This paper extends the nonparametric methods developed by Samuelson (1948), Houthakker (1950), Afriat (1973), Diewert (1973) and Varian (1982, 1983) to latently separable models. It presents necessary and sufficient empirical conditions under which data on the market behaviour of a price-taking consumer, and a hypothesised allocation across latent groups are nonparametrically consistent with latent separability (Gorman (1968, 1978), Blundell and Robin (2000)). It considers homothetic latent separability and weak separability as special cases.Separability, revelaed preference

Ian Crawford

English

Research Papers in Economics

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Discussion Papers in Economics 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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ISSN: 1749-5075 
 
 
NECESSARY AND SUFFICIENT CONDITIONS FOR 
LATENT SEPARABILITY 
 
 
 
By 
 
Ian A. Crawford 
 (University of Surrey) 
 
 
DP 07/06 
 Necessary and Sufficient Conditions for Latent Separability
Ian A. Crawford 
Abstract This paper extends the nonparametric methods developed by
Samuelson (1948), Houthakker (1950), Afriat (1973), Diewert (1973) and Varian
(1982, 1983) to latently separable models. It presents necessary and suﬃcient empiri-
cal conditions under which data on the market behaviour of a price-taking consumer,
and a hypothesised allocation across latent groups are nonparametrically consistent
with latent separability (Gorman (1968, 1978), Blundell and Robin (2000)). It con-
siders homothetic latent separability and weak separability as special cases.
Acknowledgements Financial support from the ESRC Centre for the Mi-
croeconomic Analysis of Public Policy and Leverhulme Centre for Microdata Meth-
ods and Practice is gratefully acknowledged. The author is grateful to Richard
Blundell for helpful comments, but is responsible for all errors.
What are the observable consequences of a model of consumer behaviour in which the con-
sumer’s utility function is latently separable? The notion of latent separability was introduced
in Gorman (1968) and (1978) and thoroughly developed in Blundell and Robin (2000). So far
the characterisation of the conditions for latent separability has been parametric. This paper
seeks necessary and suﬃcient nonparametric conditions for latent separability.
Deﬁnition 1. (Blundell and Robin, 2000) A direct utility function U : q ∈ RK
+ → U (q) ∈ R is
said to satisfy latent separability if
U (q)= m a x
h q1,...,h qM∈RK
+
(
u
¡
v1 ¡
e q1¢
,...,vM ¡
e qM¢¢
|
M X
m=1
e qm = q
)
where u,v1 ¡
e q1¢
,...,vM ¡
e qM¢
are regular utility functions.
Under a latently separable functional structure the consumer’s allocation problem is as follows
max
h q1,...,h qM∈RK
+
u
¡¡
v1 ¡
e q1¢
,...,vM ¡
e qM¢¢¢
subject to
M X
m=1
p0
te qm = xt
in other words they choose both the total quantity vector qt and the latent allocations {e qm
t }
m=1,...,M
in order to maximise their utility subject to a budget constraint.
Suppose that there are T observations (indexed t =1 ,...,T)o nK−vectors of prices and cor-
responding demands {pt,qt}.L e t{e qm
t }
m=1,...,M
t=1,...,T where
PM
m=1 e qm
t = qt denote a hypothesised
allocation of these quantity vectors across M latent groups. When can these data and hypoth-
esised allocations be rationalised by a latently separable structure? The following deﬁnition
makes clear what is meant by rationalise in this context.Deﬁnition 2. A latently separable utility function rationalises the data {pt,qt}t=1,...,T and the
allocation {e qm
