2 Aggregate demand in partial equilibrium static framework

Transcription

1 Unversy of Mnnesoa 8107 Macroeconomc Theory, Sprng 2012, Mn 1 Fabrzo Perr Lecure 1. Aggregaon 1 Inroducon Probably so far n he macro sequence you have deal drecly wh represenave consumers and represenave frms, meanng ha you have solved some maxmzaon problem of a fcous sngle agen or sngle frm, assumng ha hs agen or frm was represenave of he whole economy. Wha does represenave exacly mean? Obvously does no mean ha knowng some varables for he average agen (for example s consumpon, s wealh or s nvesmen) ells you everyhng abou all agens n he economy. Knowng, for example, ha average household wealh n 2006 n he US s $ does no ell you wha s he fracon of US households n 2006 wh 0 wealh. Represenave here refers o he fac ha a gven se of aggregae varables (aggregae saes) s enough o predc anoher se of aggregae varables of neres (for example prces) and o predc he evoluon of aggregae saes n he fuure. To connue wh he prevous example f knowng ha average household wealh n 2006 n he US s $ s enough o know average asse prces n 2006 and o predc average household wealh n 2007, hen, for some quesons, we do no need o worry abou keepng rack of he enre wealh dsrbuon. In hs lecure we wll dscuss n deal when and why s enough o keep rack only of averages, boh n sac and dynamc frameworks. In some sense aggregaon s he foundaon of all neo-classcal macro. 2 Aggregae demand n paral equlbrum sac framework We sar wh an mporan resul ha s dscussed n Mas-Colell e al. secon 4.B whch gves condons under whch we can gnore he dsrbuon of wealh across consumers n order o know aggregae demand. In hs smple example we ll ake prces as gven. Consder a sac economy wh I consumers and L goods and assume ha each consumer has wealh w and sandard preferences (poenally dfferen for every consumer) over consumpon bundles. Solvng a sandard maxmzaon problem yelds (vecor valued) demand funcons for each consumer x (p, w ) and aggregae demand x(p, w 1,..w I ) = x (p, w ) (1) where p s he vecor of prces of each good. Equaon (1) ells us ha n order o know aggregae demand (and hus, for example n a general equlbrum conex, prces) one needs o know he enre dsrbuon of wealh. We now ask when and f we can wre aggregae demand as X(p, w ) = x (p, w ), for every p, (w 1,.. w I ) (2) 1

2 In order for (2) o hold mus be ha redsrbung 1 dollar from consumer o consumer j does no change aggregae demand or more generally for every change n he wealh dsrbuon dw sasfyng dw = 0, for every p and for every (w 1,.. w I ) bu hs can be rue f and only f x l (p, w ) w dw = 0 for every l x l (p, w ) w = x lj(p, w j ) w j, for every l, p, (, j), (w 1,.. w I ) (3) whch means ha any wo ndvduals who face he same ncremen n wealh change he demand of any gven good n exacly he same (across ndvduals) fashon. In words hs means ha f I gve an exra dollar o each consumer, each consumer wll exacly spend he same addonal amoun on, say, cherres. Graphcally means ha wealh expanson pahs for every consumer have o be lnear n each good wh he same (across consumers) slope (see Mas-Colell fgure 4.B.1). I s easy o see ha, f hs s he case, n order o compue aggregae demand s suffcen o know oal wealh and s no necessary o know how wealh s dsrbued. Proposon 4.B. 1 n Mas-Colell e al. gves necessary and suffcen condon for equaon 3 o hold and obvously he condons have o do wh preferences. In parcular mus be ha he ndrec uly funcon of each agen V (p, w ) mus have he followng form (also called he Gorman form) V (p, w ) = a (p) + b(p)w How resrcve s hs condon? I s obvously prey resrcve and s easy o hnk of many dfferen preferences n whch hs condon s no sasfed. Bu here are wo neresng cases n whch s sasfed. One s he case n whch all consumers have dencal and homohehc preferences, where dencal guaranees ha b(p) does no depend on and homohec guaranees ha b(p) does no depend on w,.e. consumers wh dfferen wealh level have he same margnal uly from wealh. Anoher case s when preferences of all agens are quaslnear n he same good. I s sor of nuve o see why hs s he case: he Gorman form requres ha he margnal uly of wealh s ndependen of wealh.e. V (p,w ) w = b(p). A smple applcaon of he envelope heorem ells you ha V (p, w ) w where λ s he Lagrange mulpler on wealh. Takng each agen frs order condons for he good wh quas-lnear preferences yelds ha λ s equal o he (consan) margnal uly of ha good and so ndependen from wealh of ha consumer: hence he resul follows. In he nex secon we wll dscuss under whch condons he represenave agen resul apples for dynamc economes, n general equlbrum. = λ 2

