The CES is a discrete choice model?

Economics Letters North-Holland

THE CES IS A DISCRETE

CHOICE

Simon

P. ANDERSON

CEME,

UniversitP Libre de Bruxelles,

Andre

DE PALMA

Northwestern

University, Evanston,

Jacques-Franqois CORE,

139

24 (1987) 139-140

MODEL?

Brussels, Belgium

IL 60201, USA

THISSE

UniversitC Catholique de Louvain, Louvain-la-Neuve,

Received

Belgium

6 April 1987

The CES demand

function

is a special case of a nested logit model whose second-stage

is deterministic.

Consider an individual who has to choose a certain amount of one good among n possible goods. His choice is seen as a two-stage process: (i) he chooses which good to buy, and (ii) the quantity of that good. Suppose that the outcome of the first stage is good i. Then the individual’s conditional direct utility is assumed to be U,=ln

i=l

ql,

>.‘.> n,

(1)

where q, is the quantity of good i. Let y denote his available Maximizing (1) under the budget constraint yields the demand

q,L=;,

income

and

i.

p, the price of good

(2)

i=l,...,n. I

The conditional V(p,)=

-In

indirect pi+ln

y,

utility i=l

is therefore ,...,n.

(3)

Let us now describe the first stage. It is assumed that the choice utility approach used in discrete choice theory, i.e.,

0165-1765/87/$3.50

0 1987, Elsevier Science Publishers

B.V. (North-Holland)

of good

i follows

the stochastic

S. P. Anderson

140

et al. / The CES is a discrete choice model?

where U, is the stochastic utility associated with i, p is a positive constant and e, a random variable with zero mean and unit variance. The probability for the individual to choose i is given by

Assuming that the eis are identically, nomial) logit: ev(P,)/P

P,=

i=l

n

ce

vc P, )/P

independently

Gumbel distributed,

(5) becomes the (multi-

,...> n.

(6)

’

J=l

Substituting (3) into (6) yields

p, =

-l/P pz

1

p,‘/”

(7)

i=l,...,n. ’

;=1 Given (2), the expected demand of the individual for good i is -l/p-l

D,=

5 c

i=l

,...,n.

(8)

p;l/“y’

j=l

In (B), we recognize the demand system generated by a CES direct utility function

with 0 I Thus, a nested given by

p I 1, provided that p = (1 - p)/p. the demand model derived from the CES utility function (9) is equivalent to the solution of logit model in which the second stage is deterministic with conditional direct utility function (1).