General Equilibrium Multipliers of Housing Wealth Effects and Fiscal Shocks

Transcription

1 General Equilibrium Mulipliers of Housing Wealh Effecs and Fiscal Shocks Online Supplemen for Housing Wealh Effecs: The Long View Adam Guren Boson Universiy Alisdair McKay Boson Universiy Jón Seinsson Columbia Universiy June 5, 2018 Emi Nakamura Columbia Universiy 1 Overview We show how he open-economy fiscal muliplier can be used o gauge he srengh of he local general equilibrium muliplier on housing wealh effecs on consumpion. We use an esimae of he fiscal muliplier o remove hese general equilibrium effecs from an esimae of he housing wealh effec ha comes from comparing consumpion and home price changes across ciies. The end resul is an esimae of he parial equilibrium housing wealh effec and marginal propensiy o consume ou of housing wealh ha corresponds o a change in home prices holding fixed wages oher non-housing prices. We presen his argumen in he conex of a fully-specified general equilibrium model. For simpliciy we work wih a model where each region has a paien household and an impaien household raher han a model wih rich microeconomic heerogeneiy. This difference is no imporan for our argumen. In Secion 3 of his documen, we show how o creae an aggregae consumpion funcion for each region. The same argumen ha we use here would apply o a model in he Bewley-Hugge-Aiyagari radiion. 1 Finally, he argumen we make here follows from he fac ha differen demand disurbances 1 See Secion 6 of Farhi and Werning 2017 for a similar aggregae consumpion funcion in a Bewley-Hugge- Aiyagari model. 1

2 can have he same general equilibrium effecs. This poin has been made recenly by Aucler e al and our analysis bears some resemblance o heirs. 2 Model Demographics There are wo regions, home and foreign. The populaion of he enire economy is normalized o one wih a share n in he home region. Wihin each region here are wo represenaive households, which we call paien and impaien. Le ωx be he share of households ha are x {paien, impaien} in each region. Preferences =0 β x uc x, L x, Q +1 x; Ω, where he argumens are consumpion, labor supply, unis of housing Q +1, and Ω is an aggregae housing demand shock. x indexes a household s paience. Commodiies and echnology There is a final good assembled ou of inermediae inpus ha is used locally for consumpion, residenial invesmen, and governmen purchases. The producion of he final good saisfies: Y H, = φ Y F, = φ 1 η 1 η H Z η H, 1 η H 1 η + φ Z F, η 1 η η 1 F Z η F, 1 η + φf η η 1 η η 1 η 1 Z η H,, where Z H, and Z F, are he home and foreign inpus o he home producion of final goods. These inpus are composie goods produced in he home and foreign regions and described shorly. Normalize so ha φ H + φ F = 1. The bundle shows home bias if φ H > n. The cos-minimizaion problem implies: η PH, Z H, = φ H Y H, Home demand for home composie good P H, η PF, Z F, = φ F Y H, Home demand for foreign composie good P H, η ZH, PH, = φ F Y F, Foreign demand for home composie good P F, η ZF, PF, = φ H Y F, Foreign demand for foreign composie good P F, 2

3 where P r, is he price of he final good in region r {H, F }. The wo composie goods are produced in amouns Y H, and Y F, and are hemselves aggregaes of inermediae inpus: 1 Y H, = 0 1 Y F, = 0 θ y H, z θ 1 θ 1 θ dz θ y F, z θ 1 θ 1 θ dz. The usual cos-minimizaion problem resuls in price indices 1 P H, = 0 1 P F, = P H, = P F, = 0 φ H P H, φ H P F, 1 p H, z 1 θ 1 θ dz 1 p F, z 1 θ 1 θ dz + φ F P 1 F, + φ F P H, 1 Each inermediae good is produced linearly ou of labor y H, z = L H, z. Housing supply The supply of housing saisfies: Q H, = 1 δq H, 1 + I α H H, M 1 α H H, Q F, = 1 δq F, 1 + I α F H, M 1 α F F,. where I H, is resources in goods devoed o residenial invesmen in he home region and M H, is unis of consrucion permis sold by he federal governmen. The region-specific parameer α allows he consrucion echnology o differ across regions. Markes The wo regions share he same money, which serves as he numeraire. Composie inermediae goods markes are compeiive and compleely inegraed across regions and each region faces he price P H, for he home good and P F, for he foreign good. Inermediae variey firms face Calvo price-seing fricions wih probabiliy of adjusing heir price of 1 χ. The labor markes are local o each region and compeiive wih nominal wages W H, and W F,T. Households rade a nominal bond a nominal ineres rae i. Unis of housing rade a price J r, in region r {H, F }. 3

