Analyzing Consumer s Behaviour in Risk and Uncertainty Situations

Transcription

1 Internatonal Journal of Economc Practces and Theores, Vol, No, (October), e-issn wwweporg Analyzng Consumer s Behavour n Rsk and Uncertanty Stuatons by Danela Elena Marnescu, Dumtru Marn, Ioana Manaf The Bucharest Academy of Economc Studes E-mal: danelamarnescu@hotmalcom, dumtrumarn@hotmalcom, oanamanaf@gmalcom Abstrac In the paper we wll generalze the Slutsky Equaton n rsk and uncertanty stuatons usng the compensated and uncompensated demand and some local measures of rsk averson We wll obtan a nonlnear optmzaton problem of maxmzng the expected utlty; ths problem wll be solved usng the Kuhn-Tucker method We use the results to analyze the ncome and substtuton effects of prce changes on demand n rsk and uncertanty condtons Key words: compensated demand, rsk averson, Slutsky Equaton, uncertanty, uncompensated demand JEL classfcaton: C6, D, D53 Introducton The rsk and uncertanty concepts have been rather recently ntroduced n the economc feld They were frst used n the economc theory n 944 by von Neumann and Morgenstern, n the paper Theory of Games and Economc Behavour But before that, t was Knght who had nferred the mportance of the rsk and uncertanty concepts for the economc analyss n the paper Rsk, Uncertanty and Proft n 9 A very nterestng aspect s that the margnal utlty n the context of choces n rsky condtons was proposed by Bernoull n 738 Ths s consdered the startng pont for the neo-classcal economy theory After the World War II the concept of rsk averson had been studed by Fredman and Savage (949), and also by Markowtz (95) Pratt (964) and Arrow (965) developed a measure for rsk averson, later mproved by Ross (98) Yaar (968), Khlstrom and Mrman (974) defned the rsk averson n multple cases Rothchld and Stgltz (97, 97), Damond and Stgltz (974) studed modaltes to measure the rsk propensty Knght was the frst to dstngush rsk from uncertanty n 9 He consdered that the rsk refers to the stuatons n whch the prncpal could attrbute probabltes to random events The uncertanty refers to the stuatons n whch one could not attrbute and calculate specfc probabltes (Keynes) In 95, Arrow noted that the most challengng aspect was to specfy how rsk and uncertanty affect the economc decsons Therefore, t s very mportant to establsh a connecton between the ncrease or decrease of the uncertanty and the prncpal s behavour Also t s mportant to know how prncpals consder the rsky stuatons when the ncomes are random To fnd the answers, new concepts such as choces n rsky stuatons or n uncertanty needed to be ntroduced Hcks (93) and Marschak (938) consdered that the preferences should be formulated over a probablty dstrbuton, but they could not separate the atttude towards rsk or uncertanty by the pure preferences over the results The startng pont was random to order randomly the speculatons takng nto account the mean and the varance Arrow (953) and Debreu (959) explaned the uncertanty usng the preferred states Ths was used by Hrshlefer (959, 966) n the nvestment theory and further developed by Radner n the fnancal feld and n the general equlbrum theory Nowadays, the rsk and uncertanty concepts are largely used n theoretcal and emprcal studes analysng economc agents behavour, the lterature on these topcs beng really huge 5

