In this paper, we show that, when combining revealed (RP) and stated (SP) data, for marginal changes in quality of environmental goods, rationality implies consistency, as the consistency conditions coincide with a subset of the conditions for rationality.combined (RP and SP) individual data; rationality; data consistency

Cunha-e-Sa, Maria A.

Ducla-Soares, Maria M.

Nunes, Luis C.

English

Research Papers in Economics

COMBINING CV AND RP DATA: A NOTE ON THE RELATIONSHIP BETWEEN 
CONSISTENCY AND RATIONALITY  
by 
 
Maria M. Ducla-Soares, Maria A. Cunha-e-Sá, and Luis C. Nunes 
 
Faculdade de Economia, Universidade Nova de Lisboa, 
Travessa Estevão Pinto, 1099-032 Lisboa, Portugal  
 
 
July, 2000 
 
 
 
 
 
 
Tel: (351)(21)(3801600); Fax: (351)(21)(3886073);     Email: mdsoares@fe.unl.pt 
                                                  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   2 
Abstract 
In this paper, we show that, when combining revealed (RP) and stated (SP) data, for marginal changes 
in quality of environmental goods, rationality implies consistency, as the consistency conditions coincide 
with a subset of the conditions for rationality. 
 
Keywords: combined (RP and SP) individual data; rationality; data consistency. 
 
1.  Introduction 
In most empirical work, researchers assume data consistency when combining data of different origins, 
by developing methods to jointly utilize demand data revealed (RP) and stated (SP) by individuals. 
However, Huang, Haab and Whitehead (1997) found that RP and SP data should not be combined unless 
both observed and hypothetical decisions imply the same change in future decisions induced by a given 
change in quality of an environmental good, that is, should be consistent. 
In Cunha-e-Sá and D ucla-Soares (1999), the conditions for the existence of an underlying rational 
preference structure based on a system of mixed demand functions are derived, which include symmetry of 
the diagonal blocks and skew symmetry of the off-diagonal blocks of the m atrix  c S
)
. 
1 Besides, testable 
conditions for data consistency were also obtained for discrete changes in the quality of environmental 
goods. These involve the relationship between the direct compensated demand functions of the private 
goods and the indirect compensated demands of the public goods. 
The purpose of this paper is to show that for marginal changes in quality: (i) a subset of the above 
referred consistency conditions coincide with the skew-symmetry of the off-diagonal blocks of the matrix 
c S
)
; (ii) the remaining consistency conditions are trivially satisfied. Therefore, in this context, consistency 
is necessary to rationality.  
 
2. The relationship between consistency and rationality 
Let us consider a system of mixed demand functions for the general case of r private goods and 
                                                                   
1  c S
)
is the matrix of substitution effects in the case of mixed demand systems. See Cunha-e-Sá and Ducla-
Soares (1999), pg.227. 
   3 
 n-r public (environmental) goods, and the corresponding Hicksian compensating variation 
function (.)
j
CV s for a change in the quality of the environmental good j˛{r+1,…,n}, from  j q  to  ' j q , 
respectively, the degraded and the improved levels of quality. If the reference level of utility is  
) , ,..., ,..., , ,..., ( 1 1 y q q q p p v u n j r r + = ,  
then, 
 
 
 
where  y is income,  i p  and 
m
i q , i=1,…,r, represent the price and compensated demand of good i, 
respectively, and qi  and  ,
n
i p  i=r+1,…,n, are the fixed (nonnegative) quantities and virtual price of the 
public goods.
2 
  Differentiating  (.)
j
CV s with respect to the price of the private good i,  i p , and income,  y , and 
making use of well-known theoretical results, it follows that:
3 
 
for all i=1,…r, where  1 -
¶
¶
=
y
s
j
CV q , and 
i
j
CV
i p
s
¶
¶
= b .  
                                                                   
