Consumer theory. A. The preference ordering B. The feasible set C. The consumption decision. A. The preference ordering. Consumption bundle

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1 Föreläsgsderlag för Gravelle-Rees. Del. Thomas Soesso Cosmer theory A. The referece orderg B. The feasble set C. The cosmto decso A. The referece orderg Cosmto bdle ( 2,,... ) Assmtos: Comleteess 2 Trastvty 3 Reflevty 4 No-satato 5 Cotty 6 Strct covety Every bdle (comleteess) ca be t to oe (reflevty) ad oly oe dfferece set (trastvty) To gve these sets a artclar strctre (fgre 2.2) No-satato movg betwee bdles the dfferece set by sbstttg oe good for aother (the dfferece crve has a egatve sloe) Cotty o holes the dfferece set (however small the redcto of oe good s, we ca always fd a crease ay other good whch wll eactly comesate the cosmer) Strct covety (of the better set) (whe we redce the qatty of oe good the comesato has to be larger ad larger to kee the cosmer o the dfferece crve)

2 Föreläsgsderlag för Gravelle-Rees. Del. Thomas Soesso Assmtos - 6 The referece orderg ca be rereseted by a set of cotos cove-to-the-org dfferece crves/srfaces where Each cosmto bdle les o oe ad oly oe dfferece crve/srface Bdles o a hgher dfferece crve/srface are referred to those o a lower The tlty fcto () A mercal reresetato of the referece orderg Ay fcto satsfyg the assmtos below s a tlty fcto for the cosmer (a) (b) ( ) ( ) f ad oly f s dfferet to ( ) > ( ) f ad oly f s referred to A tlty fcto reflects oly the orderg of cosmto bdles there are a fte mber of ermssble trasformatos of a secfc tlty fcto The tlty fcto s strctly qas-cocave To choose the referred alteratve to mamze the tlty fcto Assmto 7: dfferetablty ossble to defe the margal rate of sbsttto 2

3 Föreläsgsderlag för Gravelle-Rees. Del. Thomas Soesso The margal rate of sbsttto the egatve of the sloe of the dfferece crve d j MRS j d d j, j the margal tlty of goods ad j resectvely The margal tlty deeds o the secfc tlty fcto chose to rereset the cosmers refereces The ratos of margal tltes are varat to ermssble trasformatos of the tlty fcto B. The feasble set Bdget costrat M... The feasble set s boded, closed, cove ad oemty Together wth the assmto made for the refereces (tlty fcto) we kow that a qe global otmm s to be fod o the bdget le Bdget le M... 3

4 Föreläsgsderlag för Gravelle-Rees. Del. Thomas Soesso C. The cosmto decso ma (, 2,... ) s. t. M... The bdget costrat ca be wrtte as a eqalty costrat the Lagrage fcto L L,,... ) + λ [ M ( 2 ] Assme a solto where > (... ) L λ (... ) L λ M ) j j MRS betwee two goods shold eqal the rato of ther rces. See fgre 2.8 for the two-goods case; tagecy solto 2)... λ The margal tlty of eedtre o 2 2 the margal tlty of eedtre o 2 ad so o 3) d λ dm M The Lagrage mltler ca be terreted as the margal tlty of moey come 4

5 Föreläsgsderlag för Gravelle-Rees. Del. Thomas Soesso Corer soltos Assme a solto where t s ossble that for some goods L λ (... ) ( λ ) margal tlty for good λ oortty cost ( terms of tlty) for good f < λ (for ) f λ > Aother way to formlate ths: λ f λ > for If the margal tlty of eedtre o good s less tha the margal tlty of moey at the otmal ot, the good wll ot be boght (fgre 2.9) f > λ 5

