Mathematical
Social Sciences
287
25 (1993) 287-293
North-Holland
Note
Mixed diamond goods and anomalies
consumer theory
Upward-sloping
compensated
in
demand curves with unchanged
diamondness
Yew-Kwang
Department
Ng
ofEconomics, Monash University, Claylon, Melbourne, Victoria 3168, Australia
Communicated
by F.W. Roush
Received 2 June 1992
Mixed diamond
a changing
marginal
goods are valued both for their intrinsic consumption
utility of income, the Marshallian
share of the intrinsic marginal
observable
utility in aggregate
marginal
presence of the diamond
effect. More surprisingly,
even if the degree of the diamond
taxation
until its intrinsic consumption
surplus has to be weighted by the
utility. Unfortunately,
from market data even given idealized information.
may be upward-sloping,
consumed
effects and their values. Even ignoring
measure of consumer
Roy’s equality
this share is not ‘directly
(identity) does not hold in the
demand curves for mixed diamond
effect is negative,
effect is unchanged.
making
goods with no inferiority
A mixed diamond
good may be
a negative burden (not excess burden)
of
possible.
Key words: Marshallian
measure,
intrinsic
marginal
utility, Roy’s equality,
negative burden.
1. Introduction
In Ng (1987), I argued that some goods (such as gold and diamonds) are valued
for their values, not their intrinsic consumption effects. These are defined as pure
‘diamond goods’. The basic results obtained are: (1) a change in the price of such
a good leaves its own value and the amounts of all other goods consumed and hence
the utility levels of consumers unchanged; (2) the demand curve for a pure diamond
good is a rectangular hyperbola; and (3) it is optimal to place very high taxes on
pure diamond goods which impose not only no excess burden but no burden at all.
Correspondence to: Y.-K. Ng, Department
ria 3168. Australia.
0165-4896/93/$06.00
0
1993-Elsevier
of Economics,
Science
Monash
Publishers
University,
Clayton,
B.V. All rights
reserved
Melbourne,
Victo-
288
Y.-K. Ng / Mixed diamond goods
While some goods may be close to being pure diamond goods, most diamond
goods are impure or mixed, i.e. they are valued both for their intrinsic consumption
effects and for their values. This paper explores the implications of the diamond
effect on some traditional results of consumer theory. Section 2 shows that the
correct measurement of consumer surplus for pure and mixed diamond goods is
quite different from the traditional method even ignoring well-known ambiguities
associated with a changing marginal utility of money or income. An upward-sloping
compensated demand curve is not surprising since it can be explained by a changing
degree (to be defined precisely below) of the diamond effect. (That is, the degree
of diamond effect decreases as the price decreases. A specific example is available
from the author.) However, Sections 3 and 4 show that, even if the degree of the
diamond effect is unchanged, the compensated demand curve may be upwardsloping. This is somewhat surprising since the compensated demand curves for both
pure diamond goods and pure intrinsic goods are downward-sloping.
A mixed
diamond good may be consumed until its intrinsic consumption effect is negative,
making possible a negative burden (not just negative excess burden) of taxation.
This paper remains in the atemporal model of traditional welfare economics; the
stock-flow question was briefly touched on in Ng (1987).
2. Diamond
effect and consumer
surplus measurement
For the case of a pure diamond good, it is obvious that the Marshallian area (the
area bounded by the demand curve and the vertical axis between the two price lines)
does not measure the change in consumer surplus. The area may show a substantial
amount for a big price change, but consumers remain indifferent; consumer surplus
is always zero. (Government revenue may increase if the price increase is due to a
tax on diamond goods.) We thus have the following obvious proposition.
Proposition
1. The consumer surplus associated with any change in the price of a
pure diamond good is exactly zero.
Since the utility level remains unchanged as the price changes, we have -aU/ap’=O.
Thus,
-7
I- =
au au
apI aM
OfX’,
where xi is a pure diamond good. A corollary
Corollary
Corollary
of Proposition
1 is thus,
1. Roy’s equality does not hold for a diamond good.
2. The ordinary (Marshallian) demand curve and the compensated
(Hicksian) demand curves for a pure diamond good are identical.
289
Y.-K. Ng / Mixed diamond goods
Though worth pointing out, the case of a pure diamond good is too obvious to
be interesting. Let us now consider impure diamond goods or mixed diamond
goods, which are also practically much more important, including precious metals
and stones, expensive cars, fur coats, luxurious yachts, high-class hotels and
restaurants, selective wines, etc. In general, top-priced brands of most types of
goods are likely to possess some degree of diamond effect.
