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II III I II V I Ihll l I II IIIII II III IIIN IIIII IIII III Ifaculteit der economische wetenschappen
RESEARCH MEMORANDUM
LBURG UNIVERSITY
-PARTMENT OF ECONOMICSstbus 90153 - 5000 LE Tilburg~therlands
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330. ~~S3~~ ~
Comment on 'Microeconometric Demand Systemswith Binding Non-Negattvity Constraints:The Dual Approach'~
Arthur van SoestPeter Kooreman
June 1986 Department of EconometricsTilburg UniversityP.O. Box 901535000 LE TilburgThe Netherlands
~ Helpful comments by Arie Kapteyn are gratefully acknowledged.
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Contents
1. Introduction
2. Derivation of the likelihood function
3. Discusaion
Reference
Pa e
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6
7
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1. Introductíon
In a recent issue of Econometrica Lee and Pitt (1986) advocatethe virtual price approach to the problem of binding non-negativity con-straints in demand analysis. For a wide range of flexible specificationsfor demand systems, including the Almost Ideal Demand System and theIndirect Translog model, the direct utility function cannot be derivedexplicitly, which precludes the direct use of Kuhn-Tucker conditions toderive conditional demand equations when some non-negativity conatraintsare binding. The apparent advantage of the dual approach is that it doesnot require a direct utility function to derive conditional demand equa-tions.
To obtain the likelihood functíon for a sample of independentobservations Lee and Pitt derive the likelihood contributions correapon-ding to all possible demand regimea (defined as the set of poaitivelyconaumed goods). However, theae regimes are only guaranteed to be uni-quely characterized íf the cost function underlying the demand eyatem iaglobally concave in prices. In Section 2 we show that for the IndirectTranslog specification proposed by Lee and Pitt the lack of global con-cavity yields the anomaly that the probabilities of observing each ofthe regimes do generally not sum to unity. Consequently, the likelihoodfunction is not well-defined and parameter estimates will be inconsis-tent.
2. Derivation of the likelihood function
The share equatíons of the propoaed Indirect Translog apecifica-tion are:
(1) viqi '
Ka t E s jRnvjfE1 1a1 i i
i ~ 1,...,KD
K Kwhere D ~-1 f E E 91j Rn vj. Here vi i s a normalized price, ai and
1~1 j-1Bij are parametera and Ei is a random variable representing etochastic
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preferences; qi is the so-called notional demand (i,j - 1,...,K). ForK K
normalization Lee and Pitt set E a~-1, E e~ 0.i-1 1 i~l i
The demand regime where the first A, goods are not consumed ischaracterized by the conditions
(2a) Ri(v) ~ vi
(2b) xi ~ 0
i ~ 1,...,Jt
i ~ Rf1,...,K
Here IIi(v) is the virtual price of the i-th good and v is the aet ofmarket prices of the positively consumed gooda; xi is the demand for thei-th good (i ~ Rf1,...,K) given that the first R goods are not conaumed.These conditions follow from the Kuhn-Tucker conditions and the conceptof virtual prices, assuming that the direct utility function is conti-nuously differentiable, quasi-concave and increasing.
In the following, we only consider three goods, where the thirdgood is always consumed in positive amounts. (These assumptions do notaffect the generality of our argument.)
The conditione for observing positive demands for both good 1and good 2(regime I) are
(3)
,~ 3 1)where ei ~ ai t E Bij Rn vj f ei.
j~l
The virtual price of good 1 if xl ~ 0 and x2 ~ 0 is given by
3(4) Rn R1 ~-(alt E 813 Rn v~tel)~sllj~2
From (2a) and (2b) it follows that the conditione for observing zerodemand for good 1 and positive demand for good 2 ( regime II) are
1) Lee and Pitt implicitly assume D C 0.
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(5a)
(5b)
Kwhere B ' E B1 . The left hand side of (5b) is obtained by replacing
.j ial j1Cn vl in ( 1) by Rn II1 given in (4).
Analogously, the conditions for observing positive demand forgood 1 and zero demand for good 2(regime III) are
~ 0~11
~ s12 ~E2 - Sll E1
s.l ~
D - BL1 el
~ 0
e(6a) 2 ~ 0
s22
(6b)
~ s12 ~E1 - sz2 ~2s ~o~D - - e
s2 2 2
The virtual prices of goods 1 and 2 if both xl ~ 0 and x2 ' 0
are given by
Rn II1 al } s13Rn v3 E1~ -B-1 - B-1
Rn II2 a2 } s23 kn v3 e2
where B s(sll s12 . The conditions for observing zero demanda for both`s12 s2z
goods (regime IV) are
(8a)
C Rn 1I1~ En vl~Rn II ~ R.n v ~ which can be written as2 2
~ ~S22E1 S12E2-~- ~ 2811s22 - s12 Slls22 - s12
and
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(8b)~ ~
s11Ez ~ S12E12
511~22 - S12 slls22 - S12
For each regime the conditions correspond to a partícular reglon~ ~
in the ( el,e2)-plane. In figure 1 these regions are índicated for D ~ 0,Sl i ~ 0 and 0 ~ S.1 ~ 612 ~ B22 ~ 6, 2.
