Relationship between the Uncompensated Price Elasticity and the. Income Elasticity of Demand under Conditions of Additive Preferences

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1 Relatonshp between the Uncompensated Prce Elastcty and the Income Elastcty of Demand under Condtons of Addtve Preferences Lorenzo Sabatell, PhD GLOBMOD Health, Market Analyss Unt, Barcelona, Span Keywords: consumer choce, demand, ncome elastcty, prce elastcty, preference ndependence, forecastng, mathematcal model, mcroeconomcs, prcng, fnancal nstruments of polcy

2 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 ncome elastcty values and the uncompensated own and cross prce elastcty of demand are here derved 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 used 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. Introducton A change n the prce of a market good determnes a change n the purchasng power of consumers (ncome effect), and a change n the relatve prce of goods (substtuton effect). The aggregate consumer responsveness to changes n prce and n ncome s measured usng the (own and cross) prce elastcty and the ncome elastcty of demand, respectvely. 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. Contngency studes, e.g. wllngness-to-pay studes, are often used to elct the potental response of consumers, nevertheless they are not always fnancally and logstcally feasble, or consstent [1][2]. Unlke the prce elastcty, the ncome elastcty of demand can often be estmated from routnely collected data (e.g. from household 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. 2

3 The mathematcal relatonshp between demand and prce can be modeled usng the neoclasscal consumer theory, assumng a representatve economc agent wth preferences over consumpton goods, captured by a utlty functon [3]. The Rotterdam model, frst proposed by Barten (1964) [4] and Thel (1965) [5], bulds on ths approach, allowng for the estmaton of substtutes and complements, and separablty of preferences. The Rotterdam model produces constant margnal shares, a problem that can be avoded usng a demand functon called the almost deal demand system (AIDS) model [6], whch was subsequently extended by Thel, Chung, and Seale [7][8]. 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 Florda model [7][8]. 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, mathematcal relatonshps that allow the estmaton of the uncompensated own prce elastcty and of the cross prce elastcty of demand for ndependent bundles of goods are obtaned followng the dfferental approach used to derve the Florda model. The proposed equatons requre three nputs: the ncome-elastcty of demand, the 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 of Demand The defntons used throughout ths paper are reported n Table 1. The followng assumptons are made: 3

4 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. Theorem: Gven the assumptons I-II-III-IV, the mean value of the budget share (w) spent on each bundle, and the elastcty of the margnal utlty of ncome (r), 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 (h): η = ωε + ω ε ρ ρ () 1 and that the cross-prce elastcty (y) of demand for a bundle of goods () wth respect to the prce of a bundle (j) s: 1 ψ j = ωεε j j ωε j ρ ( 2) The parameter ρ s estmated analyzng surveys of subjectve happness [9], and ts value appears to be qute stable across dfferent geographc areas and populatons groups, wth an average value equals to and a standard devaton equals to 0.1. w s estmated from household surveys, and from standard market-research data. To take nto account the effect of parametrc uncertanty on model estmates, 4

5 credble ntervals for the estmates of the prce elastcty of demand are calculated va Monte-Carlo smulaton, drawng random model parameter values from normal (N) and unform (U) dstrbutons:: ( ρ ρ) ρ ~N μ, σ ( ) ω ~U ω, ω mn max ( 3) The Proof The dervaton 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: a) the relatve mportance of dfferent goods to the consumer; and (b) 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 s [7] and Barten s [4] 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 quanttes of goods that the consumer purchases from each bundle: 5

6 ( ) q= q1, q2..., qn and the vector contanng the average prces pad for the goods n each bundle : ( ) p = p1, p2..., pn. The followng vector notaton s used: d =,..., dq q qn n q p q p = = 1 Assumpton (I) mples that frst and second order dervatves of the utlty functon u exst, and that the Hessan matrx of u s symmetrc negatve. Assumpton II mples that: - Gven the budget constrant E = p q ( 4) The followng Lagrangan functon can be defned: ( ) μ ( ) ( 5) F = u q p q E - u can be maxmzed, subject to the budget constrant (4), by usng the Lagrangan multpler method, whch conssts n searchng the values of q such that the gradent of F s null and the Hessan of F s negatve defned. Dfferentatng u wth respect to q : du dq = μp ( 6) Combnng (6) wth the budget constrant (4), and followng the Barten s approach [4] the result can be wrtten n the followng parttoned matrx form: 6

7 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 [7], the general form of the dfferental demand system s obtaned: n 1 p ωdlog q = θd log Q + θjd log 8 * ρ j = 1 p j ( ) ( ) where q s the margnal share, and the followng equaton holds for the (Frsch) prce deflator: 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 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 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 log Q + θd log 10 * ρ p ( ) ( ) Step 2 Dfferentatng the budget constrant (4) yelds: ( log ) = ( log ) + ( log ) ( 11) d E d Q d P where: 7

