Income Elastcty and Functonal Form Tmothy Beatty and Jeffery T. LaFrance January 2004 Workng Paper Number: 2004-04 Food and Resource Economcs, Unversty of Brtsh Columba Vancouver, Canada, V6T Z4 http://www.agsc.ubc.ca/fre
Income Elastcty and Functonal Form Tmothy K.M. Beatty and Jeffrey T. LaFrance Department of Agrcultural and Resource Economcs Unversty of Calforna, Berkeley Berkeley, CA 94720-330 Abstract: A smple, utlty theoretc, demand model whch nests the both the functonal form of ncome and prces s presented. Ths model s used to calculate the ncome elastctes of twenty-one food tems over the course of the last century. Keywords: Functonal Form; Income Elastcty; PIGLOG; Quadratc Utlty JEL Classfcaton: C3;C5
Introducton The purpose of ths paper s to emphasze the mpact of functonal form on estmates of ncome elastcty for twenty-one foods over the course of the last century. We use a theoretcally consstent emprcal model of household food consumpton that: () nests the functonal form of the ncome terms n demand equatons; (2) nests the functonal form of the prce term n demand equatons. We wll then show that exstng models, whch ntegrate prces and ncome ether lnearly or n logarthmc form, tend to overstate the sze and the varablty of the ncome elastcty for most of the twenty-one foods. 2 Data In order to answer the queston posed above, we wll employ three dfferent tme seres data sets. The frst s data on per capta consumpton of food tems and ther correspondng prces. Currently, ths data set conssts of annual tme seres observatons over the perod 909-995. Per capta consumpton of twenty-one food tems and correspondng average retal prces for those tems were constructed from several USDA and Bureau of Labor Statstcs sources. The second data seres are demographc factors that help explan the evolvng pattern of demands. These demographc factors nclude the frst three central moments (mean, varance, and skewness) of the age dstrbuton and the proportons of the U.S. populaton that are Black and nether Black nor Whte. The thrd data seres nvolves the U.S. ncome dstrbuton. The Bureau of the Census publshes annually quntle ranges, ntra-quntle means, the top fve-percentle lower bound for ncome, and the mean ncome wthn the top fve-percentle range for all U.S. famles. 3 Modelng the demand for food We start wth a theoretcally consstent reduced form econometrc model of n q -vector of
demands for food tems wth condtonal mean gven by, Eq ( pmd,, ) = hpmd (,, ), where q s an n q -vector of food quanttes, p s an n q -vector of food prces, m s ncome and d s a k-vector demographc characterstcs. Let x denote the scalar varable for total consumer expendtures on all nonfood tems. Assume that each of the prces for ndvdual food tems and ncome are deflated by a prce ndex measurng the cost of nonfood tems. Consder the Gorman Polar Form (Gorman 96) for the (quas-) ndrect utlty functon generated by a quadratc (quas-)utlty functon, ( m α( d) p α0( d)) v( p, m, d) =, pbp +γ 0 where α(d) sann q -vector of functons of the demographc varables, α 0 (d) s a scalar functon of the demographc varables, B s an n q n q matrx of parameters and γ 0 s a scalar parameter. For dentfcaton purposes, we choose the normalzaton γ 0 =. Applyng Roy s dentty to ths (quas-) ndrect utlty functon generates a system of demands. ( m α( d) p α0( d)) Eq ( pmd,, ) =α+ Bp. () ( pbp + ) Next, we defne Box-Cox transformatons for m and p by m( κ)=(m -)/ κ and p ( λ ) = ( ) λ,for =,,n q, wth p(λ) [p (λ),, p n (λ)], and replace m and p p λ wth m(κ) andp(λ), respectvely, n (). Applyng Roy s dentty to the resultng (quas-) ndrect utlty functon then gves a demand system that can be wrtten n expendture form as, κ λ κ m( ) ( d) p( ) 0( d) Ee ( pmd,, ) = Pm α ( d) + κ α λ α Bp( λ), (2) p( λ) Bp( λ ) + 2