Impurity may occur in different ways. For simplicity, let us start with the case
where the good in question serves simultaneously as a diamond good and as an
ordinary item of consumption yielding intrinsic consumption effect. For example,
pricey paintings are valued for their values but usually also yield intrinsic consumption utility by being looked at. It is true that the utility derived from looking
at a painting may also be affected by the price of the painting. This is an intertwining of the intrinsic and the diamond effects. For simplicity we assume that we
may remove all diamond effects to be included under the diamond effect, leaving
the intrinsic consumption effect untainted. Considering one mixed diamond good
x and one composite pure ordinary good y, we may then write the utility function as
U(c, 0, Y)9
(1)
where c=x is the intrinsic consumption component and D=px (using y as a
numeraire with price = 1) is the diamond-effect component of the mixed diamond
good x. The consumer maximizes (1) subject to
px+y
= M,
(2)
where income A4 is taken as given. The first-order
Uc+PU,
conditions
are
(34
= AP,
UY = A,
(3b)
where a subscript denotes differentiation, e.g. U,=aU/aD,
L is the Lagrangian
multiplier, and aU/ax (aggregate marginal utility of x) = UC (intrinsic MU only) +
pUD (MU of x through the diamond effect).
Combining (3a) and (3b):
u, +pu,
-pu,
The differentiation
dU=
Substituting
-x-pax/ap
= 0.
(4)
of (1) yields, at given M,
ci,$dp+U~~dp+ci,$dp.
(5)
in aD/ap =x+pax/ap
(from the definition of D) and ay/ap =
(from the budget constraint, eqn. (2), holding A4 unchanged),
dU=
(Uc+pU~-pUY)~+x(U~-UY)
dp=
-(xUJp)dp.
(6)
Y.-K. Ng / Mixed diamond goods
290
The last equation follows from (4). Dividing the LHS (dU) by 1 and the RHS of
the last equation by U, (this is permissible from eqn. 3(b)) and integrating,
(7)
where s= U,/(U,+pU,)
is the share of the intrinsic marginal utility of the mixed
diamond good in its aggregate marginal utility. The second equation in (7) follows
from (4).
For the case of a non-diamond good, s= 1 and the RHS of (7) collapses into
--j xdp, which is the traditional measure of consumer surplus. For the case of a
pure diamond good, s= 0, confirming Proposition 1. Thus, eqn. (7) is the generalization of consumer surplus measurement to cover the possible presence of the
diamond effect. However, s is not (at least not directly) observable from market
data and has to be estimated by other methods (such as introspection and interviews).
Nevertheless, eqn. (7), by relating the measure of consumer surplus for a mixed
diamond good to the traditional measure by just adding a weighting (or discounting)
factor s (0~s~ 1 for a mixed diamond good), provides a simple guidance on the
correct estimation.
2. The consumer surplus associated with a change in the price of a mixed
diamond good is measured by the traditional measure (-x) weighted by the share
of intrinsic marginal utility in aggregate marginal utility of the mixed diamond
good. This share is not directly observable from market data even given idealized
information.
Proposition
3. Upward-sloping
compensated
demand
curve with unchanged
diamondness
It is well known that the compensated demand curve for any good (no diamond
effect) is downward-sloping. This is not necessarily so for a mixed diamond good,
even if the degree of diamondness is unchanged. The slope of the compensated demand curve may be derived by minimizing expenditure subject to a constant utility
level and deriving the comparative statics on the resulting first-order conditions.
This is done below for a simple model of a single mixed diamond good x and a composite of all other good y, as in Section 2 above.
Let the consumer minimize px+ y subject to
U(c, 0, Y) = 0,
(8)
where c=x is the intrinsic consumption component and Dspxis the diamond-effect
component of the mixed diamond good. In a specific example available from the
author, D is taken to equal I(p)px, where Z is the index of diamond effect, with
aI/ap positive over some range. Here, we take Z= 1 throughout, thus abstracting
away upward-slopingness in the demand curve due to a variable diamond effect.
291
Y.-K. Ng / Mixed diamond goods
The first-order
P =
conditions are
(94
~tUc+p”Dh
(9b)
1 = ClUy,
where p is the Lagrangian multiplier and a subscript denotes partial differentiation,
e.g. u,= awac.
