Nonessentiality of a good is often used in welfare economics, cost-benefit analysis and applied work. Various definitions of this property are presented in the literature on public and environmental economics. This note clarifies their relationship.Preferences, nonessentiality

Udo Ebert

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 3 
  
On the definition of nonessentiality 
  
 
 
Udo Ebert  
University of Oldenburg 
Abstract 
Nonessentiality of a good is often used in welfare economics, cost-benefit analysis and applied work. Various 
definitions of this property are presented in the literature on public and environmental economics. This note clarifies 
their relationship.
I thank an Editor of the Bulletin for helpful comments. 
Citation: Udo Ebert, (2009) ''On the definition of nonessentiality'', Economics Bulletin, Vol. 29 no.3 pp. 2045-2049. 
Submitted: Jul 21 2009.   Published: August 20, 2009. 
 
       - 1 -
1.  Introduction 
A good is nonessential (for a consumer) if there are situations under which the consumer is 
willing to do without it. E.g., luxury goods and goods which are not vitally necessary can be 
nonessential. In the literature three different definitions of this concept have been presented 
and are used independently. This note investigates the connections between them. 
Two definitions are based on demand functions. In this case a good is called nonessential if 
the demand for it can be zero. The demand function can be either uncompensated 
(Marshallian) or compensated (Hicksian). A third definition proposed by Willig requires that 
the consumer can attain any (feasible) utility level without consuming the nonessential good. 
Nonessentiality of a good is often introduced and discussed in theoretical and applied welfare 
economics. If demand can be zero, the demand curve intersects the price axis and the 
corresponding consumer surplus (CS) measure is always well defined. The property is 
therefore important whenever welfare measures are to be calculated since it guarantees
1 the 
finiteness of the Marshallian CS and, respectively, the Hicksian measures. Moreover, the 
property of nonessentiality is employed in public and environmental economics when the 
concept of weak complementarity is used. A private good and a nonmarket good are weakly 
complementary if the marginal willingness to pay for the nonmarket good is zero when the 
private good is not consumed (cf. e.g. Freeman 2003). In this case the private good must be 
nonessential. The concept allows to evaluate changes in nonmarket goods (like environmental 
and public goods) by observations on the related private good. 
This note reconsiders the alternative definitions of nonessentiality and completely clarifies 
their relationship. It turns out that the definitions are not equivalent. The strongest one is 
based on the compensated demand function. It implies the other ones which are in turn 
independent from one another. 
2.  Alternative definitions of nonessentiality 
The framework and the notation are introduced first. We assume that there are only two 
(private) goods, X and Y. Here we denote by X the good which is nonessential and by Y a 
Hicksian composite which could be replaced by  2 m ≥  commodities without difficulties in the 
analysis below: A commodity bundle is denoted by ( )
2 , XY + ∈\ . We consider one consumer 
possessing a preference ordering \ over bundles ( ) , X Y . Its domain D is a subset of 
2
+ \  and 
contains  ++ + × \\ , i.e. the ordering is also defined for bundles ( ) 0,Y  where Y ++ ∈\ . It is 
assumed that () 0,0 D ∉ . The ordering \ is supposed to be continuous and convex on D and 
increasing in both commodities on 
2
++ \ . Furthermore it is assumed that the demand 
correspondence is single-valued and the ordering is represented by a (direct) utility function 
() , UX Y which is continuous, increasing and concave on D. Prices are given by 
()
2 , XY pp ++ ∈\  and the range of U by  () ( ) () { } :, , UU X Y X Y D =∈ R . Let M ++ ∈\  be the 
consumer’s (exogenous) income. The ordinary (Marshallian) and, respectively, compensated 
(Hicksian) demand functions are denoted by  ( ) ,, XY X pp M ,  () ,, XY Yp pM and 
() ,,
c
XY X pp u ,  () ,,
c
XY Ypp u  where  ( ) uU ∈R  is a utility level.  
                                                 
