Mehrene Larudee
Economics Department
University of Kansas
July 2001
JEL codes: D31, D60, D69
Keywords: intertemporal utility, welfare, relative deprivation, envy, new good
Utility from possession of a good, and utility from not possessing it when others do, are analyzed separately from utility from consumption. The implications of two main assumptions are explored: (1) that having a new good evokes pride that decreases monotonically as a function of the proportion of the population that has it; and (2) that not having a good, once some people have it, evokes a feeling of relative deprivation or envy that is more intense, the greater the proportion of the population that has the good. It turns out that the intertemporal aggregate utility over an infinite horizon may be positive, zero or negative, depending on parameters governing the shapes of the pride and envy functions, as well as parameters governing the pattern of acquisition of the good over time. For some shapes of the pride and envy functions, intertemporal aggregate utility is more likely to be positive if, once a majority of the population owns it, the rest quickly acquire it. For other shapes of these functions, intertemporal aggregate utility is more likely to be positive if the first half of the population acquires it quickly, and then the remainder acquire it slowly.
Thanks to Elizabeth Asiedu, Nathan Berg, Steve Burks, and Mohammed El-Hodiri for useful discussions of these ideas.
*University of Kansas Economics Department, 213 Summerfield Hall, Lawrence, KS
66405, mehrene@ukans.edu, 785-864-2859.

I. Introduction
Robert Frank suggested in his 1999 Eastern Economic Association Presidential address that increasing income inequality can be a problem even if all individual incomes are growing (Frank (1999); see also Frank (1985)). A key argument was that the lowerincome members of any social group feel embarrassed or humiliated because they cannot afford the same clothes or shoes or restaurant meals that their peers enjoy. News stories about one school-age child killing another for a pair of shoes remind us that such feelings experienced by the “have-nots” in society may be intense.
The idea that human beings habitually compare their own living standard to that of others, and are unhappy if their own living standard is low relative to others, figures into several areas of economic theory. For one, it occupies a central place in the theory of poverty (Sen (1984) and sources cited therein). In macroeconomic theory it is present in
Duesenberry’s (1949) relative income hypothesis. And in the theory of income distribution, the idea of envy – its definition, measurement and implications – has been discussed, though not extensively, by various authors (Frank (1985); Easterlin (1974);
Podder (1996); and see brief survey in Solnick and Hemenway (1998)).
The main purpose of the present paper is to explore the implications of this idea for the welfare effect of introduction of a new good. Duesenberry (1949) analyzed some positive aspects of the effect of introducing a new good in the presence of interdependent preferences such as envy or feelings of relative deprivation. Recently, modelers have tended to make assumptions which guarantee that introduction of a new variety of an existing type of good is unambiguously welfare-improving. For example, in models of intra-industry trade, a new variety increases the number of varieties of a good available, and consumers’ Spence-Dixit-Stiglitz utility functions imply a love of variety. However, the key feature of these models that guarantees that new goods are welfare-improving is that utility functions are not interdependent. But if, instead, one person’s utility is a negative function of acquisition of a good by others, via envy or feelings of relative deprivation, then it is no longer obvious that the introduction of a new good will be welfare-improving. This paper defines conditions under which the introduction of a new good will indeed increase (or decrease) that component of intertemporal welfare which flows from interdependent preferences.
1

Further, Easterlin cites surveys show that, over time, although people with higher income or wealth report being happier than poor people, the proportion of people reporting that they are happy has remained about the same despite the introduction of many new goods. This paper also contributes a partial explanation of these facts, by exploring how the presence of interdependent utility affects the intertemporal welfare change that results from introducing a new good.
Interdependent utility, we assume, takes two forms. One is envy, a feeling whose existence has been recognized by economists such as Marshall, Veblen, and Knight, but not formally incorporated into their theories (Duesenberry 1949: 14-15). Envy, or feelings of relative deprivation, are described extensively in Frank (1985). Thus one consequence of introducing a new good, which initially only a small fraction of the population will own, is the disutility felt by those who do not yet own it, the “have-nots”.
Interdependent utility can also be manifested among the “haves”, in a feeling we might call pride: the utility from being a “have”, that is, from being among the first to possess a good, when many people do not yet own it because they cannot afford it. If I am the first person in town to own a CD player, this makes me especially proud. Even if I am among the first half of the population to have a CD player, I may well feel happy that
I have an advantage that some others do not have. The motive for conspicuous consumption is evidently this sort of pride.
This idea is relevant not only to the standard sorts of goods but also to certain kinds of services. In fact, it is perhaps most striking with respect to permanent alterations in personal appearance, such as cosmetic surgery or orthodontic services (Frank 1985).
Having teeth straightened, getting a breast implant or a facelift or tummy tuck and so on can be thought of as purchasing a service, but it could also be seen as buying a (more or less) durable good, and that is how we view it here. There is clearly an advantage to
“possessing” the good when others (say, in one’s age group) do not, since it gives one an edge in the dating game. And, as Hamermesh and Biddle (1994) found, attractiveness not only provides utility in itself, but also affects income, for example through employment and promotion. Feelings of relative deprivation may be especially acute in matters involving personal appearance. When the poor can be identified as poor by their crooked teeth, they are all the more humiliated.
2

