The estimation of mixed demand systems

Tilburg University

The estimation of mixed demand systems

Barten, A.P.

Publication date:

1992

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Barten, A. P. (1992). The estimation of mixed demand systems. (Reprint Series). CentER for Economic

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87

The Estimation of Mixed

Demand Systems

by

Anton P. Barten

Í~ ,~Q

~~"~~~ ,

,'

~`

_{ti~OJQ~ ,}

1

i

Q~~Í

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

~ C I N 0 0 8 8 2~

Reprinted from R. Bewley and T. Van Hoa

(eds.), Contributions to Consumer Demand and

Econometrics, Essays in Honour of Henri Theil,

Basingstoke: The Macmillan Press Ltd., 1992

Research Staff Helmut Bester Eric van Damme Board

Helmut Bester

Eric van Damme, director

Arie Kapteyn

Scientific Council Eduard Bomhoff Willem Buiter Jacques Drèze

Theo van de Klundert Simon Kuipers Jean-Jacques Laffont Merton Miller Stephen Nickell Pieter Ruys Jacques Sijben Residential Fellows Svend Albaek Pramila Kríshnen Jan Magnus Eduardo Siandra Hideo Suehiro Doctoral Students Roel Beetsma Hans Bloemen Sjeak Hurkens Frank de Jong Pieter Kop Jansen

Erasmus University Rotterdam Yale University

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Groningen University

Université des Sciences Sociales de Toulouse University of Chicago

University of Oxford Tilburg University Tilburg University

European University Institute San Francisco State University Tilburg University

UCLA

Kobe University

The Estimation of Mixed

Demand Systems

by

Anton P. Barten

Reprinted from R. Bewley and T. Van Hoa

(eds.), Contributions to Consumer Demand and

Econometrics, Essays in Honour of Henri Theil,

Basingstoke: The Macmillan Press Ltd., 1992

3 The Estimation of Mixed

Demand Systems

Anton P. Barten

3.1

INTRODUCTION

In applied demand analysis the quantities demanded are usually

explained as a function of prices and total expenditure. Complete

systems of such demand functions describe in this way all (groups of)

commodities in the budget of the consumer. T1~ey reflect the basic

assumptions about utility maximizing behaviour of the consumer by

the parametrization of the functions. The Rotterdam demand system

of Theil (1965) may serve as an example of such a regular system.

Actually, the first law of demand as formulated by D'Avenant in 1699 explained the price of corn as a function of the available quantity of corn. Such `inverse' demand functions underly the work of Antonelli (1886). For more recent theoretical treatment see Katzner (1970) or Anderson (1980). Theil (1976) estimated an inverse de-mand system under its mode as a regular system. Salvas-Bronsard, Leblanc and Bronsard (1977) estimated cuch a system directly. A complete system of inverse demand functions displays properties analogous to those of a regular demand system. These properties are of considerable use in estimation. Inverse demand functions appear to be specifically suitable for the explanation of the price formation of quickly perishable goods, like fish - for example, Barten and Betten-dorf (1989).

Regular and inverse demand systems are both extreme cases with either all quantities or all prices endogenous. One can think of a situation where of some commodities the prices are endogenous and the corresponding quantities exogenous while the reverse holds for the other commodities. Such a mixed system has been put forward by Samuclson (1965). Bronsard and Salvas-Bronsard (1980) and Chavas (1984) analyse it extensively from a theoretical point of view. Bron-sard and Salvas-BronBron-sard also provide estimates of a mixed system for seven Canadian consumer categories with food and clothing as

the price endogenous goods.

32 The Estimation of Mixed Demand Systenu

A natural way to estimate a mixed demand system is to first write it in reduced form and thcn estimate the equations. This is usually done tor regular and inverse dcmand systems. One starts from the first-order conditions for a maximum of the utility function subject to the budget constraint and solves these for the quantities in the regular case and for the prices in the inverse case. In the mixed context one would solve for the endogenous prices and the endogenous quan-tities. The disturbance terms of such `reduced forms' can be taken to be independent of the right-hand side variables. There is then no problem of inconsistency of estimation due to `simultaneity'.

ln the case of the regular and inverse system it is fairly simple to select a parametrization which reflects the constraints implied by the structural formulation in a way which is easy to take into account when estimating the system. This property does not hold to the same extent for mixed systems.

Another approach is possible: estimate either a regular or an inverse demand system taking into account the endogenous nature of some of the right-hand side variables. One can then fully benefit from the simple parametrization properties of those systems without being inconsistent in estimation. For estimation one can use an instrumen-tal variables approach with the exogenous quantities and prices as instruments. Theil (1976) estimated an inverse demand system for meat (USA) in its mode as a regular system with the quantities as instruments. Meyermans (s.a.) used such an approach for the esti-mation of a mixed system for meat and vegetables (Belgium) with the exogenous quantities and prices as the instruments. It is not so simple to impose the negativity condition of demand in this way.

One can also apply a maximum likelihood estimation procedure

with a pcoperly specified Jacobian transformation determinant. This

is the approach taken here. It builds upon the maximum likelihood

approach to estimate demand systems put forward in Barten (1969)

and Barten and Geyskens (1975). The parametrization used is one of

a regular Rotterdam system.

otherwise preserved vegetables the prices are taken to be set by the producer and the quantity will adjust.

In the next section the theory of mixed demand systems is briefly reviewed. The issue of the parametrization of such systems is taken up in Section 3.3. It prepares the way for the formulation of the likelihood function. 1fie application to the market for vegetables in Belgium, first quarter 1975-1ast quarter 1984, follows in Section 3.5. The chapter ends with some concluding remarks.

3.2 SOME THEORY OF MIXED DEMAND FUNCTIONS

Let q E IRf be the vector of quantities of n commodities. We assume

that a preference order is defined on all admissible q vectors that can

be represented by a well-behaved real-valued utility function.

u(q)

(3.1)

i.e. a strongly quasi-concave, monotone increasing function, at least

twice differentiable with the second-order derivatives being

continu-ous functions of q.

The n commodities have positive prices per unit: p. The product of prices and quantities adds up to the given budget m:

P~q - m ,

Defining n-(llm)p one can write (3.2) also as: n'q - 1

(3.2)

(3.3)

with n being the positive vector of `normalized' prices.