t }
m=1,...,M
t=1,...,T if u
¡
v1 ¡
e q1
t
¢
,...,vM ¡
e qM
t
¢¢
≥ u
¡
v1 ¡
e q1¢
,...,vM ¡
e qM¢¢
for all alterna-
tive allocations {e qm}
m=1,...,M such that p0
tqt ≥
PM
m=1 p0
te qm.
The ﬁrst Theorem presents the conditions under which there exists a latently separable model
of consumer preferences which rationalises the data and the hypothesised allocation.
Theorem 1. The following statements are equivalent:
(U) There exists a latently separable utility function, where u(v) and vm (e qm) are nonsatiated,
monotonic, concave and continuous functions, which rationalises the data {pt,qt}t=1,...,T and
the allocation {e qm
t }
m=1,...,M
t=1,...,T .
(A) There exist numbers {Ut,λ t}t=1,...,T and M−vectors {Vt,ρt}t=1,...,T such that for all t,s,m
Us ≤ Ut + λtρ0
t (Vs − Vt) (A.1)
V m
s ≤ V m
t +
1
ρm
t
p0
t (e qm
s − e qm
t ) (A.2)
(G) The data {pt,e qm
t }
m=1,...,M
t=1,...,T satisfy the Generalised Axiom of Revealed Preference (GARP)
and the data {Vt,ρt}t=1,...,T also satisfy GARP for some choice of {Vt,ρt}t=1,...,T which satisfy
(A.2)
Proof
(U) ⇒ (A):First consider the implications of optimising behaviour and the ﬁrst order condi-
tions from the consumer’s problem. Continuity ensures that suitable subgradients exist such
that ∇u(e qm
t ) ≤ λtpt where ∇u(e qm
t )=∇u(vm
t )∇vm (e qm
t ).D e ﬁne λtρm
t = ∇u(vm
t ).T h e n
∇vm (e qm
t ) ≤ (ρm
t )
−1 pt. Now consider the concavity conditions for this structure
u(vs) ≤ u(vt)+∇u(vt)
0 (vs − vt)
vm (e qm
s ) ≤ vm (e qm
t )+∇vm (e qm
t )
0 (e qm
s − e qm
t )
Substituting in ∇vm (e qm
t ) ≤ (ρm
t )
−1 pt and λtρm
t = ∇u(vm
t ) preserves the inequalities and gives
u(vs) ≤ u(vt)+λtρ0
t (vs − vt)
vm (e qm
s ) ≤ vm (e qm
t )+
1
ρm
t
p
0
t (e qm
s − e qm
t )
which are conditions (A.1) and (A.2).
(A) ⇒ (U):S u p p o s ew eh a v en u m b e r s{Ut,λ t}t=1,...,T and M−vectors {Vt,ρt}t=1,...,T such
that condition (A) holds. Consider some arbitrary {e qm}
m=1,...,M such that p0
tqt ≥
PM
m=1 p0
te qm.
We need to show that there exists a latently separable utility function, with the stated properties
such that u
¡
v1 ¡
e q1
t
¢
,...,vM ¡
e qM
t
¢¢
≥ u
¡
v1 ¡
e q1¢
,...,vM ¡
e qM¢¢
. Using (A.2) we can construct T
upper bounds on vm (e qm) and if we take the minimum of these as the function vm (e qm) then
we have a piecewise linear, nonsatiated, monotonic, concave and continuous utility function
vm (e qm)=m i n
s
½
V m
s +
1
ρm
s
p
0
s (e qm − e qm
s )
¾
s=1,...,T
≤ V m
t +
1
ρm
t
p
0
t (e qm − e qm
t )Summing this inequality over m gives
ρ0
tVt − p
0
tqt ≥ ρ0
tV − p
0
te q
where Vt =
£
V 1
t ,...,V M
t
¤0, V =
£
V 1,...,VM¤0 , (V m = vm (e qm))a n dρt =
£
ρ1
t,...ρM
t
¤
.T h e n
since p
0
tqt ≥ p
0
te q we have ρ0
tVt ≥ ρ0
tV.U s i n g (A.3) we can similarly construct the following
macro-utility function
U (V)=m i n
s {Us + λsρ0
s (V − Vs)}s=1,...,T ≤ Ut + λtρ0
t (V − Vt)
Since λtρ0
t (V − Vt) ≤ 0 we have U (V) ≤ Ut as required.
(A) ⇐⇒ (G):Follows from Afriat’s Theorem.¥
One special case of particular empirical interest is the one in which the latent utility functions
v(e qm) are homothetic (Gorman (1968), Blundell and Robin (2000)).
Theorem 2. The following statements are equivalent:
(U) There exists a homothetically latently separable utility function, where u(v) and vm (e qm)
are nonsatiated, monotonic, concave and continuous functions and vm (e qm) are homothetic,
which rationalises the data {pt,qt}t=1,...,T and the allocation {e qm
t }
m=1,...,M
t=1,...,T .
(A) There exist numbers {Ut,λ t}t=1,...,T and M−vectors {Vt,ρt}t=1,...,T such that for all t,s,m
Us ≤ Ut + λtρ0