For any new durable good, there is a period, possibly long, during which the fraction of the population that possesses it is growing, but remains less than 1. Suppose the good is what Frank (1985) (following Hirsch (1976)) calls a “positional good”, that is, it is the type of good about which people tend to have feelings of relative deprivation.
Then it follows that during this period, from the standpoint of society as a whole, the utility enjoyed by those who already have acquired the good is offset to some degree by the disutility of those who do not have it, and who may hence feel excluded and deprived.
This naturally leads to the question whether, over time, the disutility for the havenots of not-having a particular good could outweigh the utility for the haves of having it, so that, with respect to this good, the aggregate intertemporal utility over all of society could be negative. That is the question which this paper seeks to answer: under various assumptions, what is the intertemporal aggregate utility that results from the introduction of a new good? More precisely, the aim is to characterize a set of assumptions which makes the aggregate intertemporal utility equal to zero, and so forms a boundary between net positive and net negative intertemporal utility from introduction of a new good.
Since the focus is on possession, to simplify matters let us consider a single good which we assume is perfectly durable: although the good is “consumed” it does not deteriorate. Think of a diamond or a work of art, whose “consumption” consists of seeing it and having others see it. Also to simplify the presentation, we look only at the portion of utility or disutility which is specifically attached to being a “have” or a “have-not” at a point in time. That is, we leave out of the picture the utility that flows from consumption, in the sense of actually using the good. If total utility is additively separable in consumption and possession, then we can obtain the aggregate intertemporal utility taking account of both by simply adding the utility from consumption to the result we will derive here. Since this paper makes no attempt to contribute to the theory of utility from consumption per se, it henceforth focuses exclusively on utility from possession, or lack of possession.
II. The Model
We define the size of the whole population to be unity. We further define time t = 0 as the moment in which the first consumer acquires the good. We denote by h ( t ) the
3

fraction of “haves”: the fraction of the population that owns the good at time t . We assume h is non-decreasing in t so that once a unit of the good is acquired, it continues to exist and to be owned by the same person. Of course (1 – h ( t )) is then the fraction of
“have-nots” at time t . We assume also that all members of society have identical cardinal utility functions, but different levels of income (and zero or limited access to credit); this is why richer people acquire the good sooner, and poorer people acquire it later. We define a “pride function” which is the instantaneous utility of being a “have” at time t as v ( h ) = v ( h ( t
)). And we define an “envy function” which is the instantaneous utility of being a “have-not” as w ( h ) = w ( h ( t )), where of course w ≤ 0.
1 Actually, what we hope to capture is better described as relative deprivation or humiliation than “envy”, but we will nevertheless use the word “envy” as a kind of shorthand to refer to this idea. We assume that both curves are monotonically decreasing in h ( v

< 0 , w

< 0): as an increasing proportion h of the population acquires the good, possessing the good makes a person less proud and not possessing it makes a person more unhappy. Then the utility of society as a whole at time t from having or not-having this good is a weighted average of the utilities (positive or negative) of the haves and have-nots:
U ( h ) = hv ( h ) + (1 – h ) w ( h ) where h is understood to mean h ( t ).To explore the possible outcomes it is useful to
(1) suppose that v and w as functions of h take on very simple linear shapes. Later we will choose general quadratic functional forms for both v ( h ) and w ( h ).
In this section of the paper we consider an infinite time period to simplify the algebra, treating each household and its descendants as a single infinitely-lived individual, and ask whether over this infinite period the aggregate utility of the haves and have-nots is positive, zero or negative. The analysis can easily be extended to finite time periods; the algebra is just messier.
1 While we define this as a one-good world, in order for the model to be useful in a multi-good world we must impose the additional assumption that utility or disutility from having one good is unaffected by possession, or lack of possession, of other goods. In other words, total utility needs to be additively separable in the utility/disutility of having or not-having each good, as well as in utility derived from consumption of this basket of goods.
4

Case 1: v(h) and w(h) both linear
Suppose v ( h ) = v
0
(1 – h ) (2a) w ( h ) = – w
1 h where v
0

v (0) > 0 is the satisfaction of being the first to possess the good and
(2b) w
1

| w (1)| > 0 is the absolute value of the dissatisfaction from being the last person not to possess the good. We want these functions to be decreasing in h and to satisfy v (1) = 0 and w (0) = 0, and they do meet these conditions. U ( h ) is depicted in the top portion of
Figure 1, drawn for v
0
= 1, w
1
= 1.5. Then U = h ( t
)· v
0
(1 – h ) + (1 – h )(– w
1 h ), or
U ( h ) = ( v
0
– w
1
) h (1 – h ) (3)
In the case of linear v and w , U ( h ) has the same sign as ( v
0
– w
1
) over the open interval, since h and (1 – h ) are both positive. This parabola is depicted in the bottom half of
Figure 1 for the same values of v
0
and w
1
. If instead we had v
0
= w
1
, then U ( h ) would be identically 0.
Now we choose a functional form for h ( t ) which makes h (0) = 0, h
(∞) = 1 and h

≥ 0, and also yields a function whose improper integral over the infinite interval t

[0,

) exists (i.e., is finite) and is relatively easy to calculate. A simple function that serves nicely is: h ( t ) = 1 – 1/(1 + kt ) n with n > 1 and k > 0. The parameter k provides a convenient way to adjust our
(4) assumption about how rapidly acquisition of the good spreads through the population.
When k is large, h rises rapidly over time; when k is small, acquisition of the good spreads more slowly. Variation of n has a similar but not identical effect, which will be discussed later. Figure 2 shows h ( t ) for two values of k and two values of n . For now we use n = 2. Then equation (3) becomes
U

( v
0
 w
1
)


1
(1
 kt )
2

1
(1
 kt )
4 
(5)
5

Since in the linear case U has the same sign as ( v
0
– w
1
), the same is true of V , the expression for the intertemporal utility obtained by integrating over the infinite time interval
2
:
 
V

U ( h ( t )) dt

0

0

( v
0
 w
1
)

1
(1
 kt )
2

1
(1
 kt )
4

 dt
Performing the integration yields:
(6)
V = (2/3)(1/ k )( v
0
– w
1
) (7)
We have shown that in the linear case, if the satisfaction of being the first to possess a good ( v
0
) is greater in magnitude than the dissatisfaction of being the last to possess it ( w
1
), intertemporal utility is positive; otherwise it is negative. This result, however, is clearly sensitive to the functional forms of v , w and h . If either v or w is nonlinear, U ( h ) may change sign over the interval h

[0,1], although it does not have to do so. We now consider a somewhat more general case.
A General Case: Both v(h) and w(h) Quadratic
Suppose both v ( h ) and w ( h ) are quadratic in form. We choose the most general form which satisfies the following conditions, already stated above: v