1  If the demand curve does not intersect the price axis CS can be finite or infinite depending on the underlying 
preference ordering.   - 2 -
Given this framework we are able to introduce the concept of nonessentiality precisely. We 
present three different definitions:
2 
Definition 
a)  Good X is Marshall-nonessential (M-nonessential)    
:⇔  For all ()
2 , Y pM ++ ∈\  there is a minimal price  ( ) , XY p pM   such that 
() () ,, , 0 XY Y Xp pM pM=  . (M) 
b)  Good X is Hicks-nonessential (H-nonessential)    
:⇔  For all () ( ) , Y p uU ++ ∈× \ R  there is a minimal price  () ,
c
XY p pu   such that 
() () ,,, 0
cc
XY Y X p pu pu=  .  (H) 
c)  Good X is Willig-nonessential (W-nonessential)    
:⇔  For all () , X YD ∈  there is a quantity  ( ) , YX Y   such that  ( )( ) () ,0 , , UX Y U YX Y =  . (W) 
The minimal price  X p   and, respectively, 
c
X p   for which consumption of X is zero is often 
called choke price. The alternative definitions are suggested in the literature: A good is 
M-nonessential if one can find a choke price under which its consumption is driven to zero 
(Just, Hueth, and Schmitz 2004, p. 212). The choke price  X p   corresponds to the price where 
the (Marshallian) demand curve intersects the price axis. A good X is called H-nonessential if 
there is a (minimum) price for it such that the Hicksian demand equals zero (Bockstael and 
Kling 1988, p. 656 and Freeman 2003, p. 112). Then the compensated demand curve for the 
nonessential good intersects the price axis at 
c
X p  . H-nonessentiality of a good X is equivalent 
to a geometrical property of the preference ordering: all indifference curves intersect the 
0 X = -axis. W-nonessentiality of a good requires ‘that it can be omitted from the 
consumption bundle without completely decimating the consumer’ (Willig 1978, p. 230), i.e. 
that any bundle including the nonessential good can be matched by a bundle excluding it 
(Johansson (1987), p. 46). In this case there is a quantity of the other good(s) which will 
compensate the consumer for the absence of (or loss of access to) the nonessential good 
(Bockstael, McConnell, and Strand 1991, p. 239, and Bockstael and McConnell 1993, 
p. 1248). The references mentioned demonstrate that nonessentiality is a relevant property in 
welfare economics and environmental economics. 
3.  Discussion and conclusion 
Now we want to investigate the relationship between the alternative definitions. At first we 
show that these properties can be satisfied simultaneously: 
(1)  Existence: There exist preference orderings satisfying (M), (H), and (W). 
Proof: Consider Example 1: The preference ordering \ is represented by 
() ( )
12 12 ,1 UX Y X Y =+  and implies the demand functions  ( ) () ,, 1 2 XY X Xp pM Mp =−  
and  () ()
12 ,, 1
c
XY Y X X pp u pp u =− . It is obvious that  ( ) , XY p pM M =  ,  ()
2 ,
c
XY Y p pu p u =   
and  () ( ) ,1 YYX X Y =+  .    
But the definitions presented are not equivalent as the following analysis demonstrates. 
                                                 
