ntroducton Calculatng Equvalent and Copensatng Varatons n CGE Models Véronque Robchaud July Most CGE odelers use these odels to assess the pacts of gven shocks or polces on a specfc econoy. Whle t s qute straghtforward to easure pacts on aggregate nonal producton and consupton levels relatve prces nonal ncoe and savngs t s less obvous to quanttatvely evaluate how uch better or worse off the households are. As drect and ndrect utlty functons are purely ordnal n nature we can only analyse the drecton of change. An nterestng alternatve s provded by usng the oney etrc utlty functon whch easures the nonal ncoe the consuer needs at one set of prces n order to be as well of at an alternatve set of prces and nonal ncoe. As such t can be used to obtan onetary easures of the welfare effects of dfferent polcy scenaros. The ost coon of these easures are equvalent varatons and copensatng varatons whch we wll present n ths note. We ght also be tepted to s and s between households. However ths s sleadng as the utlty each consuer receves fro a gven aount of ncoe dffers. n the sae way s and s should not n prncple be aggregated across households as a welfare easure. Nonetheless aggregate s and s could be nterpreted as the total lup-su transfer that s equvalent to the polcy change that s exaned. Note fnally that even "dsaggregate" s and s n CGE odels are evaluated at the se-aggregate level of household categores rather than at the ndvdual consuer level. Ths brngs us nto the larger debate of aggregaton n CGE odels and the representatve agent assupton that goes beyond ths note see Kran 99. The objectves of the present eo s to provde MMA teas wth a concse presentaton of oney etrc welfare ndcators and ther calculaton usng the ost popular utlty functons found n CGE odels. The general forulaton wll be presented as well as ts dervaton for the Cobb-Douglas and the utlty functons. Theory Let us frst defne the utlty functon µc the ndrect utlty functon and the oney etrc ndrect utlty functon µ. n these expressons C represents the vector of goods consued the vector of prces and household ncoe. Equaton A represents the utlty functon for a Cobb-Douglas specfcaton and equaton B for the specfcaton: µ C C A µ C C B
where and and represents the nu consupton level. Gven these specfcatons deand functons are derved by axzng utlty subject to the budget constrant: C A C B The ndrect utlty functon s obtaned by replacng the C n the utlty functons A and B wth the deand functons A and B respectvely: 3A 3B Solvng equaton 3A and 3B for gves the oney etrc ndrect utlty functon whch s a easure of the ncoe needed to attan utlty level v at the vector of prces : 4A 4B Equvalent and copensatng varatons are welfare easures based on the oney etrc ndrect utlty functon. Before dervng the atheatcal expressons for these easures let us quckly recall graphcally what they represent usng a two-good exaple. Suppose that ntally prces were. After soe polcy change new prces are observed assung for splcty that nonal ncoe s unchanged. Fgure shows that n ths exaple a hgher utlty level s attaned after the polcy change snce µ s further than µ fro the orgn.
Fgure : ostve shock C µ µ C Although we can sply say that the polcy had a postve pact on the consuer we ay want to quanttatvely estate the "sze" of the pact n order to be able to copare between consuers and/or scenaros. The oney etrc ndrect utlty functon allows such an evaluaton. Ths s done by easurng the dstance between the two ndfference curves at constant prces. Those prces could be the ntal ones or the fnal ones. Fgure shows the frst easure whch s called equvalent varaton. The ntuton behnd that easure s to calculate what would be the ncoe change at ntal prces that s welfare-equvalent to the observed change n prces. Fgure : Equvalent varaton C µ µ C The second easure s known as the copensatng varaton and uses the postpolcy prces. The thus easures the ncoe change necessary to copensate the consuer for the change n prces. Fgure 3 represents that easure. 3
4 Let us now derve these two easures forally n the case where both prces and nonal ncoe change. The equvalent varaton s gven by: For the Cobb-Douglas specfcaton we use equaton 3A and 4A: 5A Slarly for the specfcaton we use equaton 3B and 4B: 5B µ µ C C Fgure 3: Copensatng varaton
5 The copensatng varaton s gven by the followng expresson: For the Cobb-Douglas specfcaton and usng equatons 3A and 4A: 6A And fnally for the specfcaton usng equatons 3B and 4B: 6B References: Kran A. 99 "Who or What does the Representatve ndvdual Represent?" Journal of Econoc erspectves 6: 7-36. Varan Hal R. 99 Mcroeconoc Analyss Thrd Edton W.W. Norton & Copany New ork. Varan Hal R. 999 nteredate Mcroeconocs: A Modern Approach Ffth Edton W.W. Norton & Copany New ork.