (30 adms nfnely many soluons. hs requres an equlbrum selecon rule. In analogy o polcy expermens n sandard New-Keynesan models, where moneary polcy mplemens a zero oupu gap afer eher fne me or asympocally, we shall selec he equlbrum ha ses he ermnal oupu gap o zero, dy X = 0.2 hroughou hs secon, we wll drop he superscrp X and nsead regard (30 ndependenly of s dervaon. hs requres us o be precse abou he objecs nvolved n. In parcular, we assume ha M s a column-sochasc marx n R (+ (+, =0 M,s =, and ha Y R + has zero ne presen value (NPV, ha s, =0 Y = 0, whch we also wre as Y, usng o denoe he orhogonal complemen. An equaon of he form (30 does no necessarly adm any soluon, even wh he assumpon n place so far. For example, suppose he MPC marx M s he deny marx. In ha case, no soluon dy exss whenever Y = 0. Clearly, such an MPC marx would no be a very convncng descrpon of an aggregae economy for would mply ha any addonal ncome dy earned n some perod s enrely spen n ha perod. In oher words, here s no money beng spen across perods. We now nroduce wo resrcons on he MPC marx M ha rule ou such src whn perod spendng, he frs slghly more general han he second. Assumpon. M s non-negave and rreducble: ha s, for each s, {0,..., }, M,s 0 and here exss an m N such ha (M m,s > 0. Accordng o hs assumpon, M needs o be such ha, for each wo perods s and, an ncrease n aggregae ncome n perod s wll be parally spen n perod afer m eraons. o gve an example of m >, ake a world n whch addonal aggregae ncome n perod + s spen n perod bu no perod. Ye, snce spendng n perod s equal o ncome n perod, s rue afer wo eraons ha some addonal ncome n perod + wll be spen n perod despe he lack of a drec lnk. If here s such a drec lnk, we refer o a marx as M as prmve, as specfed n he follwng, more demandng assumpon. Assumpon 2. M s prmve: ha s, s non-negave, and here exss an m N such ha (M m,s > 0 for any s, {0,..., }. Ulzng he lnk of our mehodology o nework heory and Markov chans, we noe ha Assumpons and 2 are also commonly used n he heory of Markov chans: Vewng M as he ranson marx of a Markov chan, Assumpon mples he exsence of a unque saonary 2 In fuure versons of hs paper, we nend o provde a formal argumen o esablsh hs connecon. 2

13 dsrbuon of ha Markov chan; and Assumpon 2 mples convergence o he saonary dsrbuon sarng from any oher nal dsrbuon. Boh of hese properes wll come n handy n our characerzaon of he soluon o he neremporal Keynesan cross (30. Even hough hese wo assumpons may appear resrcve a frs glance, boh are n fac sasfed under mnmal resrcons on he envronmen presened n Secon 2, as we demonsrae n he followng lemma. Lemma 3 (Properes of he MPC marx. Suppose ha for each par of perods s, {0,..., } here exss a an agen wh posve margnal uly of consumng n perod s, Uc s > 0, wh posve ncome sensvy n perod, γ > 0, and, wh no bndng borrowng or savngs consran beween perods s and. hen, he MPC marx M s prmve, and Assumpons and 2 are sasfed. Proof. he proof s mmedae: Under he saed assumpons, M,s > 0 for each par of perods s, {0,..., }. hus, M s prmve (wh m = n he Assumpon 2. Havng nroduced he resrcons M needs o sasfy, we nex descrbe he famly of soluons o (30, before he mposon of a moneary polcy rule. heorem (Solvng he Ineremporal Keynesan Cross. Le M sasfy Assumpon. hen, here exss a marx A R (+ (+ mappng he se of zero-npv vecors no self, wh he followng propery: For any soluon dy o he neremporal Keynesan cross (30 here exss a scalar λ R such ha dy = A Y + λv, (3 and for any λ R hs s a soluon. Here, v R + s he unque and posve rgh-egenvecor of M wh respec o egenvalue ha s normalzed o v =. Under Assumpon 2, A Y can be expressed by he nfne sum A Y = Y + M Y + M 2 Y + M 3 Y (32 Proof. Snce M s column-sochasc, has an egenvalue of and adms a rgh-egenvecor v R +. By he Perron-Frobenus heorem for non-negave, rreducble, (column-sochasc marces, hs egenvalue s unque and weakly exceeds he absolue value of every oher egenvalue. Moreover, v can be chosen o be posve n all enres. We normalze henceforh so ha v =. Rewre (30 as ( MdY = Y. Usng he fac ha he Kernel of M s sngle-dmensonal, he rank-nully heorem mples ha he mage of M mus be dmensonal. Bu, as every vecor n he mage of M s orhogonal o a vecor of ones, follows ha he mage of M s exacly he space of zero NPV vecors. Moreover, noe ha R + can be decomposed no Ke( M snce v = 0. ogeher, hs means ha M defnes a lnear bjecon (auomorphsm from no self. Le A be any marx ha s equal o he nverse of hs bjecon when resrced o. here exss a one-dmensonal famly of such marces; an obvous choce s A = M + v. hen, any 3