2 Internatonal Journal of Economc Practces and Theores, Vol, No, (October), e-issn wwweporg In the frst part of the paper we wll present some classcal concepts from mcroeconomc theory regardng optmal choces of the consumer: uncompensated and compensated demand, the ndrect utlty functon and the expendture functon These concepts are used for dervng one of the most mportant results of consumer s theory the Slutsky Equaton Ths result shows the relatonshp between prce dervatves of Hcksan and Walrasan demands, relaton that helps to analyze the ncome and substtuton effects (the effect of prce changes on demand) It s also helpful n summarzng the goods propertes: f they are substtutes or complements In the second part of the paper we wll consder an nvestor (a consumer) wth an ntal nvestment captal (ntal endowment) he can nves We wll suppose there are two nvestment opportuntes: one part of the ntal captal s nvested n one rsky actve and the other part s nvested n a safe (wthout rsk) actve The randomness return of the rsky asset generates a new optmzaton problem for maxmzng the consumer s expected utlty We wll solve ths problem usng some local measures of rsk averson (the relatve and absolute rsk averson ndex, rsk premum and the certanty equvalent), to analyze the ncome and substtuton effects n rsk and uncertanty condtons and we wll derve a generalzed Slutsky Equaton The Classcal Demand Theory In the classcal approach to consumer s demand, the analyss of consumer behavour begns by specfyng the consumer s preferences over the commodtes bundles n the consumpton n set, X R The consumer s preferences are represented by a preference relaton defned on X, relaton that s consdered to be ratonal, complete, reflexve and transtve Other types of assumptons regardng ths relaton are needed for modellng consumer s behavour: contnuty, monotoncty (or local non sataton) and convexty Alternatvely and ths s the way we adopt n ths secton for analytcal purposes we can summarze the consumer s preferences by means of a utlty functon We assume throughout ths secton that the consumer has preferences represented by a contnuous utlty functon U : X R, where X s the consumpton set, contanng all possble commodtes bundles, whose prces are denoted by ( p, p,, p n ) or by the prce vector p >> Uncompensated demand functons We now turn to the study of the consumer s decson problem, whch s called utlty maxmzaton problem (Mas-Colell et al, 995) A ratonal consumer wll always choose a consumpton bundle (the most preferred) from the set X, gven the prces p >> and the wealth level (ncome level) w >, n order to maxmze her utlty level So, the problem of utlty maxmzaton can be stated as: max U ( x) (P) x s px w X, the optmal (vector) soluton of ths problem, whch s called the vector of Walrasan (or uncompensated) demand functons and has the followng propertes (Mas-Colell et al, 995): We denote by ( p ; ) satsfes Walras law: px w ; ) f U ( ) s strctly concave, the soluton of (P) s unque The optmal value of the obectve functon n problem (P) s denoted by V The functon V s called the ndrect utlty functon and has the propertes (Mas-Colell et al, 995, Varan, 984) lsted below: ) V s contnuous for all p >>, w > ; ) V s nonncreasng n p and nondecreasng n w; ) V s quas-convex n p; v) V s homogeneous of degree zero n Compensated demand functons Instead of searchng for the maxmal level of utlty that can be obtaned gven a wealth w, we can fnd the mnmal amount (level) of wealth requred to reach a gven utlty level u Ths ) s homogeneous of degree zero n ( 5

3 Internatonal Journal of Economc Practces and Theores, Vol, No, (October), e-issn wwweporg analyss conssts n dervng and solvng a dual problem to (P), namely the expendture mnmzaton problem: mn px x (D) s U ( x) u The optmal soluton of ths problem s called the vector of Hcksan (or compensated) demand functons and represents the least costly bundle that allows the consumer to obtan the utlty level u, gven the prces p >> We denote by ϕ ths soluton It satsfes the followng propertes (Mas-Colell et al, 995): ) s homogeneous of degree zero n ; ) satsfes Walras law: px w ; ) f U ( ) s strctly concave, the soluton of (P) s unque The optmal value of the obectve functon n problem (D) s denoted by C and t s called the expendture functon Its propertes are summarzed below (Mas-Colell et al, 995, Varan, 984): ) C s nondecreasng n p; ) C s concave n p; ) C s homogeneous of degree one n p >> ; v) C satsfes Shepard s Lemma: Assumng that the dervatve exsts and for p >>, then ϕ, for,,, n Relatonshps between Demand functons, Indrect Utlty functon and Expendture Functon Slutsky Equaton (Varan, 984) There are several mportant denttes relatonshps between demand functons, ndrect utlty functon and expendture functon Let us consder the two problems (P) and (D), wth ther respectve solutons We can now lst the followng propertes: ) The mnmal expendture needed to obtan the utlty u V s w: C V ) w ) The maxmal utlty obtaned wth an ncome equal to C s u: V ( ) u 3) The uncompensated demand at the ncome w s the same as the compensated demand at utlty level V : X ϕ V ),,,, n 4) The compensated demand at the utlty level u s the same as the uncompensated demand at the ncome C : ϕ X ),,,, n We can state now the followng proposton, representng the Slutsky Equaton Proposton (The Slutsky Equaton) (Varan, 984): Suppose that U ( ) s a contnuous utlty functon representng a preference relaton defned on the consumpton se Then, for all and u V we have: ϕ X, for all, Proof We consder that the consumer facng the prce vector p >> and havng an ncome equal to w obtan a utlty level u, so that V u We note that the ncome w must satsfy the frst dentty, C u ) w The solutons (uncompensated and compensated demands) of the two optmzaton problems (P) and (D) satsfy the dentty: ϕ X ), for all,,, n We dfferentate ths relaton wth respect to p and we evaluate t n u ) Therefore, we get: ϕ u ) u )) u )) u ) But, from Shepard Lemma, we have: ϕ so ths can be used n the prevous relaton: ϕ ) ) ϕ We fnally use the denttes wrtten for the values u ), e C u ) w and ϕ u ) X u )) X and rearrangng the terms we get: 53