< 0, w

< 0, v (1) = 0, w (0) = 0, and we allow the parameter n in the h ( t ) function to vary. We set v ( h ) = ( c + d )
– ch – dh
2
(2a

) w ( h ) = – ph – qh 2 (2b

) with c , d , p and q all positive constants, so that the vertical intercept of v is positive and of w is 0, and both curves are downward sloping. (The constant in v ( h ) is constrained to be ( c + d ) by the requirement that v (1) = 0.) We can transform equation (2a

) without loss of generality by defining v
0
 c + d and f
 c /( c + d ) to get v ( h ) = v
0
– v
0 f h – v
0
(1 – f ) h
2
= v
0
[1 – f h – (1 – f ) h
2
] (2a

)
This gives v (0) = v
0
, v (1) = 0 and v

< 0, so that v
0
has the same interpretation as in the linear model, namely that it is the utility from being the first to possess the good. In exactly the same way, we can transform equation (2b

) without loss of generality by defining w
1

–( p + q ) and b
 p /( p + q ), so that we can write:
2 To simplify the exposition no discounting factor is included.
6

w ( h ) = – w
1 bh – w
1
(1 – b ) h 2 = – w
1
[ bh + (1 – b ) h 2 ] expressions into equation (1) we get
U = hv
0
[1 – fh – (1 – f ) h 2 ] – w
1
(1 – h )[ bh + (1 – b ) h 2 )]
= h (1 – h ) {( v
0
– w
1 b ) + h [ v
0
(1 – f ) – w
1
(1 – b )]}
(2b

) where w
1
is defined so as to be positive, so that – w
1
is the disutility (i.e., the utility, which is negative) of being the last person who does not own the good. Putting these
(3

) and with h defined as in equation (4) we have
V
 
0
 1

(1
 kt ) n  
1

1
(1
 kt ) n   v
0
 w
1
 b

1

1
(1
 kt ) n 
 v
0
(1
 f )
 w
1
(1
 b )
 
 dt (6

)
After performing the integration and simplifying we obtain
V

 v
0

(5

2 f ) n

1
  w
1

(2
 b ) n
 b
  n
 k ( n

1)(2 n

1)(3 n

1)

(7

)
Let us look at some simple cases, choosing n = 2 and various values of b and f to get a feel for the sort of outcomes generated.
Case 2: n = 2, k = 1, f = 1, b = 0, v(h) linear, w(h) quadratic and concave
First we choose f = 1 to make v linear just as in equation (2a), and we choose b = 0 to make w = – w
1 h
2
, so that w is quadratic and concave. We take n = 2 and k = 1.
Case 2(a): v
0
= w
1
. As shown in Figure 3, drawn for v
0
= w
1
= 1, U is positive over the whole interval. V is therefore also positive. However, this need not be the case for v
0
< w
1
.
Case 2(b): v
0
< w
1
.
U changes sign at a value of h which makes the term in curly brackets in equation (3

) equal to zero. For the present case in which f = 1 and b = 0, that value is h = v
0
/ w
1
. This is the point at which the instantaneous utility of society as a whole is zero as the disutility of the have-nots just offsets the utility of the haves, and so in this case instantaneous net utility is in the process of turning from positive to negative.
So while for these parameter values and for v
0
= w
1
the U function is positive over the whole open interval, for v
0
< w
1
the same parameter values cause the U function to change sign at h = v
0
/ w
1
< 1. For f = 1 and b = 0, what choice of v
0
and w
1
gives V = 0?
We find the answer by putting the values of the parameters into (7

), which yields
V = (2/3)(1/ k )( v
0
– 0.8
w
1
) (7

)
7

and this tells us that V = 0 when v
0
= 0.8
w
1
. For this case Figure 4 shows the v ( h ) and w ( h ) curves as well as the corresponding U ( h ) curve, which changes sign at h = 0.8.
It is useful to see how U evolves with t . We could do this by defining a function
W ( t ) = U ( h ( t )) and plotting it against t . However, instead of constructing another graph we add a t -axis to Figure 4 and label the point where W ( t ) = 0. We find t* which makes
W ( t* ) = 0 by solving equation (4) for t with h = v
0
/ w
1
; this yields t = (1/ k )(1 – v
0
/ w
1
)
-1/ n
. In this case, for k = 1 and h = 0.8, we have t* = 5
1/2
or approximately t* = 2.236.
Of course in this case V = 0 implies that over any finite interval t

[0, T ] the proper integral
 T
V ( T )

U ( h ( t )) dt
0 will be positive; once this proper integral reaches its maximum at the h for which U ( h ) = 0, the integral then declines, approaching 0 as a limit from above as T  ∞. But for the case v
0
< 0.8
w
1
, V ( T ) becomes 0 at some finite value of
T , which can be calculated if desired.
Case 3: n = 2, k = 1, f = 1, b = 2, v(h) linear, w(h) quadratic and convex
Similarly, Figure 5(a) shows the case f = 1, b = 2 (drawn for v
0
= w
1
= 1), in which v is linear as in equation (2a) and w is quadratic but convex; Figure 5(b) shows the corresponding U ( h ). This gives
V = (2/3)(1/ k )( v
0
– 1.2
w
1
) (7

)
Now the case v
0
= 1.2
w
1
gives V = 0.
What shapes can U(h) take on?
Acquisition of the good may spread throughout the population quickly or slowly, as reflected in the shape of h ( t ). In general, the speed with which this occurs affects the value of V , though there are some sets of parameter values for which the sign of V is unaffected by the steepness of the h curve. Specifically, if the parameters make U always positive, as in Figure 5(a), then V is positive regardless of how steep h ( t ) is. Likewise,
U < 0 as in Figure 5(b) implies V < 0 independent of the shape of h ( t ). However, if U changes sign over h

(0,1), as in Figure 5(b) or 5(d), then the shape of h ( t ) can affect not just the magnitude but the sign of V . In other words, the speed with which acquisition of
8

the good spreads throughout the population can affect whether the intertemporal welfare effect of introducing the good is positive or negative.
We now look for parameter values that create cases for which U changes sign at some h