8 d n ( logq) ω ( 12) = = 1 dq q d n ( log P) ω ( 13) = = 1 dp p If prces are kept constant, the followng equaton holds: ( log ) = ( log ) ( 14) d E d Q Snce r does not depend on ncome (assumpton IV), the ncome elastcty of demand ε 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). The uncompensated own prce elastcty of demand h 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 r does not depend on prces (assumpton IV), and gven Eq. (9), y s gven by: ( q ) d log 1 θ ω j 1 θ ψ j = = θ j θ + δj d ρω ω ρω ( log pj ) (16) where: 8

9 δ j 1 f = 0 f = j j Step 3 Obtanng the values of q from Eq. (15) and substtutng them nto Eq. (16) yelds: 1 δj ψj = ωεε j j ωε j + ε ρ ρ (17) Eq. (1) s equvalent to Eq. (17) for =j: η = ωε + ω ε ρ ρ and Eq. (2) s equvalent to Eq. (17) for j: 1 ψj = ωεε j j ωε j ρ 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. Results and Dscusson Analytcal calculatons show that the uncompensated own prce elastcty h and cross prce elastcty of demand y of ndependent bundles of goods can be expressed as functons (Eq. (1) and Eq. (2)) of the ncome elastcty of demand, the average budget share w, and the elastcty of margnal utlty of ncome r: 9

10 1 1 η = ωε + ω ε ρ ρ 2 A A A A A 1 ψ AB = ωεε B A B ωε B A ρ For nstance, for a bundle of goods A, f w s n the range 0.01%-10%, r s drawn from a normal dstrbuton wth mean equal to and standard error equal to 0.1, and ε s equal to 1, then Eq. (1) predcts a prce elastcty equal to -0.8 (wth the 95% credble nterval: -0.96, -0.64). The statstcal error assocated wth the prce elastcty estmates ncreases wth the value of the ncome elastcty (see Fg 1). The senstvty 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 therefore so does the wdth of credble ntervals assocated wth predctons. A unvarate senstvty analyss (whose results are not dsplayed), ndcates that the prce elastcty would change by less than 7% (relatve change), as the average budget share fluctuates between 0.01% and 10%, suggestng that for bundles of goods that account for less than 10% of the total expendture an approxmate estmate of the average budget share may be suffcent to produce relatvely accurate estmates of the own prce elastcty. 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 (Fg 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 absolute value of the elastcty of the margnal utlty of ncome, an ncrease n the prce of B wll determne a reducton n the demand for A. If the ncome elastcty of B s larger than the absolute value of the elastcty of the margnal utlty of ncome, an ncrease n the prce of B wll determne an ncrease n the demand for A. Ths s the effect of a reallocaton of the budget performed by the consumer to offset the consequences of a change n the prce 10

11 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 Fgs 2 and 3), and (unlke the case of the own prce elastcty) s rather 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 for estmatng the potental mpact of fnancal nstruments of polcy, e.g. subsdes and cost-sharng schemes, usng household survey data [10] [11], and for scenaro-analyss n early stage product prcng, f the relevant target s a bundle of goods that can be treated as 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 Florda model to real world data collected n natonal surveys of consumpton [8]. 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 Fg 4. Countryspecfc data can be found n S2Fg. The results show that the data are confned wthn the (funnelshaped) 95% credble nterval regon of model predctons, close to the medan value curve. Nevertheless, there are several assumptons and caveats that need careful consderaton. The accuracy of estmates based on the model here presented 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 reles on strong separablty assumptons whch are often not met n real markets. 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 11

12 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 [7][8][12][13]. In fact, the mathematcal framework here descrbed bulds on approaches already used n those studes, but wth dfferent objectves. For nstance, Barnett and Serlets [13] 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) [12] 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 publshed study has nvestgated the analytcal relatonshp between ncome elastcty and prce elastcty of demand. When compared wth the results of studes that have concurrently estmated prce and ncome elastcty of demand [8][14][15][16], the predctons of the model here presented appear consstent wth the avalable data (see Fg 4, S1.Fg, and S2.Fg). 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 ncome elastcty of demand, under condtons of addtve preferences, the proposed model provdes an effectve shortcut to forecast the mpact of prce changes on consumpton patterns. 12

