Relatonshp Between the Uncompensated Prce-Elastcty and the Income- Elastcty of Demand Under Condtons of Addtve Preferences Author: Lorenzo Sabatell, PhD Author afflaton: GLOBMOD, Barcelona, Span (www.globmod.com) Correspondence: Lorenzo.Sabatell@globmod.com Runnng head: Keywords: consumer choce, demand, ncome-elastcty, prce-elastcty, preferencendependence, forecastng, mathematcal model Fgures: 4 November 2015 Verson
ABSTRACT Income- and prce-elastcty of demand quantfy the responsveness of markets to changes n ncome, and n prces, respectvely. Under the assumptons of utlty maxmzaton and preference-ndependence (addtve preferences), mathematcal relatonshps between ncomeelastcty values and uncompensated own and cross prce-elastcty of demand are here derved for bundle of goods, usng the dfferental approach to demand analyss. Key parameters are: the elastcty of the margnal utlty of ncome, and the average budget-share. The proposed method can be appled to forecast the drect and ndrect mpact of prce changes, and of fnancal nstruments of polcy usng avalable estmates of the ncome elastcty of demand. INTRODUCTION Market responsveness to changes n ncome and n prces of ndvdual goods, or of bundles of goods, s measured usng ncome-elastcty and (own and cross) prce-elastcty of demand. These quanttes capture the aggregate effect of consumer response to changes n prces and n ncome. A change n the prce of a good determnes a change n the purchasng power of consumers (ncome-effect), and a change n the relatve prce of goods (substtuton-effect). Knowng the uncompensated own and cross prce-elastcty of demand s essental to antcpate the mpact of prce changes, and of fnancal nstruments of polcy such as subsdes, cost-sharng schemes, and taxaton, nonetheless forecastng t requres data that s not always readly avalable. In many nstances, contngency studes, e.g. wllngness-to-pay studes, are used to forecast the response of consumer, nevertheless they are not always fnancally and logstcally 2
feasble, or consstent (Cameron, 2008, Knetsch and Snden, 1984). Unlke the prce-elastcty, the ncome-elastcty of demand can often be estmated from routnely collected populaton data (e.g. from surveys), and s therefore more readly avalable. Nonetheless, t does not contan n and of tself enough nformaton to nfer the consequences of changng prces. The relatonshp between demand and prce can be modeled and analyzed usng neoclasscal consumer theory, assumng a representatve economc agent wth preferences over consumpton goods, captured by a utlty functon (see for nstance Deaton, 2008). The Rotterdam model, frst proposed by Barten (1964) and Thel (1965), bulds on ths approach, allows the estmaton of substtutes and complements, and t also allows for separablty of preferences. The Rotterdam model produces constant margnal shares, a problem that can be avoded usng a demand functon called the Almost Ideal Demand System (AIDS) model (Deaton et al., 1980), whch was subsequently extended usng the dfferental approach of the Rotterdam model by Thel, Chung, and Seale. They added a non-lnear substtuton term to the basc lnear functon, whch allows for separablty and has fewer parameters to be estmated than n the AIDS model, creatng the so-called Workng Preference-Independence or the Florda model (Thel et al., 1989, Seale et al., 2003). If separablty holds, total expendture can be parttoned nto groups (or bundles) of goods, makng t possble to analyze the preferences for one group ndependently of other groups. In that case, the mathematcal relatonshp between prce and demand becomes amenable to analytcal calculatons. In the present study, followng the dfferental approach used to derve the Workng Preference- Independence Model, mathematcal relatonshps that allow the estmaton of the uncompensated own prce-elastcty and the cross prce-elastcty of demand for ndependent bundles of goods are obtaned. The proposed model requres three nputs: the ncome-elastcty of demand, the 3