(0,1) so that the shape of h ( t ) affects the sign of V . We use three constraints, of which the first was already imposed and applies to all cases, not just Figure 5(b) and 5(d).
(1) v

(h) < 0 and w

(h) < 0.
Differentiating equations (2a

) and (2b

), this means that f + 2(1 – f ) h > 0 and b + 2(1 – b ) h > 0 for all h in the interval (0,1). Inserting h = 0 and h = 1 into both inequalities gives a lower bound of 0 and an upper bound of 2 for both b and f .
(2) U = 0 at some h in the open interval (0,1).
We denote this value by h
0
. Then h
0
is the value of h which makes the expression in curly brackets in equation (3a

) equal to zero, and this means that to get Figure 5(b) or Figure 5(d) we need: h
0
 
X
Y

(0,1) with X
 v
0
– w
1 b and Y
 v
0
(1 – f ) – w
1
(1 – b ) (8)
(We have defined X and Y just to simplify the exposition here and below.) Equation (8) says that Y must be of the opposite sign from X , and larger in absolute value, to make
0 < h
0
< 1, implying either 0 < X < – Y or else 0 < – X < Y .
Together these two constraints give us either Figure 5(b) or 5(d). We now use an additional condition to distinguish these two cases.
3
(3) To get Figure 5(b) we need both U

(0) > 0 and U

(1) > 0. Differentiating equation (3a

), it is easy to see that
U

(0) = v
0
– w
1 b
U

(1) = w
1
– v
0
(2 – f )
(9)
(10) and hence we need [1/(2 – f )] > v
0
/ w
1
> b (equivalently, 0 < X < – Y ). To get Figure 5(d) we need U

(0) < 0 and U

(1) < 0, or[1/(2 – f )] < v
0
/ w
1
< b (equivalently, 0 < – X < Y ). The figures are drawn based on the values shown in the labels in Figure 5, and it is easy to check that they fit the conditions above.
3 As a mathematical footnote, there are several equivalent sets of conditions that could be used. For example, instead of condition (3) we could get Figure 5(b) by imposing the condition that U ( h ) is upward sloping where it crosses the axis in the interior of the interval (0,1), i.e. that U

( h
0
) > 0, and instead of (4) we could assume U ( h ) falls as it crosses the axis in the interior of the interval, i.e. that U

( h
0
) < 0. Together with condition (2) this would give the same result, as the reader is invited to check.
9

Comparative statics
How does V vary with its parameters? We have shown that under plausible assumptions it is possible for the intertemporal utility flowing from possessing or not possessing a new good to be either positive, zero or negative.
Given our assumption that n > 1, from equation (7

) we know that V increases with v
0
as long as [(5 – 2 f ) n – 1] > 0 and decreases with w
1
as long as [(2 + b ) n – b ] > 0. It is not hard to see that both of these are guaranteed by the requirements that n > 1,
b

[0,2] and f

[0,2]. It is also easy to see from equation (7

) that n > 1 also guarantees that V b and V f
are both negative, so that V decreases as f and b increase. None of this is surprising. It means that greater feelings of pride increase the likelihood that intertemporal aggregate welfare will be positive from introduction of a new good, while greater envy decreases this likelihood.
III. A finite-time h(t)
We asserted that variation over time in the speed with which the population acquires the good can affect the value of V , that is, the shape of h ( t ) affects the value and potentially the sign of V . For example, if half the population acquire the good rapidly, and the remaining half acquire it more slowly, this will make V more likely to be positive in the case of Figure 5(d) and more likely to be negative in the case of Figure 5(b).
We now consider a different functional form for h ( t ), shown in Figure 6, which is well suited to exploring this issue. Suppose the economy has two groups. Members of the group (presumed richer
4
) that first acquire the good constitute a fraction H of the population. They acquire the good at a constant rate until at time t
H
all of them have it.
Then the remaining fraction (1 – H ), the poorer group, acquire it at a different rate – more slowly in Figure 6(a), more rapidly in Figure 6(b) – until the whole population owns the good at time t = 1 (and not before, in general). The curve h ( t ) is therefore piecewise linear with its two segments joined at ( t
H
, H ), and its equation is:
4 Those in the richer group are not assumed to all be equally rich, nor those in the poorer group equally poor.
10

h ( t )
 


H t
H

H


 t if 0

1

H

1
 t
H

( t
 t
H
) if t
H
 t
 t
 t

H
1

(11)
As shown in the Appendix, if we write the integral and perform the integration, as well as define R = v
0
/ w
1
and set w
1
=12 we get
V


1

H
 t
H
 
2 x
 y



 t
H
  
1

3 H
 y

4 x

(12) with x

X / w
1
= R – b and y

Y / w
1
= R (1 – f ) – (1 – b ).
Finding V = 0 in (b, f) space
Since V = V ( t
H
, H , R , b , f ) is a function of five variables or parameters, we find it convenient to divide them into two groups and hold one group constant while we find what combinations of the others makes V = 0. We first take H and t
H
as given constants, and R as a parameter, and find the regions of ( b , f ) space in which V is negative, zero or positive. Second, we take b, f and R as given and find regions of ( t
H
, H ) space which make V negative, zero or positive. The solutions will be described here algebraically and geometrically, while proofs and details are found in the Appendix.
With H and t
H
constant, it is clear from equation (12) that V is a linear function of x and y , which in turn are linear functions of b and f , and hence V is a linear function of b and f . We can therefore set V equal to zero and solve for f as a linear function of b . This gives us the following proposition.
Proposition 1 . V = 0 along the line where f = A + (1 – 1/ R ) + (1/ R )(1 – A ) b
A



1

H
 t
H
H
 t
H


4 H H
 t

H H
 t

H
 
1

3 H
H
 

(13) and where A ≥ 1 and R > 0. As a little algebra shows, equation (12) is then equivalent to
Ax + y = 0. This downward-sloping line may be seen as specified by two points through which it passes, the first of which is specified by A and the second by R :
( b
A
, f
A
) = (1/(1 – A ), 1 + A ) (14) and
11

