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Estimating Exponential Utility Functions ,~ j,;' I) ':;l) . G rio"". 0 • 0 (? _.0 o I') Q () l% a 9(;' o Q estimatiqg "ExPC?ttenti~1 ·:U····:t·····~·I·····I:·I·+u·· ;F;";;u::·:n6',.t5"'·;~:n·'.' ·s.. ·, .' ..., .'.: :','. ..•. ".. '..'. ;. ~':,,;'V~ ..' .'. o c:::::::> ' ,;", o o (b~s. ) EconOm:ics, .Statistics'·Oftd (OOfI,ati,tS /} o i_JS~~n;'~~t'~C:'T" " o .j o i) " (> C. "French " State Un I vers" I ty, and -­ ~~ ~ from Agric~ltur;al Economic Research, Janucrry 1978" \/01. 30, No. 1 ~~'~....' .. ' .. exponential.utility fun~tion for money has 10ng att:racted attentic;m fromAtheorists~~ because it exhibits\~oni"creasing absolute, risk aversion. j\lso, under certain con- ~ ditions, it generates an expected utility functi()n that is maximizable in a quadratic 1 program. However, "this "'functiona,l form presents 'estimation problems. Logarithmic • transformation of an exponential uti 1i,t,y, function does not conform to the Von Neumann­ Horgenste~ axioms. Hence', it cannot be used as ,.a basis for best fit in statistical analysis. A criterion is described that can be used to select a best-flt exponential uti 1ity.fun.ction, and its appl.ica,ti~!!c"",L~I.J~~~er utJ1ity an~,,~J~!s. is (~en~~~.~~r.~~ed ,~-».,.....'~\ ~ ....,.,." .,-~~ '1\, ~.... jt-~ .. -~..,q,·""·"~.""""11!"~""""'"""'\.'<'..".tft4._ ... 1' ,,''-,. < ~ i\ ,c,( ~--~""..-~,."".... .... ,~. '" ~. "~:"''''.jq''K>''''~~''l;:' ", " & ' (J 11' 1 i ,~ r.~ .... ' t ' .'~ >':J ~rr 1 C >J '¥ ~r ~1 f' if! 1 t::' ~ t,n t; ~,l IG " Canputerized simulation o Computers c Economic analysis Est imates 1\, Exponential functions " ~ogarlthm functions Risk ~Statistical analysis Utility routines 1711. IdenEifier./Open-Eaded Titnu (;:::; Absolute risk aversion Compute r app,l i cat i ens' Logari thmi c traosformatian Honey (/ Von Neumann-Morgenstern axioms )) ~ 12-A, 12-8 PB .. 29Q4S8 - " --... ............_. .. ........... -- .,. I. i) 1 by St",en", T~B~ccola and Ben C. French: '[) / Jl II Reprinted from: "Agricultural'Economic Researc'h 1\ ~,_.... ... .. c... _ _ .... ....._ • • .. , . \\ . January 1978 Vol. 30 No. '1 : " . ~ ,I o U.S. I Department'of Agriculture I Economics;' Statist;~w DFJd CooP\erativ~ Service ' ,§ , Additional copies~available "from--: ._- ~ ... National Technical Information Service ·~. ,. ---,' . -. ~'1 ... "., M'_".~_~·-~··'~:----"-· "'--' --"""'.~~-''''''''''''''''--'' ESTIMATIN GEXPON ENJIALUTlllTYfU NCTIOf4S' . " 6 By St~venT. Bucco., and Ben C. French' ?'J \) () In",a ~cent study for the U.S. ~partment of Agri­ SELECTrNG A.UTILITY , ~ulture's Farmer Cooperative Service, we developed a FUNCTIONAL FORM framework in 'which a processing/marketing ~perative 'pi' other finp might evaluate alternative long-term con­ , Since the develt)pment of the Bemoullian money tract provisions for rmal product sale and raw product utility function, t~'i~e of its proper functional form purchase. The study focused 011 a Califomia coopera­ has been discu_d witJi no determinate COriclusion in tive .fnait and vegetable procenor. Selection of alter­ Iight.''''Earty theorists and practitioners preferred ~he _tive contract pricing. arrangements for tomato paste rr -quadr.itf.c utility funCtion: sales and tomato purchases Was treated as a problem in & Portfolio analysis. ' . U.-a+bM-cM2, b,c>O (1) , wh~re U is utinty ~nd M is money, for th~ reasons. ~ .