Calibrating the Modified An Implicitly Directly Additive Demand System (MAIDADS) for Applied Economic Modeling Ken Itakura∗ Dominique van der Mensbrugghe† January 31, 2011
Preliminary draft, not for citation, comments welcome Abstract The main objective of this paper is to make a good use of the recent study that modified the AIDADS (An Implicitly Directly Additive Demand System). The modified AIDADS (MAIDADS) is able to capture varying consumer behaviors across different expenditure levels and also allows minimum consumption quantities to change as utility varies. These features would attract attention from researchers conducting applied economic analysis which involves consumers at wide range of expenditure levels; consumers at developing and developed countries, households at different income levels, and evolution of expenditure patterns over the long-run simulated often under CGE models, e.g. for 50 or 100 years from the benchmark year. The parameters in the MAIDADS are estimated with the 2005 International Comparison Program dataset, and we obtained elasticities as well as the minimum consumption quantities for 147 countries and 12 expenditure categories. With these estimation results, the MAIDADS can be incorporated into applied economic models.
∗ †
Faculty of Economics, Nagoya City University, Japan (
[email protected]) DECPG, The World Bank
1
Introduction
Recent developments in the area of study on consumer demand system have enabled researchers conducting applied economic analysis to construct practical and elaborated economic models. A good example can be found in the consumer demand system implemented in Computable General Equilibrium (CGE) models. A widely used specification of the Cobb-Douglas demand system has been extended to the Linear Expenditure System (LES) by incorporating subsistence levels of consumption goods. Rimmer and Powell (1992) developed An Implicitly Directly Additive Demand system (AIDADS) which allows marginal budget shares to vary as expenditure changes. Their generalization of the LES overcomes an empirical weakness found in the LES, such as the constant marginal budget shares. Demand equation for good i in the AIDADS (Rimmer and Powell, 1992) can be expressed as: ∑ αi + βi eu xi = θi + c− pj θ j , (1 + eu ) pi
(AIDADS)
j
where xi and pi are consumption quantity and price of good i, and total consumption expenditure is c, and level of utility is u. Note that a parameter of subsistence consumption quantity, θi , is constant as well as αi and βi . Restrictions on the parameters ∑ ∑ are 0 ≤ αi , βi ≤ 1, and i αi = i βi = 1, and 0 ≤ θi < xi . MAIDADS Preckel, Cranfield, and Hertel (2010) extends the consumer demand system developed by Rimmer and Powell (1992) by allowing subsistence consumption quantities, θi , to vary along with utility, thereby referred to as Modified An Implicitly Directly Additive Demand System (or MAIDADS). The constant parameter of subsistence consumption quantity, θi in AIDADS is defined as a function of utility, u, with nonnegative parameters of δi , τi , ω. Thus, θi (u) =
δi + τi eωu . 1 + eωu
Note that the MAIDADS becomes AIDADS as δi is equal to τi as a special case.
1
2
Calibration of MAIDADS
A system of the MAIDADS demand equations is calibrated for each country, by solving the minimization problem specified as follows.
min
n ( ∑ si
u,ω,α,β,δ,τ
s.t.
sˆi
i
)2 −1
n ∑ δj + τj eωu (δi + τi eωu ) pˆi αi + βi eu si = + cˆ − pˆj (1 + eωu ) cˆ (1 + eu ) cˆ 1 + eωu j
n ∑ i n ∑
αi = 1 βi = 1
i
x ˆi >
δi + τi eωu ≥0 1 + eωu
ω≥0 αi , βi , δi , τi ≥ 0
∀i.
Prices, pˆ, and quantities, x ˆ, are obtained from the ICP 2005 data (World Bank, 2008), ∑ and then consumption expenditure, cˆ = i pˆi x ˆi , and initial shares, sˆi = pˆi x ˆi /ˆ c, are calculated. This minimization problem was solved numerically. ICP 2005 data records 12 consumption commodities for 147 counties / regions, and Table 1 and Table 2 reports their listing respectively. Table 1: Commodities from ICP 2005 Data 1 2 3 4 5 6
Food and non-alcoholic beverages Alcoholic beverages and tobacco Clothing and footwear Housing, water, electricity, gas and other fuels Furnishings, household equipment and household maintenance Health
7 8 9 10
Transport Communication Recreation and culture Education
11
Restaurants and hotels
12
Miscellaneous goods and services
Source: World Bank (2008)
2
Table 2: ICP 2005 Country / Region Listings No.