The consumer's optimum is the vector q' that maximizes u(q)

among all q satisfying (3.2) and (3.3). Under the assumptions made the q` will satisfy (3.3) and the Second Law of Gossen:

óu(q') - ~

aq

(3.4)

34

The Estimation of Mixed Demand Systems

the Lagrange multiplier associated with constraint (3.3).

Solution of (3.3) and (3.4) with respect to q' and ]~ for given n

yields the Marshallian demand functions:

q' - f(n)

(3.5)

Such a system of regular demand functions indicates the quantities

consumed for given prices n. It satisfies (3.3). Hence n'f(n) - 1 and

the Cournot aggregation condition:

n' af - -f(n)'

(3.6)

Inserting j(n) for q in (3.1) yields the indirect utility function:

v(n) - u

[I(n)1 - max(u(q) I n'

q-1)

(3.7)

9 .

This is a monotone decreasing function in n. One can maximize v(n) with respect to n subject to n'q - 1. Here q is taken to be fixed. One has:

av(J[') - _Fr9

an

(3.8)

as first-order conditions from which the inverse demand functions:

n' - 8(9)

(3.9)

can be obtained. These express the (normalized) prices one is willing

to pay for a given bundle q.

To show that inverse system (3.9) is indeed the inverse of (3.5) one

can proceed as follows. One has from (3.7), (3.4) and (3.6):

av

au

_{af - lv~' af --~f(n)'}

_(3.10)

an' - ~3q'

án'

ón'

holding for any n, so also for n- n'. In that case also (3.8) applies. Because of (3.3):

- n" av(n~) - -~

and therefore f(n') - q, which is the inverse of (3.9). Otherwise said, n' are the prices which induce a consumer to purchase the vector q out of free choice.

In view of this relationship between the regular and inverse de-mand systems they both can be derived from the solution of (3.3) and (3.4), in the regular case with respect to the quantities given prices and in the inverse case with respect to the prices given quantities.

While it may be true that for some commodities the quantities

adjust to the prices, it is very much possible that for other

commodi-ties the prices adjust to the supplied quanticommodi-ties. Think of quickly

perishable goods like fresh fish or fresh vegetables. By definition

these cannot be stored without losing some of their quality. Partition

the set of commodities into two subsets such that:

q - (q~, 9í)~ ]L - (7ii, ní)~

with q, being endogenous and qz exogenous while nz is endogenous and n, exogenous. (The exogenous variables are barred). The counterpart of (3.3) is now:

ni qi } ní 9z ` 1 (3.11)

To derive a mixed demand system explaining the endogenous

prices and quantities in terms of the exogenous quantities and prices Samuelson (1965) formulates the utility potential:

z(q„ 9z, n„ n~) - u(q„ áz) - v(n~, nz) (3.12)

with u(.) being the direct utility function and v(.) the indirect one. Clearly, z is a monotone increasing function of its arguments. For prices and quantities satisfying (3.11) its maximum is clearly zero. At this maximum:

az(q,', 9z, n~, n2 )-

au(q,`, 9~)

_{- vn,}

aq,

aq,

az(q,', 9z, n„ n~

_{- -}

av(n~, nz)

- ó9~

an2

an2

(3.13)

(3.14)

36

The Estimation of Mixed Demand Systems

q~ - h, (n„ ~2)

.

nz - : n„ ~:

(3.15)

(3.16)

Obviously, the case of all prices exogenous is a special case of (3.12) with (3.5) as the result, while the case of all quantities

exogen-ous is the opposite extreme with inverse system (3.9) as the outcome. liowever, the relation between a mixed demand syste~n and the regular and inverse system goes deeper.

Let f,(~) and f2(~) be the obvious partitions of f(-) and let g,(.) and

gz(-) be the corresponding subvectors of g(-). Then one can state:

q~ - h,(n„ qz) - f,(~„ n~) (3.17)

~ - !z(n„ nZ) (3.18)

n, - 8,(q„ 9z) (3.19)

n~ - hz(n„ 9z) - 8z(q„ 9z) (3.20)

To see this one may start from:

av(n„ n?)

au(q;, 92)

af,(n„ n?)

anZ

-

aq~

anZ

} au(qi, 9z) afz(n„ ni)

aqZ

an2

aj,

au

af2

-

v~~I

anZ } aq2 an~

au

- vn2'

afz - vf2 --vqz

-l aq2

an2

where use is made of (3.13), (3.6) and (3.14). Otherwise said:

:

s

au 9,19:

-v~Z~ a? - v(fz(n„ nz) - 9z)

a qz aJ~z

au(q~, 92) - vn2, aq2

Combining this condition with (3.13) one has the second Law of

Gossen. Together with (3.15) and (3.16) this implies (3.17) through

(3.20).

The conclusion of these various equivalences is that the Second

Law of Gossen can serve as an unifying starting point for the

deri-vation of regular, inverse and mixed demand systems. One simply

solves this condition (and n'q - 1) for the relevant endogenous

variables (and ~) in terms of the exogenous variables. Depending on

the particular choice of what is endogenous and what exogenous one

obtains the desired system. Because of this common basis one can

easily switch from one system to the other by simply relabelling an

exogenous price as endogenous and the corresponding endogenous

quantity as exogenous or vice versa. This switching property is

employed in the next section.

3.3 FUNCTIONAL SPECIFICATION

The previous section applies to any type of parametrization. Here we

will use one in particular, the Rotterdam specification first proposed

by Theil (1965) and used in many applications. A regular demand

equation of the Rotterdam variety for tim~ series is writtcn here as:

w;, Alnq;, - b; ~k wk, Olnqk, -~ ~s;; Olnp;~ t u;,

(3.21)

where

~ is the operator of taking first backward differences

q;, is the quantity of good i

p~, is the price of good j

u;, is a disturbance term

w;, - (wu f w;.~-i)l2 with

w„ - Pr~ 9r~~m„ the budget share of good i

m~ - ~kpk,9k„ total expenditure

b;, s;;, are constants

i,j,k-1,...,n

t is time subscript

38 The Estimation of Mixed Demand Systems

~;b; - 1, ~~s;; - 0

(adding-up)

(3-22)

~ s;; - 0

(homogeneity)

(3.23)

s;~ - s~;

(symmetry)

(3.24)

~;~~x;s;;x; c 0

(negativity)

(3.25)

if at least one x; is aitierent from the other x's.