• The eXponentilil utility f~i:tiOn for money "~ ..,Iongattrected attention from thlorilts bece... it If properly constrained,' the function conforms to the , a~i~,its nonincrelling o.olilta rilk aversion. Allo. riskavelter's reqj}!lrements .of a positively sloping, con­ under O!WUin co!MIitioll!!/ it ....,•• en axpecl~ cave function; when combined with linear profit func­ utility f""ctionthet Ii meximiable in a q......ic tions, it generates quadratic expected utility fUnctions program. HCMever, ttail functionel form ..,.........i­ metion probJe-. LotIIrith,nctrenaformetion of an that ue easily maximized with current programming axponenti" utHity function'" not conform to the routines; and it is easily fitted by 01.8 to utility ques­ 0 Vein Neur;'nn"'~ axioms. ~. it alllnot tionnaire data. be ....... a bali_ for beet fit in ...iiticlll ...IYlis. A Criticism of q~dratic forms began with Arrow's critwlon Ii dllcribed thet ,..n be .... to ...act a beet, and Pi..tt's identification of a coefficient of absOlute fit aJCPOfl8lltIal ~jlity ~ion. and i.. appli_ion in ........ utHity·aMIysii i1.dlmonatr8ted. risk avel'lion, Ra (M) -- U"/U', If this coefficient is a • K~.:' ElJPOnentiei. utility, risk. declinirlg function of M, ibe decision maker becomes \~ more wHling to accept a gamble with fixed p~babUi­ f/ ties of f'lXed "small" payof,s as his wealth increues (I, pp.96-96).' A ~ing coefficient implies decreased will­ t Bemoullian utUlty f.-nctions were estimated for a ingnfts, and a .:oDltant Cbefficlelit, unchanged wQling­ eooperative IIIUllllement spokesman and a b~Jd mem­ nea, ~o adopt this pmbl&~ Intult~n ....est. that ber, ad for eight tomato lrowen, to identify contract declining risk aftnion ought to ducrlbe many penons' portfoliOs that wouJd mui~ze expected' utility for behavior, but coeffICient Ra (M) In quadratic uWitiel is ~J'en or PIOCeIIOn.An important iIIUt! in this iden­ ~/(b-2cM), wbicb Inll&ead riIieI with Mo· tification proee. is .the utility functional form em­ A1temative forms that are more acceptable accord· ployed, since this form intluences the expected utility ing to the hypothesil,of 4ecUninga.,.,lute'risk ner­ formulation that is the buis for portfolio choice. lion include the .mliotlll'ithmic (, In this 'article, we revjew lOme of the questions DiIect in "Jecting a utility. functional form, suggest a U - d + lin M,I > 0, (2) methOd for fitting exponegtial forms to utility . JWIPOIlllidata, and dilCU. leveral applications ot this" ~ method. ~ I U&ilit7 functionamay refer to ~oney wealth or money profit, where th, latter reflect ebUllea frotll an initial wealth paGtion. Functiona dilcUiled in thil arti­ , I Steven 'u~eola iI ..iI~t profeuor of Alricultural cle may be applied to either wealthcw.. pl'f)fit, Tele em­ Ec:onomica~,~ Virainia Polytechnic Jnatitute and State pirical ap~Jicat~,DDI in.~" profit utilities. '. Univenity~ ltiC~burl. lien Frencil iI prof~r of aeri' altaliel:JleCl DUPlbe.- m parentbelea refer to lte~ In I . c:ultural eConQmi~ at the Univel'lity of Californ", Da~. Rftferencea at the eDd of tbill article. , r Special thana for .their he,plul ~UII..tiOila are 0"" If Riaint abaoIJlte',illk a.enaioD ia coJlliltent with • i to Robert .Ie~n and " ..ph ,,-,licek ot the. Depart. predilection for hoantlnt,an4 hen~ iI not to be ruled ......t of APiculturalEc:o.."m~, ...4 Ray~ond lIye.. of ollt,pf )land. Be,ciclea, one IIlJY ..ue that the data and I, , the P.putmentof Statistics, Viliiniapf)lyte~lmic""ti· notneearche~' ,tzpectatio~: .ouhl.