Calibrating the Modified An Implicitly Directly Additive Demand System (MAIDADS) for Applied Economic Modeling

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1 Calbratng the Modfed An Implctly Drectly Addtve Demand System (MAIDADS) for Appled Economc Modelng Ken Itakura Domnque van der Mensbrugghe January 31, 2011 Prelmnary draft, not for ctaton, comments welcome Abstract The man objectve of ths paper s to make a good use of the recent study that modfed the AIDADS (An Implctly Drectly Addtve Demand System). The modfed AIDADS (MAIDADS) s able to capture varyng consumer behavors across dfferent expendture levels and also allows mnmum consumpton quanttes to change as utlty vares. These features would attract attenton from researchers conductng appled economc analyss whch nvolves consumers at wde range of expendture levels; consumers at developng and developed countres, households at dfferent ncome levels, and evoluton of expendture patterns over the long-run smulated often under CGE models, e.g. for 50 or 100 years from the benchmark year. The parameters n the MAIDADS are estmated wth the 2005 Internatonal Comparson Program dataset, and we obtaned elastctes as well as the mnmum consumpton quanttes for 147 countres and 12 expendture categores. Wth these estmaton results, the MAIDADS can be ncorporated nto appled economc models. Faculty of Economcs, Nagoya Cty Unversty, Japan (takura@econ.nagoya-cu.ac.jp) DECPG, The World Bank

2 1 Introducton Recent developments n the area of study on consumer demand system have enabled researchers conductng appled economc analyss to construct practcal and elaborated economc models. A good example can be found n the consumer demand system mplemented n Computable General Equlbrum (CGE) models. A wdely used specfcaton of the Cobb-Douglas demand system has been extended to the Lnear Expendture System (LES) by ncorporatng subsstence levels of consumpton goods. Rmmer and Powell (1992) developed An Implctly Drectly Addtve Demand system (AIDADS) whch allows margnal budget shares to vary as expendture changes. Ther generalzaton of the LES overcomes an emprcal weakness found n the LES, such as the constant margnal budget shares. Demand equaton for good n the AIDADS (Rmmer and Powell, 1992) can be expressed as: x = θ + α + β e u c (1 + e u ) p j p j θ j, (AIDADS) where x and p are consumpton quantty and prce of good, and total consumpton expendture s c, and level of utlty s u. Note that a parameter of subsstence consumpton quantty, θ, s constant as well as α and β. Restrctons on the parameters are 0 α, β 1, and α = β = 1, and 0 θ < x. MAIDADS Preckel, Cranfeld, and Hertel (2010) extends the consumer demand system developed by Rmmer and Powell (1992) by allowng subsstence consumpton quanttes, θ, to vary along wth utlty, thereby referred to as Modfed An Implctly Drectly Addtve Demand System (or MAIDADS). The constant parameter of subsstence consumpton quantty, θ n AIDADS s defned as a functon of utlty, u, wth nonnegatve parameters of δ, τ, ω. Thus, θ (u) = δ + τ e ωu 1 + e ωu. Note that the MAIDADS becomes AIDADS as δ s equal to τ as a specal case. 1

3 2 Calbraton of MAIDADS A system of the MAIDADS demand equatons s calbrated for each country, by solvng the mnmzaton problem specfed as follows. mn u,ω,α,β,δ,τ s.t. n ( ) 2 s 1 ŝ s = (δ + τ e ωu ) ˆp (1 + e ωu ) ĉ n α = 1 n β = 1 ˆx > δ + τ e ωu 1 + e ωu 0 ω 0 α, β, δ, τ 0. + α + β e u ĉ (1 + e u ) ĉ n j δ j + τ j e ωu ˆp j 1 + e ωu Prces, ˆp, and quanttes, ˆx, are obtaned from the ICP 2005 data (World Bank, 2008), and then consumpton expendture, ĉ = ˆp ˆx, and ntal shares, ŝ = ˆp ˆx /ĉ, are calculated. Ths mnmzaton problem was solved numercally. ICP 2005 data records 12 consumpton commodtes for 147 countes / regons, and Table 1 and Table 2 reports ther lstng respectvely. Table 1: Commodtes from ICP 2005 Data 1 Food and non-alcoholc beverages 7 Transport 2 Alcoholc beverages and tobacco 8 Communcaton 3 Clothng and footwear 9 Recreaton and culture 4 Housng, water, electrcty, gas 10 Educaton and other fuels 5 Furnshngs, household equpment 11 Restaurants and hotels and household mantenance 6 Health 12 Mscellaneous goods and servces Source: World Bank (2008) 2