4 Internatonal Journal of Economc Practces and Theores, Vol, No, (October), e-issn wwweporg ϕ X The Slutsky Equaton shows a decomposton of the effect of a prce change on quantty demanded n two dfferent effects: - a substtuton effect, whch s defned as the effect of a prce change on a quantty demanded due exclusvely to the fact that ts relatve prce has changed; - an ncome effect, whch s defned as the effect of a prce change on a quantty demanded due exclusvely to the fact that the consumer s real ncome has changed The left hand sde of the above relaton represents the change n uncompensated demand holdng expendture fxed at w when p changes (change n usual demand) The rght hand sde s a sum of two ϕ terms: the frst term shows how compensated demand changes when p changes the substtuton effect and the last one, ncludng the sgn, ϕ u ) X s equal to the change n demand when ncome changes multpled by the correspondng uncompensated demand (the change n ncome to keep utlty constant) 3 A Generalzaton of Slutsky Equaton n rsk and uncertanty condtons In the classcal settngs, dfferent authors studed optmal consumer s choces that result n perfectly outcomes In realty, many mportant economc decsons nvolve some elements of rsk In ths secton we focus on the specal case n whch the outcome of a rsky choce s an amount of money (Marnescu et al, 8) We consder an economc agent representng the prvate nvestors that has an ntal ncome w (the ntal endowment) He has two choces (or nvestment opportuntes): to nvest n one safe actve (wthout rsk), wth a rate of return denoted by r or to nvest n a rsky actve whose rate of return s a random varable e wth mean and varance fnte We denote by a the proporton of ntal endowment nvested n rsky actve The agent s ncome at the end of the frst perod wll be: ) The agent nvests aw n rsky actve and at the end of the frst perod he wll have aw ( e) ) The agent nvests ( a) w (for a certan value e for e ) n actve wthout rsk and he wll have at the end of the frst perod ( a ) w ( r) So, at the end of the frst perod he wll obtan: w e w a e w a r ( ) ( ) ( )( ) or w( e) w [ ae ( a) r] We consder that the agent s rsk adverse and let U ( ) be the von Neumann-Morgenstern utlty functon wth the usual propertes: U '( ) >, U "( ) < Let e be the mean value of e and we denote by σ ts varance, neglgble wth respect to the hgher moments We consder the followng notatons: [ ] { ( )} x E x E aw e and σ x ( a w ) σ We wll use an alternatve way of analyzng consumer s choces and we consder the utlty functon B ( x, ) ( EU x ), approxmated by a dependent functon on x and x For ths, we wll use the concept of certanty equvalent, denoted by E C ( x ) The certanty equvalent satsfes the equaton: B( x, ) EU ( x ) U ( x EC ( x )) () We wll also consder the followng notatons: z e e, σ z σ Hence E ( z) holds Then: B x, x ) E [ U ( x ( z ) )] ( () We wll approxmate the functon from () by a seres expanson: x ( ) z U"( x ) U ( x x x z) U x x U '( x x ) x z!! (3) 54

5 Internatonal Journal of Economc Practces and Theores, Vol, No, (October), e-issn wwweporg The rght sde of (), U ( x x EC ( x )) x, could be approxmated (from Lagrange s Theorem) by: U ( x EC ( x ) ) U ( x ) U '( x ) EC ( x ) (4) From (3), takng the expected value of both sdes (applyng the operator E ) we wll get: U"( x x ) EU ( x σ! x ) U ( x x ) x (5) From (4) and (5) we wll obtan: U"( x ) x σ x σ EC ( x ) ra ( x ) U '( x ) where r a ( ) represent the absolute ndex of rsk averson We wll approxmate B ( x, ) by: x σ B( x, ) ( ) U x ra x (6) From the defnton of x and x, we wll get: x w e r e or x w ( e) (7) r The equaton (7) s the ncome equaton or welfare equaton p e and, then the e above equaton could be wrtten as: We defne p r r w pe x Therefore, the problem of choosng the optmal portfolo s: max B ( x, x ) x, x s pe x w or σ x maxu x ra ( x ) x, x s pe x w (8) From Gossen s Law, the optmal soluton must satsfy the frst order condton: U x r pe (9) U e pr x The welfare equaton and the condton (9) form a system of equatons Solvng ths system we * * wll obtan the optmal solutons x and x Let X ( pe, w ) be the uncompensated demand and V ( pe, w ) be the ndrect utlty functon, both havng the usual propertes Consder a fxed utlty level for the economc agent, denoted by b The goal of our analyss s to determne the mnmal ntal endowment that consumer needs to attan the utlty level b (e the utlty level assocated to one partcular socal category or for people lvng n a communty) Now we can state the followng nonlnear optmzaton problem: mn[ pe x ] x, s t B () ( x, ) b The soluton of ths problem s ( ϕ,ϕ ) or the ϕ vector ( ) Φ pe, b Then, the mnmal ϕ expendture functon s C ( pe, b) peϕ ϕ and t has the followng propertes: t s concave and ncreasng n prces and b and satsfes the Shepard s Lemma Dfferentatng the relatonshps between the functons X ( ) and Φ ( ), we wll get: Φ x,, () e e e or x ( p,, ) (,, ) e pr w Φ x p p w x e r * x () e e The equaton () shows how the prce effect (the left sde of equaton) can be decomposed nto a substtuton effect and an ncome effect (the frst and respectvely, the second term of the rght sde) Analogous, t follows: X, Φ x, * (3) r r X, Φ x, * (4) r r x ( p,, ) (,, ) e pr w Φ x p p w x e r * x (5) e e 55