mean budget share allocated to the bundle of goods of nterest, and the elastcty of the margnal utlty of ncome. METHODS Relatonshp between Income-Elastcty and Prce-Elastcty The defntons used throughout ths paper are reported n Table 1. The followng assumptons are made: I. The utlty functon s strctly concave twce contnuously dfferentable (.e. the Hessan matrx s contnuous and negatve defnte); II. The consumers have a lmted budget and they allocate t n a way that maxmzes ndvdual utlty; III. Preference-ndependence: the utlty generated by the consumpton of a bundle of goods does not depend on the consumpton of goods from other bundles. In other words, the utlty s the sum of the utltes assocated wth the consumpton of each ndvdual bundle of goods; IV. The elastcty of the margnal utlty wth respect to ncome s constant. Gven the assumptons I-II-III-IV, the mean value of the budget-share spent on each bundle (ω ), and the elastcty of the margnal utlty of ncome (), we show that for a gven bundle of goods (), a quanttatve (parabolc) functonal relatonshp exsts between the ncome-elastcty of demand (ε), and the uncompensated own prce-elastcty of demand (η): 4
σ φ ~ U n Lorenzo Sabatell 1 2 1 η = ωε + ω ε () 1 And that the cross-prce elastcty ( η j ) of demand for a bundle of goods () wth respect to the prce of a bundle (j) s: 1 ηj = ωjεε j ωjε ( 2) The parameter r can be estmated analyzng surveys of subjectve happness (Layard, 2008), and ts value has been found to be qute stable across dfferent geographc areas and populatons groups, wth an average value equals to -1.26 and a standard devaton equals to 0.1. ω can be estmated from household surveys, and from standard market-research data. To take nto account the effect of parametrc uncertanty on model estmates, credble ntervals (CrI) for the estmates of the prce-elastcty of demand can be calculated va Monte-Carlo smulaton, drawng random model parameter values from normal (N) and unform (Unform) dstrbutons: ( ) ~ μ, σ N (3) ω ~ Unform[ ω, ω ] mn MAX The Proof The proof of Eq.(1) and Eq.(2) uses Lagrange multplers and dfferental equatons, and s based on the fact that any change n the prce of a good determnes a change n the purchasng power of the consumer (ncome-effect), and a change n the relatve prce of goods (substtuton-effect). The substtuton effect depends on two elements: ) the relatve mportance of dfferent goods to 5
the consumer; and ) the deflatonary mpact that a change n the prce of a sngle good has on all market goods. The proof follows three steps: 1. A demand equaton for a bundle of goods () s derved usng the Thel (Thel et al., 1989) and Barten (Barten, 1964) approach. 2. Analytcal expressons for ncome elastcty, uncompensated own prce-elastcty, and cross prce-elastcty of demand for bundle () are derved from the demand equaton, under the assumpton of preference-ndependence. 3. The analytcal expressons obtaned n step-2 are then combned to derve Eq. (1) and Eq. (2). Step 1 Let s frst defne the vector of the purchased quanttes of bundles of goods {,..., } the vector of ther average prces {,..., } = 1 2 n q q q q and = 1 2 n p p p p. Assumpton (I) mples that frst and second order dervatves of the utlty functon uq ( ) exst, and that the Hessan matrx symmetrc negatve. Assumpton II mples that: a. Gven the budget constrant U = 2 T d u dqdq s E= p q = pq n = 1 ( 4) The followng Lagrangan functon can be defned: ( ) μ ( ) ( 5) F = u q p q E 6
b. uq ( ) can be maxmzed, subject to the budget constrant (4), by usng the Lagrangan multpler method, whch conssts n searchng the values of vectors q such that the gradent of F s null and the Hessan of F s negatve defned. Dfferentatng F wth respect to q : du dq = μ p ( 6) Combnng (6) wth the budget constrant (4), and followng the Barten s approach (Barten, 1964) the result can be wrtten n the followng parttoned matrx form: T U p dq / de dq / dp 0 μ 0 μ μ/ = I T T T 1 p d de d dp q ( 7) where I s the dentty matrx. Solvng the matrx demand equaton, Eq. (7), and followng Thel s dervaton (Thel et al., 1989), the general form of the dfferental demand system s obtaned: 1 ω log = θ log + n p d q d Q θjd log 8 * j = 1 p j ( ) ( ) where θ * * p = p s the margnal share, and the functon ( θ,,...,, 1 p1 θn pn) s the Frsch prce deflator, defned as: d * ( log ) = n p θ ( 9) = 1 dp p The frst term on the rght of Eq. (8) s the real-ncome term of demand, whch results from the change n money-ncome E, and the ncome-effect of the prce-change. The second term on the rght of Eq.(8) s the substtuton term. Under preference-ndependence (assumpton III), the term 7
