Economcs 90 Prce Theory ON THE QUESTION OF SEPARABILITY What we would lke to be able to do s estmate demand curves by segmentng consumers purchases nto groups. In one applcaton, we aggregate purchases by consumpton category, such as food, housng, etc., nstead of lookng at each tem of food purchased by a consumer. Or n another settng, as n the dog food example, we estmate demand usng the budgeted expendtures for a partcular class of products as the constrant on behavor. The theoretcal consstency of ths approach s the queston we pursue today. Consder a utlty functon where we want to combne the goods nto categores: U = U( x, x2, x3,..., x n ) y = f( x, x ) 2 = h( x3,..., x n ) 2 Let p be the prce of and p and p 2 be the prces of the goods makng up y. In general there s no problem n groupng goods. For nstance f = = 3, npx, then p can be nterpreted as a scalar for the p such that p= tpx = t px. Utlty s a functon of, whch s really an expendture value. In ths sense we can group all other goods nto one composte commodty where we examne the effect of changes the prce level of ths composte. These changes are proportonal adjustments of all component prces. Relatve prces are constant. Changes n the prce of the sngle good have an effect on the consumpton of the composte whch s really the expendture on all other goods. Ths s the framework that underles our analyss of the labor lesure tradeoff and t s perfectly acceptable. The more troublesome queston s posed by the Problem wth Dogs. We must address the queston, Are ordnary demand curves usng the budget share of y as a constrant, logcally consstent wth our standard model of consumer behavor? Ths s the problem of emprcal separablty. In other words, can we effectvely gnore n the constructon and estmaton of demand curves for the other goods, x and x 2? The utlty maxmng model usng a budget group constrant looks lke ths: [ ] max U = ( x, x, ) s.t. B= p = p x + p x 3 { x, x2} 2 2 2 whch gves ordnary demand curves that look lke: B B x = x ( p, p, B) 2 4 compared to our normal demand curves: I am conng the phrase, emprcal separablty, not to be confused wth separable utlty functons. Revsed: September, 2006
Economcs 90 Prce Theory x = x ( p, p, p, ) 2 5 The standard textbook treatment focuses on utlty functons and t s not very emprcally orented even though the queston s bascally a practcal emprcal ssue. 2 That s, when can we effectvely gnore all but a subset of consumer behavor and stll get demand curves that are logcally consstent. Snce we do ths all the tme, t s mportant to know what assumptons we are mplctly makng and what consequences follow f the world devates from those assumptons. As a purely emprcal matter, f x = 0 6 p for all, then we have emprcal separablty. That s the demand effects estmated from eqt (4) wll be dentcal to those estmated usng eqt (5). It s reasonable to ask, When s t lkely that eqt (6) holds? We know from the budget constrant condtons that: ( ε + ) S +ε ( S ) = 0 y If eqt (6) s true for all, then the cross prce elastcty of y wth respect to the prce of s ero. Ths means that the own-prce elastcty of must be. Ths says that f the set of goods s expendture neutral wth respect to a prce ndex on ths set, then the set of goods y can be treated as separate. oreover, f we estmate demand curves for the set y and fnd that at the weghted average expendture elastctes and own-prce elastctes are untary, then we can logcally nfer that ε y s ero. Emprcal separablty means that all the consumer behavor relatons that we derved for the set of all goods apply to a subset standng alone. Ths wll be true f all cross prce effects n the subset wth respect to the rest are ero. Ths wll be the case f the own prce elastcty of all other goods s untary, and t wll be known to be the case f when we estmate the demand system for the subset, the weghted average own-prce and ncome elastctes are untary. A look at the Chnese household demand curves gves us an dea of the applcaton. When I estmated the demand curves for all goods, food was