6 Internatonal Journal of Economc Practces and Theores, Vol, No, (October), e-issn wwweporg The equatons (), (3), (4) and (5) are combned nto the matrx form of Slutsky s Equaton: * * X Φ( b) X x x (6) p 4 Conclusons p w ( ) As we already saw, n ts standard form the Slutsky Equaton decomposes the change n demand due to a prce change nto a substtuton effect and an ncome effect and the ncome effect was due to a change n the purchasng power when prces changed There was a strong assumpton made n that model: the money ncome was held constan In a revsted verson of the Slutsky Equaton, Varan examnes the case where the money ncome changes snce the value of the consumer s endowment changes when the prces change; he then shows that the ncome effect can be decomposed nto an ordnary ncome effect (when the prce changes, the purchasng power also changes) and an endowment ncome effect (a prce change also affects the consumer s endowment bundle so that the money ncome changes) (Varan, ) The Slutsky Equaton was also used as a standard mcroeconomc tool to analyse the change n demand due to an nterest rate change nto ncome effects and substtuton effects (consumer s ntertemporal choces) Our approach was based on some classcal mcroeconomc concepts from consumer s theory, but we analysed the case where the consumer s choces are made n rsk and uncertanty condtons, such that we were able to derve a generalzed form of the Slutsky Equaton The model we proposed here could be adapted for analyzng the evoluton of physcal or chemcal processes (or any type of process), havng a rsk when producng Hence, the goal, of the model s to mnmze the costs to obtan a certan result, a pror fxed Acknowledgment Ths work was cofnanced from the European Socal Fund through Sectoral Operatonal Programme Human Resources Development 7-3oect number POSDRU/5/5984 Performance and excellence n postdoctoral research n Romanan economcs scence doman References Cass, D and Stgltz, JE (97), Rsk Averson and Wealth Effects on Portfolos wth Many Assets, Revew of Economc Studes, 39(3): Fshburn, PC and Burr Porter, R (976), Optmal Portfolos wth One Safe and One Rsky Asset: Effects of Changes n Rate of Return and Rsk, Management Scence, ():64-73 Laffont, JJ (999) Econome de l Incertan et de l Informaton, Economca, Pars Mas-Colell, A, Whnston, MD and Green, JR (995), Mcroeconomc Theory, Oxford Unversty Press Marnescu, D and Marn, D (9), Rsk and Uncertanty: Analyzng the Income and Substtuton Effects, Economc Computaton and Economc Cybernetcs Studes and Research, : 4-54 Marnescu, D, Ramnceanu, I and Marn, D (8), The Influence of the Income Taxaton on the Agent Savngs, Symposum on Computer Applcatons n Scence, Engneerng and Educaton, CASEE 8, Crete, Greece Varan, HR (984), Mcroeconomc Analyss, Second Edton, Norton and Co, New York Varan, HR (), Intermedate Mcroeconomcs A modern approach, Sxth Edton, Norton and Co, New York and London Authors descrpton Danela Elena Marnescu, PhD s Assocate Professor at the Economc Informatcs and Cybernetcs Department, the Bucharest Academy of Economc Studes, Romana She has a PhD n Economcs from 6, n Economc Cybernetcs feld Her research and teachng nterests refer to advanced topcs n Mcroeconomcs, Labor Economcs, General Equlbrum and Incentves Theory She s (co) author of more than 3 ournal artcles and scentfc presentatons at natonal and nternatonal conferences 56

7 Internatonal Journal of Economc Practces and Theores, Vol, No, (October), e-issn wwweporg Dumtru Marn, PhD s Professor at the Economc Informatcs and Cybernetcs Department, the Bucharest Academy of Economc Studes, Romana He has a PhD n Economcs from 98, n Economc Cybernetcs feld Hs research and teachng nterests refer to advanced topcs n Mcroeconomcs, Theory of Optmal Systems, General Equlbrum and Incentves Theory He s author of more than 5 ournal artcles and scentfc presentatons at natonal and nternatonal conferences Ioana Manaf, PhD s Lecturer at the Economc Informatcs and Cybernetcs Department, the Bucharest Academy of Economc Studes, Romana She has a PhD n Economcs from 9, n Economc Cybernetcs feld She s (co) author of more than ournal artcles and scentfc presentatons at natonal and nternatonal conferences 57