13 Acknowledgments The author wshes to thank Prof. Dean T. Jamson (UCSF) for useful dscussons. The author acknowledges partal fnancal support from the Bll and Melnda Gates Foundaton through the Dsease Control Prortes Project (Department of Global Health of the Unversty of Washngton, Seattle). References 1. Cameron TA. Contngent valuaton. The New Palgrave Dctonary of Economcs Onlne 2ed; Knetsch JL, Snden JA. Wllngness to Pay and Compensaton Demanded: Expermental Evdence of an Unexpected Dsparty n Measures of Value. The Quarterly Journal of Economcs 1984; 99 (3): The MIT Press. 3. Deaton A. Consumer expendture The New Palgrave Dctonary of Economcs 2ed; Barten AP. Consumer Demand Functons Under Condtons of Almost Addtve Preferences. Econometrcs 1964; 32: Thel H. The Informaton Approach to Demand Analyss. Econometrca 1965, 33: Deaton A, Muellbauer J. An Almost Ideal Demand System, The Amercan Economc Revew 1980; 70 (3): Thel H. Chung C-F, Seale, JL Internatonal evdence on consumpton patterns. JAI Press, Inc. Appendx B:The Dfferental Approach to Consumpton Theory; Seale JL, Regm A, Bernsten J. Internatonal evdence on food consumpton patterns, Economc Research Servce, US Department of Agrculture; Layard R, Mayraz G, Nckell S. The margnal utlty of ncome. Journal of Publc Economcs 2008; 92:

14 10. Sabatell L, Jamson, DT. A model usng household ncome and household consumpton data to estmate the cost and the effectveness of subsdes: a modellng study usng cross-sectonal survey data. The Lancet 2013; 381: S Sabatell L. A. Methodology for Predctng the Impact of Co-payments on the Utlzaton of Health Technologes. Value n Health 2013; 16(7). 12. Brown MG, Lee J-Y. Journal of Agrcultural and Appled Economcs 2002; 34: Barnett WA. Serlets A. The dfferental approach to demand analyss and the Rotterdam model. Quantfyng consumer preferences, Contrbutons to Economc Analyss. Emerald Group Publshng Lmted, 61-82; 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 1987; 77 (3): Rngel, JS, Hosek SD, Vollaard BA. Mahnovsk S. 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; Santerre RE, Vernon JA. Assessng Consumer Gans From A Drug Prce Control Polcy In The Unted States. Southern Economc Journal 2006; 73:

15 Fgures Fg 1. Relatonshp between ncome elastcty and uncompensated own prce elastcty of demand. The darker lne (n the mddle) ndcates the medan of smulated values, whle the lghter external lnes defne the 95% credble nterval calculated usng a Monte-Carlo smulaton. The average budget share was drawn from a unform dstrbuton rangng from to 0.1, and the elastcty of the margnal utlty of ncome was drawn from a normal dstrbuton wth mean equal to and standard devaton equal to

16 Fg 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 the absolute value of the elastcty of the margnal utlty of ncome. The average budget share s equal to 0.05 and the elastcty of the margnal utlty of ncome s equal to

17 Fg 3. Quantfcaton of the uncertanty affectng the estmates of the cross prce elastcty of demand for a bundle of goods A, whth respect to the prce of B. In ths example, the ncome elastcty of the bundle of goods B s equal to 0.2, and the cross prce elastcty s plotted aganst the ncome elastcty of demand for A. The darker lne (n the mddle) ndcates the medan of smulated values, whle the lghter external lnes defne the 95% credble nterval calculated usng a Monte-Carlo smulaton. The average budget share was drawn from a unform dstrbuton rangng from to 0.1, and the elastcty of the margnal utlty of ncome was drawn from a normal dstrbuton wth mean equal to and standard devaton equal to

18 Fg 4. Comparson of smulaton results wth publshed estmates of ncome elastcty and uncompensated own prce elastcty of demand for clothng and footwear, educaton, healthcare, and recreaton. The estmates (colored crcles) n the three panels refer to low-ncome, mddle-ncome, and hgh-ncome countres and were obtaned fttng the Florda model to country survey data (source: Seale JL, Regm A, Bernsten J. Internatonal evdence on food consumpton patterns. Economc Research Servce; US Department of Agrculture; 2003). The darker lne (n the mddle) ndcates the medan of smulated values, whle the lghter external lnes defne the 95% credble nterval calculated usng a Monte-Carlo smulaton. The average budget share was drawn from a unform dstrbuton rangng from to 0.1, and the elastcty of the margnal utlty of ncome was drawn from a normal dstrbuton wth mean equal to and standard devaton equal to

19 Table 1. Defntons Concept 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 ncome the partal frst dervatve of the utlty functon wth respect to ncome μ Elastcty of the margnal utlty of ncome the partal frst dervatve of the logarthm of the margnal utlty of ncome, wth respect to the logarthm of ncome ρ 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 average prce of a bundle of goods) named B ψ Preference ndependence 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 19