4 Households face an LTV consrain: B +1 s J r,+1q i. We assume his consrain is consanly binding on impaien households and never binding on paien households. Inermediae goods firms produce profis, which are owned by he paien households in he region. We use D x o denoe he nominal profis received by household of ype x. Governmen There is a federal governmen ha purchases goods in he wo regions, sells consrucion permis in he wo regions, and ses a common moneary policy. Le G H, and G F, be per capia spending in he wo regions. These are exogenous and financed by naional lump-sum axes. There is a moneary policy rule ha ses he nominal ineres rae: 1 + i = 1 βpaien + ϕ π π + ϕ y ȳ, where π = nπ H, + 1 nπ F, is he weighed average of he wo inflaion raes in he regions and ȳ is he weighed average of he log deviaions of oupu from heir seady sae values. The governmen sells consrucion permis according o he rules: γh M H, = M JH, H P H, M F, = M γf JF, F. P F, The region-specific parameer γ reflecs he fac ha regions differ in he elasiciy of supply of vacan land. Decision problems A household solves we omi he region subscrips because all prices in he household s problem are local: β x uc x, L x, Q +1 x; Ω, =0 4

5 subjec o: P C x + J Q +1 x + B +1 x = W L x T i 1 B x + J Q x + D x The Lagrangian is: B +1 x s J +1Q +1 x 1 + i. L = =0 β x {uc x, L x, Q +1 x; Ω λ x P C x + J Q +1 x + B +1 x W L x + T 1 + i 1 B x J Q x D x + ζ x B +1 x + J } +1Q +1 x P 1 + i The firs-order condiions can be rearranged o: u C = P λ u L = W P u C J+1 u Q = J u C β P 1 + i u C = β u C + ζ, π +1 P +1 u C J+1 P ζ +1 π i where π +1 P +1 /P. For paien households we have ζ = 0. For impaien households we have ζ > 0 and he LTV consrain holds wih equaliy. The real esae developer solves: max J I α M 1 α P I, I wih firs order condiion: αj I α 1 M 1 α = P. The resources invesed in housing are hen: I = αj M 1 α P α, 1 and he consrucion of new houses is: I α M 1 α = α J α 1 α M. P 5

6 Subsiue in he rule for sales of consrucion permis: I α M 1 α = α J α 1 α P γ J M. P I follows ha he elasiciy of new houses wih respec o he price of housing is α/1 α + γ. This differs across regions because α and γ differ across regions. Finally, he resources invesed in housing can be obained by subsiuing he rule for permis ino 1 o obain: γ+ 1 I = α 1 J 1 α 1 α M. 2 P Noice ha he housing supply elasiciy is α/1 α + γ, bu he elasiciy of residenial invesmen is 1/1 α+γ. Ciies wih more elasic housing supply large α will have residenial invesmen respond more srongly o a given change in he price of housing. However, i is no clear which ciies have more volaile residenial invesmen because house prices will rise more in low elasiciy ciies. The inermediae goods producer solves: where: max P 0 χ λ P0 y H, z W H, L H, z, =0 P θ y H, z = Y 0 H, P H, L H, z = y H, z. Subsiuing he consrains ino he objecive yields: wih firs-order condiions: =0 max P 0 χ λ Y H, PH, θ P0 W H, P0 θ =0 χ λ Y H, P θ H, θ 1 P θ 0 = P 0 = =0 θ θ 1 χ λ Y H, P θ H,θW H, P θ 1 0 =0 χ λ Y H, PH, θ W H, =0 χ λ Y H, PH, θ. 6