4 Table 2: ICP 2005 Country / Regon Lstngs No. Code Name No. Code Name No. Code Name 1 ZAR Congo, Dem. Rep. 50 CPV Cape Verde 99 KAZ Kazakhstan 2 GNB Gunea-Bssau 51 PRY Paraguay 100 BLR Belarus 3 NER Nger 52 GNQ Equatoral Gunea 101 MEX Mexco 4 LBR Lbera 53 BOL Bolva 102 LBN Lebanon 5 MOZ Mozambque 54 SYR Syran Arab Republc 103 LVA Latva 6 CAF Central Afrcan Republc 55 AZE Azerbajan 104 MAC Macao, Chna 7 RWA Rwanda 56 SWZ Swazland 105 HRV Croata 8 MWI Malaw 57 NAM Namba 106 POL Poland 9 AGO Angola 58 BWA Botswana 107 LTU Lthuana 10 MLI Mal 59 GEO Georga 108 SVK Slovak Republc 11 GIN Gunea 60 FJI Fj 109 EST Estona 12 BFA Burkna Faso 61 GAB Gabon 110 KOR Korea, Rep. 13 TGO Togo 62 MDV Maldves 111 BHR Bahran 14 MDG Madagascar 63 MDA Moldova 112 HUN Hungary 15 SLE Serra Leone 64 ARM Armena 113 CZE Czech Republc 16 CIV Cote d Ivore 65 EG2 Egypt, Arab Rep. (AFR) 114 KWT Kuwat 17 GHA Ghana 66 EG1 Egypt, Arab Rep. (WAS) 115 BRN Brune Darussalam 18 BEN Benn 67 PER Peru 116 PRT Portugal 19 BGD Bangladesh 68 TUN Tunsa 117 SVN Slovena 20 NPL Nepal 69 JOR Jordan 118 QAT Qatar 21 MRT Maurtana 70 ECU Ecuador 119 SGP Sngapore 22 TCD Chad 71 WAS WESTERN ASIA 120 MLT Malta 23 SEN Senegal 72 COL Colomba 121 ISR Israel 24 NGA Ngera 73 ALB Albana 122 NZL New Zealand 25 CMR Cameroon 74 THA Thaland 123 GRC Greece 26 COG Congo, Rep. 75 VEN Venezuela, RB 124 OEE OECD-EUROSTAT 27 STP Sao Tome and Prncpe 76 SAM SOUTH AMERICA 125 ITA Italy 28 KEN Kenya 77 BRA Brazl 126 ESP Span 29 DJI Djbout 78 MNE Montenegro 127 FIN Fnland 30 LAO Lao PDR 79 TUR Turkey 128 CYP Cyprus 31 SDN Sudan 80 MYS Malaysa 129 TWN Tawan, Chna 32 AFR AFRICA 81 ZAF South Afrca 130 JPN Japan 33 YEM Yemen, Rep. 82 UKR Ukrane 131 IRL Ireland 34 KHM Camboda 83 WLD WORLD 132 HKG Hong Kong, Chna 35 IND Inda 84 MKD Macedona, FYR 133 DEU Germany 36 VNM Vetnam 85 BIH Bosna and Herzegovna 134 BEL Belgum 37 CHN Chna 86 URY Uruguay 135 DNK Denmark 38 LSO Lesotho 87 CIS CIS 136 AUS Australa 39 MAR Morocco 88 CHL Chle 137 FRA France 40 PAK Pakstan 89 SAU Saud Araba 138 SWE Sweden 41 IDN Indonesa 90 SRB Serba 139 NLD Netherlands 42 PHL Phlppnes 91 ARG Argentna 140 CAN Canada 43 ASP ASIA AND THE PACIFIC 92 OMN Oman 141 CHE Swtzerland 44 TJK Tajkstan 93 ROM Romana 142 AUT Austra 45 MNG Mongola 94 IRN Iran, Islamc Rep. 143 GBR Unted Kngdom 46 IRQ Iraq 95 BGR Bulgara 144 NOR Norway 47 LKA Sr Lanka 96 RU1 Russan Federaton (OEE) 145 ISL Iceland 48 KGZ Kyrgyz Republc 97 RU2 Russan Federaton (CIS) 146 LUX Luxembourg 49 BTN Bhutan 98 MUS Maurtus 147 USA Unted States Source: World Bank (2008) 3