( b
R
, f
R
) = ( R , 2 – 1/ R ) (15)
Strictly speaking, since we have restricted b and f to the square [0, 2]

[0, 2], V = 0 is the portion of that line which falls within this square.
Although all the points ( b
A
, f
A
) are outside the admissible region, nevertheless each such point, corresponding to a specific value of A , is a pivot point from which the lines V = 0 radiate for different values of R , as shown in Figure 7. The locus of all ( b
A
, f
A
) is the upper left branch of the rectangular hyperbola f = 2 – 1/ b , also shown in Figure 7.
The points ( b
R
, f
R
) given by equation (15) are within the admissible region for
0.5 ≤ R ≤ 2 (see Figure 8). Moreover, for a given value of R , ( b
R
, f
R
) is a pivot point from which the lines V = 0 radiate for different values of A , as shown in Figure 8. The locus of such points is the lower right branch of the rectangular hyperbola f = 2 – 1/ b .
Corollary 1 . If ( b , f ) = ( b
R
, f
R
) = ( R , 2 – 1/ R ) and b , f

[0,2], that is if ( b , f ) is on the lower branch of the hyperbola f = 2 – 1/ b and within the admissible square region, then V is identically zero regardless of the values of t
H
and H .
This is the case in which x = y = 0 and so x + yh = 0, making U ( h ) = 0
 h . The linear case which we first examined, b = f = 1, would be one instance of this special case if it were also true that R = 1.
(However, Figure 1 graphs a case in which R = 2/3 and b = f = 1).
Corollary 2 . If H = t
H
, or H = 0, or H = 1, then A = 2, so that the point (1/(1A ),1+ A ) = (–
1, 3) on the upper branch of the hyperbola is one point on the straight line f = (3 – 1/ R ) – (1/ R ) b in ( b , f ) space, and this line is the locus V = 0.
More generally, from equation (12):
H = t
H

V = 2 x + y .
H = 0

V = (1 – t
H
)(2 x + y ), and
H = 1

V = t
H
(2 x + y ).
(16)
12

Proposition 2 . V b
< 0 and V f
< 0, and this ensures that in the region below and to the left of the line V = 0 we have V > 0, while in the region above and to the right of the line we have V < 0.
The case H = t
H
is the case in which we are on the diagonal of the box in Figure 6: everyone acquires the good at the same rate until all have it (regardless of the value of H or t
H
). For example, if R = 1 (maximum pride is of the same magnitude as maximum envy) then V = 0 along f = 2 – b , and this is graphed in Figure 7. The cases H = 0 and
H = 1 are formally identical to the case H = t
H
since in each case everyone acquires the good at the same rate, but the time period over which they acquire it is compressed, from
[0, 1] to [0, t
H
] if H = 1, and from [0, 1] to [ t
H
, 1] if H = 0. If ( t
H
, H ) = (1, 0) or (0,1) then
A is undefined since it is 0/0, but we do not need A , since equation (12) becomes identically 0, and so V = 0 regardless of the values of b , f and R .
Proposition 3 . For each value of A there is a value of R , namely R
L
= 1/(1 + A ), below which V < 0 for all ( b , f ) within the admissible square region [0, 2]

[0, 2]. Similarly, for each value of A there is a value of R , namely R
H
= 2 – 1/(1 – A ), above which V > 0 for all
( b , f ) within the admissible square region.
For A = 2 these bounds are therefore R
L
= 1/3 and R
H
= 3. The lines V = 0 are shown in
Figure 7 for both these cases, and they just touch the corners (0, 0) and (2, 2) of the admissible square region. Thus if the maximum “pride from having” is less than 1/3 the maximum “envy from not-having” (and v , w and h have the functional forms postulated), then the intertemporal aggregate utility is necessarily negative, independent of how rapidly any part of the population acquires the good. Similarly, if the maximum pride is more than 3 times the maximum envy, the intertemporal aggregate utility is necessarily positive, independent of the rate of acquisition of the good. This completes the first part of the analysis, telling us what happens when ( t
H
, H ) is given.
13

Finding V = 0 in (t
H
, H) space
We now turn to the second part of the analysis: given b , f and R , what does the locus V = 0 look like in ( t
H
, H ) space? We answer by solving equation (13) for a value of
A , and then finding the ( t
H
, H ) combinations that correspond to that A . This procedure is equivalent to specifying a point ( b , f ) in the admissible region and a point ( b
R
, f
R
) =
( R , 2 – 1/ R ) on the lower branch of our hyperbola. The line through these points intersects the upper branch of the hyperbola in ( b
A
, f
A
) = (1/(1 – A ), 1 + A ), whose A -value is the one we want. We solve equation (13) for A to get A = – y / x , which holds true when t
H

H


2 x
 y
2 x
 y
 
( y

4 x ) H

3 yH
2
(17)
We established earlier that t
H
= H implies V = 2 x + y , so that when the values of b , f and
R make 2 x + y = 0, the diagonal t
H
= H in ( t
H
, H ) space is the locus V = 0. Shortly we will look at less symmetrical cases, but first let us formalize a claim which we made earlier, and which was the motivation for introducing the piecewise linear finite-time h ( t ):
Proposition 4 . Suppose H = h
0

(0, 1) (where as before, h
0
is defined by U ( h
0
) = 0), and suppose also that the parameters b , f and R take on values such that if H = t
H
then
V = 0. Then exactly one of the following three is true: either
(1) U ( h ) is identically zero over the whole interval h

[0, 1], in which case V = 0 regardless of the values of t
H
and H , or
(2) U

( h
0
) > 0 (equivalently b > R ), in which case V has the same sign as H
– t
H
, or
(3) U

