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 Thomas Soesso Mcoecoomcs Lecte Cosme theoy A. The efeece odeg B. The feasble set C. The cosmto decso A. The efeece odeg Cosmto bdle x ( 2 x, x,... x ) x Assmtos: Comleteess 2 Tastvty 3 Reflexvty 4 No-satato 5 Cotty 6 Stct covexty Evey bdle (comleteess) ca be t to oe (eflexvty) ad oly oe dffeece set (tastvty) To gve these sets a atcla stcte (fge 2.2) No-satato movg betwee bdles the dffeece set by sbstttg oe good fo aothe (the dffeece cve has a egatve sloe) Cotty o holes the dffeece set (howeve small the edcto of oe good s, we ca always fd a cease ay othe good whch wll exactly comesate the cosme) Stct covexty (of the bette set) (whe we edce the qatty of oe good the comesato has to be lage ad lage to kee the cosme o the dffeece cve)
2 Thomas Soesso Mcoecoomcs Lecte Assmtos - 6 The efeece odeg ca be eeseted by a set of cotos covex-to-the-og dffeece cves/sfaces whee Each cosmto bdle les o oe ad oly oe dffeece cve/sface Bdles o a hghe dffeece cve/sface ae efeed to those o a lowe The tlty fcto (x) A mecal eesetato of the efeece odeg Ay fcto satsfyg the assmtos below s a tlty fcto fo the cosme (a) (b) ( x ) ( x ) f ad oly f x s dffeet to x ( x ) > ( x ) f ad oly f x s efeed to x A tlty fcto eflects oly the odeg of cosmto bdles thee ae a fte mbe of emssble tasfomatos of a secfc tlty fcto The tlty fcto s stctly qas-cocave To choose the efeed alteatve to maxmze the tlty fcto Assmto 7: dffeetablty ossble to defe the magal ate of sbsttto 2
3 Thomas Soesso Mcoecoomcs Lecte The magal ate of sbsttto the egatve of the sloe of the dffeece cve dx j MRS j dx d j, j the magal tlty of goods ad j esectvely The magal tlty deeds o the secfc tlty fcto chose to eeset the cosmes efeeces The atos of magal tltes ae vaat to emssble tasfomatos of the tlty fcto B. The feasble set Bdget costat x M x... The feasble set s boded, closed, covex ad oemty Togethe wth the assmto made fo the efeeces (tlty fcto) we kow that a qe global otmm s to be fod o the bdget le Bdget le x M x... 3
4 Thomas Soesso Mcoecoomcs Lecte C. The cosmto decso max ( x, x2,... x ) s. t. x M x... The bdget costat ca be wtte as a eqalty costat the Lagage fcto L L ( x, x2,... x) + λ [ M x ] Assme a solto whee x > (... ) L x λ (... ) L λ M x ) j j MRS betwee two goods shold eqal the ato of the ces. See fge 2.8 fo the two-goods case; tagecy solto 2)... λ The magal tlty of exedte o 2 2 x the magal tlty of exedte o x 2 ad so o 3) d λ dm M The Lagage mltle ca be teeted as the magal tlty of moey come 4
5 Thomas Soesso Mcoecoomcs Lecte Coe soltos Assme a solto whee t s ossble that fo some goods x L x λ x (... ) x ( λ ) magal tlty fo good λ ootty cost ( tems of tlty) fo good f < λ (fo x ) x f x λ > Aothe way to fomlate ths: λ f λ > fo x x If the magal tlty of exedte o good s less tha the magal tlty of moey at the otmal ot, the good wll ot be boght (fge 2.9) f x > λ 5
6 Thomas Soesso Mcoecoomcs Lecte The Mashalla demad fcto D Fom the solto to the cosme s otmzato oblem: x D (, 2,..., M ) D (, M ),... D s homogeeos of degee zeo x D ( k, km ) k D (, M ) D (, M ) Icome ad (ow) sbsttto effects fom a ce chage Hck s defto: costat tlty (fge 2.2) Comesatg vaato (CV) the chage M whch wll make the cosme as well off afte the ce chage as he was befoe Sltsky s defto: costat chasg owe (fge 2.4) - The ow sbsttto effect s always of ooste sg to the ce chage - The come effect s of ooste sg to the ce chage fo omal goods, ad of the same sg fo feo goods - Fo omal goods the total effect s always of ooste sg to the ce chage - Fo feo goods the total effect ca be of the same sg f the come effect s stog eogh to moe tha offset the ow sbsttto effect (Gffe goods) 6
7 Thomas Soesso Mcoecoomcs Lecte Demad cves (fge 2.5) - The Mashalla (costat moey come) demad cve (DD) shows the effect of chages wth M (ad 2 ) held costat (fom the ce cosmto cve) - The Sltsky costat chasg owe demad cve (gg) - The Hcksa costat tlty demad cve (hh) Fo omal goods: hh steee tha gg steee tha DD Fo feo goods: DD steee tha gg steee tha hh Offe cves ad et demad cves The cosme has a fxed amot of commodtes, a tal edowmet stead of a gve moey come Bdget costat / (wealth costat) x x W x, x... xˆ ( x x ) xˆ x... W Maket vale of the tal edowmet (wealth) x ˆ ( x x ) et demad Offe cve the locs of otmal bdles taced ot as / 2 vaes wth x fxed (fge 2.7) 7
8 Thomas Soesso Mcoecoomcs Lecte Otmal cosmto ove tme (chate ) Assmtos: - Two tme eods, ad - The cosme s edowed wth a come tme-steam M, M - A gve ce fo boowg ad ledg, the teest ate the sloe of the bdget le - / - (+) f the + - Pefeeces ae eeseted by a tlty fcto ( M, M) the sloe of the dffeece cve - / - (+ρ), whee ρ the cosme s sbjectve ate of teest - Postve magal tltes of cet ad fte cosmto Wealth costat: M M + M + + M + V Bdget costat: (et demad) dä + ( M M) ( M M ) + + At the otmm (fge.2): + + ρ + ρ 8
Consumer theory. A. The preference ordering B. The feasible set C. The consumption decision. A. The preference ordering. Consumption bundle
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