Code
Name
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
ZAR GNB NER LBR MOZ CAF RWA MWI AGO MLI GIN BFA TGO MDG SLE CIV GHA BEN BGD NPL MRT TCD SEN NGA CMR COG STP KEN DJI LAO SDN AFR YEM KHM IND VNM CHN LSO MAR PAK IDN PHL ASP TJK MNG IRQ LKA KGZ BTN
Congo, Dem. Rep. Guinea-Bissau Niger Liberia Mozambique Central African Republic Rwanda Malawi Angola Mali Guinea Burkina Faso Togo Madagascar Sierra Leone Cote d’Ivoire Ghana Benin Bangladesh Nepal Mauritania Chad Senegal Nigeria Cameroon Congo, Rep. Sao Tome and Principe Kenya Djibouti Lao PDR Sudan AFRICA Yemen, Rep. Cambodia India Vietnam China Lesotho Morocco Pakistan Indonesia Philippines ASIA AND THE PACIFIC Tajikistan Mongolia Iraq Sri Lanka Kyrgyz Republic Bhutan
No.
Code
Name
No.
Code
Name
50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98
CPV PRY GNQ BOL SYR AZE SWZ NAM BWA GEO FJI GAB MDV MDA ARM EG2 EG1 PER TUN JOR ECU WAS COL ALB THA VEN SAM BRA MNE TUR MYS ZAF UKR WLD MKD BIH URY CIS CHL SAU SRB ARG OMN ROM IRN BGR RU1 RU2 MUS
Cape Verde Paraguay Equatorial Guinea Bolivia Syrian Arab Republic Azerbaijan Swaziland Namibia Botswana Georgia Fiji Gabon Maldives Moldova Armenia Egypt, Arab Rep. (AFR) Egypt, Arab Rep. (WAS) Peru Tunisia Jordan Ecuador WESTERN ASIA Colombia Albania Thailand Venezuela, RB SOUTH AMERICA Brazil Montenegro Turkey Malaysia South Africa Ukraine WORLD Macedonia, FYR Bosnia and Herzegovina Uruguay CIS Chile Saudi Arabia Serbia Argentina Oman Romania Iran, Islamic Rep. Bulgaria Russian Federation (OEE) Russian Federation (CIS) Mauritius
99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147
KAZ BLR MEX LBN LVA MAC HRV POL LTU SVK EST KOR BHR HUN CZE KWT BRN PRT SVN QAT SGP MLT ISR NZL GRC OEE ITA ESP FIN CYP TWN JPN IRL HKG DEU BEL DNK AUS FRA SWE NLD CAN CHE AUT GBR NOR ISL LUX USA
Kazakhstan Belarus Mexico Lebanon Latvia Macao, China Croatia Poland Lithuania Slovak Republic Estonia Korea, Rep. Bahrain Hungary Czech Republic Kuwait Brunei Darussalam Portugal Slovenia Qatar Singapore Malta Israel New Zealand Greece OECD-EUROSTAT Italy Spain Finland Cyprus Taiwan, China Japan Ireland Hong Kong, China Germany Belgium Denmark Australia France Sweden Netherlands Canada Switzerland Austria United Kingdom Norway Iceland Luxembourg United States
Source: World Bank (2008)
3
3
Elasticities in MAIDADS
Consumption expenditure elasticities, ηi , and uncompensated price elasticities, εij , can be obtained from the MAIDADS demand equation and an implicit function defining the relationship between total consumption expenditure and utility: f (u, c) =
∑
( µi (u) ln
i
µi (u) pi
)
+ ln c∗ (u, c) − ln A − u ≡ 1,
(1)
αi + βi eu , (1 + eu ) ∑ c∗ (u, c) = c − pi θi (u) , µi (u) =
i
δi + τi eωu , θi (u) = 1 + eωu where A is a constant. Following the derivation steps in Rimmer and Powell (1992) and van der Mensbrugghe (2001), firstly the marginal budget share, ρi , is calculated from the MAIDADS demand equation. xi = θi (u) + ρi = pi
∂xi ∂c
µi (u) ∗ c (u, c) pi
(MAIDADS)
∂θ ∂µi ∂u ∂θi ∂u µi µi j ∂u + + − pj = pi pi ∂c pi ∂u ∂c ∂u ∂c pi ∂u ∂c j ∑ ∂θj ∂u ∗ ∂µi ∂θi = µi + c + pi − µi pj ∂c ∂u ∂u ∂u j ωu u ωu ∑ ωe (τj − δj ) ∂u ∗ e (βi − αi ) ωe (τi − δi ) = µi + c + pi − µi pj 2 2 u ωu ∂c (1 + e ) (1 + e ) (1 + eωu )2 j u ωu ∑ ∂u e (βi − αi ) ωe pi (τi − δi ) − µi = µi + c∗ + pj (τj − δj ) 2 u ∗ ∂c (1 + e ) c (1 + eωu )2 j ∂c∗
= µi + c∗
c∗
∂u Ψi . ∂c
∑
(2)
Total consumption expenditure and utility are combined in an implicit form, and thus the term, ∂u/∂c, in the equation (2) is calculated by applying the implicit function
4
theorem to the MAIDADS identity defined in (1). [ ] ∂u ∂f ∂f −1 =− ∂c ∂c ∂u [ ] 1 ∂f −1 =− ∗ c ∂u ]−1 [ ( ( ) ) ∑ ∑ µi 1 ωeωu eu (βi − αi ) ln pi (τi − δi ) = − ∗ −1 + +1 − c pi (1 + eu )2 i c∗ (1 + eωu )2 i ]−1 [ ∑ ∑ 1 eu ωeωu = − ∗ −1 + (βi − αi ) ln (xi − θi ) − pi (τi − δi ) c (1 + eu )2 i c∗ (1 + eωu )2 i 1 = − ∗Ω c Therefore, we can obtain the expression for the marginal budget share and the consumption expenditure elasticity as: ρi = µi − Ψi Ω, ∂xi c ∂c xi ∂xi c pi = ∂c xi pi ρi = si 1 = (µi − Ψi Ω) . si
ηi =
(3)
Uncompensated price elasticities, εij , are also derived from the MAIDADS demand equation.