contanng the Frsch-deflated prce of good s the only non-zero term n the substtuton term. Therefore Eq. (8) becomes: 1 p ωdlog q = θd logq + θd log 10 * p ( ) ( ) Step 2 Dfferentatng the budget constrant (4): ( log ) = ( log ) + ( log ) ( 11) d E d Q d P where: d ( log ) = n Q ω ( 12) = 1 dq q d ( log ) = n P ω ( 13) = 1 dp p If prces are kept constant, the followng equaton holds: ( log ) = ( log ) ( 14) d E d Q Snce does not depend on ncome (assumpton IV), the ncome-elastcty of demandε s: ( log q ) ( log E) d θ ε = = d ω ( 15) whch s obtaned substtutng Eq. (14) n Eq. (10), and then dfferentatng wth respect to d( log E ). 8
The uncompensated own prce-elastcty of demand η can be derved substtutng Eq (9) n Eq. (10), and usng the equaton: n dp d( logq) = d( log E) d( log P) = d( log E) ω p = 1 Snce does not depend on prces (assumpton IV), and gven Eq.(9), dfferentatng log q wth respect to log p yelds: ( q ) d log 1 θ ω j 1 θ ηj = = θ j θ + δj d ω ω ω ( log pj ) (16) where δ j 1 f = 0 f = j j Step 3 Obtanng θ and θ from Eq. (15) and substtutng them nto Eq. (16) yelds: j 1 δj ηj = ωεε j j ωε j + ε (17) Eq.(1) s equvalent to Eq. (17), for = j : 1 2 1 η = ωε + ω ε and Eq.(2) s equvalent to Eq. (17), for j : 1 ηj = ωεε j j ωε j 9
As an example, the dervaton of Eq. (1) for a hypothetcal market, tradng only n two ndependent bundles of goods, s outlned n the S1.text.docx. RESULTS AND DISCUSSION Analytcal calculatons show that the uncompensated own prce-elastcty η and cross prceelastcty of demand η j of ndependent bundles of goods can be expressed as functons of the ncome-elastcty of demand, the average budget shareω, and the elastcty of margnal utlty of ncome r: 1 1 = + 2 η ωε ω ε () 1 1 ηj = ωjεε j ω jε ( 2) For nstance, f ω s n the range 0.1%-10%, and r s drawn from a normal dstrbuton wth mean equal to -1.26 and standard error equal to 0.1, and ε=1, then Eq. (1) predcts an expectaton valueη equal to -0.8 (wth the 95% credble nterval: - 0.96, -0.64). The statstcal error assocated wth η ncreases wth the value of ε (see Fgure 1). The senstvty of predctons to model parameter values (and especally to the value of the elastcty of the margnal utlty of ncome) ncreases wth the ncome-elastcty of demand, and so does the uncertanty (the wdth of credble ntervals) assocated wth predctons based on uncertan parameter values. A 10
unvarate senstvty analyss (whose results are not dsplayed), ndcates that η would change by less than 7% (relatve change), asω fluctuates between 0 and 0.1, suggestng that for bundles of goods that account for less than 10% of the total expendture an approxmate estmate ofω may be suffcent to produce relatvely accurate estmates of η. Accordng to Eq.(2), the cross prce-elastcty of demand for a bundle of goods A wth respect to a bundle B grows lnearly wth the ncome-elastcty of demand for A (Fgure 2). The slope depends on the ncome-elastcty of demand for B, on the budget share of B, and on the elastcty of the margnal utlty of ncome. If the ncome-elastcty of B s smaller than the elastcty of the margnal utlty of ncome, an ncrease n the prce of A wll determne a reducton n the demand for B. If the ncome-elastcty of B s larger than the elastcty of the margnal utlty of ncome, an ncrease n the prce of A wll determne an ncrease n the demand for B. Ths s the effect of a re-allocaton of the budget performed by the consumer to offset the consequences of a change n the prce of B on real ncome and on the relatve prces of market goods. The statstcal uncertanty assocated wth cross prce-elastcty estmates ncreases wth the ncome-elastcty of B (see Fgure 2 and Fgure 3), and (unlke the case of the own prce-elastcty) s senstve to the actual value of the budget share. Eq.(1) can be used to forecast the mpact of a change n the own prce of bundle A on the demand for A, whle Eq. (2) can be used to assess how the demand for a bundle A changes f the prce of an (ndependent) bundle B changes. As such, they provde a theoretcal bass to estmate ex-ante the potental mpact of fnancal nstruments of polcy, e.g. subsdes and cost-sharng schemes, usng household survey data (see for nstance Sabatell, 2013), and to nform earlystage product prcng, when the relevant target s a bundle of goods that can be treated as 11