,~ S 12 ~
E2-s11 E1
III ~ III
IV ~ IV
,~ S22 ~E2-S12 E1
IIIII
III
,~ S22- E2-D.a:z
~ ~Figure 1. Regimes in (el,e2)-plane
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Clearly, in this example the conditions define sets in the~ ~
(el,e2)-plane which are neither mutually exclusive nor exhaustive. As aconsequence the probabilities of observing each of the regimes do notadd up to uníty.
In figure 2 the regions are indicated for D~ 0 ~ s12 ~ S11 ~s22~s1~s2.
Figure 2.
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In this case [he regimes are properly defined in the sense that the con-ditions define mutually disjoint and exhaustive sets.
It is straigh[forward to show that, for D~ 0, the demand regi-mes are uniquely characterized if and only if the matrix B is positivedefinite and 8.1 ~ 0 and s.2 ~ 0.
Apparently, the likelihood function is only well-defined for arestricted subset of the parameter space and a restricted range of exo-genous variables (note that D depends on the normalized pricea, whichdiffer per observation).
A unique characterization of the regimes for all possible valuesof the exogenous variables will be guaranteed if the cost function un-derlying the demand aystem is globally concave. For the tranalog modelconcavity is satiafied íf the matrix defined by
(9) ci~ ~ D{-dijviqiD f gij-viqi8.~ - v~q~B.i f viqivjqj(E B.~fD)}~
is negative semi-definite (ái~ ~ 1 if i a j and 0 otherwise). Note thatfor vlql a v2q2 - 0(regime IV) this requires the matrix B to be posi-tive semi-definite if D~ 0. As is easily seen from (9) global concavi[ycannot be imposed by restricting parameters without completely sacrifi-cing flexibility.
3. Discussion
The advantage of the approach proposed by Lee and Pitt ia thatit in principle allows for consiatent estimation using all observations.Moreover, ín contrast to the approach based on the direct uae of theKuhn-Tucker conditiona, Lee and Pít['s framework allows for the use offlexible functional forma.
However, as we have shown in section 2, reatrictions on the pa-rameters and exogenoua variablea are required Eor the model to makesense. Since these reatrictiona alao are a condition aine qua non forthe likelihood function to be well-defined, they have to be imposedduring eatimation.
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Reference
Lee, L.F. and M.M. Pitt (1986), "Microeconometric Demand Systems with
Binding Non-Negativity Constraints: The Dual Approach", Econo-
metrica.
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IN 1985 REEDS VERSCHENEN
168 T.M. Doup, A.J.J. TalmanA continuous deformation algorithm on the product epace of unitsimplices
169 P.A. BekkerA note on the identification of reatricted factor loading matrices
170 J.H.M. Donders, A.M. van NunenEconomische politiek in een twee-sectorecrmodel
171 L.H.M. Bosch, W.A.M. de LangeShift work in health care
172 B.B. van der GenugtenAsymptotic Normality of Least Squares Estimatora in AutoregressiveLinear Regresaion Models
173 R.J. de GroofGeYsoleerde versus gecoArdineerde economische politiek in een twee-regiomodel
174 G. van der Laan, A.J.J. TalmanAdjuatment processes for finding economic equilibria
175 B.R. MeijboomHorizontal mixed decomposition
176 F. van der Ploeg, A.J. de ZeeuwNon-cooperative strategies for dynamic policy gamea and the problemof time inconsistency: a comment
177 B.R. MeijboomA two-level planning procedure with respect to make-or-buy deci-sions, including cost allocatione
178 N.J. de BeerVoorapelprestaties van het Centraal Planbureau in de periode 1953t~m 1980
178a N.J. de BeerBIJLAGEN bij Voorspelprestaties van het Centraal Planburesu in deperiode 1953 t~m 1980
179 R.J.M. Alessie, A. Kapteyn, W.H.J. de FreytasDe invloed van demografiache factoren en inkomen op consumptieveuitgaven
180 P. Kooreman, A. KapteynEstimation of a game theoretic model of household labor supply
181 A.J. de Zeeuw, A.C. MeijdamOn Expectations, Information and Dynamic Game Equilibria
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182 Cristina PennavajaPeriodization approaches of capitalist development.A critical survey
183 J.P.C. K1 eijnen, G.L.J. Kloppenburg and F.L. MeeuwsenTesting the mean of an asymmetric population: Johnson's modified Ttest revisited
184 M.O. Nijkamp, A.M. van NunenFreia versus Vintaf, een analyse
185 A.H.M. GerardsHomomorphisms of graphs to odd cycles
186 P. Bekker, A. Kapteyn, T. WanebeekConsistent sets of estimates for regressions with correlated oruncorrelated measurement errors in arbitrary subsets of allvariables