2 See Cunha-e-Sá and Ducla-Soares (1999), pg. 216, for the definition of virtual price. 
3 See Cunha-e-Sá and Ducla-Soares (1999) for all the derivations. In alternative, we could have used the 
expenditure function to obtain the same results. 
i i i p q u , p q b q - = + + +   u) , q ,..., q ,..., q , p ,..., (     ) q ,..., ' q ,..., q , p ,..., ( n j 1 r r 1
m
n j 1 r r 1
m
￿
￿
„
+ =
+ +
+ +
=
+ +
-
- -
- - =
n
j i
r i
n j r r n j r r
v
i
n j r r n j r r
v
j
r
i
n j r r n j r r
m
i i
j
CV
y q q q p p v q q q p p p
y q q q p p v q q q p p p
y q q q p p v q q q p p q p y s
1
i 1 1 1 1
j 1 1 1 1
1
1 1 1 1
q   )) , ,..., ,..., , ,..., ( , ,..., ' ,..., , ,..., (
' q   )) , ,..., ,..., , ,..., ( , ,..., ' ,..., , ,..., (
)) , ,..., ,..., , ,..., ( , ,..., ' ,..., , ,..., (   (.)  4 
Dividing both sides of the above equation by  j j q q   '- , we obtain: 
   
j
i
j
i i
q q
q q
-
- =
-
+
' q ' q
  ,.) q (.,     ,.) ' q (.,
j j
j
m
j
m
b q
.    (1) 
 
Since            
j
i
q q
j
i i
q q q q
q q
j j j j -
- =
-
+
ﬁ ﬁ ' q
lim
' q
  ,.) q (.,     ,.) ' q (.,
lim
j
'
j
j
m
j
m
'
b q
 
 
and, given that  1 lim
' - =
ﬁ q
j j q q
, conditions (1) collapse into the following marginal ones:  
 
for all i=1,…,r and j=r+1,…,n. These conditions coincide with the skew-symmetry conditions derived in 
the rationality context, according to which individuals should be consistent when classifying the goods 
(private and public goods) as substitutes or complements in each data set. 
Differentiating  (.)
j
CV s with respect to income,  y , and the degraded quality level, j q ,
4 and making use 
of well-known theoretical results, the n-r remaining conditions are obtained, as follows: 
 
  for j=r+1,…,n, where  .
j
j
CV
q
s
¶
¶
= m  Taking the limits as  ' j j q q ﬁ , the above conditions collapse into 
the following identities: 
 
 
 
) , ,..., ,..., , ,..., ( p     1 1
v
j u q q q p p n j r r + =
q
m
). , ,..., ' ,..., , ,..., ( p ) , ,..., ' ,..., , ,..., ( p 1 1
v
j 1 1
v
j u q q q p p u q q q p p n j r r n j r r + + ”
) ' q (
p   
p   
) ' q (
q   
q   
j
i
v
j
j
j
m
i
¶
¶
¶
¶
- =  5 
3. Conclusions 
The use of combined data has implications for empirical work. In this paper we show that, for  
marginal changes in quality, the skew-symmetry conditions included in the set of the rationality conditions 
for a mixed world, are actually the consistency conditions that evaluate the effect of a marginal change in 
the price of the private good on the variation function. Therefore, rationality implies consistency.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
                                                                                                                                                                                                      
4 All the derivations below can be identically obtained for a change in the improved quality level.   6 
References 
M. Cunha-e-Sá , and M. Ducla-Soares, Specification Tests for Mixed Demand Systems with an Emphasis 
on Combining Contingent Valuation and Revealed Data, Journal of Environmental Economics and 
Management 38, 215-233 (1999). 
J. Huang, T. Haab, and J. Whitehead, Willingness to Pay for Quality Improvements: Should Revealed and 
Stated Preference Data Be Combined?, Journal of Environmental Economics and Management 34, 
240-255 (1997). 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Specification Tests for Mixed Demand Systems with an Emphasis on Combining Contingent Valuation and Revealed Data,

Willingness to Pay for Quality Improvements: Should Revealed and Stated Preference Data Be Combined?,

Combining CV and RP data: a note on the relationship between consistency and rationality

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Combining CV and RP data: a note on the relationship between consistency and rationality

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