To simplify notation we will use:

Dlnyu - w;, Olnq;r

We will define:

A1nQ, -~kw,~Olnq~, - ~k A1nYk,

(3.26)

(3.27)

which may be seen as the relative change in real total expenditure, or

in the average quantity or in the quantity index. In what follows

AInQ~ is taken to be exogenously determined. In view of

homogen-eity condition (3.23) one may replace the p;~ in (3.21) by n;~ - p;~lm,.

With all these notational conventions we rewrite (3.21) as:

Olny, - b;A1nQ; f~s;; t11nn;, f uu_r (3.28)

In obvious matrix notation we write the full system as:

Alny~ - bOlnQ~ f SAlnn, f u~

(3.29)

with the following properties following from (3.22}-{3.25):

~'b - 1

~'S - 0

(adding-up)

(3.30)

S~ - 0

(homogeneity)

(3.31)

S - S'

(symmetry)

(3.32)

x'Sx G 0, t!x ~ a~, a real scalar

(negativity)

(3.33)

way that all principal minors of (n - 1) by (n - 1) are non-zcro. It follows from (3.30) and from (3.27) that:

~'u, - 0 (3.34)

The covariance matrix S2 - E(u,u,) is then a singular matrix satisfying:

~'S2 - 0'

(3.35)

The Rotterdam specification is attractive because it allows one to

express the constraints on the system in terms of the estimated

parameters.

As explained in the preceding section a mixed demand system can be obtained from a regular demand system by treating some of the prices as endogenous and the corresponding quantities as exogenous. Let this be the case for the first n, G n commodities. For the n- n, remaining goods the quantities are endogenous and the prices ex-ogenous. (This order is the reverse of that in the preceding section in the interest of convenient exposition.) Dropping the time subscript and using the subscript 1 or 2 to indicate the category (3.29) is rewritten as:

~Iny,

b,

S„ S1z

Olnn,

u,

-

~InO f

f

DIny2 _{b2 SZ, SZZ} Olnn2 u2 (3.36)

Here ~Inn, and ~1ny2 are endogenous, OInn2, ~Iny, and OInQ are exogenous.

Since S„ is non-singular one has:

Olnn, - -S;;b, OInQ -~ S;; Olny,

- S;;S,Z ~Inn2 - S;;u,

(3.37)

This result can be used to eliminate ~Inn, on the right-hand side of

thc relation for OIny2 in (3.26):

~Iny:

-(6Z - S21 S;;b,) ~1nQ f SZ, S;;~Iny,

t(SZ, - SZ,S ;;S,2) ~lnnZ f ii2 -SZ,S ;;u, (3.38)

40 Tl:e Esti~nation oj Mixed Demand Systems ~ - ~~ ~

~InQ

b-S21S ;;b,

S;; -S ;;S12 Olny, v, f f SZ,S;; SZZ-SZ,S;;S,2 ~InnZ vz

(3.39)

where the random components are defined by:

v,

-S;;

0

u,

(3.40)

vZ

_-SZ,S;;

_I

_uZ

An alternative version is:

Olnn,

c,

R„

R12

Alny,

-

O1nQ f

Dlny~

c2

R11

R22

Alnn2

f

v, v2 (3.41)

with the following properties on the parameters:

r'c2

- 1,

r'R~, --r,

r'RZ, - 0'

(adding-up)

(3.42)

R,Z~ - ~,

R22~ - 0

(homogeneity)

(3.43)

R„

- R;,,

RZZ - RZ2, R,Z --Ru

(symmetry)

(3.44)

x'R„x c 0 tlx ~ 0,

(negativity)

x'RZZx c 0 dx ~ ar

(3.45)

- see also Bronsard and Salvas-Bronsard (1980). Clearly, the

con-straints are less easy to impose on the direct estimation of mixed

demand system (3.41) than on that of regular demand system (3.29).

Before turning to the issue of estimation first some attention

should be paid to the random components. As is readily verified from

(3.40), the counterpart (3.34) for the mixed demand system is:

with j' -(0', ~'). The adding-up condition applies only to the second,

the quantity endogenous part of the mixed system. Let ~ be E(v,v,').

Then (3.46) means:

j'E - 0'

(3.47)

implying singularity of ~.

The relation between v and u is given by (3.40). Let:

-S;;

0

C--S21S;;

I

Note j'C - ~'. The reverse of (3.40) is:

u - C-'v

with

(3.48)

(3.49)

-S„

0

C-~ -

(3.50)

-SZ,

I

Clearly, F- C S2 C', S2 - C1 EC'-' ~ (3.51)

42 The Estimation of Mixed Demand Systenu

estimate (3.36) with properly accounting for the endogenous nature

of ~Inn2. That is the topic of the next section.

3.4

MAXIMIZING THE LIKELIHOOD

Mixed demand system (3.41) is the natural starting point for the formulation of the likelihood function. It is assumed that the vector of disturbance terms, v„ is normally distributed with the zero vector as mean and ~ as covariance matrix. We also assume that E(v,v~) - 0 for s~ t and that, in keeping with the exogenous nature of O1nQ„ Olny„ and ~Inny, the v~ are distributed independently of the expla-natory variables in (3.41).

Because of (3.47) the covariance matrix E is singular and the joint density of v, is not defined. Delete one of the equations for endogen-ous quantities, i.e. an equation of the second part of the system. The reduced disturbance vector G~ will now have a covariance matrix ~ of full rank n- 1. Let the deleted equation be the last. The joint density for a sample of T independent realization of the endogenous vari-ables can then be written as:

InL„ - -[T(ii - 1)In2n f TIn~É-'~-~ ~~v; É-'v,,12 (3.52)

which is the likelihood function when v, is expressed in observations and unknown parameters. Following Barten ( 1969), mutatis mutan-dis, one can express ( 3.52) also as:

- r T(n - 1)ln2n - Tln n2 -~ TIn~F

~-f ~ vi ( E f_~

1

-ll~I

n2

(3.53)

(3.42), (3.46) and (3.47) will enable one to reconstruct the required information for the deleted equation.