7 Preference specificaion The preferences of he home households ake he form: u = 1 κ 1 σ C ψ L1+ν Q Ω 1 κ. 1 σ 1 + ν We hen have: κ σ κ 1 u C = C ψ L1+ν Q Ω 1 κ κ C ψ L1+ν Q Ω 1 κ 1 + ν 1 + ν κ σ κ 1 u L = C ψ L1+ν Q Ω 1 κ κ C ψ L1+ν Q Ω 1 κ ψl ν. 1 + ν 1 + ν Using hese derivaives, he labor supply curve is: ψl H, x ν = W H, P H,. 3 Equilibrium definiion Le lower-case denoe real variables i.e. normalized by P. For r {H, F } and x {paien, impaien}, he equilibrium variables are j r,, w r,, π r,, Y r,, Y r,, b r, x, i, Q r, x, Q r,, C r, x, L r, x, M r,, G r,,, Z r,, Z r,, d r, paien, I r,, E P H, /P F,, where Y r, is foreign demand for he composie good produced in r. The equilibrium condiions are: Q r, = x ωxq r, x r 4 1 ph, z θ Y r, dz = ωxl r, x r 5 0 P x ψl r, x ν = w r, r, x 6 Y r, = x ωxc r, x + G r, + I r, r 7 Y H, = nz H, + 1 nz H, 8 Y F, = nzf, + 1 nzf, 9 Z H, = φ H Y H, φ H + φ F E η 10 Z F, = φ F Y H, φ H E η + φ F 11 ZH, = φ F Y F, φ H E ZF, = φ H Y F, φ H + φ F E + φ F η 12 η. 13 7

8 Addiionally here are wo producion funcions for he final goods, wo Phillips curves, four budge consrains for he paien and impaien household in each region, he moneary policy rule, four housing firs-order condiions, wo Euler equaions for he paien agens, wo LTV consrains for he impaien agens, wo exogenous G sequences, wo housing permi supply rules, wo dynamic equaions for Q, one governmen budge, he definiions of dividend for each region, he firs-order condiion of he real esae developer in each region, and he evoluion of he real exchange rae E given by: P H, = P F, = π H, = π F, = π H, π F, = φ H + φ F E φ H + φ F E φ H + φ F E φ H + φ F E 1 φ H + φ F E φ H + φ F E 1 1 P H, 1 P F, 1 P H, 1 P H, φ H + φ F E φ H + φ F E P F, P F, 1 φ H + φ F E 1 φ H + φ F E 1 1 E E 1. 3 Preliminaries Le s consider perfec foresigh ransiions lasing T daes in a linearized version of he model. For simpliciy we will assume houses are creaed ou of permis alone α 0 and complee home bias φ F 0, which implies Y r, = Y r,. 3.1 A dynamic consumpion funcion The firs sep is o esablish ha consumpion and housing demand a dae along he ransiion can be wrien as a funcion of curren and fuure prices and preference shocks and an endogenous sae ha summarizes he asse holdings of he paien and impaien households. Le A 1+i 1 B +J Q be he wealh of an individual a dae. For an individual paien or impaien household, consumpion a dae is he soluion o: V A = max uc, L, Q +1 ; Ω + βv i B +1 + J +1 Q +1, C,Q +1,L,B +1 8

9 subjec o: P C + J Q +1 + B +1 = W L T + A + D B +1 s J +1Q i. V T +1 is he seady sae value funcion associaed wih seady sae prices and preference shifer Ω, which are fixed. C T hen depends on A T and he prices a T. Recursing backwards, C depends on A and he prices a fuure daes. Similarly for he choices Q +1 and B +1. For L we already know ha labor supply only depends on he curren real wage due o GHH preferences. Given he iniial, seady sae value A 0, we have already esablished ha A 1 depends on he whole pah {P, J, W, i, D, Ω } T =1. We hen recurse forwards and say A s+1 depends on curren and fuure prices {P, J, W, i, D, Ω } T =s and A s, which iself is a funcion of {P, J, W, i, D, Ω } T =1 so we can wrie A s+1 as a funcion of he whole pah of prices. Finally, subsiue hese soluions for asse holdings ino he soluion for C o wrie C as funcion of he full pah of prices. The same argumen applies o Q +1 and B Price funcions Nex we esablish ha here are funcions ha map pas, curren and fuure demand in region r, {Y r,s } T s=1 o he curren inflaion rae, π r,, price level, P r,, wage, W r,, and dividend, D r,. We use he linearized aggregae producion funcion: Y r, = x ωxl r, x. From he labor supply curve 3 we see ha paien and impaien ypes work he same hours so he producion funcion can be wrien as: Y r, = L r,, for he common labor supply L r,. Using he labor supply curve 3 his becomes: Y r, = Wr, ψp r, 1/ν, 14 and we can use his o solve for he real and nominal wages as a funcion of curren demand and he curren price level. The definiion of he dividend gives D r, in erms of Y r,, W r,, and 9

