A NOTE ON ELASTICITY ESTIMATION OF CENSORED DEMAND

Transcription

1 Octobe 003 B A NOT ON ASTICITY STIATION OF CNSOD DAND Dansheng Dong an Hay. Kase Conell nvesty Depatment of Apple conomcs an anagement College of Agcultue an fe Scences Conell nvesty Ithaca New Yo

2 It s the polcy of Conell nvesty actvely to suppot equalty of eucatonal an employment oppotunty. No peson shall be ene amsson to any eucatonal pogam o actvty o be ene employment on the bass of any legally pohbte scmnaton nvolvng but not lmte to such factos as ace colo cee elgon natonal o ethnc ogn sex age o hancap. The nvesty s commtte to the mantenance of affmatve acton pogams whch wll assue the contnuaton of such equalty of oppotunty.

3 A note on elastcty estmaton of censoe eman systems Dansheng Dong Depatment of Apple conomcs an anagement Conell nvesty Ithaca NY 4853 SA mal: Fax: Hay. Kase Depatment of Apple conomcs an anagement Conell nvesty Abstact: stmatng censoe eman systems usng mco-level ata has become moe pevasve n ecent yeas. Howeve not enough attenton has been pa to the evaluaton of the elastctes fom the censoe systems an the exstent methos use n lteatues ae usually ncoect. Ths note poposes a pactcal poceue on how to obtan the elastctes fom a censoe AIDS moel. J Classfcaton: C34 June 00

4 A note on elastcty estmaton of censoe eman systems I. Intoucton In a ecent pape by Golan Peloff an Shen 00 the metho of maxmum entopy was ntouce to estmate a censoe AIDS moel. Howeve n the posteo analyss of the pape the pce an expentue elastctes wee evaluate usng the fomula fo the uncensoe systems. In ths note we show that the way to evaluate elastcty usng such a fomula s nappopate fo censoe moel an an appopate metho s evelope theeafte. II. lastcty of ncensoe AIDS moel e efne an uncensoe empcal AIDS moel as: Y α γ ln P β ln P whee s a column vecto of expentue shaes P s a column vecto of commoty pces equaton paametes ae: α [ x ] γ [ x ] an β [ x ]. s a [ x ] vecto of equaton eo tems Y s total expentue an P s a tanslog pce nex efne by: ln P α 0 α' ln P ln P' γ ln P whee α s a scala paamete. 0 If t s assume that the eo tem s stbute nomal wth a mean vecto of zeos the followng expecte buget shae s eve:

5 Y α γ ln P β ln. 3 P Then the uncompensate ashallan pce elastcty s gven by: γ β α γ ln P Ε 4 whee s a [ x ] matx of coss an own pce elastctes; s a [ x ] agonal matx of ones. The ey ssue n evng elastctes s to use expecte values of obseve shaes. III. lastcty of Censoe AIDS moel xpecte values of obseve expentue shaes can be obtane fom the censoe eman system by summng the poucts of each egmes pobablty an expecte contonal shae values ove all possble egmes. Suppose the censoe ule fo equaton s efne as: / f > 0 S 5 0 f 0 whee an ae latent an obseve shaes espectvely an S s a set of all postve shae s subscpts. Ths mappng maes : le between 0 an an sum to unty ales an oolan. et epesent the th eman egme an efne t as: > 0; > That s the egme of the fst s ae zeos an the est ae postve. Gven zeo s othe possble egme can be tansfome to ths patten by eaangng the oeng of the s so that the fst ae zeos. Then we have the expecte shae fo commoty as:

6 α 7 whee α s the pobablty of egme occung an > > 0 0 0; pob pob φ α 8 whee φ s the multvaate nomal pf wth a mean vecto of zeos. The expecte shae value contonal on puchase egme can be epesente as: > ; 0 f f 9 wth α α. s the th ow of gven n an. φ α 0 The calculaton of elastcty s base on equaton 7 whch nvolves the evaluaton of 8 an 0. quatons 8 an 0 ae -fol ntegals an we may appoxmate them by numecal poceue Gauss quaatue. Gven that thee ae - puchase egmes one nees to evaluate 8 an 0 tmes. Ths woul be vey tme consumng. Howeve a smulaton poceue can be use nstea to evaluate elastctes. 3

7 Assume we have eplcates of the [] eo tem vecto e n. The th smulate latent shae evaluate at sample means of the exogenous vaables ncate by a ba ove a vaable s: Y α γ ln P β ln e P whee e s the th eplcate of e. The th eplcate of obseve shae gven by 5 then s / f > 0 S 0 f 0 whee the subscpt of epesents the th element n the vecto of. The expecte obseve shae vecto fo eplcates s then calculate as smple aveage of these smulate values:. 3 Suppose we have a small change n pce P the elastcty vecto wth espect to ths pce change s: 4 P P / η δ P / whee δ s a vecto of 0 s wth the th element an s the change of the smulate gven the change of pce P. 4

8 FNCS Golan A. J.. Peloff an Shen. Z. 00. stmatng a Deman System wth Nonnegatvty Constants: excan eat Deman. The evew of conomcs an Statstcs ales T. J. an oolan A. D "stmaton of Consume Deman Systems wth Bnng Non-Negatvty Constants." J. conometcs

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