2  In order to distinguish between the alternative concepts we extend ‘nonessential’ by an appropriate term.   - 3 -
(2)  Implications of (M): (M) does not imply (H) or (W).  
Proof: We introduce Example 2: In this case the preference ordering \ is represented by 
()( ) ,l n1 1 UX Y X Y =+ − . It leads to the demand function  () ,, XY Xp pM= 
() () 0.5 0.25 YX Y X M pM p p p ⎡⎤ +− + + ⎢⎥ ⎣⎦
 and the choke price  ( )
2 , XY Y p pM M p =  . Thus 
(M) is satisfied. On the other hand  ( ) 0, 1 UY Y =−  is always strictly negative. Therefore it is 
obvious that (W) is violated since there is no  ( ) , YX Y   whenever  ( ) ,0 UX Y≥ . For the same 
reason, for  0 u ≥  the (compensated) demand for X cannot become zero. Thus (H) is also not 
satisfied.    
In Example 2 some indifference curves cut the  0 X = -axis (those corresponding to a strictly 
negative utility level), others do not (the remaining ones). Therefore (H) and (W) are not 
fulfilled. 
(3)  Implications of (W): (W) does not imply (M) or (H). 
Proof: We present Example 3: Consider  ( ) , UX Y X Y =+ . We get 
() ()
2
, YX Y X Y =+  , i.e. (W) is satisfied. The corresponding demand functions are given 
by  () () ( ) ,, XY Y X X Y X pp M ppMp p =+  and  () ()
2 2 ,, 1
c
XY XY X pp u u pp =+ . Demand 
for good X is always strictly positive since  , Y pM ++ ∈\  and  ( ) U ++ = \ R . Therefore the 
properties (M) and (H) are violated.    
In Example 3 every indifference curve cuts the  0 X = -axis, but the marginal willingness to 
pay for the first unit of X is infinite if X is equal to zero (the marginal rate of substitution 
between X and Y is equal to  ( ) , XY MRS X Y Y X = ). Thus there is no finite price system 
() , XY p p  for which the consumer chooses  0 X = . 
Finally we examine the implications of (H): 
(4a)  Implication of (H): (H) implies (M).  
Obviously (H) and (M) are not independent. One would expect this result for the case in 
which X is a normal good. Since then the Hicksian demand curve intersects the Marshallian 
demand curve from above. But normality is not used in the proof: The result is also true if X is 
(locally) inferior.  
Proof: Choose any U satisfying (H). Then define  ˆ :0 X = ,  ˆ : Y YM p = , and  () ˆˆ ˆ:, uU X Y =  for 
given  ()
2 , Y pM ++ ∈\ . (H) implies that there is  ( ) ˆ ,
c
XY p pu   such that 
() () ˆ ˆˆ ,,, 0
cc
XY Y X pp u p u X = =   and  ( ) ( ) ˆ ˆˆ ,,,
cc
XY Y Y Y p pu pu Y Mp ==  . Then 
() ( ) ˆ ,,
c
XY XY p pM p pu =  .   
(4b)  Implication of (H): (H) implies (W). 
The properties (H) and (W) are also related. 
Proof: In order to prove this claim we choose any U satisfying (H). Then define 
() ˆ:, uU X Y =  for any () , X YD ∈ . Setting  ˆ :1 X p =  and  () ˆ :, YX Y p MRS X Y =  we obtain 
() ( ) () ˆˆˆ ˆˆˆ ˆ ,,, ,,
cc
XY XY UX p pu Y p pu u = . Because of (H) there exists  () ˆˆ ,
c
XY p pu   leading to   - 4 -
() () ˆ ˆˆˆˆ :, , , 0
cc
XY Y XX p p u p u ==   and  ( ) ( ) ˆ ˆˆˆˆ :, , ,
cc
XY Y YYpp u p u =  . Then 
()( ) () ˆˆ ˆ ˆ ,0 , , UX Y U Y u UX Y == = . Thus  ( ) ˆ , YX Y Y =  .   
These investigations clarify the relationship between (M), (H), and (W). Collecting the results 
we obtain 
Proposition 
(H) implies (M) and (W). Neither (M) nor (W) implies (H). (M) and (W) are independent.  
It turns out that Hicks-nonessentiality is the strongest property. It has to be emphasized that 
the proposition holds for all preference orderings satisfying the basic assumptions. No further 
properties, like e.g. normality or inferiority of good X, are required. 
Given the main result Hicks-nonessentiality seems to be preferable to Marshall- and Willig-
nonessentiality since it implies (M) and (W). But more important is the fact that – given (H) – 
any kind of welfare measurement with respect to changes of the nonessential good can be 
performed: The areas behind the Marshallian and Hicksian demand functions which measure 
the compensation required by the consumer for an elimination of this good are always finite. 
Thus consumer surplus and the Hicksian welfare measures are also well-defined and finite for 
an arbitrary change in (the price of) the nonessential good, i.e. they can be employed without 
any difficulties in applied work. 
References 
Bockstael, N.E. and C.L. Kling (1988) “Valuing environmental quality: weak 
complementarity with sets of goods” American Journal of Agricultural Economics  70, 
654-662. 
Bockstael, N.E. and K.E. McConnell (1993) “Public goods as characteristics of non-market 
commodities” Economic Journal 103, 1244-1257. 
Bockstael, N.E., K.E. McConnell, and I.E. Strand (1991) “Recreation” in Measuring the 
demand for environmental quality by J.B. Braden and C.D. Kolstad, Eds., Elsevier Science 
Publishers, Amsterdam, Chapter 3, 41-76. 
Freeman, A.M. III (2003) The measurement of Environmental and Resource Values, 2
nd ed., 
Resources for the Future: Washington D.C. 
Johansson, Per-Olov (1987) The economic theory and measurement of environmental 
benefits, Cambridge University Press: Cambridge. 
Just, R.E., D.L. Hueth, and A. Schmitz (2004) The welfare economics of public policy, 
Edward Elgar: Cheltenham. 
Willig, R.D. (1978) “Incremental consumer's surplus and hedonic price adjustment” Journal 
of Economic Theory 17, 227-253. 
 

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