5 3 Elastctes n MAIDADS Consumpton expendture elastctes, η, and uncompensated prce elastctes, ε j, can be obtaned from the MAIDADS demand equaton and an mplct functon defnng the relatonshp between total consumpton expendture and utlty: f(u, c) = where A s a constant. µ (u) = α + β e u (1 + e u ), c (u, c) = c p θ (u), θ (u) = δ + τ e ωu 1 + e ωu, ( ) µ (u) µ (u) ln + ln c (u, c) ln A u 1, (1) p Followng the dervaton steps n Rmmer and Powell (1992) and van der Mensbrugghe (2001), frstly the margnal budget share, ρ, s calculated from the MAIDADS demand equaton. x = θ (u) + µ (u) p c (u, c) (MAIDADS) x ρ = p c = p µ c p c + c µ p c + θ c µ p = µ + c = µ + c = µ + c c θ j p j c j c µ + p θ µ θ j p j j c eu (β α ) ωe ωu (τ δ ) (1 + e u ) 2 + p (1 + e ωu ) 2 µ ωe ωu (τ j δ j ) p j (1 + e j ωu ) 2 eu (β α ) ωe ωu (1 + e u ) 2 + p c (1 + e ωu ) 2 (τ δ ) µ p j (τ j δ j ) = µ + c c Ψ. (2) Total consumpton expendture and utlty are combned n an mplct form, and thus the term, / c, n the equaton (2) s calculated by applyng the mplct functon j 4

6 theorem to the MAIDADS dentty defned n (1). c = f c [ f = 1 c [ f ] 1 ] 1 = 1 e [ 1 u c + (1 + e u ) 2 ( (β α ) ln ( µ p ) ) + 1 ωe ωu c (1 + e ωu ) 2 p (τ δ ) = 1 e [ 1 u ωe ωu c + (1 + e u ) 2 (β α ) ln (x θ ) c (1 + e ωu ) 2 p (τ δ ) = 1 c Ω Therefore, we can obtan the expresson for the margnal budget share and the consumpton expendture elastcty as: ρ = µ Ψ Ω, η = x c = x c = ρ s c x c x p p ] 1 = 1 s (µ Ψ Ω). (3) ] 1 Uncompensated prce elastctes, ε j, are also derved from the MAIDADS demand equaton. ε j = x p j p j = p j x x [ ( µ c p 2 φ j µ ) ] θ j + c µ + 1 θ µ θ k p k p p p j p j p p j k p 2 φ j µ ) ( ) θ j + p j c µ p x p j p + 1 θ µ θ k p k p k ) ( + p j c x p j = p j ( µ c x = p j ( µ c x p 2 φ j µ θ j p p e u (β α ) (1 + e u ) 2 + ωeωu (τ δ ) (1 + e ωu ) 2 µ p k ) ωe ωu (τ k δ k ) p k (1 + e ωu ) 2 5

7 = p j ( µ c x p 2 ( µ c = p j x p 2 φ j µ ) θ j + c p j p p x p j φ j µ θ j p [ e u (β α ) (1 + e u ) 2 + ( )] ωe ωu c (1 + e ωu ) 2 p (τ δ ) µ p k (τ k δ k ) ) + c p j p x p j Ψ (4) k where 1 = j φ j = 0 j. as; Applyng the mplct functon theorem on / p j, we wll have the expresson such = f p j p j = [ f ( µj + θ j p j c ] 1 ) [ f Therefore, the uncompensated prce elastctes are ] 1 = s jc Ω. (5) p j c 4 Calbraton Results ε j = p j ( µ c x p 2 φ j µ ) θ j + c p j s j c p p x p j c Ψ Ω = µ s c (p jθ j + φ j c ) + s j Ψ Ω s = µ c s c (µ j φ j ) s j η (6) Consumpton expendture shares of 12 commodtes for 147 countres / regons are calculated from the ICP 2005 data, and the Fgure 1 reports the results. The panel (a) plots the ponts for food and non-alcoholc beverages, alcoholc beverages and tobacco, and clothng and footwear. It s remarkable that the share of food expendture rapdly declnes as total consumpton expendture ncreases along the horzontal axs. smoothed lnes n the panels are drawn to trace ther statstcal trend. Once numercally solved the mnmzaton problem for each country / regon, all the parameters as well as the level of utlty are obtaned to compute the consumpton expendture elastctes. Fgure 2 report the results. Note that there are 7 countres The 6