14 soluon o (30 s of he general form wh λ R. dy = A Y + λv If M s a prmve marx (Assumpon 2, we know ha any non-un egenvalue of M has an absolue value srcly below. Snce M maps no self, any vecor w can be decomposed no a lnear combnaon of egenvecors of M, w = = λ w, where Mw = µ w, w = 0, and µ s a (possbly complex egenvalue wh µ <. herefore, he nfne sum w + Mw + M 2 w +... has he fne (and real lm w lm = = µ λ w, whch clearly sasfes ( Mw lm = w. hs concludes our proof of heorem. Accordng o heorem, he general soluon (3 o he neremporal Keynesan cross consss of wo erms: A zero-npv erm A Y and an egenvecor erm λv. We descrbe each n urn. As equaon (32 llusraes n he specal case where M s prmve, he frs erm s analogous o he MPC sum when solvng he radonal Keynesan cross. he key dfference, however, s ha he elemens n our MPC sum are vecors: Y s he drec paral equlbrum mpac; M Y s he second order mpac, ncludng he fac ha one agen s spendng s anoher agen s ncome; M 2 Y s he hrd order mpac, and so on. All hese effecs are zero-npv as M preserves zero-npv vecors. As a sde noe, hs s precsely why he nfne sum (32 converges when M s prmve: In ha case, M n x approaches κv for any vecor x R + where he scale κ of v s jus he sum of he elemens of x, κ = x. In he case a hand, Y has a zero NPV, ha s, κ = 0, so M n Y 0. hs s a requremen for he nfne sum (32 o be well-defned, whch also urns ou o be suffcen. he fac ha he frs erm A Y s zero NPV rases an mporan queson: Does our neremporal Keynesan cross mply ha all paral equlbrum shocks Y, whch have o respec he budge consrans and are herefore zero NPV, can only have zero NPV general equlbrum effecs? Effecvely, hs would mean ha any posve demand shock oday s necessarly followed by a negave demand shock n he fuure. ha hs s no he case s due o he second erm λv n our expresson for dy, (3. hs second erm capures how he equlbrum selecon rule (or n oher words, he moneary polcy rule can crucally nfluence he dynamcs of consumpon. Our approach clarfes ha any such equlbrum selecon rule acs by shfng he level of dy up or down proporonal o v. hs vecor can be regarded as a measure of (egenvecor cenraly n he Markov chan descrbed by ranson marx M: For each perod gves a value v of ncome ha corresponds o he addonal ncome earned n perod f spendng n all oher perods ncreases accordng o v. In ha sense, v s self-susanng. As all of s elemens are posve, s naurally posve NPV and we normalze s NPV o, mplyng ha λ corresponds o he ne presen value of he oal 4