187 P. Bekker, J. de LeeuwThe rank of reduced dispersion matricea
188 A.J. de Zeeuw, F. van der PloegConsistency of conjectures and reactions: a critique
189 E.N. KertzmanBelastingstructuur en privatisering
190 J.P.C. KleijnenSimulation with too many factors: revíew of random and group-acreening designs
191 J.P.C. KleijnenA Scenario for Sequential Experimentation
192 A. DortmansDe loonvergelijkingAfwenteling van collectieve lasten door loontrekkers?
193 R. Heuts, J. van Lieshout, K. BakenThe quality of some approximation formulas in a continuous reviewinventory model
194 J.P.C. KleijnenAnalyzing simulation experiments with common random numbers
195 P.M. KortOptimal dynamic investment policy under financial restrictions andadjustment costs
196 A.H. van den Elzen, G. van der Laan, A.J.J. TalmanAdjustment processes for fiuding equilibria on the simplotope
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197 J.P.C. KleijnenVariance heterogeneity i n experimental design
198 J.P.C. KleijnenSelecting random number seeds i n practice
199 J.P.C. KleijnenRegressíon analyais of simulation experiments: functional softwarespecification
200 G. van der Laan and A.J.J. TalmanAn algorithm for the linear complementarity problem with upper andlower bounde
201 P. KooremanAlternative apecification tests for Tobit and related models
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IN 1986 REEDS VERSCHENEN
202 J.H.F. SchilderinckInterregional Structure of the European Community. Part III
203 Antoon van den Elzen and Dolf TalmanA new strategy-adjustment process for computing a Nash equilibriumin a noncooperative more-pereon game
204 Jan VingerhoetsFabrication of copper and copper semis in developing countries.A review of evidence and opportunities.
205 R. Heuts, J. v. Lieshout, K. BakenAn inventory model: what is the influence of the sha..e of the leadtime demand distríbution?
206 A. v. Soest, P. KooremanA Microeconometric Analysis of Vacation Behavior
207 F. Boekema, A. NagelkerkeLabour Relations, Networks, Job-creation and Regional DevelopmentA view to the consequences of technological change
208 R. Alessie, A. KapteynHabit Formation and Interdependent Preferencea in the Almost IdealDemand System
209 T. Wansbeek, A. KapteynEstimation of the error components model with incomplete panels
210 A.L. HempeniusThe relation between dividends and profits
211 J. Kriens, J.Th. van LieshoutA generalisation and some properties of Markowitz' portfolioaelection method
212 Jack P.C. Kleijnen and Charlea R. StandridgeExperimental design and regression analysis in simulation: an FMScase study
213 T.M. Doup, A.H. van den Elzen and A.J.J. TalmanSimplicial algorithms for solving the non-linear complementarityproblem on the simplotope
214 A.J.W. van de GevelThe theory of wage differentials: a correction
215 J.P.C. Kleijnen, W. van Groene~aalRegression analyeis of factorial designs with sequential replica-tion
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V
216 T.E. Nijman and F.C. PalmConsistent estimation of rational expectations modela
217 P.M. KortThe firm's investment policy under a concave adjuetment cost func-tion
218 J.P.C. KleijnenDecision Support Syatems (DSS), en de kleren van de keizer ...
219 T.M. Doup and A.J.J. TalmanA continuoua deformation algorithm on the product space of unitsimplices
220 T.M. Doup and A.J.J. TalmanThe 2-ray algorithm for solving equilibrium problema on the unitsimplex
221 Th. van de Kluntlert, P. PeteraPrice Inertia in a Macroeconomic Model of Monopolistic Competition
222 Chriatían MulderTeating Korteweg's rational eapectationa model for a amall openeconomy
223 A.C. Meijdam, J.E.J. PlasmanaMaximum Líkelihood Estimation of Econometric Models with RationalExpectations of Current Endogenoua Variablea
224 Arie Kapteyn, Peter Kooreman, Arthur van SoeatNon-convex budget aete, institutional conatrainte and impoaition ofconcavity in a flexibele houeehold labor supply model.
225 R.J. de GroofInternatioaale coàrdínatie van economieche politiek in een twee-regio-twee-sectoren model.
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