Constant Elastct of Substtuton n Appled General Equlbru The choce of nput levels that nze the cost of producton for an set of nput prces and a fed level of producton can be epressed as n sty.. f Ltng for eposton purposes the set of nputs to and focusng on the constant elastct of substtuton (CES) producton functon elds n + +,, ( ) sty.. + + Forng the Lagrangan for the optzaton proble L + + + Y + + The frst-order condtons for ths nzaton proble are then λ λ + + 0 Gventhat and λ + + 0 λ + + 0 λ + + 0 Y + + 0 λ The rato of the frst-order condton for the frst nput to the frst-order condton for the second nput elds
AEB 684 Appled General Equlbru Notes + + λ + + λ Spl replacng the prce and constant for elds Substtutng these results nto the frst-order condton th respect to the Lagrange ultpler elds Y + + 0 Focusng on the ter nsde the brackets + + + + Frst note that +. Thus, the above epresson can be splfed to Net, e solve + + z z z z substtutng ths soluton nto the frst ter elds + + Fnall, e ultpl the frst ter b a for of one Aprl, 00
AEB 684 Appled General Equlbru Notes ths elds + + Factorng out the ters nvolvng (-)/,, and elds The frst-order condton fro the Lagrange ultpler then becoes Y 0 Y Y No f e assue that land, labor, and captal are appled to produce agrcultural and anufacturng goods, e can specf a producton sste: cf (c-coodt and f-factor) Land ( c ) Labor ( c ) Captal ( c ) c Y a Agrculture 0.70 0.0 0.0 0.95 Y Manufacturng 0.05 0.45 0.50.05 Gven these paraeters, e have to producton functons Ya.6870 +.0886 +.88.056.056.056.0476.0476.0476 Y.0577 +.4675 +.567 The nput deands condtonal on the output level can be defned based on these paraeters 9 Aprl, 00
AEB 684 Appled General Equlbru Notes.70Y a a 9.95.05.05.05.70 +.0 +.0.0Y a a.95.05.05.05 9.70 +.0 +.0.0Y a a.95.05.05.05 9.70 +.0 +.0.05Y.05.05.05.05.05 +.45 +.50.45Y.05.05.05.05.05 +.45 +.50.50Y.05.05.05.05.05 +.45 +.50 It s obvous that gven output levels and nput prces, the deand equatons defne the quantt of nput deanded. If e assue that each factor s gven soe ntal endoent, then the ecess deand for each nput can be defned as (,, ) (, ) (, ) ξ Ya Y a Ya + Y E Takng the eaple a step further, e could rte the epresson n an object for as ξ pi,, Y, pi, + Y, pi, E ( ) ( [ ]) ( [ ]) a a To coplete the pcture, e have to forulate the deand equatons for the outputs. Agan, f e assue the CES for, the utlt azaton proble becoes + β a, β stp.. + p I Forng the Lagrangan of the azaton proble L β + β + µ I p p [ ] p β β + β µ 0 p β β + β µ 0 L β β + β L I p p 0 µ µ p 0 Aprl, 00 4
AEB 684 Appled General Equlbru Notes Agan takng the rato of the frst and second frst-order condtons elds + µ p µ p + β β β β β β β β β β p p p p p β p β Substtutng ths soluton nto the frst-order condton th respect to the Lagrange ultpler elds p β I p p 0 p β β I p p p 0 β β I p p p β Iβ pβ+ p pβ I β pβ+ p p β To reconcle ths result th prevous results, e factor p out of the denonator eldng I β p p β+ p β or I β p pj β j j It s no possble to specf the ecess deand for an output prce/ncoe cobnaton Aprl, 00 5
AEB 684 Appled General Equlbru Notes ξ j I β j ( pi, ) ( pi,, ) k kj j p pj β j j j j Lettng β β a.5, β β.65, and.75 the deand for each good becoes.5i a.75.5.5 p a.5 pa +.65p.65I.75.5.5 p.5 pa.65 p + Aprl, 00 6