Because no lack of information is involved we will work with (3.52). As said earlier we will not estimate the coefficients of the mixed demand system directly, but rather those of the regular sys-tem. Likelihood function (3.52) has then to be expressed in terms of S2 and u„ where S2 is S2 with the last row and column deleted and u, is the u, vector without the last element. It follows from (3.40) that:

v, - C~u,

(3.54)

C. is the matrix C, defined by (3.48), with the last row and column

deleted. Consequently,

É - C~S2C~

Then

V~ E-'V, - U~CÍC~-'S2 'C~ ~C~U, - UíS2-'Ur

and

(3.56)

ln~~~ - In~S2~ f 21n~C~~

(3.57)

Wc can then rewrite (3.52) as:

,

InL„ --[ T(n - 1)ln2n ~- Tln~S2~ f 2TIn~C~~

f ~ u~S2-'u, ] l2

_~

(3.58)

which differs from the likelihood function of a system with all prices exogenous by the presence of TIn~C~~.

It is useful to look somewhat closer into ~C~~. The matrix C~ is a lower block-triangular matrix. Its determinant is then given by ~-S;;~. Since S„ is a negative definite matrix -S;; is a positive definite matrix. One clearly has In~C-~ - ln~-S„~. Let S~ be the matrix S of (3.29) with last row and column deleted. Barten and Geyskens (1975) use the Cholesky decomposition

44

The Estimation of Mixed Demand Systems

where B is a lower triangular matrix with ones on the diagonal and H is a(n - 1) x(n - 1) diagonal matrix with the Cholesky values h„ ..., h„-, as diagonal elements. From (3.59) it follows that:

S„ - - B, H, B; (3.60)

where B, is the n, x n, leading block of B and H, is the n, x n, diagonal matrix of the first n, Cholesky values. It follows from the nature of B, that ~B,~ - 1. Consequently:

ln~-S„~ - 21n~B,~ -~ ln ~H,~ -

ln h;

(3.61)

n,

~

;,~

We now rewrite (3.58) as:

n,

InL„ --[ T(n - 1)Iri2n f 71n ~52~ - 2T ~

_;-~

Inh;

f ~~u;S2-'u,, l2

(3.62)

One can next follow the same path as outlined in Barten and Gey-skens except that the first- and second-order derivatives of lnL„ with respect to the h; have to be adjusted. It turns out that these adjust-ments involve only a minor change in the computer package DEMMOD which was originally designed for regular demand systems.

In most of the earlier experiments the Cholesky values h; were less

than one in absolute value. Their logarithm is then negative. For

fixed h; the likelihood of a mixed demand system will be less than that

of a regular demand system.

One can expect that for the mixed demand system the estimates of the h;, i- 1, ..., n, will be somewhat hígher, i.e. closer to one, than for the regular system in order to reduce the maximum in the least way. This means that also the estimates of Slutsky matrix S will tend to be higher for the mixed case than for the regular case.

3.5

THE VEGETABLE MARKET

addítional supplies. One can expect that the price is set to absorb the given supply of fresh vegetables. The possibility of destroying part of the supply to maintain a minimum price exists but is rarely used. Imports of fresh vegetables are relatively unimportant. Canned or otherwise preserved vegetables are easy to store without losing their quality. The difference between demand and supply can be bridged by changes in inventories rather than by price adjustments. Of course, canned and fresh vegetables are mutual substitutes. The price formation of fresh vegetables takes into account the prices set for canned ones. The seasonal variations in the supply of fresh veg-etables will be partly compensated by opposite variations in the demand for canned vegetables. The market for vegetables appears to be well suited for description by a mixed demand system.

The models of the preceding sections express individual consumer

rationality. We will assume that they are also valid in the aggregate,

for the whole market.

These models also apply to the full consumer allocation problem. To what extent can they be used for vegetables only? Under weak separability of the preferences in vegetables and various other com-modity groups (meat, clothing, etc.) the demand for the group of vegetables as a whole can be described as a function of total available means and the price indexes of the groups. The demand for veg-etables as a group or rather its log-change AInQ, acts as the explanat-ory variable of the subsystem for a particular market. For this market only relative prices matter, not the general Price index of the group. If all prices go up by the same factor also m, total expenditure for the group goes up by that factor and the n; - p;lm remain unchanged. The endogeneity of some of the relative prices in the subsystem is not in contradiction with the exogenous nature of DInQ,. The models presented earlier can be meaningfully applied to the market for vegetables.

The data to which the mixed demand system is applied are quar-terly data collected by the Agricultural Economic Institute of the Belgian Ministry of Agriculture. This Institute observes the purchas-ing behaviour for foodstuffs of a shiftpurchas-ing panel of about 300 families and publishes quarterly average prices and quantities. Our data start with the first quarter of 1975 and ends the last quarter of 1984. The time series cannot be easily extended after this last observation because the format of the published data changed.

46 The Estimation of Mixed Demand Sys1e~u

Brussels sprouts, beans, frozen spinach, canned tomatoes, canned peas and carrots and frozen beans. The first eight form the category

of fresh vegetables, the last four are of the preserved kind.

There is in principle no major difficulty in handling a system of 12 types of vegetables. For the purpose of a numerical illustration,

however, a system of lower dimension suffices. The 12 kinds of vegetables have therefore been aggregated to a set of eight composed as follows:

1. CFSS (0.12): cauliflower, Brussels sprouts

2. LTSP (0.14): lettuce, spinach

3. CTBN (0.13): carrots, beans

4. TOMA (0.26): tomatoes

5. BEND (0.26): Belgian endives

6. PCBC (0.04): canned peas and carrots, frozen beans

7. SPIF (0.02): frozen spina~h

8. TOMC (0.03): canned tomatoes

The numbers in parenthesis are the shares of expenditure on the type of vegetables in the total budget for vegetables averaged over the sample period. Obviously the fresh vegetables dominate the pre-served ones.

The data for the fresh vegetables display considerable seasonal variability. Tomatoes, for example, are low in quantity in the first quarter and high in the third one. Their prices show an opposite pattern. The compensating price variation is not enough to eliminate seasonal effects from the expenditure shares, which range from 7 per cent of the first quarter of 1976 to 50 per cent in the third quarter of

1980. To allow for the possibility that the seasonal variability in

supply of fresh vegetables is not completely absorbed by price changes seasonal dummies have been added to the equations of the system. As the system is in first differences of the variables also first differences of the seasonal dummies have been taken. This means that, for example, the dummy for the first quarter has a one in quarter 1 and minus one in quarter 2 and zero in quarter 3 and 4. Four of such quarter dummies are fully collinear. The one for the second quarter has therefore been deleted. The coefficients of the remaining season dummies measure the difference with respect to the second quarter.

the fresh vegetables. The results for the b; and the s;; are given in Table 3.1.