15 general equlbrum demand response ha was caused by he paral equlbrum shock Y. We summarze hs nsgh n he followng corollary. Corollary. he NPV of he general equlbrum aggregae demand response o paral equlbrum shock Y s gven by dy = λ, where λ s he unque scalar n he decomposon (3. Of course, when evaluang he general equlbrum ransmsson of a paral equlbrum shock, he equlbrum selecon rule needs o be aken no accoun. As we menoned above, we assume moneary polcy o mplemen a zero oupu gap n he ermnal perod. hs means λ adjuss endogenously o Y n order o ensure dy = 0, gvng he followng resul. Corollary 2. When ncludng he equlbrum selecon rule dy = 0, he soluon (3 becomes dy = ( ve A Y (33 v where e = (0,...0,. In parcular, he oal demand generaed by he paral equlbrum shock Y s gven by λ = (A Y v. he expresson n (33 combnes he zero NPV feedback nsde A Y wh he nonzero NPV shf comng n hrough he equlbrum selecon rule. One way o hnk abou he combnaon s ha A Y mples a ceran pah for consumpon, whch s hen lfed up or down along he vecor v, proporonal o he value of (A Y. 4 Examples o llusrae he power of he decomposon provded n he prevous secon, we now provde specfc examples of economes and paral equlbrum shocks Y. We sar wh a smple 2 perod model ha provdes an neremporal formalzaon of he radonal Keynesan cross. hen, we consder a more general economy n whch all agens are unconsraned, and lasly move o a sudy wha happens when hand-o-mouh agens ener he pcure. hroughou, we wll assume ha M sasfes Assumpon (. 4. Ineremporal and radonal Keynesan cross In hs subsecon, we consder a specal case of he analyss n Secon 3, where here are only wo perods, ha s, =. One may vew hs as he mos sraghforward mcrofoundaon of a radonal Old-Keynesan cross. We wll prove ha n hs case, any nonzero paral equlbrum equlbrum response wll be amplfed no a general equlbrum one of he same sgn, wh a 5

16 facor ha precsely corresponds o he one n he Old-Keynesan cross, /( MPC, wh MPC beng he MPC o spend ncome earned n he nal perod on he nal perod. Wh wo perods and due o s sochascy propery, he MPC marx M only has wo degrees of freedom and can be wren as M = [ MPC 0,0 MPC, MPC 0,0 MPC, where we assume MPC 0,0 and MPC, o le srcly nsde (0,, ensurng ha M s ndeed a prmve marx (see Assumpon 2. As one can easly verfy, he egenvecor v n hs case s [ ] MPC, v =. 2 MPC, MPC 0,0 MPC 0,0 Moreover, here s a second egenvecor w, whch wll help n dervng he marx A. I s gven by w = [ and has egenvalue η = MPC 00 + MPC <. Noce ha w s zero NPV and herefore any paral equlbrum response Y wll be proporonal o w. Fnally, we fnd A by compung he nfne sum (32 usng w as zero NPV vecor, Aw = ( + η + η w = η w = 2 MPC, MPC 0,0 w. We summarze hese seps n he followng proposon. Proposon 2 (Ineremporal and radonal Keynesan Cross.. Le = and assume M s of he form n (34 wh MPC 0,0, MPC, (0,. hen, our general decomposon n (3 akes he form [ dy = η Y + λ η ] ] MPC, MPC 0,0 where η MPC 00 + MPC <. When s mposed ha he ermnal oupu gap s zero, dy = 0, hs gves dy 0 = ] (34 MPC 0,0 Y 0. (35 Proposon 2 s he analogue of heorem and Corollary 2 for he wo perod case. A very smple resul emerges: he frs perod oupu response behaves precsely as predced by a radonal Old-Keynesan cross. One may wonder how hs s possble gven ha he Old-Keynesan cross s neher mcrofounded nor closed? he logc s slghly dfferen n he neremporal verson: Any paral equlbrum response Y by self s zero NPV. Ye f for example he second perod response Y s negave, hen he equlbrum selecon rule requres an nervenon (n 6