3 3 The represenave agen resul n dynamc general equlbrum economy The objecve of hs secon s o provde an example n whch aggregae varables (n hs case prces) a each pon n me do no depend on he dsrbuon of resources among agens of he economy a he momen n me, bu only on average resources avalable a ha pon n me. So n oher words ells us under whch condons one can explan aggregae prces (such us, for example, neres raes or sock prces) usng only aggregae varables, such as aggregae consumpon or GDP. Noe ha hs s boh a dynamc and general equlbrum economy, as opposed o a sac, paral equlbrum case consdered n he prevous secon. Consder an economy populaed by N ndvduals, ndexed by I = {1, 2,... N}. Each ndvdual s nfnely lved. In each perod here s one nonsorable consumpon good. Each ndvdual household has a sochasc endowmen process {y} of hs consumpon good. Le s and s = (s 0,... s ) he even and even hsory of hs economy, respecvely. and π (s ) denoe he objecve probables of even hsores. Agens subjecve probably belefs are assumed o concde wh hese objecve probables. Noe ha here s no need of addonal assumpons abou he sochasc process; n parcular he ndvdual processes need no be Markov processes nor o be ndependen across agens. Noce hough ha s s a prey large se as encompasses all hsores of ncome of all agens n he economy. Assume ha y Y, a fne-dmensonal se of cardnaly M and defne he aggregae sae smply as s = (y 1,..., y N ). Also ake he even s 0 as gven. A consumpon allocaon {(c (s )) I } T =0,s S maps aggregae even hsores s no consumpon of agens I a me. Preferences are dencal across agens, addvely me-separable and ha agens dscoun he fuure a common subjecve me dscoun facor β (0, 1), so ha he uly funcon akes he form u (c ) = β π (s )U (c (s )) (4) =0 s S Defnon 1 A consumpon allocaon {(c (s )) I } T =0,s S s feasble f c (s ) 0 for all,, s (5) N N c (s ) = y(s ) for all, s (6) =1 =1 A consumpon allocaon s Pareo effcen f s feasble and here s no oher feasble consumpon allocaon {(ĉ (s )) I } T =0,s S such ha u (ĉ ) u (c ) for all I (7) u (ĉ ) > u (c ) for some I (8) 3

4 4 Complee Markes and he Represenave Agen The key assumpon ha wll allow as o ge he represenave agen resul n hs dynamc conex s he one of complee markes. Complee markes means ha here exs a full se of conngen clams (.e. one clam for every possble realzaon of uncerany) ha are raded a me 0 before any uncerany has been revealed. The ndvdual Arrow-Debreu budge consrans ake he form =0 s S p (s )c (s ) =0 s S p (s )y (s ) (9) where p(s ) s he perod 0 prce of one un of perod consumpon, delvered f even hsory s has realzed. Arrow-Debreu Equlbrum Defnon 2 An Arrow Debreu compeve equlbrum consss of allocaons {(c (s )) I } T =0,s S and prces {p(s )} T =0,s S such ha 1. Gven {p(s )} T =0,s S, for each I, {c (s )} T =0,s S maxmzes (4) subjec o (5) and (9) 2. {(c (s )) I } T =0,s S sasfes (6) for all, s. Now le us make he followng assumpon Assumpon 1: The perod uly funcons U are wce connuously dfferenable, srcly ncreasng, srcly concave n s frs argumen and sasfy he Inada condons lm c(c) c 0 = (10) lm c(c) c = 0 (11) I s hen sraghforward o prove he frs welfare heorem for hs economy (n fac, for hs resul we only need ha he uly funcons are srcly ncreasng). Hence any compeve equlbrum allocaon s he soluon o he socal planners problem of max {(c (s )) I } T =0,s S N α u (c ) (12) subjec o (5) and (6), for some Pareo weghs (α ) N =1 sasfyng α 0 and N =1 α = 1 (see, e.g. MasColell e. al.; hs resul requres pars of assumpon 1). Aachng Lagrange mulplers λ(s ) o he resource consran and gnorng he non-negavy consrans on consumpon we oban as frs order necessary condons for an opmum =1 α β π (s )U c(c (s ), s ) = λ(s ) (13) for all I. Hence for, j I U c(c (s )) U j c (c j (s )) = αj α (14) 4