( h
0
) < 0 (equivalently b < R ), in which case V has the same sign as t
H
– H .
The proof is given in the Appendix, and we note that under these assumptions we must have H = h
0
= 1/2 as shown in Figures 5(b) and 5(d). Let us restate Proposition 4.
Suppose we have a piecewise linear h ( t ) of two segments, with parameter values such that V = 0 when everyone in the population acquires the good at the same rate (that is, along the diagonal). Suppose also that U ( h ) is not identically zero for all h . Then if U > 0 while the first half of the population acquires the good, V will be positive if this group acquires the good at a more rapid constant rate than the second half, while V will be negative if the first half acquires it at a slower constant rate than the second half. In
14

contrast, if the parameters make U negative while the first half of the population acquires the good, then V will be negative if the first half group acquires it at a slower constant rate than the second half, and will be positive if the first half acquires it at a faster constant rate than the second half.
Finally, let us consider the case in which the value of V along the diagonal of the
( t
H
, H ) box, namely 2 x + y , is not zero. In that case it turns out that V = 0 is a curve that looks as shown in Figure 10.
IV. Factors that influence the parameters: charity, advertising, size of reference group
We may note several factors which influence the parameters of the model. First, the speed with which the population acquires a good may be altered by the operation of charitable feelings, possibly driven by guilt. As a result the shape of the h ( t ) curve may be altered.
Second, globalized advertising and communication may broaden the reference group within which both pride and relative deprivation are felt; and in a worldwide population the growth of h over time is likely to be far slower than in a single nation with a narrower range of income and wealth.
First we look at the influence of charity. We have assumed that those who have a good feel only positive utility, which we have called “pride,” from possessing it.
However, in many societies there is a strong enough ethic of equality that, once a large fraction of the population has a good, it comes to be considered a necessity of life, so that the non-poor collectively feel obliged to provide some of the good to the poor through charitable organizations.
5
A few examples are: shelter for the homeless; programs channeling donations of money to subsidy of poor households’ utility bills; pro bono work performed for the poor by lawyers; the expectation that hospitals will provide emergency care to the poor; and a program in Kansas City which channels donations to provide electric fans to poor people who have neither air conditioning nor fans. Survey evidence that a significant number of the non-poor are uncomfortable with inequality is cited, for example, by Fong (2001, forthcoming). It is possible, then, that once h reaches a high value, forces such as charitable activities are set in motion that accelerate h toward
5 Of course some of these programs may also be motivated by the self-interest of the non-poor in reducing crime or increasing tourism.
15

1. In cases where the parameters imply that U < 0 for h close to 1, the charitable donations would appear to increase the likelihood that intertemporal aggregate utility V is positive. However, at the same time if inequality causes the “haves” to feel uneasiness
(the feelings which motivates them to charity) these feelings must also be tallied as part of the utility calculus. It would probably make sense to model this in a more complex way, distinguishing goods by their age, and modeling utility maximization by the “haves” as they donate their older good and buy a newer one; this, however, is beyond the scope of this paper.
Second, the shape of the w ( h ), and the size of w
1
, may be strongly influenced by global advertising. Advertising is capable of creating the desire for new goods among people, including have-nots, in developing countries, and creating or exacerbating the disutility of being a have-not. Helena Norberg-Hodge has described this vividly:
…In 1975, I was shown around the remote village of Hemis Shukpachan by a young Ladakhi named Tsewang. It seemed to me that all the houses we saw were especially large and beautiful. I asked Tsewang to show me the houses where the poor people lived. Tsewang looked perplexed a moment, then responded, “We don’t have any poor people here.”
Eight years later I overheard Tsewang talking to some tourists. “If you could only help us Ladakhis,” he was saying, “we’re so poor.” (Norberg-Hodge 1996]
Dependency theorists have long expressed concern over the creation in poor countries of the desire for goods imported from rich countries, although their main concern has been that the rich tend to consume imports and thereby worsen the balance of payments. Wide access to television has greatly accelerated this trend. This compels us to ask what should be the reference group for which the parameters of the model are set: should it be local? national? or global? The advance of global communication strongly suggests that the reference group should be the whole world, since have-nots throughout the world now are, more than ever before, made constantly aware of their relative deprivation.
At the same time, the disutility felt by have-nots is unlikely to intrude into public discussion with the same force as the utility felt by “haves”, simply because have-nots (at least involuntary have-nots) are poor, and the poor have little political voice. The utility of acquiring goods is announced to the world in paid advertisements, and so gets voice not only – in fact not mainly – from consumers but from sellers. But the disutility from being unable to acquire desired goods has far fewer channels for expression.
16

Conclusion
There is substantial evidence that a consumer’s utility is affected not just by whether he/she possesses a good, but by how many others possess it. We have explored the implications of assuming both that the utility of possessing a durable good is nonnegative and declining function of the fraction of the population that possess it, and the disutility of not yet owning it increases as the fraction of the population that owns it increases.
These assumptions have been shown to imply that society’s intertemporal welfare from introduction of a new good, derived from having or not-having the good, can be either negative or positive depending upon the shapes of the utility and disutility functions as well as on the speed with which acquisition of the good spreads throughout the population.
Allowing these two types of interdependent utility gives results that, for some shapes of these functions, differ from the more commonly told story that the introduction of a new good, or a new variety of an existing good, is unambiguously welfareimproving. The analysis of this paper may therefore help to explain why the overall level of happiness in the population does not seem to increase over time. Where feelings of relative deprivation or envy exist, introduction of new goods is not guaranteed to be welfare-increasing over the long run, and not necessarily even over the short run.
17