17 pracce hs mgh happen hrough a aylor rule, as menoned above, lfng boh perods aggregae demand upwards. hrough hs mechansm, he zero NPV paral equlbrum response Y s urned no a nonzero NPV general equlbrum response, wh an amplfed frs perod response dy 0. Remarkably, hs amplfcaon facor s he exac same as n he radonal Keynesan cross, /( MPC 0,0. We wsh o pon ou ha hs dscusson reaed he paral equlbrum response Y as exogenous. Ye, as we shall see n he followng subsecons, o evaluae he general equlbrum responses from specfc shocks or nervenons, such as a rse n governmen spendng, he specfc assumpons on he underylng economc envronmen (preferences, consrans, ec wll maer. No jus does he paral equlbrum response Y o a gven shock depend on he envronmen, bu he MPCs hemselves mgh also nfluence Y, possbly undong he amplfcaon we found n ( Unconsraned agens wh consan ncome sensves We sar wh an economy of N unconsraned agens whose ncome sensves γ are consan over me, ha s, for each agen, γ = γ for some γ. In ha case, all agens only care abou he presen value of her ncome, and herefore choose o spend ncome n he same way, no maer n wha perod was earned. ogeher wh he fac ha ncome sensves are equal across perods, he defnon of he MPC marx M n equaon (24 shows ha n hs case he columns of M are he same. hs urns ou o smplfy our decomposon a lo. Proposon 3 (Equal columns n M.. Suppose ha all columns n M are alke and posve, ha s, for all s, holds ha M,s = M for some M > 0. hen, A = and v = M n he decomposon n heorem (. In parcular, he general equlbrum response o a shock Y s gven by dy = Y Y M v. (36 Proof. Snce M has equal columns, maps zero NPV vecors o zero. herefore, A Y = Y for all zero NPV Y, usng (32. hs means A = s a feasble choce for he decomposon n (3. Moreover, s sraghforward o prove ha he column vecor (M self s a rgh-egenvecor of M f M has equal columns. he proposon esablshes a smple general equlbrum correcon n he case of unconsraned agens and consan ncome sensves. here are no hgher order effecs n A, because he frs order effec Y has no effec he presen value of ncome, whch here s he only hng ha maers for any hgher order effecs. he oal NPV of he general equlbrum demand response s gven by Y /M, so, for example, backloaded fnancng of a smulave polcy ends o ncrease he NPV of demand more han fronloaded fnancng. he NPV of dy s also larger f agens are more mpaen and end o spend her margnal ncome n earler perods, lowerng M. Fnally, any moneary polcy rule changes dy accordng o he spendng propenses n v = (M. 7

18 o brng hs decomposon o more lfe, we now consder wo specfc polcy examples: A governmen spendng shock and a moneary polcy shock. ax-fnanced governmen spendng shock. Suppose he governmen leves lump-sum axes o pay for an ncrease n governmen spendng by G > 0 n he frs perod. For smplcy, assume ha he ax pad by agen s proporonal o s ncome sensvy γ. Snce agens reduce her consumpon accordngly, he paral equlbrum shock s gven by Y 0 = G( M 0 > 0 n he frs perod, and by Y = GM hereafer. By desgn, Y = 0. Accordng o (36, he general equlbrum mpac s hen G for = 0 dy = Y + GM =. (37 0 for > 0 Moreover, he oal demand creaed s precsely equal o G. hs resul seems o sand n remarkable conras o our prevous wo-perod resul n (35. Afer all, our governmen spendng resul (37 also holds n wo perods, so why are hey dfferen? he answer o hs queson les n he way he paral equlbrum response Y s deermned. For our Old-Keynesan resul (35, we assumed a fxed paral equlbrum response Y. Ye, as we llusrae n hs smple example, Y afer a governmen spendng shock wll depend on MPCs self. In parcular, Y 0 = G( M 0 s smaller f he agens average MPC o spend n perod 0, M 0, s larger. Inuvely, hs s because (unconsraned hgh MPC agens end o reduce her consumpon more n response o fuure ax cus. hs effec precsely undoes he amplfcaon effec hghlghed n (35. 3 he effec of governmen spendng here s also remnscen of he analogous resul n Woodford (200, where he sudes governmen spendng n a New-Keynesan model whle keepng real raes consan. Our approach emphaszes ha hs response comes abou because he paral equlbrum response Y s canceled o zero n every perod afer he frs, by an endogenous response of moneary polcy n he fuure. As we wll show below, a key drvng force behnd he resul n (37 one ha s shared by he exbook New-Keynesan model s he assumpon of unconsraned agens. hs assumpon s also wha caused he mulpler n he radonal Keyensan cross (35 o be exacly canceled by he M 0 erm n he paral equlbrum response. Moneary polcy shock. o racably llusrae he PE and GE mechancs of a moneary polcy shock, we shall make furher resrcons on he agens uly funcons U. In parcular, we assume ha all preferences are equal U = U, and have a consan elascy of neremporal subsuon σ. Leng β be he share of ncome spen on perod, we hen have M = β = v. Le d log R be he exogenous neres rae change beween perods = 0 and =. Fnally, 3 We shall noe ha he governmen could alernavely also fnance s expendure by cung fuure governmen spendng, n whch case he paral equlbrum response s no mgaed by any MPC erms and he Old Keynesan cross formula (35 fully apples. 8