5 for all daes and all saes s. Hence wh a complee se of conngen consumpon clams he rao of margnal ules of consumpon of any wo agens s consan across me and saes. Also agens, ceers parbus (.e. f hey have he same uly funcon), wh hgher relave Pareo weghs wll consume more n every sae of he world because he uly funcon s assumed o be srcly concave. Assumpon 2: All agens have dencal CRRA uly, U (c) = c1 σ 1 1 σ (15) wh σ 0 (for σ = 1 s undersood ha uly s logarhmc). Here σ s he coeffcen of relave rsk averson. Wh hs assumpon (14) becomes c (s ) c j (s ) = ( α α j ) 1 σ (16).e. he rao of consumpon beween any wo agens s consan across me and saes. Ths, n parcular, mples ha here exs shares (θ ) I wh θ 0 and I θ = 1 such ha c (s ) = θ y(s ) θ y (s ) = θ c (s ) (17) I where y (s ) = I y (s ) s he aggregae ncome n he economy and c (s ) = I c (s ) s aggregae consumpon. In fac, he shares are gven by 1 α 1 σ θ = ( ) 1 = 0 (18) N αj σ N j=1 j=1 α 1 σ j α Noe ha when σ ends o (.e. a very concave uly), for any dsrbuon of Pareo weghs a, we have ha θ 1 N reflecng he fac ha he benef ha he planner ges from ransferrng resources from a low wegh agen o a hgh wegh one s very quckly ouweghed by he loss of uly due o concavy, so regardless of weghs an equalaran allocaon would emerge. Wh logarhmc preferences (σ = 1) we have ha θ = α,.e. he share of aggregae consumpon an agen s allocaed corresponds exacly o he Pareo wegh he planner aaches o hs agen. preferences) we have ha θ = { 1/J f α = max α 0 oherwse Wh σ approachng 0 (lnear where J s he number of agens go whch α = max α. The dea here s ha when preferences are close o lnear he benef he planner ges from ransferrng resources from one ndvdual o anoher no ouweghed by concavy, hence even a small dfference n weghs leads o very large dfferences n allocaon. Ths dscusson helps us undersand ha opmal or socally desrable 5

6 amoun of redsrbuon depends on ndvdual preferences as well as on socal preferences oward redsrbuon. Suppose for example dsrbuon of wealh s very unequal. Ths would lead, for all preferences, o hghly unequal dsrbuon of consumpon. Suppose now ndvduals have very curved preferences: n hs case even a small degree of socal preference oward equaly (.e. egalaran dsrbuon of he α ) would call for large redsrbuon of resources from rch o poor. To sum up, wh separable CRRA uly complee markes mply ha ndvdual consumpon a each dae, n each sae of he world s a consan fracon of aggregae ncome (or consumpon). Noe ha does no mply ha ndvdual consumpon s consan across me and saes of he world, because sll vares wh aggregae ncome (a varaon agans whch no muual nsurance among he agens exss). I also does no mply ha consumpon among agens s equalzed. From (16) we see ha he level of consumpon of agen wll depend posvely on he Pareo wegh of ha agen. Le s urn o he characerzaon of effcen (and hence equlbrum) allocaons. The growh rae of consumpon beween any wo daes and saes s gven from (17) as ( c log +1 (s +1 ) ( ) c (s +1 ) ) c = log (s ) c (s ) ha s, f agens have CRRA uly ha s separable n consumpon and we have complee markes, hen ndvdual consumpon growh s perfecly correlaed wh and predced by aggregae consumpon growh. In parcular, ndvdual ncome growh should no help o predc ndvdual consumpon growh once aggregae consumpon (ncome) growh s accouned for. Ths dea s he bass of all emprcal ess of perfec consumpon nsurance, see e.g. Mace (1991), Cochrane (1991), among ohers. To oban Arrow Debreu prces assocaed wh equlbrum allocaons we oban from he consumer problem of maxmzng (4) subjec o (9) ha a (19) p(s +1 ) p(s ) = β π +1(s +1 ) Uc(c +1(s +1 )) π (s ) Uc(c (s )) (20) Under assumpon 2 hs becomes p(s +1 ) p(s ) = β π +1(s +1 ) π (s ) = β π +1(s +1 ) π (s ) ( c +1 (s +1 ) σ ) c (21) (s ) ( c+1 (s +1 ) σ ) (22) c (s ) and hence equlbrum Arrow Debreu prces can be wren as funcons of aggregae consumpon only. We hen have he followng Proposon 3 Suppose allocaons {(c (s )) I } T =0,s S and prces {p(s )} T =0,s S are an Arrow- Debreu equlbrum. Then under assumpon 2 he allocaon {c (s )} T =0,s S defned by c (s ) = c (s ) (23) I 6