@terpaine fun,:" ~ ~te and State Uliiftnity. R.pO~"i1ity,~o~ the article', ti!Hlal.form. W~,do DotpropP!ie to jlld" tbelDeri-. (Jf t' .tent ~If)nl toibe authorS; " theh)'POth,tsia of decreUJnlabioluter.iaJt ave"on, .~ ---===; 37 '0 o \\ where Ra (M) .., 11M; and the negative inverse expo­ THEORY OF ESTIMATING nential (hereafter called exponential) THE EXPONENTIAL PARAMETER In general only utility parameters encountered as U ~ K -9 exp [-AM), K,e,}..>o, (3) coefficients of income probability moments in an expected. utility model have ultimate importance to with Ra (M) = A, a co~stant. In addition, Lin and the decision theorist 'or researcher. This observa· Chang propose in this issue of the journal a polynomial tion may be infened from the fact that the specification with variable transformations onU or M; maximized expected utility model is the hypothesized in their article, Ra (M) coefficients depend upon the basis of choice under risk, and that, under known pro­ valul!s(~ken by transformation constants. Among the gramming methods, only coefficients of income proba­ more traditional forms, (2) and (3) have not be(;'n bility moments aCfect optimal variable levels in these widely used because they are not, as with the quad­ models. Since neither K nor e are coefficients of prob­ ratic, associated with a quadratic and thus tractable 'ability moments ~,02 in (4), they are in themselves expt!cted utility function. irrelevant to decision making. Conversely, a decision is This presumably .exclusive advantag~ of the quad­ uniquely determined once )., ~, and 0 2 are known.6 ratic was, however, undermined as early as 1956. At It would seem reasonable that a regression approach that time, R. Freund demonstrated that exponential to estimating A in (3) would first require A's removal utility, linear profit function, and normally distribu~~d Crom the ei,tponent. Experience wjth Cobb-Douglas and c~profit generate an expected utility model that is maXi­ other variabie exponent functions suggests expressing mizable by operating with an associated quadratic (3) in log form to llccomplish this. Subtracting K from function. Following Wiens' notation, exponential both sides of (3), c f.' utility (3) and normally distrlhuted profit M ~ N(~, . 0 2) produce expected utility 5 • \\ U-K = -eexp[-AM). (3)' \\ lJ :fj E[UJM)) = K-ee~p(-741+(}..2/2)a21. (4) Taking natural logs of both sides, i '.\ .; Expression (,.) is maximized by minimizing the expo­ nent, a quadratic function. No such tractable solution In (U - K) = In (- e exp [-AM]). (5) procedure, other than use of the Taylor expansion with its associated error telm, has been offered for the semi logarithmic form. For empirical r.eselt()hers, this is If utility in (3) is positive, K> e eltp [-AM], so 61n important diSildvantage which overrides the hypo­ that ,I{ > U. Thus (U-K) < 0, In (U-K) does not exist, thetical superiority of the ~miJog's declining. absolute and (5) cannot be estimated. However, multiplying (3)' risk aversion. by ·-1, There is no difficulty fitting quadratic or semiloga-. rithmic forms to utUity questitlnnaire data. In the 1.~/.·. ,0";' - U + K = e exp [-XM 1 (3)" y. latter ~, for example, one merely expresses money L "-.,') value.s (positive only) in logs and regresses utility obser­ vations G~t these logs.
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