Adding-up conditions (3.22) are met automatically. The homogen-eity and symmetry conditions are imposed. Negativity condition (3.25) is satisfied freely. The estimated coe~icients characterize equi-librium relationships between prices and quantities. They do not represent a pure impulse-response effect. One may observe that TOMA and BEND have b; values larger than their average expendi-ture shares. Specifically endives have a strong b value. This vegetable is commonly considered a luxury. The other (than TOMA and BEND) vegetables, fresh or not, have all rather low marginal pro-pensities to consumers. That for SPIF, frozen spinach, is even nega-tive, but not significantly so.

The Slutsky coefficients s;; are in absolute value somewhat larger than one usually finds in a system of this size. Of the 28 i~depen-dently estimated ones 19 are twice their asymptotic standard errors, in absolute value, which is also better than usual given that there are 39 observations. Of the 21 pairs of different goods seven s;; have the negative sign of Hicksian complementarity. Of these two are signifi-cantly negative, namely that for CTBN (carrots and beans) and CFSS (cauliflower and Brussels sprouts) and for SPIF (frozen spinach) and PCBC (peas, carrots and beans, preserved). Domination of substi-tution is plausible, not only because of the mathematical properties of the matrix S, but also because of the nutritional properties of these vegetables.

Given the point estimates of Table 3.1 `one can calculate the

coefficients of (3.39) or (3.41), the mixed demand system. The results

are given in Table 3.2. Under the assumptions made these

coef-ficients correspond with impulse-response effects. They are `reduced

form' coefficients. No asymptotic standard errors have been

calcu-lated. The results satisfy properties (3.42) to (3.45).

Note that in Table 3.2 the first five equations have the log-change

in (normalized) prices as dependent variables, while the last three the

log-change in quantities (multiplied by the wi). After OInQ, the first

five exogenous variables are the log-change in quantities (multiplied

by the w;) and the last three the log-change in prices. Only the

coefficients of c, and R,Z are elasticities. All the others would have to

be divided andlor multiplied by the relevant expenditure shares to

turn them into elasticities.

TabJe 3.1 Estimates of b and S with endogenous prices for fresh vegetables, Belgium 1975-84`

i. type b,

CFSS

LTSP

CTBN

TOMA

BEND

PCBC

SP1F

TOMC

l. CFSS

0.016

-0.412

0.216

-0.053

0.030

0.177

0.037

-0.009

0.014

0.056

0.039

0.034

0.019

0.042

0.044

0.011

0.006

0.007

2. LTSP 0.060 0.216 -0.275 0.063 0.054 -0.063 -0.006 0.018 -0.007 0.059 0.034 0.037 0.017 0.040 ' 0.043 0.008 0.005 0.005 3. CTBN 0.013 -0.053 0.063 -0.167 0.028 0.121 0.006 -0.006 0.009 0.029 0.019 0.017 0.013 0.023 0.025 O.OOS 0.005 0.003 4. TOMA 0.298 0.030 0.054 0.028 -0.511 0.321 0.029 0.029 0.019 0.073 0.042 0.040 0.023 0.058 0.055 0.011 0.006 0.007 5. BEND 0.587 0.177 -0.063 0.121 0.321 -0.591 0.036 0.002 -0.003 0.090 0.044 0.043 0.025 O.OSS 0.077 0.011 O.OOS 0.006 6. PCBC 0.017 0.037 -0.006 0.006 0.029 0.036 -0.073 -0.023 -0.007 0.011 0.011 0.008 0.005 0.011 0.011 0.011 0.006 0.007 7. SPIF -0.001 -0.009 0.018 -0.006 0.029 0.002 -0.023 -0.026 0.014

O.OOS 0.006 O.OOS 0.003 0.006 0.005 0.006 0.007 O.OOS

8. TOMC 0.009 0.014 -0.007 0.009 0.019 -0.003 -0.007 0.014 -0.040

0.009 0.007 0.005 0.003 0.007 0.006 0.007 0.005 0.00?

i. type c;

CFSS LTSP CTBN TOMA BEND PCBC SPIF TOMC

SO The Estimation of Mixed Demand Systen~s

is positive, even strongly positive for the five price formation equa-tions. Part of L11nQ is due to Alny, - see (3.27). One can separate that part out from OInQ and attribute it to Alny,. L.et:

AInQ, - W„OInQ„ f Wu01nQu

with

r,1nQu - ~ (Wi,~Wu) Alnqu - (llW,r) ~ Alnyir

i-t i-~

OlnQv - ~ (wir~Wu) ~lnq;, - (1IWy) ~ Alnyir

i~6 i~6

W ~r - (r Wir. wu - (. Wir

i~l ` i~6

Here, ~lnQ,r ts the averagc ~og-change for the goods of the quantity exogenous groups. A similar definition holds for AInQ,,. The AlnQ,r part of AInQr is already exogenous in its own right, because the relevant Olny;, are ezogenous. The exogenous nature of A1nQr im-plies then the additional exogeneity of A1nQy. One can therefore replace c~lnQ, by

cWy AlnQy f c ~ Alny;,

i-l

For our sample Wu is in the mean 0.09, which scales down the c

vector considerably. The ~;; with j- 1, ..., 5 have to be increased by

c;, which reduces their absolute value also substantially. Note that the

diagonal elements of R„ t c,~' still remain negative. An increase of

the supply of a good will depress its price but it might increase the

price or quantity of another good. This becomes clear from Table 3.3

which states the total effects of exogenous quantity changes.

i. rype

Tnble 3.3 Total effects of exogenous quantities

r;~ f- c; CFSS LTSP CTBN TOMA BEND 1. CFSS -3.083 -2.121 0.455 0.344 -0.104 2. LTSP -1.984 -5.279 -0.819 0.251 0.259 3. CTBN 1.205 -0.206 -6.286 0.396 -0.334 4. TOMA 0.899 0.669 0.201 -0.885 0.102 5. BEND 1.020 1.246 0.040 0.671 -0.771 6. PCBC -0.015 0.033 0.010 0.029 -0.015 7. SPIF 0.011 -0.055 0.019 -0.025 0.008 8. TOMC 0.005 0.023 -0.029 -0.004 0.007

Ta61e 3.4 Seasonal effects for the vegetables market Belgium 1975-84

Type Regular Mixed

Q,

Q,

ia.