10 L r,. For he inflaion rae and price level we work wih he log-linearized Phillips curve, π = βπ βχ 1 χ χ log Wr, P r, θ log.. 1 θ Solving his forward and using 14 gives π = T s= s 1 βχ 1 χ β log ψyr, ν θ log. χ 1 θ We can hen accumulae he inflaion rae o find he pah of he price level P = s=1 π P Summary The key poins from his secion are i for any and region r, we can wrie he equilibrium soluions for C r, and Q r,+1 as funcions of {P r,, J r,, W r,, i, D r,, Ω } T =1 and ii we can wrie he equilibrium soluions for W r,, P r,, D r, as funcions of {Y r, } T =1. 4 Our argumen We consider a perfec foresigh ransiion lasing T daes. The previous secion esablishes ha here are funcions ha map he pahs for {P r,, J r,, W r,, i, D r,, T, Ω } T =1 o C,r and Q,r for each r and. A firs-order approximaion of hese funcions gives: Ĉ r = C J Ĵ r + C P ˆPr + C i î + C W Ŵ r + C D ˆDr + C T ˆT + CΩ ˆΩ ˆQ r = Q J Ĵ r + Q P ˆPr + Q i î + Q W Ŵ r + Q D ˆDr + Q T ˆT + QΩ ˆΩ, where Ĉ is a T vecor of consumpion deviaions from seady sae for he T daes, C J is a T T marix where he i, j elemen gives he effec of Ĵ +j on Ĉ+i, and so on. We assume ha he model is linearized around a symmeric seady sae so he C marices are he same for he wo regions. While he wo ciies differ in heir housing supply elasiciies, he consequences of hese differences affec consumpion hrough he pahs of prices. Similarly, governmen purchases affec consumpion hrough he pahs of prices and axes. Nex, he previous secion esablished ha here are funcions of {Y r, } T =1 ha give P r,, W r,, 10

11 and D r, a each dae. The linearized version of hese funcions gives: ˆP r = P Y Ŷ r Ŵ r = W Y Ŷ r ˆD r = D Y Ŷ r, where P, W, and D, are T T marices. The aggregae resource consrain is: Ŷ r = Ĉr + Ĝr. Combining all hese equaions we have: Ĉ r = C J Ĵ r + C P P Y + C W W Y + C D D Y Ĉr Ĝr + + C }{{} i î + C T ˆT + CΩ ˆΩ 15 m I m Ĉr = C J Ĵ r + mĝr + C i î + C T ˆT + CΩ ˆΩ Ĉ r = I m 1 }{{} M C J Ĵ r + mĝr + C i î + C T ˆT + CΩ ˆΩ. 16 Finally, consider he effec on oupu: Ŷ r = Ĉr + Ĝr = M C J Ĵ r + mĝr + C i î + C T ˆT + CΩ ˆΩ + Ĝr = M C J Ĵ r + C i î + C T ˆT + CΩ ˆΩ + Mm + I Ĝr = M C J Ĵ r + Ĝr + C i î + C T ˆT + CΩ ˆΩ, 17 where he las equaliy follows from he fac ha Mm+I = I +m+m 2 + = I m 1 = M. This makes clear ha he consumpion-income muliplier ha applies o he parial equilibrium effecs of he general equilibrium home price change is he same as he muliplier ha applies o he governmen purchases shock. 5 Empirical specificaion To relae 16 o our empirical specificaion i is convenien o simplify by assuming all he marices in 16 are diagonal wih a consan elemen on he diagonal. One inerpreaion 11

