EC3075 Mathematal Aroahes to Eonoms etures 7-8: Dualt and Constraned Otmsaton Pemberton & Rau haters 7-8 Dr Gaa Garno [Astle Clarke Room 4 emal: gg44]
Dualt n onsumer theor We wll exose the rmal otmsaton roblem on the HS and the dual one on the RHS of eah age detalng eah ste of the roess: Maxmse [mnmse] the Objetve Funton wth reset to the Choe Varables onsumton of goods and : max tlt Funton Subjet to the Constrant: mn + Exendture Funton + Budget Constrant Parameters are: Fxed tlt level Indfferene Curve Pres and Inome agrangean Funton s: Pres and Fxed tlt level + where s the agrange Multler.
3 We then wrte the frst order ondtons FOC s; that s the frst dervatves of the agrangean wth reset to the hoe varables and the agrange Multler and equate them to zero for a so alled nteror soluton : 0 0 ; 0 ; Ths s vald for both rmal and dual. Solvng the FOC s then gves us the OPTIMA soluton values of the hoe varables and of the agrange multler all exressed as funtons of the arameters n eah roblem: Marshallan Demands Hksan Demands Substtutng the above hoe varable solutons bak nto the objetve funtons then gves us: V V Indret tlt Funton E E + + Otmal Exendture Funton urveand budget lne ndfferene between tangen onts of are the
On the other hand the otmal value of the agrange Multler has the followng eonom nterretaton n eah roblem: dv > 0 d Margnal tlt of Mone de > 0 d Margnal Cost of tlt Smlarl n eah roblem when dfferentatng the ndret utlt and the otmal exendture funton wth reset to an of the res we fnd the followng results: dv < 0 de > 0 d d Ro s Identt Shehard s emma The four results on ths age are the dret onsequene of a more general result alled the Enveloe Theorem whh ou wll stud n more detal f ou do an ostgraduate or advaned mro. The enveloe theorem basall looks at the hange n the OPTIMA value of the objetve funton ndued b a hange n an of the arameters. It s ver useful for the so alled omaratve stats analss n eonoms whh looks for examle at how a onsumer equlbrum ont hanges when arameters suh as nome utlt or res hange. 4 It has a dret use n ol desons.
Ro s dentt means that gven that nome has ostve margnal utlt a rse n the re of a good has a negatve effet on utlt whh s roortonal to the otmal amount of the good onsumed; Shehard s lemma means that f the re of a good rses then to mantan the same level of utlt the onsumer needs an amount of extra nome exatl equal to hs demand. Taken together these two results ml an dentt between Marshallan and Hksan demands: ; E Dfferentatng the above dentt wth reset to res then gves rase to the Slutzsk equaton or law of demand that we have seen n earler letures.. Havng omleted the desrton of the theor we an now look at a ratal examle of dual otmsaton roblems n onsumer theor based on log utlt a handout wll be rovded.. The method that I suggest for solvng the frst order ondtons an be aled to ANY utlt form and n fat also to an roduton funton as we wll see. The method onssts of the followng stes: Wrte the 3 frst order ondtons FOC s; Take the rato of the frst FOC s to obtan a smle relatonsh between the hoe varables; then derve an one varable as a funton of the other; Substtute the above relatonsh nto the thrd FOC and solve sequentall for all hoe varables and the agrange multler. 5
Produer theor Agan we wll exose the rmal otmsaton roblem on the HS and the dual one on the RHS of eah age. However as we wll see dualt does not qute work out n roduer theor. Ths s beause the dual roblem ost mnmsaton does not onsder the re of outut amongst ts arameters. On the other hand the rmal roblem roft maxmsaton does onsder outut re but turns out to be equvalent to an unonstraned maxmum. So we do not get the one to one orresondene of onets that we got n onsumer theor. Maxmse [mnmse] the Objetve Funton wth reset to the Choe Varables nuts to roduton atal and labour : max w r Proft Funton mn w + r Subjet to the Constrant: f Produton Funton f w r Pre of outut wage rental of atal π f w r Parameters are: agrangean s: NOT a agrangean but just a roft funton here denoted π sne we ould dretl substtute the onstrant through wthout a agrange multler. Produton Cost Funton Fxed evel of Outut Isoquant of Produton w r Fxed level of outut wage rental of atal a w + r f A roer agrangean here denoted a to avod onfuson wth labour. 6
Agan we wrte the frst order ondtons FOC s; that s the frst dervatves of the roft funton π [and of the agrangean a] wth reset to the hoe varables [and the agrange Multler] and equate them to zero for an nteror soluton : π π 0 ; 0 a a a 0 ; 0 ; Solvng the above FOC s then gves us the OPTIMA soluton values of the hoe varables [and of the agrange multler] all exressed as funtons of the arameters n eah roblem: w r w r onts between are of the tangen soquant w and soost lne The above are generall known as otmal fator demands ondtonal fator demands n the ost mnmsaton roblem. You an see the dfferene between the two otmsaton roblems here: one nludes the re of outut the other a fxed level of outut amongst ther arameters. nlke onsumer theor where A res are onsdered n both roblems. Substtutng the otmal fator demands bak nto the orresondng objetve funton n eah roblem gves us: C w + r Π f Π w r w r f w r w r w w r r w r Maxmum Proft Funton w w r r w w r + r C w r r w r Mnmum Cost Funton 7 also known as ondtonal ost 0
The otmal value of the agrange Multler has the followng eonom nterretaton n the roduton ost mnmsaton roblem: dc w r d > 0 Margnal Cost of Produng a gven Outut whle dfferentatng the mnmum ost funton wth reset to nut res w and r gves agan the followng result: dc w r > 0; dw dπ dc w r dr w r < 0; dw > 0 Agan all the above results derve from the enveloe theorem. Shehard s emma Fnall n the roft maxmsaton unonstraned roblem we fnd that: dπ w r < 0 dr We an now look at a ratal examle of otmsaton roblems n roduer theor based on a log roduton funton and on a Cobb Douglas roduton funton a handout wll be rovded. The soluton method s exatl the same as for onsumer theor and generall easer. 8