∂xi pj ∂pj xi [( ] ) pj µi θj µi c∗ c∗ ∂µi ∂u ∂θi ∂u µi ∑ ∂θk ∂u = − 2 φij − + +1 − pk xi pi pi ∂u ∂pj ∂u ∂pj pi ∂u ∂pj pi k ( ) ( ) pj µi θ j pj ∂u c∗ ∂µi µi c∗ ∂θi µi ∑ ∂θk = − 2 φij − + +1 − pk xi pi xi ∂pj pi ∂u ∂u pi ∂u pi k ( ) ( ) µi θ j pj ∂u c∗ eu (βi − αi ) ωeωu (τi − δi ) µi ∑ ωeωu (τk − δk ) pj µi c∗ − 2 φij − + pk = + − xi pi xi ∂pj pi (1 + eu )2 pi pi (1 + eωu )2 (1 + eωu )2 k
εij =
5
pj = xi pj = xi
( (
µi θ j µi c∗ − 2 φij − pi pi µi θ j µi c∗ − 2 φij − pi pi
) )
c∗ pj ∂u + pi xi ∂pj +
[
eu (βi − αi ) ωeωu + (1 + eu )2 c∗ (1 + eωu )2
(
c∗ pj ∂u Ψi pi xi ∂pj
k
(4)
1 φij = 0
where
pi (τi − δi ) − µi
i=j i 6= j.
Applying the implicit function theorem on ∂u/∂pj , we will have the expression such as; [ ] ∂u ∂f ∂f −1 =− ∂pj ∂pj ∂u ( ) [ ]−1 θj µj ∂f + ∗ = pj c ∂u sj c = Ω. pj c∗
(5)
Therefore, the uncompensated price elasticities are
pj εij = xi
(
µi θj µi c∗ − 2 φij − pi pi
) +
c∗ pj sj c Ψi Ω pi xi pj c∗
sj µi (pj θj + φij c∗ ) + Ψi Ω si c si ∗ µi c =− (µj − φij ) − sj ηi si c =−
4
(6)
Calibration Results
Consumption expenditure shares of 12 commodities for 147 countries / regions are calculated from the ICP 2005 data, and the Figure 1 reports the results. The panel (a) plots the points for food and non-alcoholic beverages, alcoholic beverages and tobacco, and clothing and footwear. It is remarkable that the share of food expenditure rapidly declines as total consumption expenditure increases along the horizontal axis. The smoothed lines in the panels are drawn to trace their statistical trend. Once numerically solved the minimization problem for each country / region, all the parameters as well as the level of utility are obtained to compute the consumption expenditure elasticities. Figure 2 report the results. Note that there are 7 countries
6
∑
)] pk (τk − δk )
excluded from the figure because of their elasticities becoming negative (Guinea-Bissau, Madagascar, Bangladesh, Sudan, Pakistan, Bhutan, and Syrian Arab Republic). In the panel (a), expenditure elasticities of food indicate a increasing trend at the higher expenditure levels. Figure 3 reveals that the ratio of the minimum consumption quantity to the level of consumption becomes lower for most of the commodities, as expenditure increases. However, there are exceptions for housing, health, miscellaneous goods. Having calibrated the MAIDADS and computed elasticities, it is possible to project the expenditure shares. To make simple projections, we set the path of the total consumption expenditure as: ct+1 = (1 + r/100)ct , for t = 1, ..., 99, r = 3. Figure 4 shows the results for selected counties; Vietnam, China, and Japan.