Q,

Q,

Q.

1. CFSS -5.37 -9.20 -9.76 -18.2 -38.3 -33.3 1.71 5.42 2.75 2. LTSP -8.07 -1.65 -4.75 -56.7 -44.1 -66.6 1.78 5.38 2.81 3. CTBN -4.41 0.211 -5.90 -0.136 10.2 -10.8 0.920 2.94 1.45 4. TOMA -15.1 -17.1 -32.0 -0.359 -24.7 -33.6 2.15 7.06 3.61 5. BEND 33.8 24.9 50.0 57.7 24.1 61.5 2.68 8.65 4.33 6. PCBC -0.362 0.105 0.968 1.37 -0.851 1.29 0.420 1.33 0.678 7. SPIF -0.478 1.72 0.717 -1.26 0.539 -0.994 0.211 0.687 0.350 8. TOMC -0.068 1.01 0.626 -0.112 0.312 -0.297 0.232 0.757 0.388

52

The Estimation of Mixed Demand Systems

appear to be very strong for Belgian endives for which supply goes up

in the fourth quarter to reach a peak in the first quarter. The second

quarter sees a decline and the bottom is reached in the third quarter.

The prices move inversely. Apparently in the fourth and first quarter

the price decrease is not enough and in the third quarter the price

increase is too strong as follows from the positive coefficients of the

quarter dummies. Non-price factors appear to pick up the extra

supply.

This interpretation is confirmed by the transformation of the

dummy coefficients by matrix C, defined by (3.48). This

transform-ation gives the values for the dummy coefficients in the mixed mode

of the system. They are given in the last three columns of Table 3.4.

The first five rows show the seasonal effects on the price formation.

Positive signs mean that the prices are not low enough in comparison

to the situation of the second quarter. Negative signs indicate the

opposite.

~

The seasonal effects are quite important in terms of explanatory

power of the model. Still they are somewhat puzzling. Their

at-tribution to shifts in preferences is disputable. The data refer to

household demand and are net of the effects of market interventions

or importlexport fluctuations. The seasonal effects refiect an

undeni-able seasonal pattern in the part of consumer behaviour not

ex-plained by exogenous quantity and price changes.

It is of some interest to compare the results given in Table 3.1 with

those obtained under the assumption that all quantities are

endogen-ous and all prices (and AInQ) are exogenendogen-ous. Table 3.5 presents the

point estimates, The b; display roughly the same pattern as in Table

3.1. The s;, are in absolute value usually smaller. This corresponds

with the higher Cholesky values obtained for the mixed case as was to

be expected. Table 3.6 gives the two sets of Cholesky values together

with their standard errors, calculated as if the model in question were

the correct one. The first five Cholesky values correspond with the

S„-part of matrix S. They are all substantially higher in the price

endogenous case than for the price exogenous situation.

The asymptotic standard errors are usually smaller in the

exogen-ous case. Because of the lower absolute values of the s;; also here 19

of thé 28 independent coefficients are in absolute value more than

twice their standard error.

exogen-i. type b;

CFSS LTSP CTBN TOMA BEND PCBC SPIF TO~ti1C

1. CFSS 0.039 -0.130 0.005 -0.029 0.040 0.017 0.024 -0.000 0.015 0.019 0.024 0.013 0.010 0.020 0.019 0.009 0.005 0.006 2. LTSP 0.030 0.005 -0.034 -0.002 -0.020 0.026 0.014 0.007 0.003 0.01 S 0.013 0.011 0.007 0.014 0.014 0.006 0.003 0. 003 3. CTBN 0.027 -0.029 -0.002 -0.115 0.019 0.061 0.002 -0.001 0.006 0.018 0.010 0.007 0.009 0.014 0.015 0.004 0.002 0.00.1 4. TOMA 0.282 0.040 -0.020 0.019 -0.136 0.072 0.017 0.009 -0.001 0. 03S 0. 020 0. 014 D. 014 0. 038 0. 033 0. 009 0. 005 0. DOS 5. BEND 0.613 O.OI7 0.026 0.061 0.072 -0.181 0.008 0.007 -0.008 0. 046 0. 019 0. 014 0. OlS 0. 033 D. 044 0. 008 0. 004 0. 005 6. PCBC 0.006 0.024 0.014 0.002 0.017 0.008 -0.051 -0.016 0.002 0.008 0.009 0.006 0.004 0.009 0.008 0.012 0.007 0.007 7. SPIF -0.000 -0.000 0.007 -0.001 0.009 0.007 -0.016 -0.024 0.017 0.004 0.005 0.003 0.002 0.005 D.004 0.007 0.007 0.005 8. TOMC 0.004 0.015 0.003 0.006 -0.001 -0.008 0.002 0.017 -0.035 0.005 0.006 0.003 0.003 0.005 O.OOS 0.007 0.005 0.007

54

The Estimation of Mixed Demand Systems

Table 3.6 Cholesky values of S with and without endogenous prices'

h; 1 2 3 4 S 6 7 With five 0.412 0.162 0.153 0.468 0.141 0.010 0.016 endogenous 0.039 0.016 0.010 0.049 0.012 0.008 0.009 prices With all 0.130 0.033 0.108 0.107 0.062 0.010 0.018 prices 0.024 0.011 0.012 0.044 0.012 0.010 0.009 exogenous

' Asymptotic standard errors are given in italicx.

ous and obtains the value of -65.670. Its value, given the b and S of

Table 3.1 and, of course, the dummy coefficients, is -60.215, which

is larger, as is to be expected. The conversion of ln~S2~ to 1n~É~ can be

achieved by subtracting 2 E; lnh;. For the price endogenous case one

obtains for ln~É~ the value of -28.067, while the price exogenous

variant yields a value of -22.999, clearly larger. As a very rough

goodness of fit test this comparison fails to reject. Each variant

produces the least variance for the case for which it is appropriate. It

is beyond the scope of the present chapter to develop a more refined

test procedure which could sort out for which goods the prices are

exogenous and for which the quantities are given.