19 noe ha aggregae consumpon n perod s jus gven by Y. he shock hen nduces a paral equlbrum response equal o ( β 0 Y 0 σ Y = d log R for = 0. β Y 0 σ d log R for > 0 Agan, noe ha C = 0. he oal amoun of demand generaed s now gven by λ = Y σ d log R, and he general equlbrum demand response s gven by Y 0 σ dy = d log R for = 0, 0 for > 0 where we noe ha, agan, he moneary polcy rule a = mples ha he smple paral equlbrum response s offse by (λv = β Y 0 σ d log R n any perod > 0. Even f hs resul, accordng o whch only he nal perod s affeced, looks analogous o he prevous one, s less robus. I heavly depends on he fac all agens preferences are he same and homohec, whch s bascally equvalen o he assumpon of a represenave agen. An neresng nsgh ha can be gleaned from hs decomposon s ha he general equlbrum mpac dy 0 /Y 0 of he moneary polcy shock s ndependen of he agens mpaence β 0, even hough hgher mpaence reduces he paral equlbrum response Y 0 /Y 0. he reason s an offseng general equlbrum mulpler: Whenever β 0 s larger, agens are more lkely o spend any addonal demand creaed by he moneary polcy rule n perod = 0. Forward gudance. We can also urn o our framework o decompose he effec of forward gudance as well. o do hs, consder he envronmen we used for moneary polcy. In hs envronmen, an neres rae shock beween perods τ and τ + (our smplfed forward gudance polcy d log R τ+ causes he followng paral equlbrum response, Y Y = ( s>τ β s σ d log R τ+ for τ. ( s>τ β s σ d log R τ+ for > τ I s sraghforward o check ha hs response s ndeed zero NPV. Clearly, s percenage mpac on he nal perod falls n τ, and s herefore smaller, he laer he nervenon s announced. o calculae he general equlbrum mpac of forward gudance, recall ha he moneary polcy rule dcaes ha dy = 0 and herefore, he oal amoun of demand generae s λ = Y 0 s>τ β s β 0 σ d log R τ+. 9

20 he general equlbrum demand response s hen dy Y = σ d log R τ+ for τ. 0 for > τ hs s a remarkable resul. However far n he fuure we announce a moneary polcy nervenon of a gven sze, hs baselne model predcs he same nal response. hs s an ncarnaon of he forward gudance puzzle. 4 Our approach llusraes ha he share of he response beng due o paral equlbrum effecs s gven by s>τ β s and falls o zero as τ becomes larger. In ha sense, he forward gudance puzzle s enrely due o general equlbrum effecs. 4.3 Hand o mouh agens Consder any economy lke he one nroduced n Secon (??, wh MPC marx M and a paral equlbrum shock Y. We now explore wha happens o dy f hand o mouh agens are added o hs economy, so ha hey consue a fracon µ > 0 of he oal populaon and have a consan ncome sensvy γ h2m. We have he followng formal resul. Proposon 4 (Addng hand o mouh agens.. If a measure µ of hand o mouh agens wh ncome sensvy γ h2m s added o an economy wh populaon sze µ and MPC marx M, holds ha: M new = ( µγ h2m M + µγ h2m I, where I s he + dmensonal deny marx. Moreover, n he decomposon n heorem (, we now have A new = µγ h2m A and v new = v. he decomposon s herefore gven by dy new = µγ h2m A Y + λnew v = dy. µγh2m Proposon 4 shows ha addng hand o mouh agens scales up he general equlbrum response by a facor /( µγ h2m, where µγ h2m (0, s he share of ncome gong o hand o mouh agens n each perod. I s mporan o recognze ha hs resul holds condonal on he same paral equlbrum response Y. Of course, n general one would expec Y o change as well. o llusrae he role of hand o mouh agens more concreely, we assume for he remander of hs subsecon ha he economy s populaed by µ unconsraned agens wh consan ncome sensves as n Secon 4.2 and µ > 0 hand o mouh agens. We repea he wo expermens from before. Governmen spendng shock. Agan, suppose he governmen ncreases s spendng by G > 0 n he nal perod bu now leves lump-sum axes n fuure perods on boh unconsraned and hand o mouh agens. In parcular, suppose unconsraned agens pay a share of axes χ [0, ], 4 See also Del Negro Gannon Paerson (205, McKay Nakamura Sensson (206, Angeleos Lan (206, ce more? 20