References van der Mensbrugghe, Dominique (2001), “Linkage Technical Reference Document,” Economic Policy and Prospects Group, The World Bank, http://www.worldbank. org/prospects/pubs/TechRef.pdf. Preckel, Paul V., J.A.L. Cranfield, and Thomas W. Hertel (2010), “A Modified, Implicit, Directly Additive Demand System,” Applied Economics, 42(2), 143–155. Rimmer, Maureen T. and Alan A. Powell (1992), “An Implicitly Directly Additive Demand System: Estimates for Australia,” IMPACT Project Working Paper, (OP73). World Bank (2008), Global Purchasing Power Parities and Real Expenditures – 2005 International Comparison Program, Washington, D.C.
7
0.6 0.25 0.5
0.20 0.4 Comm.
Comm.
s1 0.3
s4
0.15
s2
s5
s3
s6
0.2
0.10
0.1
0.05
5
6
7
8
9
10
5
(a) 1: Food, 2: Alcoholic bev., 3: Clothing
6
7
8
9
10
(b) 4: Housing, 5: Furnishings, 6: Health
0.15
0.15
0.10
Comm.
0.10
Comm.
s7
s10
s8
s11
s9
s12
0.05 0.05
0.00 0.00 5
6
7
8
9
10
5
(c) 7: Trans., 8: Communication, 9: Recreation
6
7
8
9
10
(d) 10: Education, 11: Restaurants, 12: Misc.
Source: World Bank (2008) and authors’ calculations
Figure 1: Observed shares and expenditures (points), and smoothed (lines)
8
4 4
3 3
Comm.
Comm.
eta1
eta4
eta2 2
eta5
2
eta3
eta6
1
1
5
6
7
8
9
10
5
(a) 1: Food, 2: Alcoholic bev., 3: Clothing
6
7
8
9
10
(b) 4: Housing, 5: Furnishings, 6: Health 4
5
4 3
3 Comm.
Comm.
eta7
eta10
2
eta8
eta11
eta9
eta12
2
1 1
0 5
6
7
8
9
10
5
(c) 7: Trans., 8: Communication, 9: Recreation
6
7
8
9
10
(d) 10: Education, 11: Restaurants, 12: Misc.
Source: Calibration results
Figure 2: Expenditure elasticities (points), and smoothed (lines)
9
1.0
0.8
0.8
0.6
0.6 Comm.
Comm.
theta1/x1
theta4/x4
theta2/x2
0.4
0.2
0.2
0.0
0.0
5
6
7
8
9
theta5/x5
0.4
theta3/x3
10
theta6/x6
5
(a) 1: Food, 2: Alcoholic bev., 3: Clothing
6
7
8
9
10
(b) 4: Housing, 5: Furnishings, 6: Health
1.0
0.8
0.8
0.6 0.6 Comm.
Comm.
0.4
theta7/x7
theta10/x10
theta8/x8
0.4
theta11/x11
theta9/x9
theta12/x12 0.2
0.2 0.0
0.0
-0.2
5
6
7
8
9
10
5
(c) 7: Trans., 8: Communication, 9: Recreation
6
7
8
9
10
(d) 10: Education, 11: Restaurants, 12: Misc.
Source: Calibration results
Figure 3: θi (u)/xi and expenditures (points) and smoothed (lines)
10
0.14
0.30 s1 s2 s3 s4 s5 s6
0.25 0.20 0.15
s7 s8 s9 s10 s11 s12
0.12 0.10 0.08 0.06
0.10
0.04
0.05
0.02
1000
2000
3000
4000
5000
6000
7000
1000
2000
(a) Vietnam s1- s6
3000
4000
5000
6000
7000
(b) Vietnam s7 - s12 0.14 s1 s2 s3 s4 s5 s6
0.20
0.15
s7 s8 s9 s10 s11 s12
0.12 0.10 0.08
0.10
0.06 0.04
0.05 0.02 0
2000
4000
6000
8000
10 000
12 000
14 000
0
2000
4000
(c) China s1- s6
6000
8000
10 000
12 000
14 000
(d) China s7- s12 0.14
0.20
0.12
s1 s2 s3 s4 s5 s6
0.15
0.10
s7 s8 s9 s10 s11 s12
0.10 0.08 0.06 0.04
0.05
0.02
0
100 000
200 000
300 000
400 000
0
(e) Japan s1- s6
100 000
200 000
300 000
400 000
(f) Japan s7- s12
Note: s1: food, s2: alcoholic bev., s3: clothing, 4: housing, 5: furnishings, 6: health, 7: trans., 8: communication, 9: recreation, 10: education, 11: restaurants, 12: misc. Source: Authors’ calculations
Figure 4: Projected shares and expenditures for selected counties
11