References
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Dilip Mookherjee and Debraj Ray (eds.), Economic Theory and Policy: Essays in
Honour of Dipak Banerjee , Bombay: Oxford University Press.
Duesenberry, James. (1949). Income, Saving and the Theory of Consumer Behavior .
Cambridge: Harvard University Press.
Easterlin, Richard A. (1974). Does Economic Growth Improve the Human Lot? Some
Empirical Evidence, in: Paul A. David and Melvin W. Reder (eds.), Nations and
Households in Economic Growth , New York: Academic Press, 89-125.
Fong, Christina. (2001). Social Preferences, Self-Interest, and the Demand for
Redistribution. Journal of Public Economics , forthcoming.
Frank, Robert H. (2000). Does Growing Inequality Harm the Middle Class? Eastern
Economic Journal 26(3):253-264 (Summer)
Frank, Robert H. (1991). Positional Externalities. In Zeckhauser (ed.), Strategy and
Choice , Cambridge: MIT Press.
Frank, Robert H. (1985). Choosing the Right Pond . New York: Oxford University Press.
Hamermesh, Daniel and Jeff E. Biddle. (1994). Beauty and the labor market. American
Economic Review 84: 1174-1194.
Hirsch, Fred. (1976). The Social Limits to Growth . Cambridge: Harvard University Press.
Norberg-Hodge, Helena. (1996). The Pressure to Modernize and Globalize, in: Jerry
Mander and Edward Goldsmith (eds.), The Case Against the Global Economy ,
San Francisco: Sierra Club Books, 33-46.
Podder, Nripesh. (1996). Relative Deprivation, Envy and Economic Inequality. Kyklos
49: 353-376.
Sen, Amartya. (1984). Poor, Relative Speaking, in: Sen, Resources, Values and
Development , Cambridge, Mass.: Harvard University Press, 325-345.
Solnick, S. J. and Hemenway, D. (1998). Is More Always Better? A Survey on Positional
Concerns. Journal of Economic Behavior and Organization 37: 373-383
(November).
18

v
h
h
w
h
h
h
v
h
w
h
U
h
v
0
w
1
v
0
w
1
U
19

k
h
t
n
20

f
b
v
0
w
1
v
h
w
h
U
V
21

v
0
U
h
h
t
V
22

U
h
k
n
v
0
w
1
a
V
c
b
V
d
V
23

1
H
0
1
H
0 t
H
h(t)
h(t)
(a)
(b) t
H
1
1 t t
h
t)
H
H
a
b
t
24

V
b
f
f
b
b
A
f
A
A
A
A
H
t
H
A
b
R
f
R
R
R
R
v
0
w
1
A
R
V
b
A
f
A
25

R
A
V
b
R
f
R
R
R
26

1
V < 0
V = 0
1
V > 0
V = 0
V > 0 V < 0
0
(a) 2x + y = 0, b > R
1 t 0
(b) 2x + y = 0, b < R
H
t
H
V
t
H
H
b
f
R
V
x
y
x
y
1
V
a
b
R
b
b
R
V
t
27

a
b
f
R
b
b
f
R
V
t
H
H
b
f
V
V
t
H
H
a
V
b
V
28

Appendix
Derivation of equation (12)
First we have equations (11), repeated from the text: h ( t )
 

 t
H
H

H

 t if 0

1

H

1
 t
H


( t
 t
H
) if t
H
 t
 t
 t

H
1

We then write the integral V :
(11)
 1
V

U ( h ( t )) dt

0

0 t
H
U ( h ( t )) dt
  t
H
1
U ( h ( t )) dt (A.1)
We insert values for h ( t ):
V

U

0 t
H
H
 t
H t

 dt
  t
1
H

U H


1

H
1
 t
H


 t
 t
H
  dt
Then we use a change of variables with w

V
 t
H
H

0
H   dw
 
1
 t
H

1

H
H t
H


 1
H
  dz
(A.2) t and z

H


1

H

1
 t
H


 t
 t
H

, and get:
(A.3)
Inserting the expression for U ( h ) from equation (3

), we have
V
 t
H
H

0
H  w (1
 w )

X

Yw
  dw



1
 t
H
1

H


 1
H
 z (1
 z )

X

Yz
  dz (A.4)
Integrating, we get
V
 t
H
H

X
2 w
2 
Y

X
3 w
3 
Y
4 w
4

0
H
 
1
 t
H

1

H
 
X
2 z
2 
Y

X
3 z
3 
Y
4
1 z
4

H
(A.5) which becomes
V
 t
H
H

X

2
H
2 
Y

X
3
H
3 
Y
4
H
4



1
 t
H

1

H
 
X
 
2

1

H
2


Y

X
3

1

H
3


Y
4

1

H
4



Canceling H
’s in the first term and the (1 –
H
)’s in the second term, and combining terms in X and terms in Y , we get an equation
V
 1
12
 
2 X

Y


H
 t
H


H H
 t
H
 
Y

1

3 H
 
4 X
   which becomes equation (12) if we define R = v
0
/ w
1
and set w
1
=12
29

V


1

H
 t
H
 
2 x
 y



 t
H
  
1

3 H
 y

4 x

(12) with x

X / w
1
= R – b and y

Y / w
1
= R (1 – f ) – (1 – b ).
Proof that A ≥ 1
A ≥ 1 is equivalent to:


H
 t
H


4 H H
 t


1

H
 t
H H


 t

1

3 H
H
 
(A.6) as long as both sides of this inequality are positive. Combining terms, this is equivalent to:
1 + H – t
H
≥ ( H – t
H
)[ H (1 – 3 H ) + 4 H ] or:
1 ≥ ( H – t
H
)[ H (5 – 3 H ) – 1] (A.7)
Since –1 ≤ H – t
H
≤ 1, this inequality can only be violated if one of two alternative necessary conditions is true. Either
(1) Both factors on the RHS are negative, and the second factor is less than –1, or
(2) Both factors on the RHS are positive, and the second factor is greater than 1.
It is easy to see that the minimum of [ H (5 – 3 H ) – 1] over the interval [0, 1] is –1, reached when H = 0, since this is a parabola which reaches a maximum of 13/12 at
H = 5/6, and since its value is 1 at H = 1. Hence case (1) will not lead to a violation of the inequality.
As for case 2, we want to consider the maximum of the RHS, and we have H – t
H
> 0, so we take t
H
= 0, and find the maximum of the whole RHS, namely of [–3 H 3 + 5 H 2 – H ].
The maximum value of this expression on the interval [0,1] is 1, reached at H = 1, and hence case (2) does not violate the inequality either.
We must also consider the possibility that both sides of inequality (A.6) are negative. However, it turns out that neither side of the inequality can in fact be negative.
The proof is along the same lines as the proof above, and is omitted, but can be obtained from the author. Alternatively, by plotting A over the unit square in Mathematica , we can confirm that A ≥ 1.
Proof of Proposition 1:
In equation (12), we collect the terms in x and the terms in y to get
30