where e= [ pq pnq n] s the n q -vector of (deflated) expendtures on the food tems q and P= dag[ p ]. Equaton 2 forms the bass of our analyss of the effect of functonal form on the ncome elastctes of food groups over the course of the last century. The fundamental questons addressed wll concern the estmated values of the Box-Cox parameters. In partcular, how these departures from the PIGLOG ( κ= 0, λ= 0) and quadratc utlty form ( κ=, λ= ) affect the estmates of the ncome elastctes of the twenty-one food tems. 4 Instruments for the Moments of the U.S. Income Dstrbuton The demand model descrbed above s nonlnear n ncome. Therefore, the demand equatons do not aggregate drectly across ndvduals to average ncome at the market level. The advantage of usng the Gorman class of Engel curves s that to generate a theoretcally consstent, aggregable model of demand, only a lmted number of statstcs concernng the ncome dstrbuton are needed. The demand model proposed n ths paper requres two moments of the ncome dstrbuton, specfcally those assocated wth and m. m κ For the ncome dstrbuton defned by the densty functon f( m ), m R +,wewant to calculate the smplest possble nformaton theoretc densty for ncome condtonal on the nformaton that ncome falls wthn a gven range, say, m ( l, l ],suchasthe th quntle wth gven probablty Pr{ m ( l, l ]} =π, and wth condtonal mean ncome Emm { ( l, l ]} =µ. To do so, we choose two equal subntervals n each range, so that the probablty densty functon has a jump at the mdpont of that range, l = ( l + l )/2 as well as each boundary pont, l.on ( l, l] ths densty functon 3
satsfes, f m ( ) 3 ( l + l ) 4 4 µ, m ( l, ( ) l + l 2 ( l l ) 2 π =. ( l ) 2 l 3 ( ) µ l + l 4 4, m ( ), ( l + l 2 l ( l + l ) 2 The formal dervaton of ths ncome densty (among others) and ts propertes are derved n Lafrance, Beatty, Pope and Agnew. 5 Emprcal Results We estmate equaton 2 usng a two stage SUR procedure usng nonlnear least squares. Of crucal mportance are the pont estmates for the Box-Cox terms on ncome κ and the Box-Cox term on prces λ. Table shows us that the Box-Cox coeffcents on ncome and prces are both sgnfcantly dfferent from zero. Addtonal hypothess tests show that each coeffcent s sgnfcantly dfferent from one, jontly dfferent from zero and jontly dfferent from one. All of these tests had p-values numercally equal to z.ero. Table. Estmates of the Box-Cox Parameters Box Cox Coeffcent Pont Estmate Standard Error P-Value Income ( κ ).88649.06988 0.0000 Prces ( λ ).794752.08073 0.0000 The man result of ths paper can clearly be seen n Fgure. If the system of food demands were to be estmated usng κ= 0, λ= 0, whch results n a PIGLOG specfcaton, one would erroneously conclude that the ncome elastcty of mlk has declned pre- 4
cptously over the course of the last century. Conversely f the system were to be estmated usng the κ=, λ=, whch results n a quadratc utlty specfcaton, one would conclude that the ncome elastcty of mlk had n fact ncreased over the course of the last century. Ether of these models mght lead a researcher to conclude that there has been some form of structural change n the demand for mlk over the course of the last century. However, the model proposed n ths paper shows that the ncome elastcty of mlk has only changed slghtly over the perod movng from slghtly postve to slghtly negatve. Fgure. Income Elastcty of Mlk..0 Income Elastcty of Mlk 99-94, 947-995 0.8 PIGLOG 0.6 Elastcty 0.4 0.2 Quadratc Utlty 0.0 Box-Cox -0.2 War Years -0.4 920 930 940 950 960 970 980 990 Year In general, both the PIGLOG and quadratc utlty specfcatons tend to overstate the sze of the ncome elastctes of food. In addton, the PIGLOG and quadratc utlty 5