6 Föreläsgsderlag för Gravelle-Rees. Del. Thomas Soesso The Marshalla demad fcto D From the solto to the cosmer s otmzato roblem: D (, 2,..., M ) D (, M ),... D s homogeeos of degree zero o moey llso D ( k, km ) k D (, M ) D (, M ) Icome ad (ow) sbsttto effects from a rce chage Hck s defto: costat tlty (fgre 2.2) Comesatg varato (CV) the chage M whch wll make the cosmer as well off after the rce chage as he was before Sltsky s defto: costat rchasg ower (fgre 2.4) - The ow sbsttto effect s always of ooste sg to the rce chage - The come effect s of ooste sg to the rce chage for ormal goods, ad of the same sg for feror goods - For ormal goods the total effect s always of ooste sg to the rce chage - For feror goods the total effect ca be of the same sg f the come effect s strog eogh to more tha offset the ow sbsttto effect (Gffe goods) 6

7 Föreläsgsderlag för Gravelle-Rees. Del. Thomas Soesso Demad crves (fgre 2.5) - The Marshalla (costat moey come) demad crve (DD) shows the effect of chages wth M (ad 2 ) held costat (from the rce cosmto crve) - The Sltsky costat rchasg ower demad crve (gg) - The Hcksa costat tlty demad crve (hh) For ormal goods: hh steeer tha gg steeer tha DD For feror goods: DD steeer tha gg steeer tha hh Offer crves ad et demad crves The cosmer has a fed amot of commodtes, a tal edowmet stead of a gve moey come Bdget costrat / (wealth costrat) W,... ˆ ( ) ˆ... W Market vale of the tal edowmet (wealth) ˆ ( ) et demad Offer crve the locs of otmal bdles traced ot as / 2 vares wth fed (fgre 2.7) 7

8 Föreläsgsderlag för Gravelle-Rees. Del. Thomas Soesso Otmal cosmto over tme (chater ) Assmtos: - Two tme erods, ad - The cosmer s edowed wth a come tme-stream M, M - A gve rce for borrowg ad ledg, the terest rate r the sloe of the bdget le - / - (+r) f the + r - Prefereces are rereseted by a tlty fcto ( M, M) the sloe of the dfferece crve - / - (+ρ), where ρ the cosmer s sbjectve rate of terest - Postve margal tltes of crret ad ftre cosmto Wealth costrat: M M + M + + r M + r V Bdget costrat: (et demad) där + r ( M M) ( M M ) + + r At the otmm (fgre.2): + r + ρ + r ρ r 8

9 Föreläsgsderlag för Gravelle-Rees. Del. Thomas Soesso Cosmer theory:dalty ma (,... ) s. t. M... Marshalla demad fctos D (, M) The eedtre fcto m M s. t. (,... )... The tlty costrat ca be wrtte as a eqalty costrat the Lagrage fcto L L + μ [ (, 2,... )] L μ (... ) L (, 2,... ) μ ) 2) j 2 2 j... μ Idetcal to the codtos for the tlty mamsg roblem (wth μ /λ) (fgre 3.) 3) μ dm d 9

10 Föreläsgsderlag för Gravelle-Rees. Del. Thomas Soesso m s. t (,... )... Hcksa (comesated) demad fctos H (, ) H - the (ow) sbsttto effect of a rce chage - Sbstttg the otmal vales of the gves the eedtre fcto m(,) the mmm level of eedtre ecessary to acheve a gve tlty level as a fcto of rces ad the reqred tlty level Σ M H (, ) m(, ) (a) The eedtre fcto s cocave rces (see fgre 3.3 whe the rce vector dffers oly resect of oe rce) (b) Shehard s lemma: The sloe of the eedtre fcto at a ot s eqal to the comesated demad for good at rce (fgre 3.3) m(, ) H (, ) - Sose creases. To a frst aromato to mata the same tlty level eedtre mst crease by H Δ - For fte rce chages H Δ overstates the reqred crease eedtre as the cosmer sbstttes away from the good qesto