t (Vs − Vt) (A.1)
V m
s ≤ V m
t +
1
ρm
t
p0
t (e qm
s − e qm
t ) (A.2)
ρm
t V m
t = p0
te qm
t (A.3)
(G) The data {pt,e qm
t }
m=1,...,M
t=1,...,T satisfy the Homothetic Axiom of Revealed Preference (HARP)
and the data {Vt,ρt}t=1,...,T also satisfy GARP for some choice of {Vt,ρt}t=1,...,T which satisfy
(A.2) and (A.3)
Proof
Analogous to Theorem 1 noting that (A.2) and (A.3) are necessary and suﬃcient for the existence
of M homothetic utility functions V m (e qm
t ) such that for any e qm with that p0
te qm
t ≥ p0
te qm then
V m (e qm
t ) ≥ V m (e qm) (Varian (1983), Theorem 2).¥
Another special case concerns the situation when the allocations of goods across latent utilities
is exclusive as this corresponds to weak separability.
Theorem 3. When the hypothesised allocations are such that e q
k,m
t = qk
t for all k and t and
some m (that is the allocations to latent groups are exclusive and constant over observations),
then the conditions in Theorems 1 and 2 are equivalent to those for weak separability and
homothetic weak separability respectively.Proof
Obvious from a comparison with Varian (1983) Theorems 3 and 5. ¥
I cannot think of any way in which either the number or the composition of the latent groups
can be nonparametrically identiﬁed from the observed data. Instead the Theorems presented
are simply a way of testing the data and any hypothesised allocation for consistency with the
model. In this respect the results here are similar to Varian’s (1983) for weak separability,
where the number and makeup of separable subgroups are not identiﬁed from data, but where
the conditions under which any hypothesised grouping are rationalisable with weak separability
are set out. Note that the results here can also be applied to the analysis of production functions
with a suitable change in the interpretation of the notation and by requiring the top-level data to
satisfy the weak axiom of proﬁt maximisation rather than GARP (see Varian (1984), Theorem
9, for the conditions for weak separability in production functions).
References
[1] Afriat, S.N. (1973),“On a system of inequalities in demand analysis: An extension of the
classical method”, International Economic Review, 14, 460-472.
[2] Blundell, R. W. and J-M Robin (2000), “Latent separability: grouping goods without weak
separability”, Econometrica, 68, 53-84.
[3] Diewert, W.E. (1973), “Afriat and revealed preference theory”, Review of Economic Studies,
40, 419-426.
[4] Gorman, W. M. (1968), “Measuring the quantities of ﬁxed factors”, in Value Capital and
Growth: Essays in Honour of Sir John Hicks, ed. by J,N, Wolfe. Edinburgh: Edinburgh
University Press. Also in Separability and Aggregation, Vol 1, Collected Works of W. M.
Gorman, C. Blackorby and A. Shorrocks (ed.s), Oxford: Clarendon Press.
[5] Gorman, W. M. (1978), “More measures for ﬁxed factors,”, in Separability and Aggregation,
V o l1 ,C o l l e c t e dW o r k so fW .M .G o r m a n , C. Blackorby and A. Shorrocks (ed.s), Oxford:
Clarendon Press.
[6] Houthakker, H. S. (1950), “Revealed preference and the utility function”, Economica,M a y ,
159-174.
[7] Samuelson, P. (1948), “Consumption theory in terms of revealed preference”, Economica,
November, 243-253.
[8] Varian, H. (1982), “The nonparametric approach to demand analysis”, Econometrica, 50,
945-973.
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Studies, L, 99-110.
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Necessary and Sufficient Conditions for Latent Separability

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