NECESSARY AND SUFFICIENT CONDITIONS FOR LATENT SEPARABILITY

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1 NECESSARY AND SUFFICIENT CONDITIONS FOR LATENT SEPARABILITY Ian Crawford THE INSTITUTE FOR FISCAL STUDIES DEPARTMENT OF ECONOMICS, UCL cemmap working paper CWP02/04

2 Neceary and Sufficien Condiion for Laen Separabiliy Ian A. Crawford Iniue for Fical Sudie Abrac Thi paper exend he nonparameric mehod developed by Samuelon (1948), Houhakker (1950), Afria (1973), Diewer (1973) and Varian (1982, 1983) o laenly eparable model. I preen neceary and ufficien empirical condiion under which daa on he marke behaviour of a price-aking conumer, and a hypoheied allocaion acro laen group are nonparamerically conien wih laen eparabiliy (Gorman (1968, 1978), Blundell and Robin (2000)). I conider homoheic laen eparabiliy and weak eparabiliy a pecial cae. Acknowledgemen Financial uppor from he ESRC Cenre for he Microeconomic Analyi of Public Policy and Leverhulme Cenre for Microdaa Mehod and Pracice i graefully acknowledged. The auhor i graeful o Richard Blundell for helpful commen, bu i reponible for all error. Wha are he obervable conequence of a model of conumer behaviour in which he conumer uiliy funcion i laenly eparable? The noion of laen eparabiliy wa inroduced in Gorman (1968) and (1978) and horoughly developed in Blundell and Robin (2000). So far he characeriaion of he condiion for laen eparabiliy ha been parameric. Thi paper eek neceary and ufficien nonparameric condiion for laen eparabiliy. Definiion 1. (Blundell and Robin, 2000) A direc uiliy funcion U : q R K + U (q) R i aid o aify laen eparabiliy if ( U (q) = max u v 1 eq 1,..., v M ) eq M MX eq m = q q 1,..., q M R K + where u, v 1 eq 1,..., v M eq M are regular uiliy funcion. Under a laenly eparable funcional rucure he conumer allocaion problem i a follow max u v 1 eq 1,..., v M eq M q 1,..., q M R K + ubjec o MX p 0 eqm = x m=1 in oher word hey chooe boh he oal quaniy vecor q and he laen allocaion {eq m } m=1,...,m in order o maximie heir uiliy ubjec o a budge conrain. Suppoe ha here are T obervaion (indexed =1,..., T )onk vecor of price and correponding demand {p, q }. Le {eq m } m=1,...,m where P M m=1 eqm = q denoe a hypoheied allocaion of hee quaniy vecor acro M laen group. When can hee daa and hypoheied allocaion be raionalied by a laenly eparable rucure? The following definiion make clear wha i mean by raionalie in hi conex. m=1

3 Definiion 2. A laenly eparable uiliy funcion raionalie he daa {p, q } and he allocaion {eq m } m=1,...,m if u v 1 eq 1,..., v M eq M u v 1 eq 1,..., v M eq M for all alernaive allocaion {eq m } m=1,...,m uch ha p 0 q P M m=1 p0 eqm. The fir Theorem preen he condiion under which here exi a laenly eparable model of conumer preference which raionalie he daa and he hypoheied allocaion. Theorem 1. The following aemen are equivalen: (U) There exi a laenly eparable uiliy funcion, where u (v) and v m (eq m ) are nonaiaed, monoonic, concave and coninuou funcion, which raionalie he daa {p, q } and he allocaion {eq m } m=1,...,m. (A) There exi number {U,λ } and M vecor {V, ρ } uch ha for all,, m U U + λ ρ 0 (V V ) (A.1) V m V m + 1 ρ m p 0 (eq m eq m ) (G) The daa {p, eq m } m=1,...,m aify he Generalied Axiom of Revealed Preference (GARP) and he daa {V, ρ } alo aify GARP for ome choice of {V, ρ } which aify Proof (U) (A) : Fir conider he implicaion of opimiing behaviour and he fir order condiion from he conumer problem. Coninuiy enure ha uiable ubgradien exi uch ha u (eq m ) λ p where u (eq m )= u(vm ) vm (eq m ). Define λ ρ m = u (v m ). Then v m (eq m ) (ρ m ) 1 p. Now conider he concaviy condiion for hi rucure u (v ) u (v )+ u(v ) 0 (v v ) v m (eq m ) v m (eq m )+ v m (eq m ) 0 (eq m eq m ) Subiuing in v m (eq m ) (ρ m ) 1 p and λ ρ m = u (v m ) preerve he inequaliie and give u (v ) u (v )+λ ρ 0 (v v ) v m (eq m ) v m (eq m )+ 1 ρ m p 0 (eq m eq m ) which are condiion (A.1) and. (A) (U) :Suppoewehavenumber{U,λ } and M vecor {V, ρ } uch ha condiion (A) hold. Conider ome arbirary {eq m } m=1,...,m uch ha p 0 q P M m=1 p0 eqm. We need o how ha here exi a laenly eparable uiliy funcion, wih he aed properie uch ha u v 1 eq 1,..., v M eq M u v 1 eq 1,..., v M eq M. Uing we can conruc T upper bound on v m (eq m ) and if we ake he minimum of hee a he funcion v m (eq m ) hen we have a piecewie linear, nonaiaed, monoonic, concave and coninuou uiliy funcion ½ v m (eq m )=min V m + 1 ¾ ρ m p 0 (eq m eq m ) V m + 1 ρ m p 0 (eq m eq m ) =1,...,T