12 of his assumpion is ha he economy is hi by a permanen shock o ˆΩ and undergoes an immediae ransiion o a new seady sae so ha a any dae along he ransiion he curren value of, say, W is sufficien for he W τ for all τ 1, T. I hen follows ha a version of 16 holds period by period along he ransiion he only difference being ha he marices are now inerpreed as scalars. Anoher way of puing i is ha T = 1 so he marices are in fac scalars. Differencing 16 hen yields: Ĉr, = MC }{{ J Ĵr, + Mm Ĝr, + MC } i î + MC T }{{} ˆT + MC Ω ˆΩ. }{{} housing wealh effec ciy-specific shock ime fixed effec 6 Relaionship beween M and measured fiscal muliplier M is likely smaller han wha is measured as he open-economy fiscal muliplier. Par of he impac of he purchases shock on consumpion comes hrough is effec on local house prices. This asse-price channel is omied from M. We now explain his issue in deail alhough quaniaively i urns ou o be a small difference because he effec of purchases on consumpion ha comes hrough home prices is small given our esimaes of he housing wealh effec. Wha we need o add o he preceding analysis is ha Ĵr is an equilibrium oucome. To a firs-order approximaion, marke clearing for housing requires: Q J Ĵ r + Q P ˆPr + Q i î + Q W Ŵ r + Q D ˆDr + Q T ˆT + QΩ ˆΩ = Q S r,j Ĵ r + Q S r,p ˆP r, where Q S r,j and QS r,p deermine he linearized supply curve. The argumen ha he supply curve can be wrien his way is as follows: a each dae he governmen sells an amoun of permis given by he real price of housing so he curren supply of housing is discouned sum of he curren and pas real housing prices. Rearranging and subsiuing we have: QJ Q S r,j Ĵr + Q P Q S r,p PY Ŷ r + Q W W Y Ŷ r + Q D D Y Ŷ r + Q i î + Q T ˆT + QΩ ˆΩ = 0 Q S r,j Q J Ĵr = Q P P Y Q S r,p P Y + Q W W Y + Q D D Y Ŷr + Q i î + Q T Ĝ r + Q Ω ˆΩ Ĵ r = J r,y Ŷ r + Q S 1 r,j Q J Qi î + }{{} J r,i Q S 1 r,j Q J QT ˆT + Q S 1 r,j Q J QΩ }{{} J r,t ˆΩ, 18 }{{} J r,ω 12

13 where: J r,y Q S r,j Q J 1 QP P Y Q S r,p P Y + Q W W Y + Q D D Y. Now we subsiue his expression for Ĵr ino 17 o obain: Ŷ r = M C J J r,y Ŷ r + J r,i î + J r,t ˆT + Jr,Ω ˆΩ + Ĝr + C i î + C T ˆT + CΩ ˆΩ = I MC J J r,y 1 M Ĝr + C i + C J J r,i î + C T + C J J r,t ˆT + C Ω + C J J r,ω ˆΩ There are several hings o noe here. i The fiscal muliplier is I MC J J r,y 1 M, which exceeds M if consumpion is increasing in home prices so C J > 0 and home prices increase in income so J r,y > 0. ii The fiscal muliplier differs across regions because he asse-price muliplier differs across regions. ii The measured fiscal muliplier does no necessarily fully conrol for moneary policy and naional axes because he regions are differenially exposed o hese aggregae variables hrough he differenial responses of home prices across regions. The fiscal muliplier exceeds he muliplier of he housing wealh effec by a facor I MC J J r,y 1. Given our esimae of he housing wealh effec, which implies a value of C J, his scaling facor is close o one. To show his we proceed from he following facs: Our esimaed housing wealh effec is J C MC J Consumpion is abou 2/3 of GDP according o NIPA daa. The open economy fiscal muliplier is abou 1.5 according o Nakamura and Seinsson Local general equilibrium effecs will be sronger if more of he income gains are spen locally. Nakamura and Seinsson s esimae is a he sae level and our analysis is a he CBSA level so he relevan local governmen spending muliplier for our analysis is likely somewha smaller han 1.5. A final crucial ingredien o he calculaion is he income elasiciy of house prices is Y J J Y x. We leave his as a variable for he ime being in order o consider a range of values. The measured fiscal muliplier is I MC J J Y 1 M = 1.5, where we have omied he erm MI P P Y, which capures he response of he price level o income, because will rea J as he relaive price of housing. Sep 1. have JQ C Le Q be he quaniy unis of housing such ha he value of housing is JQ. We hen = JQ Y Y C = 3 JQ 2 Y. 13

