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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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.

Contents

1. Introduction

2. Derivation of the likelihood function

3. Discusaion

Reference

Pa e

1

1

6

7

1

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

2

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.

3

(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

4

(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

5

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.

6

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.

7

Reference

Lee, L.F. and M.M. Pitt (1986), "Microeconometric Demand Systems with

Binding Non-Negativity Constraints: The Dual Approach", Econo-

metrica.

i

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iii

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V

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