3.6

CONCLUDING REMARKS

Mixed demand systems are in between the polar cases of regular

demand systems with all prices exogenous and inverse demand

sys-tems with all quantities exogenous. They are realistic when for some

commodities the inventory costs are substantial and prices adjust to

available supply while for other goods one can let the quantities

demanded adjust to the prices and absorb eventual differences

be-tween demand and supply by rather cheap inventory changes.

All these modes of demand systems reftect the basic consumer

equilibrium consisting of the budget identity and the Second Law of

Gossen. This means that one can start from any mode and solve it for

the appropriate set of endogenous variables.

parametrization has served as a starting point for the formulation of a mixed demand system. One of the attractions of the Rotterdam specification is the ease by which one can take into account theoreti-cal constraints on the coe~icients. This property is to a certain extent lost in the transition to a mixed demand system.

One way to have your cake and eat it is to estimate the system in its

regular mode while taking into account the endogenous nature of

some of the prices. One can easily incorporate the various constraints

while avoiding inconsistencies of estimation. Following a maximum

likelihood approach this turns ou: to require only a minor adjustment

of the estimation procedure for a regular system with all prices

exogenous.

The market for vegetables in Pelgium provided quarterly data for

the period 1975-84. Eight (groups of) vegetables were selected, five

of them fresh, the remaining three frozen, canned or otherwise

preserved. The prices of the fresh vegetables were taken to be

endogenous, those of the others as exogenous.

Seasonal dummies were added to absorb the obvious seasonal pattern in the residuals. Given the fact that the seasonal variation in the supply of fresh vegetables should have been fully reflected in their prices the presence of an unexplained season is puzzling.

The results show that taking into account the endogenous nature of

some of the prices tends to increase the absolute value of the price

effects in the equilibrium relations. The results are as a whole

reasonable but intuition is lacking to serve as the touchstone of

plausibility.

~

One further step could be the use of Allais coefficients to obtain an

idea of the pattern of complementarity and substitution implied by

the estimates.

One would also like to obtain standard errors of some nature for

the various derived coe~icients. A Monte Carlo procedure could be a

possibility.

Another line of further research is the setting-up of tests for the

selection of the commodities for which prices are endogenous and

quantities exogenous and for which the reverse holds.

S6

The Estimation of Mixed Detnand Systents

Empirical work usually answers some questions but raises at the same time a host of other ones left unanswered. The present chapter is no exception.

Acknowledgements

The topic of mixed demand systems has been explored in the course of recent years by the author in cooperation with several researchers: Henri Delval, Eric Meyermans and Luc Dresse. Eric Meyermans also supplied the data for the present chapter. The author is in debt to all three. They cannot be blamed for any shortcomings of the present chapter. Rick van der Ploeg is thanked for his comments on an earlier draft. The debt of the author to Henri Theil is not easy to measure. It was Theil who set him on the track of consumer demand systems and with whom initial developments were shared. Geographical distance prevented close cooperation later on. As the work of Theil and his students show applied demand theory has turned out to be a very fruitful research area. The author is grateful to have been able to contribute his shara

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No. 44 E. van Damme, R. Selten and E. Winter, Alternating bid bargaining with a smalle.st money unit, Cama and Economic Behavior, voL 2, no. 2, 1990, pp. 188 - 201.

No. 4S C. Dang, The D,-triangulation of R' for simplicial algorithms tor computing solutions o[ nonlinear equations, Matlumatics of Opetiations Reseatrh, voL 16, no. 1, 1991, pp. 148 - 161.

No. 46 'Ilt. Nijman and F. Palm, Predictive accuracy gain from disaggegate sampling in ARIMA modeLc, Jounw! ojBusiness dc Ecotwttuc Sratisticr, voL 8, no. 4, 1990, pp. 40S - 415.

No. 47 J.R. Magnus, On certain moments relating to ratios of quadratic torms in normal variables: further results, Sankhra.. Tlu IrdinttJoutnol ojStatistict, voL S2, series B, part. 1, 1990, pp. 1- 13.

No. 48 M.FJ. Steel, A Bayesian analysis of simultaneous equation models by combining

recursive analytical and numerical approaches, Jouma! oj Econometrits, voL 48,

no. 1~2, 1991, pp. 83 - 117.

No. 49 F. van der Ploeg and C. Withagen, Pollution control and the ramsey problem,

Environnunta! mid Ruounce Economiu, voL 1, no. 2, 1991, pp. 21S - 236.

No. SO F. van det Ploeg, Money and capital in interdependent ooonomies with overlapping generations, Economica, vol. S8, no. 230, 1991, pp. 233 - 256. No. S1 A. Kapteyn and A. de Zeeuw, Changing incentives for economic research in the

Netherlands, European Economic Review, voL 3S, no. 2~3, 1991, pp. 603 - 611. No. 52 C.G. de Vries, On the relation between GARCH and stable processes,loturral

oj Econometrics, voL 48, no. 3, 1991, pp. 313 - 324.

No. Sq W. van Grcenendaal and A. de Zeeuw, Control, ooordination and conIIict on

intunational commodiry markets, Ecor~ornic ModeUing, voL 8, no. 1, 1991, pp. 90 - 101.

No. SS F. van der Plceg and AJ. Mackirilt, Dynamic poliry in linear models with rational ezpectations of future events: A oomputer package, Compura Scitnce in

Economicr and Managemau, voL 4, no. 3, 1991, pp. 17S - 199.

No. 56 HA. Keuzenkamp and F. van der Pbeg, Savings, investment, government finance, and the current aocount: The Dutcb ezperience, in G. Alogoskoufis, L

Papademos and R Portes ( eds.), Euemal Cautrnints on Macmecovromic PoGcy:

77u European F.rpr.riutce,Cambridge: Cambridge University Press, 1991, pp. 219

- 263.

No. S7 Th. Nijman, M. Verbeek and A. van Soest, The efficiency of rotating-panel

designs in an analysis-ot-variance model, loumal ojEca~nonutrics, voL 49, no. 3, 1991, pp. 373 - 399. ,

No. 38 M.FJ. Steel and J: F. Richard, Bayesian multivariate ezogeneiry analysis - aa

application to a UK money demand equation, lountol oj Ecanomtpiu, voL 49,

no. 1~2, 1991, pp. 239 - 271.