4 Summing hi inequaliy over m give ρ 0 V p 0 q ρ 0 V p 0 eq where V = 0, V 1,..., V M V = V 1,...,V M 0, (V m = v m (eq m ))andρ = ρ 1,...ρ M. Then ince p 0 q p 0 eq we have ρ0 V ρ 0 V. Uing (A.3) we can imilarly conruc he following macro-uiliy funcion U (V) =min{u + λ ρ 0 (V V )} =1,...,T U + λ ρ 0 (V V ) Since λ ρ 0 (V V ) 0 we have U (V) U a required. (A) (G) : Follow from Afria Theorem. One pecial cae of paricular empirical inere i he one in which he laen uiliy funcion v (eq m ) are homoheic (Gorman (1968), Blundell and Robin (2000)). Theorem 2. The following aemen are equivalen: (U) There exi a homoheically laenly eparable uiliy funcion, where u (v) and v m (eq m ) are nonaiaed, monoonic, concave and coninuou funcion and v m (eq m ) are homoheic, which raionalie he daa {p, q } and he allocaion {eq m } m=1,...,m. (A) There exi number {U,λ } and M vecor {V, ρ } uch ha for all,, m U U + λ ρ 0 (V V ) (A.1) V m V m + 1 ρ m p 0 (eq m eq m ) ρ m V m = p 0 eqm (A.3) (G) The daa {p, eq m }m=1,...,m aify he Homoheic Axiom of Revealed Preference (HARP) and he daa {V, ρ } alo aify GARP for ome choice of {V, ρ } which aify and (A.3) Proof Analogou o Theorem 1 noing ha and (A.3) are neceary and ufficien for he exience of M homoheic uiliy funcion V m (eq m ) uch ha for any eq m wih ha p 0 eqm p 0 eqm hen V m (eq m ) V m (eq m ) (Varian (1983), Theorem 2). Anoher pecial cae concern he iuaion when he allocaion of good acro laen uiliie i excluive a hi correpond o weak eparabiliy. Theorem 3. When he hypoheied allocaion are uch ha eq k,m = q k for all k and and ome m (ha i he allocaion o laen group are excluive and conan over obervaion), hen he condiion in Theorem 1 and 2 are equivalen o hoe for weak eparabiliy and homoheic weak eparabiliy repecively.

5 Proof Obviou from a comparion wih Varian (1983) Theorem 3 and 5. I canno hink of any way in which eiher he number or he compoiion of he laen group can be nonparamerically idenified from he oberved daa. Inead he Theorem preened are imply a way of eing he daa and any hypoheied allocaion for coniency wih he model. In hi repec he reul here are imilar o Varian (1983) for weak eparabiliy, where he number and makeup of eparable ubgroup are no idenified from daa, bu where he condiion under which any hypoheied grouping are raionaliable wih weak eparabiliy are e ou. Noe ha he reul here can alo be applied o he analyi of producion funcion wih a uiable change in he inerpreaion of he noaion and by requiring he op-level daa o aify he weak axiom of profi maximiaion raher han GARP (ee Varian (1984), Theorem 9, for he condiion for weak eparabiliy in producion funcion). Reference [1] Afria, S.N. (1973), On a yem of inequaliie in demand analyi: An exenion of he claical mehod, Inernaional Economic Review, 14, [2] Blundell, R. W. and J-M Robin (2000), Laen eparabiliy: grouping good wihou weak eparabiliy, Economerica, 68, [3] Diewer, W.E. (1973), Afria and revealed preference heory, Review of Economic Sudie, 40, [4] Gorman, W. M. (1968), Meauring he quaniie of fixed facor, in Value Capial and Growh: Eay in Honour of Sir John Hick, ed. by J,N, Wolfe. Edinburgh: Edinburgh Univeriy Pre. Alo in Separabiliy and Aggregaion, Vol 1, Colleced Work of W. M. Gorman, C. Blackorby and A. Shorrock (ed.), Oxford: Clarendon Pre. [5] Gorman, W. M. (1978), More meaure for fixed facor,, in Separabiliy and Aggregaion, Vol1,CollecedWorkofW.M.Gorman, C. Blackorby and A. Shorrock (ed.), Oxford: Clarendon Pre. [6] Houhakker, H. S. (1950), Revealed preference and he uiliy funcion, Economica, May, [7] Samuelon, P. (1948), Conumpion heory in erm of revealed preference, Economica, November, [8] Varian, H. (1982), The nonparameric approach o demand analyi, Economerica, 50, [9] Varian, H. (1983), Nonparameric e of conumer behaviour, Review of Economic Sudie, L, [10] Varian, H. (1984), The nonparameric approach o producion analyi", Economerica, 52,