found to have a untary prce and ncome elastcty. Cross prce effects wth all other goods were not statstcally dfferent from ero. In ths settng, f food as a group has no cross prce effects, then food s separable. Hence, I could have estmated demand for the ndvdual food groups by lookng only at the expendtures on food, and gnorng the expendtures on all other goods. Probably the most mportant applcaton of the separablty result s recognng what we cannot do and why. Consder the data that BLo collects when you make purchases there and use your BLo dscount card. The company knows all of your purchases, tem by tem, and your total expendture for each vst. Can they estmate demand curves for you for dfferent tems? (Could they target you for partcular dscounts usng only these data?) 2 See Layard & Walters pp. 65-67. Slberberg (990) (2000) secton 0.7, secton, Varan (992) Chapter 9.3, Intrlgator, Bodkn, Hsao (996)-sectons 7.-7.7. These texts all talk about the condtons under whch the utlty functon s separable. Revsed: September, 2006 2
Economcs 90 Prce Theory The answer s No. The problem s that of the tems that you buy at BLo, you may and probably do shop around for the best prce among several stores. BLo only sees your purchases when ts prce on some or all of the tems purchased s most attractve. When you don t buy somethng at BLo, t does not necessarly mean that you consume ero quantty of that tem. Techncally, the BLo data gve us a based sample n multple ways. Obvously, consumpton s based, but so s total expendture. Snce expendtures are hgher at BLo when t has favorable prces on select tems, purchases of those tems wll soak up some (all?) of the ncome effect. 3 Some dervatons: A. Weghted sum of own and cross prce elastctes should equal. It has to be true because f prces ncrease and ncome doesn t, total expendtures are constant even though the optmal bundle may change. Add the budget constrant condtons and collect terms: S ( ε + ) + S ε + S ( ε + ) + Sε = 0 2 2 2 22 2 Sε + S ε + S ε + Sε = S S 2 2 2 22 2 2 S ( ε +ε ) + S ( ε +ε ) = 2 2 22 2 B. Estmated budge elastcty from category expendtures s equal to estmated ncome elastcty when emprcal separablty holds. Start wth the equalty of the unrestrcted, restrcted, and compensated demand curves: x ( p, p, p, ˆ ) = x ( p, p, Bˆ) = x ( p, p, p, U) B U 2 2 2 The hat markers on ncome and the subset budget for y sgnfy an ncome level that satsfes dualty. Dfferentate wth respect to P : x x x x x p B p p B U + x = p = 3 In some smulatons usng Cobb-Douglas preferences, I found the BLo cross prce effect to completely erase the ncome effect when estmated n unrestrcted form. The homogenety condton was not satsfed. Imposng the homogenety condton reduced the bas. The estmated ncome elastcty was.88, own prce was.3, and cross prce was.25. These, of course, should be, -, 0. The estmated budget share was greater than one. Revsed: September, 2006 3
Economcs 90 Prce Theory Recall that ˆ ˆ x B = px, whch accounts for the mddle term; and = 0 by the defnton of p separablty. 4 Hence, we have: B x x x p = B p x and because the own-prce elastcty of s, we have the result. oreover, f the ncome elastcty are equal, the prce elastctes are as well. The Labor-Lesure odel It s worth lookng at the labor-lesure model rght now because t s an applcaton of the separablty queston. That s, we commonly look at systems of demand equatons wthout lookng at the labor-lesure choce. For nstance, n the Chnese demand estmates we held constant total expendtures on goods and not total potental ncome. The standard labor-lesure model takes the form: max { Cl, } U = U( C, l) + λ ( E + ( T l) w C) where C s the composte consumpton good, l s lesure, T s total tme, E s endowment ncome, and w s the wage rate. Labor (unlabeled) s T - l. Endowment ncome can be ero or even negatve. However t s both ntutvely pleasng and parsmonous to model the budget constrant so that consumpton of goods, C, s equal to endowment ncome plus labor ncome. We wll see why shortly. By the way, t can be seen drectly from the budget constrant that the wage rate s the opportunty cost of lesure because a change n the dollar value of consumpton s equal to the negatve of a change n lesure tmes the wage rate. The FOC and SSOC of the maxmaton problem mply demand curves that we are famlar wth: * * C = C ( w, E, T) * * l = l ( w, E, T).e., Next, let's take a look at the dual. Here we wll mnme E subject to a utlty constrant, E = C+ lw wt+ µ( U U( C, l)) The FOC are: 4 [ ˆ p x ] ˆ x x = x p = x x p p p p p Revsed: September, 2006 4