No. S9 Th. Nijman and F. Palm, Generalized least squares estimation of linear models

containing rational future expoctations, Intemariorw! Econonric Review, voL 32, no. 2, 1991, pp. 383 - 389.

No. 60 E. van Damme, Equilibrium seledion ia 2 x 2 games, Revitra Eqpanola dt

Ecoiwmia, voL 8, no. 1, 1991, pp. 37 - 52.

No. 61 E. Bennett and E. van Damme, Demand commitment bargaining-. the case of apex games, in R Selten (ed.), Came Equilibriwn Modelc II! - Saategic Bargnirting, Berlin: Springer-Verlag, 1991, pp. 118 - 140.

No. 62 W. Guth and E. van Damme, Gorby gamu - a game theoretic analysis of disarmament campaigns and the detense etfuienry - hypothesis -, in R Avenhau; H. Karkar and M. Rudnianski (eds.), Dcfurtt Decirion Making

-Matyrical Support and Girir Managcmatt, Berlin: Springer-Verlag, 1991, pp. 21S

- 240.

No. 63 A. Rcell, Dual-capaciry trading and the quality of the market, lourna! oj

Financia! Inremrediarion, voL 1, no. 2, 1990, pp. lOS - 124.

No. 64 Y. Dai, G. van der Iaan, AJJ. Talman and Y. Yamamoto, A simplicial algorithm for the nonlinear stationary point problem on an unbounded polyhedron, Siarn Journal oj Opru~irntion, voL 1, no. 2, 1991, pp. 151 - 165. No. 6S M. McAleer and C.R McKenzie, Keynesian and new classical models of

unemployment revisited, Tht Economic Joumal, voL 101, no. 406, 1991, pp. 3S9 - 381.

No, 68 F. van der P{ceg. Macroeconomic policy coordination issues duringthe various phases of economic and monetary integtation in Europe, European Economy

-The Economict oj E6fU, Commission of the European Communities, special

edition no. 1, 1991, pp. 136 - 164.

No. ó9 H. Keuzenkamp, A precutsor to Muth: Tinbergen's 1932 model of rational expectations, The Economiclounwl, voL 101, no. 408, 1991, pp. 12JS - 1253. No. 70 4 Zou, The target-incentive system vs. the price-incentive system under adverse

selection and the ratchet ettect,Jounta! ojPublic Economiu, voL 46, no. 1, 1991, PP. 51.- 89.

No.71 E. Bomhoff, Between price reform and privatization: Eastern Europe in transition, Fina~wnarkt tutd Porrfolio Management, vo1 S, no. 3, 1991, pp. 241 -251.

No. 72 E. Bomhotf, Stabiliry of velociry in the major industrial countries: a Kalman filter approach, Intemational Monetary Fund StafjPapers, vol. 38, no. 3, 1991, pp. 626

- 642.

No. 73 E. Bomhoff, Currenry convertibility: when and how? A contrlwtion to the Bulgarian dcbate, KrcJit und Kapital, vol. 24, no. 3, 1991, pp. 412 - 431. No.74 H. Keuzenkamp and F. van der Plceg, Perceived constrainu for Dutch

unemployment policy, in C. de Neubourg (ed.), 7Trt Art oj FuU Employrttent

-Unemployment Policy in Open Economies, Contributions to Economic Analysia

203, Amsterdam: Elsevier Science Publishers B.V. (North-Holland), 1991, pp. 7 37.

No. 75 H. Peters and E. van Damme, Characterizing :he Nash and Raiffa bargaining solutior.s by disagreement point axions, Mathematics ojOpaations Rrsearch, voL 16. :o. 3, ;991, pp. 447 - 461.

No.75 P1. Dexhamps, On the estimated variances of regression ooefficients in misspecified error componenu modeLs. Econometric Theory, voL 7, no. 3, 1991, pp. 369 - 384.

No. 77 A. de Zeeuw, Note on 'Nash and Stackelberg solutions in a differential game modei of capitalism', Jourtta! of Economic Dytsatnicr and Control, voL 16, no. 1,

1992, pp. 139 - 145.

No. 78 J.R Magnus, On the fundamental bordered matrix of linear eatimation, in F. van der Plceg (ed.), Admnced Lectutrs in Quantitotive Economicr, l.ondon-Orlando: Academic Press L,td., 1990, pp. 583 - 604.

No. 79 F. van der Plceg and A. de Zeeuw, A differential game of international pollution

control, Systemr attd Contro! Letters, vo4 17, no. 6, 1991, pp. 409 - 414.

No. 80 Tit. Nijman and M. Verbeek, The optimal choice ot controls and

No. 81 M. Verbeek and Th. Nijman, Can cohort data be treated as genuine panel data?,

Etnpiricd F,conomirs, vol. I7, no. 1, 1992, pp. 9- 23.

No. 82 E. van Damme and W. Giith, EquiLbrium selection in the Spence signaling game, in R. Selten (ed.), Came Equilibritun Modelr I! - Method~ MoroLt, and Mrlrfrcu, Berlin: Springer-Verlag, 1991. PP. 263 - 288.

No. 83 R.P. Gilles and P.H.M. Ruys, Charaderi7ation of economic agents in arbitrary communiration structures, Nieuw Atrhief voor Wiskurate, vol. 8, no. 3, 1990, pp. 325 - 345.

No. 84 A. de Zeeuw and F. van der Plceg, D'Jference games and policy evaluation: a conoeptual tramework, t~jotd Ecottornic Pgpers, vol. 43, no. 4, 1991, pp. 612 -636.

No. 85 E. van Damme, Fair division under asymmetric information, in R. Selten (ed.),

Rotionn! Interruction - F-ssays in Honor ojJohn C Harsanyi, Berlin~Heitielberg:

Springer-Verlag, 1992, pp. 121 - 144.

No. 86 F. de Jong, A. Kemna and T. Klcek, A contribution to event study methodolop~y with an application to the Dutch stock market, Jounut! oj Batdcing and Finance, vol. 16, no. 1, 1992, pp. 11 - 36.

No. 87 A.P. Barten, The estimation of mixed demand systems, in R. Bewley and T. Van Hoa (eds.), Contributions to Consumtr Demand and Econometrics, Esrays in

Bibliotheek K. U. Brabant

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