preference-ndependent. The face valdty of ths approach was tested usng publcly avalable data. Model predctons from Eq.(1) were compared wth publshed estmates of ncome- and prce-elastcty for dfferent bundles of goods and servces that had been calculated fttng the Workng Preference-Independence Model to real world data collected n natonal surveys of consumpton (Seale et al., 2003). To ensure comparablty, bundles accountng for an average budget share smaller than 10% were selected. Specfcally, the bundles consdered were: clothng and footwear, educaton, healthcare, and recreaton. Average elastcty values for low-, mddle-, and hgh-ncome countres are dsplayed n Fgure 4. Country-specfc data can be found n S2.Fgure.docx. The results show that the data are confned wthn the (funnel-shaped) 95% CrI regon of model predctons, close to the medan value. Nevertheless, there are several assumptons and caveats that need careful consderaton. The accuracy of model estmates may be lmted by the assumpton of nstantaneous maxmzaton of consumer utlty, whch n practce requres maxmzaton of utlty to be performed n a relatvely short tme. The proposed approach s approprate for studyng the consequence of changng the prce of bundles of goods and servces whose utlty can be assumed ndependent of the consumpton of goods belongng to a dfferent bundle (.e. addtve). In addton, preferences exhbt non-sataton,.e. goods are assumed to be avalable n all quanttes, and a consumer may choose to purchase any quantty of a good she desres. Whlst these assumptons allow a useful smplfcaton of the calculatons nvolved, they mght also be sources of bas n the results. Nonetheless, smlar caveats do also apply to other models (e.g. Barnett and Serlets, 2008, Brown and Lee, 2002, Seale et al., 2003 and Thel et al., 1989). In fact, the mathematcal framework here descrbed bulds on approaches already used n those studes, but wth dfferent 12
objectves. For nstance, Barnett and Serlets used the dfferental approach to demand analyss, and then mplemented the Rotterdam parameterzaton to move from a model based on nfntesmal nstantaneous changes to a dscrete model, where prces, ncome, and demand change over fnte tme ntervals (days, months, years). Subsequently they ftted the dscrete model to tme seres of prces and ncome, to calculate the parameter values of the demand system. Brown and Lee (2002) followed an approach often used when modelng advertsng effects n the Rotterdam model, and ntroduced preference varables n the utlty functon. They then studed the effect of mposng restrctons on preference varables. Nevertheless, to-date no publcly avalable study has nvestgated the relatonshp between ncome-elastcty and prceelastcty of demand. When compared wth the results of studes that have concurrently estmated prce- and ncome-elastcty of demand, (Santerre & Vernon 2006, Seale et al. 2003, Rngel et al. 2002, & Mannng et al. 1987), the predctons of the model presented appear consstent wth the avalable data (see Fgure 4, S1.Fgure.docx, and S2.Fgure.docx). In concluson, based on theoretcal consderatons and on the avalable evdence, the estmates of prce-elastcty of demand obtaned usng the analytcal relatonshps here proposed are comparable wth those generated usng other models based on the dfferental approach to demand analyss. If used to nfer the prce-elastcty from avalable estmates of the ncomeelastcty of demand, the proposed approach can provde a readly avalable and effcent tool to forecast the mpact of prce changes on consumpton patterns. 13
REFERENCES Barnett, W. A., & Serlets, A. (2009) The dfferental approach to demand analyss and the Rotterdam model. Quantfyng consumer preferences, Contrbutons to Economc Analyss. Emerald Group Publshng Lmted, 61-82 (http://mpra.ub.un-muenchen.de/12319/). Barten, A. P. (1964) Consumer Demand Functons Under Condtons of Almost Addtve Preferences. Econometrcs, 32: 1-38. Brown, M. G., & Lee, J.-Y. (2002) Journal of Agrcultural and Appled Economcs 34, 17-26. Cameron, T. A. (2008) Contngent valuaton. The New Palgrave Dctonary of Economcs Onlne 2ed. Deaton, A., Muellbauer, J. An Almost Ideal Demand System, The Amercan Economc Revew, Vol. 70, No. 3. (1980), pp. 312 326. Deaton, A. (2008) Consumer expendture The New Palgrave Dctonary of Economcs 2ed. Layard, R., Mayraz, G., Nckell, S. (2008) The margnal utlty of ncome. Journal of Publc Economcs, 92, 1846-1857. Mannng, W., Newhouse, J., Duan, N., Keeler, E., Lebowtz, A. (1987) Health nsurance and the demand for medcal care: evdence from a randomzed experment. The Amercan Economc Revew 77 (3), 251 277. Knetsch J.L., and Snden J.A. (1984) Wllngness to Pay and Compensaton Demanded: Expermental Evdence of an Unexpected Dsparty n Measures of Value. The Quarterly Journal of Economcs. 99,3, 507-521. The MIT Press. Rngel, J.S., S.D. Hosek, B.A. Vollaard, and S. Mahnovsk. (2002) The Elastcty of Demand for Health Care: A Revew of the Lterature and Its Applcaton to the Mltary Health System. Santa Monca, CA: RAND Health. 14