8 excluded from the fgure because of ther elastctes becomng negatve (Gunea-Bssau, Madagascar, Bangladesh, Sudan, Pakstan, Bhutan, and Syran Arab Republc). In the panel (a), expendture elastctes of food ndcate a ncreasng trend at the hgher expendture levels. Fgure 3 reveals that the rato of the mnmum consumpton quantty to the level of consumpton becomes lower for most of the commodtes, as expendture ncreases. However, there are exceptons for housng, health, mscellaneous goods. Havng calbrated the MAIDADS and computed elastctes, t s possble to project the expendture shares. To make smple projectons, we set the path of the total consumpton expendture as: c t+1 = (1 + r/100)c t, for t = 1,..., 99, r = 3. Fgure 4 shows the results for selected countes; Vetnam, Chna, and Japan. References van der Mensbrugghe, Domnque (2001), Lnkage Techncal Reference Document, Economc Polcy and Prospects Group, The World Bank, org/prospects/pubs/techref.pdf. Preckel, Paul V., J.A.L. Cranfeld, and Thomas W. Hertel (2010), A Modfed, Implct, Drectly Addtve Demand System, Appled Economcs, 42(2), Rmmer, Maureen T. and Alan A. Powell (1992), An Implctly Drectly Addtve Demand System: Estmates for Australa, IMPACT Project Workng Paper, (OP- 73). World Bank (2008), Global Purchasng Power Partes and Real Expendtures 2005 Internatonal Comparson Program, Washngton, D.C. 7

9 s1 s2 s s4 s5 s (a) 1: Food, 2: Alcoholc bev., 3: Clothng (b) 4: Housng, 5: Furnshngs, 6: Health s7 s8 s9 s10 s11 s (c) 7: Trans., 8: Communcaton, 9: Recreaton (d) 10: Educaton, 11: Restaurants, 12: Msc. Source: World Bank (2008) and authors calculatons Fgure 1: Observed shares and expendtures (ponts), and smoothed (lnes) 8

10 eta1 eta2 eta3 2 eta4 eta5 eta6 1 1 (a) 1: Food, 2: Alcoholc bev., 3: Clothng (b) 4: Housng, 5: Furnshngs, 6: Health eta7 eta8 eta9 2 eta10 eta11 eta (c) 7: Trans., 8: Communcaton, 9: Recreaton (d) 10: Educaton, 11: Restaurants, 12: Msc. Source: Calbraton results Fgure 2: Expendture elastctes (ponts), and smoothed (lnes) 9

11 theta1/x1 theta2/x2 theta3/x3 0.4 theta4/x4 theta5/x5 theta6/x (a) 1: Food, 2: Alcoholc bev., 3: Clothng (b) 4: Housng, 5: Furnshngs, 6: Health theta7/x7 theta8/x8 theta9/x theta10/x10 theta11/x11 theta12/x (c) 7: Trans., 8: Communcaton, 9: Recreaton (d) 10: Educaton, 11: Restaurants, 12: Msc. Source: Calbraton results Fgure 3: θ (u)/x and expendtures (ponts) and smoothed (lnes) 10

12 s1 s2 s3 s4 s5 s s7 s8 s9 s10 s11 s (a) Vetnam s1- s6 (b) Vetnam s7 - s s1 s2 s3 s4 s5 s s7 s8 s9 s10 s11 s (c) Chna s1- s6 (d) Chna s7- s s1 s2 s3 s4 s5 s s7 s8 s9 s10 s11 s (e) Japan s1- s6 (f) Japan s7- s12 Note: s1: food, s2: alcoholc bev., s3: clothng, 4: housng, 5: furnshngs, 6: health, 7: trans., 8: communcaton, 9: recreaton, 10: educaton, 11: restaurants, 12: msc. Source: Authors calculatons Fgure 4: Projected shares and expendtures for selected countes 11

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