7 and prces {p(s )} T =0,s S s an Arrow Debreu equlbrum for he sngle agen economy wh I = 1 n whch he represenave agen has endowmen y (s ) = y(s ) (24) I and CRRA preferences. Obvously he conen of hs proposon les n he prcng par. I shows ha o derve Arrow- Debreu prces (and hence all oher asse prces), wh complee markes s suffcen o sudy he represenave agen economy. Ths nsgh s he deparure of he consumpon-based asse prcng leraure as developed n Lucas (1978). The dscusson also suggess he seps ha are needed o esablsh a represenave agen resul. 1. Solve he plannng problem wh many agens. The frs welfare heorem and he Negsh- Manel algorhm (whch you wll dscuss n deal n he revew sesson) enable us o solve a plannng problem ha yeld allocaons ha concde wh equlbrum allocaon. 2. Use allocaons from planner s problem o consruc aggregaes (such us aggregae consumpon or prces). In he smple example above aggregae consumpon can be solved easly usng feasbly and prces are solved usng frs order condons of any agen. 3. Show ha he same aggregaes can be derved solvng he equlbrum problem of a fcous sngle agen. Laer we wll use hese hree seps explcly o esablsh a RA resul n a more complex envronmen. Noce he wo key assumpons for he proposon (and for exendng he represenave agen resul n dynamc economes): dencal-homohec preferences and complee markes. The complee markes assumpon guaranees ha he margnal rae of subsuon beween consumpon a any wo daes s equalzed across all consumers. The dencal-homohec preferences guaranees ha he common MRS s also equal o he MRS of he represenave agen,.e. he agen ha has consumpon equal o he aggregae. If boh assumpon are sasfed redsrbung across consumers whou changng he ncome of he represenave agen does no change he MRS of he RA and so does no change he MRS of any consumer and hence does no change neremporal prces. To make hngs more concree consder a very smple example wh wo consumers whch have me separable CRRA uly have he deermnsc ncome sequences ploed below n he lef panel of he fgure. Usng he resul above we can show ha consumpon for each consumer s a fxed fracon of aggregae ncome. Noe ha even hough consumpons are no equalzed MRS are (wh CRRA consumpon margnal rae of subsuon s proporonal o consumpon growh). In he rgh panel we consder perurbaon whch leaves oal ncome unchanged bu redsrbue ncome from consumer 1 o consumer 2 n he frs par of he sample. Obvously hs perurbaon does no change oal consumpon (smply due o feasbly) and n equlbrum ncreases ncome and consumpon of consumer 2 whle reduces reduces ncome and consumpon of consumer 1. 7

8 The key pon hough s ha hs perurbaon leaves MRS of each consumer unchanged and equal o he MRS of he represenave agen. Now consder he same perurbaon n an envronmen whou complee markes,.e. wh borrowng consrans. To make hngs smple assume ha consumers are no allowed o borrow. In hs world s easy o see ha consumpon s equal o ncome for boh consumers. Bu hs mples ha he perurbaon analyzed n he pcure does change he MRS of consumers, n parcular changes he MRS of consumer 2, whch beng he unconsraned one, has s MRS equalzed o he neres rae. Hence he perurbaon changes he neres rae and he represenave agen resul does no hold. y, c y, c c 2 y 2 y 1 c 2 y 2 y 1 c 1 c 1 Fgure 1: Consumpon and ncome n complee markes In one problem you wll have o sudy wha happens when you relax he dencal preferences assumpon bu keep he complee markes assumpon. The key nsgh s ha n complee markes margnal ules of each and every agen are proporonal and hus, for asse prcng purposes, he margnal uly of any agen can be used. Noce hough ha when preferences are no dencal across agens or are no homohec equalzaon of margnal ules no longer mples ha consumpon of each agen s a fxed share of aggregae endowmen (or ha he rao of consumpon beween agen and agen j s consan) and hus n order o compue he margnal uly of any agen we need o explcly compue he consumpon of ha agen, whch wll n general depend on he srucure of preferences across he economy. So when preferences are no dencal across consumers and no homohec n general he represenave consumer resul fals and n order o characerze prces or he evoluon of aggregae varables one needs o know he enre dsrbuon of resources across he economy. See also Guvenen (2012), Theorem 2, for more general condons under whch aggregaon apples. 8

9 5 Recommended readngs: Mas Colell e al., Secon 4B Ljungqvs and Sargen, Secons 8.1 hrough 8.11 Guvenen F. (2011), Macroeconomcs Wh Heerogeney: A Praccal Gude, NBER Workng paper 17622, Secons 2.1 and 2.2 9