models mply that the ncome elastcty of food has vared consderably over the last century. Table 2 reports summary statstcs of the ncome elastctes of food over the entre sample perod. We see that for ffteen of twenty-one foods the standard devaton of the ncome elastcty of the approxmate PIGLOG and the quadratc utlty models are greater than the standard devaton for the model where κ and λ are estmated. In addton we note that the range of the ncome elastctes s greater for the PIGLOG and quadratc utlty models than the case n whch κ and λ are estmated n most cases. 6
Table 2. Summary Statstcs of the Income Elastctes. Mlk 0.38397 0.32437 Butter 0.36399 0.529 Cheese.34438 0.6952 Frozen Dary.05307 0.0546 Powdered Mlk 0.88573 0.6333 Beef and Veal 0.82642 0.0652 Pork 0.90688 0.056958 Other Red Meat.3058 0.2523 Fsh.0534 0.083945 Poultry 0.8565 0.0772 Fresh Ctrus 0.90935 0.0902 Fresh Nonctrus 0.45747 0.26233 Fresh Vegetables 0.5342 0.0877 Potatoes 0.704 0.087866 Processed Frut 0.74486 0.07678 Processed Vegetables 0.6469 0.392 Fats and Ols.4403 0.094695 Eggs 0.9422 0.0988 Flour and Cereals 0.27848 0.27454 Sugar 0.7622 0.06856 Coffee and Tea 0.92895 0.083933 Mean/Standard Devaton Mnmum/Maxmum κ= 0, λ= 0 κ=, λ= κ=κλ=λ ˆ, ˆ κ= 0, λ= 0 κ=, λ= κ=κλ=λ ˆ, ˆ 0.4482 0.095-0.32342 0.329 0.05893 0.074345 0.07459 0.6223 0.25009 0.28085 0.20637 0.049699 0.29355 0.466 0.457 0.09886 0.60535 0.22384 0.35384 0.592 0.46889 0.26023 0.60384 0.2648 0.29957 0.298 0.009859 0.05029 0.30097 0.09024 0.37785 0.049383 0.25932 0.044004 0.20666 0.908 0.03499 0.0072 0.2829 0.589 0.32466 0.5473 0.00679 0.052534-0.34457 0.39083 0.0045295 0.07964 0.33203 0.3094 0.2605 0.744 0.826 0.03963 0.24526 0.0485 0.05363 0.063446 0.4796 0.687 0.36388 0.236 0.4435 0.694 0.33679 0.025 0.5007 0.025032-0.08652 0.3224 0.29877 0.05568 0.32865 0.05788 0.23624 0.027624 0.029444 0.225-0.5582 0.7565 0.24308 0.052 0.27907 0.04908-0.28006 0.73674 -.24634 0.92355.08009.8245 0.7374.360 0.45098.6342 0.5957 0.9775 0.70.0364 0.78382.97565 0.83872.26699 0.62402.0459 0.69823.09875-0.090433 0.9827 0.36878 0.72089 0.4928 0.92539 0.50233 0.8966 0.4048 0.8537 0.88275.3305 0.70294.5332-0.5982 0.66572 0.62273 0.95909 0.6496.348 0.026664 0.3686 -.05828 0.05047-0.95 0.6945-0.28875 0.38462-0.08365 0.98386 0.0475 0.33554 0.237 0.57878-0.030233 0.34024 0.24857 0.9825 0.08209 0.58804 0.2647 0.9994 0.965.0865 0.099527 0.47776-0.09943 0.096552 0.7093 0.60043 0.24987 0.5007 0.549 0.33565 0.06077 0.4336 0.0054256 0.063042 0.064675 0.4378 0.6 0.6079-0.078287 0.62 -.785 0.493-0.22536 0.2749 0.08254 0.73254 0.026205 0.55959 0.098776 0.26238 0.525 0.34626-0.873 0.5522 0.27846 0.65558 0.068 0.54282 0.9297 0.80564 0.068466 0.50325 0.0807 0.2046-0.23709 0.5203 0.2824 0.52287 0.20279 0.47374 0.7844 0.3475-0.245 0.6394-0.404 0.0902 0.6882 0.32794 0.9995 0.38547 7
6 Concluson The demand model proposed n ths paper s a straghtforward but powerful generalzaton of currently used models. Usng ths approach we test and reject the restrctons that exstng models mplctly place on the Box-Cox parameters on ncome and prces. Our results show that the exstng models of food demand sgnfcantly overstate the sze and varablty of the ncome elastcty of most food groups. 8
REFERENCES Gorman, W.M. On a Class of Preference Felds, Metroeconometrca 3 (96): 53-56. LaFrance, J.T., T.K.M. Beatty, R.D. Pope and K. Agnew, Informaton Theoretc Measures of the Income Dstrbuton n Demand Analyss, Journal of Econometrcs, Forthcomng. Muellbauer, J. Aggregaton, Income Dstrbuton and Consumer Demand. Revew of Economc Studes 42 (975): 525-543. 9