21 agan n proporon o her respecve ncome sensves, whle hand o mouh agens pay share ( χϕ n perod, where = ϕ =. For smplcy, assume ha ϕ = 0. Here, he dsrbuon across perods s mporan as hand o mouh agens canno consoldae her budge consrans and herefore he specfc ncome sream maers, raher han jus s presen value. hen, he paral equlbrum spendng shock s gven by Y = G χgm ( χϕ G n he frs perod, and by Y = χgm ( χϕ G hereafer. Applyng Proposon 4, he general equlbrum response s dy = ( µγ h2m Y + Y v M and he oal NPV of demand creaed s λ = (G ( χgϕ µγ = h2m for = 0, ( χgϕ µγ h2m for > 0 χg. µγh2m hese expressons carry a few mporan nsghs. Frs, assume he ax share on hand o mouh agens s zero, χ =. In ha case, dy s only nonzero n he nal perod, bu now he nal mpac s scaled up by /( µγ h2m. Governmen spendng herefore has a mulpler n excess of, dfferen from he exbook New-Keynesan model. hs shows ha he presence of hand o mouh agens can brng back an Old-Keynesan feaure ha s ofen used as a crcsm of he New-Keynesan approach. Second, assume he ax share on hand o mouh s close o,.e. χ 0. Somehwa surprsngly, he oal NPV of demand creaed hen ends zero as well. o explan why noce ha he paral equlbrum response has Y = 0 n ha lm. herefore he moneary polcy rule enforces dy = 0 wh λ = 0. In oher words, such a polcy mples a emporary boom n he nal perod, followed by a bus. Moneary polcy shock. o sudy moneary polcy, we use he same seup ha we used o sudy moneary polcy n Secon 4.2. In parcular, we assume ha all unconsraned households share he same preferences U = U wh a consan elascy of neremporal subsuon σ and share β of ncome spen on consumpon n perod. Agan we le d log R be an exogenous neres change beween perods = and = 2. Moreover, le ψ h2m be he ncome share earned by a hand o mouh agen on average, ne of possble deb paymens. Noce ha µψ h2m s hen he average share of ncome gong o hand o mouh agens, whle µγ h2m s he margnal share of any addonal ncome gong o hand o mouh agens. Snce hand o mouh agens do no respond o moneary polcy, we are n fac n he exac suaon of Proposon 4. herefore, he oal nal response afer a moneary polcy shock s hen gven by dy = µψh2m µγ h2m Y σ d log R, 2

22 and zero n all oher perods. Here, we already expressed he ncome earned (and spen by unconsraned agens Y unc as share of he oal ncome, ( µψ h2m Y, afer neres paymens. hs leads us o he followng nsghs. Frs, conrary o her role as amplfer for governmen spendng, hand o mouh agens have o opposng effecs on he effecveness of moneary polcy: A hgher share of hand o mouh agens reduces s effecveness snce hose agens do no respond o moneary polcy, bu also amplfes any demand ncreases or decreases n he nal perod. A case where hey exacly balance s when ψ h2m = γ h2m, a case ha s sasfed f hand o mouh agens do no have any nal asses and where her average ncome s equal o her margnal ncome. 5 he wealher hand o mouh agens are, he larger s her ncome share (afer neres paymens ψ h2m, and herefore he weaker s he aggregae moneary polcy response. Second, our approach sheds lgh on he role of GE amplfcaon for dy. Even n he case where ψ h2m = γ h2m, a hgher share µ of hand o mouh agens wll lead o a smaller paral equlbrum response, Y scales wh µψ h2m, bu a larger general equlbrum response, scalng wh /( µγ h2m. hs decomposon can be used o nerpre he recen fndngs n Kaplan Moll Volane ( Infne-horzon sochasc model In hs secon, we generalze he model of he prevous secons o an nfne-horzon envronmen, allowng for unnsured dosyncrac rsk and nroducng nvesmen and oher elemens o make he model more quanavely realsc. We show ha he key seps of our analyss carry over o hs more general envronmen, as he neremporal Keynesan cross can be solved o oban a general equlbrum mulpler operaor ha carres paral equlbrum mpulses { Y } o general equlbrum oucomes {dy }. here are, however, a few key dfferences ha come wh he rcher model. Frs, here s no longer necessarly he same mulplcy of soluons, a leas f we requre hese soluons o be bounded: alhough he marx M sll has an egenvalue of one, hs ofen corresponds o an egenvecor ha grows explosvely as. We show ha hs s, n parcular, he case when he moneary polcy feedbacks embedded n M ake a form convenonally assocaed wh deermnacy, such as a aylor rule. Second, when a paral equlbrum shock drecly affecs he supply sde of he economy as wh, for nsance, he capal accumulaon response o a moneary shock s necessary o frame he analyss n erms of ne demand mpulses { Y }. Alhough many of he seps reman he same, he nerpreaon becomes more suble. Alhough our prmary analyss s of unancpaed shocks a dae 0, hs also characerzes o frs order he mpulse responses wh respec o aggregae shocks n a sochasc equlbrum. We observe ha he paral equlbrum oupu volaly n response o a shock s drecly ed 5 he logc here s relaed o he benchmark case n Wernng (206, where borrowng consrans relax wh oupu. 22