14 Sep 2. The housing wealh effec hen implies: J C MC J = JQ C MC J = 0.071Q MC J = 0.071Q C Y Y JQ MC J = 0.071Q 2 Y 3 JQ. Sep 3. Similarly, he income elasiciy of house prices implies Y JQ J Y = xq 1 so J Y = xq 1 JQ Y. Sep 5. Now pu he pieces ogeher: I MC J J Y 1 M = 1.5 I 0.071Q 2 Y JQ 1 xq 1 M = JQ Y I x M = 1.5 M = x. If house prices do no respond o income x = 0 hen we have M = 1.5, which is he same as he fiscal muliplier. Lamon and Sein 1999 provide esimaes of he shor-run income elasiciy of house prices, which imply x < 0.8 and more likely near 0.3. For x = 0.3 we have M = 1.48 and for x = 0.8 we have M = Incorporaing residenial invesmen So far we have assumed ha houses are consruced wihou using any real resources. In he more general case wih α > 0, residenial invesmen consumes resources and his has wo consequences for our analysis. Firs, par of he housing wealh effec ha we measure comes no from he response of consumpion o home prices direcly, bu indirecly from he income gains associaed wih housing consrucion. Second, par of he measured fiscal muliplier reflecs he consumpion response o he consrucion response o he purchases shock. The laer consideraion is relaed o he asse-price channel discussed in he previous secion in ha i concerns how we relae he measured fiscal muliplier o he muliplier for he housing 14

15 wealh effec. Saring from 15 we now have: Ĉ r = C J Ĵ r + m Ĉr + Ĝr + Îr + C i î + C T ˆT + CΩ ˆΩ, where we have used he resource consrain Ŷr = Ĉr + Ĝr + Îr. Solving for Ĉr, his becomes: Ĉ r = M C J Ĵ r + m Ĝr + Îr + C i î + C T ˆT + CΩ ˆΩ. From 2, we can see ha we can wrie: Î r = I r,j Ĵ r + I r,p P Y Ŷ r, 19 and we subsiue his in for Îr o arrive a: Ĉ r = M C J + mi r,j Ĵr + mĝr + mi r,p P Y Ŷ r + C i î + C T ˆT + CΩ ˆΩ shows he firs issue: par of he consumpion response o home price changes comes from mi r,j, which is he consumpion response o he income change generaed by he change in residenial invesmen. Now plug 20 in for Ĉr in he aggregae resource consrain: Ŷ r = M C J + mi r,j Ĵr + mĝr + mi r,p P Y Ŷ r + C i î + C T ˆT + CΩ ˆΩ, +Ĝr + I r,j Ĵ r + I r,p P Y Ŷ r where we have also used 19. Now collec erms involving Ĵr and subsiue wih 18: Ŷ r = M mĝr + mi r,p P Y Ŷ r + C i î + C T ˆT + CΩ ˆΩ + Ĝr + I r,p P Y Ŷ r + M C J + mi r,j + I r,j J r,y Ŷ r + J r,i î + J r,t ˆT + Jr,Ω ˆΩ. Ŷ r = I Mm I r,p P Y + I r,j J r,y I r,p P Y I r,j J r,y MC J J r,y 1 { } MĜr + M C i î + C T ˆT + CΩ ˆΩ + M C J + mi r,j + I r,j J r,i î + J r,t ˆT + Jr,Ω ˆΩ Ŷ r = I MI r,p P Y MI r,j J r,y MC J J r,y 1 { } MĜr + M C i î + C T ˆT + CΩ ˆΩ + M C J + mi r,j + I r,j J r,i î + J r,t ˆT + Jr,Ω ˆΩ 15