Economcs 90 Prce Theory E = µ UC C = 0 E l = w µ UL = 0 7 whch along wth the SSOC mply demand curves that take the form: C = C ( w, U ) l = l ( w, U ) 8 Based on the dualty theorem we can equate the two sets of demand curves by properly choosng the parameters that demand s a functon of. Thus, the dualty theorem says: Dfferentatng, (, ) * l w U = l ( w, E ) = * + * l l l E w w E w Recogne that the demand curves n (6) mply an optmed expendture level: Thus, by the envelope theorem, E = C (.) + l (.) w Tw = * * l l l * ( w w E T l ) 9 Ths result s a lttle odd. It s the Slutsky equaton for the labor-lesure tradeoff. It says that the pure substtuton effect for lesure (the left-hand sde) s equal to the ordnary prce effect mnus the weghted ncome effect. The mnus sgn come n because of the way the budget constrant s specfed. The Slutsky equaton n (8) makes sense nonetheless. We know that the pure substtuton effect must be negatve. Ths can be derved from the comparatve statc analyss of the FOC of the endowment mnmaton equatons gven n (7). Notce that the wage rate only shows up n one place by tself so the matrx of comparatve statcs s symmetrc. The rght-hand sde of (8) s a prce effect and an ncome effect. If the prce effect s negatve, then lesure s a normal good. As wth all normal prce effects, the ncome effect can be postve or negatve (just not too bg). On the other hand, f the prce effect s postve, we have what s known as the "backward bendng supply curve of labor." Ths s the equvalent of a Gffen good n lesure. The more lesure costs, the more the consumer chooses. If the prce effect of lesure s postve, equaton (8) says that the ncome effect must also be postve. That s, f Revsed: September, 2006 5
Economcs 90 Prce Theory the consumer chooses more lesure when the wage rate goes up and t costs more not to work, then the consumer must choose more lesure when endowment ncome goes up. Indeed that effect (weghted) must be even stronger than the postve prce effect. Havng thought about the labor-lesure problem n ths way, let's consder the separablty queston. The ssue s essentally the followng: Is t legtmate to gnore the labor-lesure tradeoff when estmatng demand curves (sngly or systems of equatons)? The answer s yes, almost by constructon. Consder the model: max x x l λ u = U( x, x, l) +λ ( Px + Px E ( T l) w) {, 2,, } 2 2 2 From ths we get ordnary and compensated demand curves. The endowment ncome level that equates them for a gven set of parameters s labeled wth a hat. E x ( P, P, w, Eˆ ) = x ( P, P, w, U) U 2 2 Next consder demand curves that come from a model that just looks at expendtures called ncome n the normal parlance: max x x λ u = U( x, x ) +λ( Px Px ) {, 2, } 2 2 2 where = Px + Px E E 2 2 and from whch we get our standard ordnary demand curves: x = x ( P, P, ) 2 In order for these standard demand curves estmated by gnorng lesure and wage rate to be consstent, t must be true that the dervatves of the standard demand curves be dentcal wth the full model at the pont where the ordnary and compensated demand curves ntersect. That s, at: E (,,, ˆ U x P P w E) = x ( P, P, w, U) = x ( P, P, ˆ ) 2 2 2 the dervates of x E and x wth respect to the P must be equal and x E E x = Ths s true by constructon: ˆ E E E E = Px + Px 2 2 ( T l ) w and ˆ ˆ ( E = E+ T l ) w. Hence, dfferentatng wth respect to E yelds: Revsed: September, 2006 6
Economcs 90 Prce Theory = = E E E x x x If the ncome effects are equal, then the prce effects wll be also. Revsed: September, 2006 7