Sabatell, L. (2013) A Methodology for Predctng the Impact of Co-payments on the Utlzaton of Health Technologes. Value n Health. 16(7). Santerre, R. E., and Vernon, J.A. (2006) Assessng Consumer Gans From A Drug Prce Control Polcy In The Unted States. Southern Economc Journal, v73(1,jul), 233-245. Seale, J. L., Regm, A., Bernsten, J. (2003) Internatonal evdence on food consumpton patterns, Economc Research Servce, US Department of Agrculture. Thel, H. The Informaton Approach to Demand Analyss, Econometrca. 1965, Vol. 33, pp. 67-87. Thel, H., Chung, C.-F., Seale, J. L. (1989) Internatonal evdence on consumpton patterns. JAI Press, Inc. Appendx B:The Dfferental Approach to Consumpton Theory. Fgures Fgure 1: Relatonshp between ncome-elastcty and uncompensated own prce-elastcty of demand. The gray area ndcates the 95% credble nterval (CrI) calculated usng a Monte-Carlo 15
smulaton. ω was drawn from a unform dstrbuton rangng from 0.001 to 0.1; and was drawn from a normal dstrbuton wth mean equal to -1.26 and standard devaton equal to 0.1; Fgure 2. Relatonshp between ncome-elastcty of two (preference) ndependent bundles of goods A and B, and the cross prce-elastcty of demand for a bundle of goods A wth respect to B. The cross prce-elastcty s negatve, null or postve, dependng on whether the ncome elastcty of B s smaller of, equal to, or larger of. Parameters values: ω = 0.05 ; B and = 1.26 ; 16
Fgure 3: Quantfcaton of the uncertanty affectng the estmates of the cross prce-elastcty. In ths example, the ncome-elastcty of the bundle of goods B s equal to 0.2. The purple lne ndcates the medan value; whle the grey area ndcates the 95% credble nterval (CrI) calculated usng a Monte- Carlo smulaton. ωb was drawn from a unform dstrbuton rangng from 0.001 to 0.1; and was drawn from a normal dstrbuton wth mean equal to -1.26 and standard devaton equal to 0.1. 17
Fgure 4: Comparson of smulaton results wth estmates of ncome-elastcty and uncompensated own prce-elastcty of demand for clothng and footwear, educaton, healthcare, and recreaton. The dark grey lne (n the mddle) ndcates the medan of the smulated values, the lateral lght grey lnes ndcate the 2.5% and 97.5% quntles of the smulated values. The colored crcles refer to the values estmated fttng real country-consumpton data to the workng preference ndependence model. The bundles consdered are: clothng and footwear, educaton, healthcare, and recreaton. The data (Seale et al., 2003) refer to low-ncome (A), mddle-ncome (B), and hgh-ncome (C) countres. Smulaton parameters: the budget share ω was drawn from a unform dstrbuton rangng from 0.001 to 0.1; and the elastcty of the margnal utlty of 18
ncome was drawn from a normal dstrbuton wth mean equal to -1.26 and standard devaton equal to 0.1. Table 1: Defntons Quantty Defnton Symbol Utlty Functon a postve defned, functonal relatonshp between purchased quanttes (q) of market goods and the welfare (utlty) of consumers. u(q) Margnal Utlty of Income the partal frst dervatve of the Utlty Functon wth respect to ncome; μ Elastcty of the Margnal Utlty of Income the partal frst dervatve of the logarthm of the Margnal Utlty of Income, wth respect to the logarthm of ncome; Margnal Share of a Good the partal frst dervatve of the share of consumer-ncome allocated to the purchase of a good (or bundle of goods) wth respect to the consumer-ncome; θ Consumer Demand the quantty of a market good (or of a bundle of goods) that the consumer purchases; q Income-Elastcty of Demand the partal frst dervatve of the logarthm of the Demand for a gven good (or bundle of goods), wth respect to the logarthm of ncome; ε Uncompensated Own Prce-Elastcty of Demand the partal frst dervatve of the logarthm of the Demand for a gven good (or bundle of goods) wth respect to the logarthm of the own prce of the good (or the average prce of a bundle of goods); η Uncompensated Cross Prce-Elastcty of Demand the partal frst dervatve of the logarthm of the Demand for a good (or bundle of goods) named A wth respect to the logarthm of the prce of a good (or the η j 19
average prce of a bundle of goods) named B. Preference-Independence the condton that the utlty assocated wth the consumpton of goods belongng to a gven bundle does not depend on the consumpton of goods belongng to a dfferent bundle; No symbol 20