23 o he 2-norm Y of s paral equlbrum mpulse response, and he general equlbrum oupu volaly correspondngly depends on he 2-norm dy of he resulng general equlbrum sequence. he relaonshp beween Y and dy he exen o whch he propagaon mechansms n he economy amplfy he volaly from paral equlbrum shocks depends on he srucure of he general equlbrum mulpler, whch we explore quanavely. We show how hs and oher feaures of he general equlbrum mulpler can be decomposed beween he varous componens of he model, ncludng heerogenous agens and dfferen componens of he marx M. We parcularly emphasze he ransmsson of fscal and moneary polcy, and how he general equlbrum amplfcaon of boh becomes larger when dosyncrac rsk and consrans are severe. he paral equlbrum mpulse from moneary polcy, however, generally s aenuaed under he same crcumsances, mplyng ha he general equlbrum effec from fscal polcy grows n relave erms. (o be compleed. A Proofs A. Proof of lemma 2 In hs secon, we defne vecors of paral equlbrum responses Y X o dfferen shocks X, and prove ha hey each sasfy a fundamenal NPV-0 propery (29, hereby esablshng lemma 2. A.. Preference shocks Dfferenang he neremporal budge consran (9 wh respec o θ, we see ha for all, c ({ } y Q, {R } ; θ = 0 (38 =0 θ In he case of preference shocks, we have Y θ = Q c µ,d θ. Hence, by (38 and (27, we oban Y θ = =0 =0 c Q ( µ θ ( c = µ Q = 0 =0 θ A..2 Governmen spendng shocks Consder a change G s n governmen spendng a dae s. he paral equlbrum effec s defned as Y G s = Q ( µ c G s + {=s} 23

24 Now, for each ndvdual, we have, usng he defnon (22 ogeher wh (27, =0 c = Q G s u = =0 u=0 y u G s c,d Q u=0 Q u u G s Q c =0 Q u y u = u=0 Q u u G s =0 MPC,u = In oher words, he presen value reducon n consumpon of ndvdual resulng from an ncrease n governmen spendng a dae s s equal o he presen value of he ncrease n axes for ha ndvdual. Bu, dfferenang he governmen budge consran (5 u=0 Q u ( n =I µ u = Q s G s ogeher, hese relaonshps mply ha he dae-s value of he aggregae consumpon response o an ncrease n governmen spendng a dae s s one: =0 Q ( µ c,d = Q s G s u=0 Q u u G s and hence Y G s = Q s + Q s = Q s + Q s = 0 =0 A..3 Moneary polcy shocks Consder a change n he nomnal neres rae a dae s, nducng a change n he real neres rae R s by remark. he paral equlbrum effec s defned as Y G s = Q ( µ c R s From (22, he response of ndvdual a me has wo componens: s drec response o R s and hs ndrec response hrough he effec of he change n neres raes on ranfers. c,d R s = c R s u=0 c u y u R s Hence, usng he same seps as above, Q c,d R s c = Q R s u=0 Q u u R s Now, for he frs par, dfferenang he neremporal budge consran (9 wh respec o R s, we fnd 24

25 hs mples ha, aggregang across agens, Y G s = =0 c Q ( Q = =0 R s y c =0 R s ( Q =0 R s µ (y c u=0 Q u ( We follow he same seps as n secon A..2, dfferenang (5 o fnd Noce also ha u=0 Q u ( ( µ y = µ Combnng, we herefore oban Y G s = =0 µ u = R s W,n P n n,n Q u (G u u u=0 R s µ u R s = F l = Y Q (Y C G + =0 R s Q = (Y C G =0 R s = 0 where we used goods marke clearng a each dae (4 o oban he fnal equaon. 25