11 Föreläsgsderlag för Gravelle-Rees. Del. Thomas Soesso The drect tlty fcto ma (,... ) s. t. M... Marshalla demad fctos D (, M) - Sbstttg the otmal vales of the (,.. ) gves the drect tlty fcto (,M) the mamm level of tlty ossble as a fcto of rces ad moey come (,... ) [ D (, M ),... D (, M )] (, M ) - M λ λ the Lagrage mltler from the tlty mamsg roblem - λ M A crease M by $ creases tlty by aromately λ A crease decreases tlty by aromately λ as ths s aromately eqvalet to a decrease M by $ Roy s detty M

12 Föreläsgsderlag för Gravelle-Rees. Del. Thomas Soesso The drect tlty fcto (, M) s mootocally creasg come we ca vert the drect tlty fcto, ad let M be a fcto of ad, to obta the eedtre fcto M m(, ). The two fctos are dal to each other ad cota the same formato! Smmary Ma ( ) s. t. M M M s. t ( ) Marshalla demad fctos Hcksa demad fctos D (, M ) H Idrect tlty fcto Iverse fctos (, ) Eedtre fcto ( ) (, M ) M Roy s detty m(, ) Shehard s lemma M D (, M ) m(, ) H (, ) 2

13 Föreläsgsderlag för Gravelle-Rees. Del. Thomas Soesso Measrg the beefts of rce chages - To be sed cost-beeft aalyss, where tlty s eressed terms of moey ad t s assmed that a etra $ of beeft to a dvdal has the same socal sgfcace whchever dvdal t accres to beefts (ad losses) ca be added to form a measre of the aggregate beeft (loss) to all cosmers The effect of a fall rce (fgre 3.6.a): CV (comesatg varato): the mamm amot the cosmer wold be reared to ay for the oortty of byg the good at the ew rce rather tha at the old rce the amot of moey whch mst be take from the cosmer the ew stato order to make hm as well off as he was the tal stato EV (eqvalet varato): the amot of moey whch wold have to be gve to the cosmer whe he faces the tal rce, to make hm as well off as he wold be facg the ew rce wth hs tal come - If rce rses CV s the amot that wold have to be gve to the cosmer to make hm as well off as the tal stato ad EV the amot that mst be take from the cosmer to make hm as well off as he wold be the ew stato 3

14 Föreläsgsderlag för Gravelle-Rees. Del. Thomas Soesso CV ad EV ca be defed from the eedtre fcto - To measre CV yo focs o the tlty the tal stato ( ) ), ( ), ( ), ( d H d m m m CV [ ] ), ( ), (, ( A Area m m m d m m Hcksa (comesatg) demad fcto wth tlty Reslt: CV ca be fod from the Hcksa (comesatg) demad fcto wth tlty (fgre 3.6.b): H m A 4

15 Föreläsgsderlag för Gravelle-Rees. Del. Thomas Soesso - To measre EV yo focs o the tlty the ew stato ( ) EV m(, ) m(, ) m d H (, ) d Reslt: EV ca be fod from the Hcksa (comesatg) demad fcto wth tlty (fgre 3.6.b): If the come effect CV EV ad they ca both be measred from the Marshalla demad fcto D (, M) Fgre 3.7 shows wth a smlar aalyss the effect of the trodcto of a ew good sold at a fed rce 5

16 Föreläsgsderlag för Gravelle-Rees. Del. Thomas Soesso The cosmer as a labor sler ma (, L) s. t. M wz + M - assme a comoste cosmto commodty y wth rce - rereset refereces by v(y, z), where z T - L M ma v ( y, z) s. t. y wz + M eller y + L v( y, z) + λ [ M + wz y] w z L L vz + w v z λ y y λ v v z y w The margal rate of sbsttto betwee cosmto ad labor sly mst be eqal to the real wage w/ From the solto follows the Marshalla cosmto demad fcto y (w,, M ) ad the Marshalla labor sly fcto z ( w,, M ) The effect of a rse (real) wage (fgres 4.2 ad 4.3): - the sbsttto effect creases the sly of labor - f lesre s a ormal good the come effect decreases the sly of labor - f the come effect more tha offsets the sbsttto effect there wll be a backward bedg, egatvely sloed sly crve of labor 6