16 This expression shows ha second issue: he measured fiscal muliplier is larger han M no only because of he erm MC J J r,y, which is he asse-price channel discussed in he previous secion, bu also because of he erm MI r,p P Y + MI r,j J r,y, which capures he residenial invesmen response o he change in real home prices induced by he fiscal shock. In his case, our esimaed housing wealh effec as an elasiciy is J C M C J + mi J. We would like o remove he general equilibrium muliplier and he omied variable componen coming from consrucion income o recover J C C J, which is he parial equilibrium housing wealh effec wrien as an elasiciy. We proceed from he following addiional facs in addiion o hose used in he previous secion: In an pooled IV regression for , we esimae he response of consrucion employmen o home prices o be This esimae uses he same specificaion as he pooled specificaion in Column 1 of Table 1 of Guren e al wih consrucion and real esae employmen as he oucome variable. Residenial invesmen is abou 5 percen of GDP according o NIPA daa. The measured fiscal muliplier is now I MI J J Y MC J J Y 1 M 1.5, where we have again omied he erm MI P P Y, which capures he response of he price level o income, because will rea J as he relaive price of housing. Sep 1. have JQ C Le Q be he quaniy unis of housing such ha he value of housing is JQ. We hen = JQ Y Y C = 3 JQ 2 Y. Sep 2. We assume he elasiciy of residenial invesmen o home prices is he same as he elasiciy of consrucion employmen so we have JQ I I J = 0.343Q, which implies: JQ Y Y I I J = 0.343Q I J = Q Y JQ. 16

17 Sep 3. The housing wealh effec hen implies: J C M C J + mi r,j = JQ C M C J + mi r,j = 0.071Q M C J + mi r,j = 0.071Q 2 Y 3 JQ MC J = 0.071Q 2 Y 3 JQ MmI J MC J = 0.071Q 2 Y Y Mm Q 3 JQ JQ. Sep 4. Similarly, he income elasiciy of house prices is defined as Y JQ J Y = x, which implies Y JQ J Y = xq 1 so J Y = xq 1 JQ Y. Sep 5. Now pu he pieces ogeher o find M: I MI J J Y MC J J Y 1 M = 1.5 I MI J + MC J J Y 1 M = 1.5 I M Q Y JQ Q2 Y Y 1 JQ 1 Mm Q xq M = JQ JQ Y I M x Mm M = 1.5. M and m are relaed by M = 1 m 1 so M I = Mm and we have: I M x M I M = 1.5 I x M = 1.5 Solving his for M yields M = x. For x = 0.3 we have M =

18 Sep 6. Now remove he omied variable effec. M = 1.48 implies m = The esimaed housing wealh effec hen implies J C MC J + J C J C MC J + J C MmI J = Y Mm Q JQ = J C MC J + Mm Y C = J C C J = To summarize, he model presened in his appendix implies ha he local general equilibrium muliplier for he housing wealh effec is close o he measured fiscal muliplier. I is slighly lower han he fiscal muliplier because our housing wealh effec akes home prices as given while par of he measured fiscal muliplier operaes hrough changes in home prices. For reasonable values of he income elasiciy of home prices, he wo are very close in magniude. The model presened in his appendix also implies ha par of he measured housing wealh effec comes from income changes relaed o consrucion. Our pooled IV regression for finds a response of consrucion employmen o home prices o be 0.343, and using his we can remove he consrucion effecs from he housing wealh effec. Making boh of hese adjusmens yields a housing wealh effec elasiciy of

19 References Aucler, Adrien; Rognlie, Mahew; and Sraub, Ludwig. The Ineremporal Keynesian Cross. January 2017 working paper. Farhi, Emmanual and Werning, Ivan. Moneary Policy, Bounded Raionaliy, and Incomplee Markes, Sepember 2017 working paper. Guren, Adam; McKay, Alisdair; Nakamura, Emi; and Seinsson, Jón. Housing Wealh Effecs: The Long View. March 2018 working paper. Lamon, Owen, and Jeremy C. Sein. Leverage and house-price dynamics in US ciies. RAND Journal of Economics : Nakamura, Emi and Seinsson, Jón. Fiscal Simulus in a Moneary Union: Evidence from U.S. Regions. American Economic Review, 1043, , March