V

 

H
 t
H


4 H H
 t
H
  x


1

H
 t
H


 t
H
 
1

3 H
  y (A.8)
Call this Cx + Dy ; it equals 0 if Ax + y = 0, where A = C / D , and if D ≠ 0.
Now, from equation (12), we replace x and y by their expressions in b , f and R :
A ( R – b ) + R (1 – f ) – (1 – b ) = 0
Solving for f , we get equation (13): f = A + 1 – 1/ R + (1/ R )(1 – A ) b (13) the values ( b
A
, f
A
) = (1/(1A ), 1+ A ) in equation (14) do make equation (13) an identity, independent of the value of R :
1 + A = A + 1 – 1/ R + (1/ R )(1 – A )[1/(1 – A )]
Likewise, the values ( b
R
, f
R
) = ( R , 2 – 1/ R ) in equation (15) make equation (13) an identity regardless of the value of A :
2 – 1/ R = A + 1 – 1/ R + (1/ R )(1 – A ) R
Proof of Corollary 1:
If b = R and if f = 2 – 1/ R then f = 2 – 1/ b , so ( b , f ) is on this hyperbola. Then equation
(13) is satisfied, so V = 0. Since we did not use the value of A to reach this conclusion, it holds independent of the value of A , and hence independent of the values of H and t
H
.
Proof of Corollary 2:
This follows easily by substituting H – t
H
= 0 into the expression for A in equation (13) to get A = 2, and then putting this value of A into equations (14) and then (13).
Proof of Proposition 2:
Taking partial derivatives of V with respect to b and f , we get:
V b
V f

H (5

3 H )( H
 t
H
)

(1

H
 t
H
) (A.9)


(1

3 H )( H
 t
H
)

(1

H
 t
H
)

(A.10) and it is easy to verify that V b
and V f
are both negative over the admissible region by simply plotting them in Mathematica . but let us also prove analytically that V f
is negative.
This is true if

H
(1

3 H )

1

( H
 t
H
)

1 (A.11)
31

cannot be positive. To make this expression positive would require either that ( H – t
H
) and H – 3 H
2
– 1 both be positive, or that they both be negative. But H – 3 H
2
– 1 cannot be positive for 0 ≤ H ≤ 1, since its maximum over this interval is –11/12, reached at H =
1/6. Hence the only way expression (A.9) can be positive is if ( H – t
H
) and H – 3 H
2
– 1 are both negative. Expression (A.9) is therefore most likely to be positive if ( H – t
H
) is negative and also has an absolute value which is as large as possible. We should therefore take t
H
= 1. If we do this, then the expression becomes:

H
(1

3 H )

1

( H

1)

1 = H [–3 H
2
+ 4 H – 2] (A.12)
Since H is non-negative, this expression can only be positive if (–3 H
2
+ 4 H – 2) > 0. But for 0 ≤ H ≤ 1 its maximum value is –2/3, reached at H = 2/3. Hence V f
< 0 for all H , t
H between 0 and 1. author.
The proof that V b
< 0 is similar and is omitted, but can be obtained from the
Proof of Proposition 3
If the line V = 0 given by equation (13) lies entirely outside and above the admissible square region [0,2]

[0,2], then V is positive in that region, by Proposition 2. For given A , the lowest value of R for which this holds is the one which makes the line f = A + 1 – 1/ R
+ (1/ R )(1 – A ) b pass through the corner of the box, ( b , f ) = (2, 2), so that
2 = A + 1 – 1/ R + 1/ R (1 – A )(2)
Solving for R gives R
H
= 2 – 1/(1 – A ).
Similarly, the highest value of R for which V < 0 over the whole square ist he one which makes the line go through the corner ( b , f ) = (0, 0) so that
0 = A + 1 – 1/ R + 1/ R (1 – A )(0) or R
L
= 1/(1 + A ).
Proof of Proposition 4
The hypothesis says that taking H = t
H
must give us V = 0. Since H = t
H
makes the second term of V vanish, and since one of the factors of the first term, namely (1 + H – t
H
), must be positive (except in the case H = 0, t
H
= 1 which we have already analyzed), then
32

necessarily 2 x + y = 0, which in turn implies y = –2 x . Therefore the general expression for
V becomes:
V = 0 + H ( H – t
H
)[ y (1 – 3 H ) – 4 x ]
= H ( H – t
H
)[ y (1 – 3 H ) + 2 y ]
= H ( H – t
H
)[3 y (1 – H )]
Part of the hypothesis is that H = h
0
, the value of h at which U ( h ) = 0 in the open interval
(0, 1). We have already noted that this occurs where x + yh = 0, that is, h
0
= – x/y = H . But this is just H = – x/ (–2 x ) = 1/2. Hence
V = (1/2)( H – t
H
)[(3/2) y ] = (3/4) y ( H – t
H
)
We also need the value of U

( h ), which from equation (3

) is:
U

( h ) = (1 – h )( x + yh ) – h ( x + yh ) + yh (1 – h )
And since x + yh
0
= 0, we have
U

( h
0
) = yh
0
(1 – h
0
)
Again using the fact that H = h
0
= 1/2, we have
U

( h
0
) = (1/4) y
Finally, this establishes that under the hypotheses of the proposition, V can be written:
V = 3 U

( h
0
)( H – t
H
) which proves that when U

( h
0
) > 0, V has the same sign as H – t
H
. U

( h
0
) > 0 is equivalent in this case to y > 0, or x < 0, or b > R . In the other case, when U

( h
0
) < 0, V has the same sign as –( H – t
H
). Further, this is the case in which y < 0, so x > 0, so b < R . This completes the proof.
33