Eliminating p between (9a) and (9b), we obtain (4). Total differentiation of (8),
at 0 unchanged, gives
dy = -(UC dc+ u, dD)/u,.
dc = dx, dD
Substituting
=p
(10)
dx+ x dp into (lo), we obtain
dy = -(UC dX+pUD dx+xU,dp)/U,.
Substituting
(11)
(4) into (1 l), yields
(12)
.
Total differentiation
of (4) gives
(Uw+pUDc-pUyc~d~+(Ucy+PUDy-PUyy)dy
+CU~D+PUDD-PU~DI~=
(13)
(Uy-U~ldp-
Substituting dc=dx, dy from (12) and W=xdp+pdx
into (13), after rearrangement, taking U,, = Ucy, etc. (with 1U 1 below indicates holding utility constant), we obtain
IUI
5-g
=
Uy-
UD-x(&D-
UcyUD/Uy)
+px{UyD(l
UCC-~PUC~+P~U~~+~PUCD-~~
+ UDiuy)-
UDD-
+PIPU~D+P~UDD
UyyUD/Uy)
(14)
From the second-order condition, the denominator in the RHS of (14) is negative,
and the sign of axlap depends on the numerator. In the absence of any diamond
effect, all terms associated with D become zero and c can be replaced by x. In this
case, (14) collapses into
ax
ap=
v,
U.-2PUxy+P2Uyy’
(14’)
which is necessarily negative since WYis positive and the denominator is negative
from the second-order condition.
For the general case of a mixed diamond good, the numerator in the RHS of (14)
cannot in general be signed. It is true that, from (4), Uy - U,= UC/p> 0 if the intrinsic component of the mixed diamond good is a ‘good’, not a ‘bad’. However,
292
Y.-K. Ng / Mixed diamond goods
badness is quite possible here. For example, if I order expensive drinks to impress
my dinner guests with the amount of money I am prepared to spend on drinks, I
may consume until the intrinsic marginal utility is negative. By the way, this means
that an increase in price may actually make me better off. A tax on a mixed diamond
good could impose negative excess burden! This is consistent with the consumersurplus measurement of -js~dp
derived in Section 2, where SE U,/(U,+pU,)
is
negative if (I, is negative.
Even if UY- U, is positive, other terms in the numerator of (14) may be negative,
making the whole numerator negative without making the denominator
nonnegative (hence without violating the second-order condition). This is more likely
to be so if U, is algebraically very small (i.e. negative and large in absolute terms),
UCD, UDD, U&,big, and UC,,, U,,D small. If the numerator is negative, the denominator tends to be positive unless U,, (which does not appear in the numerator) is
very negative. This may well be so since there is really no lower bound (apart from
minus infinity) on U,,. It may be concluded that the compensated demand curve
for a mixed diamond good may be upward-sloping even with no change in the
degree of the diamond effect. A specific example confirming and illustrating this
result is given below.’
Mr. A takes Miss B out for an expensive dinner (with of course drinks). While
Mr. A enjoys the dinner, his main purpose is to impress Miss B enough for her to
spend the rest of the evening with him. Mr. A believes that, the higher the price,
the higher the probability that the objective will be met. He also believes that dinner
(with drinks) and evening activities are complementary (e.g. one has to eat more if
one is to stay later at night; drinking is conducive to an enjoyable evening). Then,
an increase in the price of the dinner, the higher is D and the higher intrinsic utility
he places on the dinner; UC. is positive.
Over the range where the degree of the diamond effect increases with the price
of the mixed diamond good, it is fairly clear that the demand curve may be upwardsloping. The fact that the compensated demand curve may be upward-sloping even
with no change in the degree of the diamond effect is much less straightforward,
though not necessarily counter-intuitive once the mechanism for this result is explained. Given the increasing importance of mixed diamond goods, our analysis
may add a bit of significant variety to our consumer theory.
r A specific numerical example may also be given. For the general case where price enters the utility
function with no restriction as to how p affects U, it is clear that one can have upward-sloping ordinary
and compensated demand curves. For the case of pure diamond goods, I have shown that demand curves
are rectangular hyperbolas and hence downward-sloping. For mixed diamond goods, upward-sloping
compensated demand curves are possible even if we assume separability such that U(x,y,D)=y+
V(x)+ W(D). For example, let V(x)=2x-x *, W(D) =2D. It is easy to check that we have ax/Q>0
with both the first- and second-order conditions satisfied, e.g. at p= 4, x= 3, ax/ap = j, UC= V,= -4,
confirming also the possibility of a negative intrinsic consumption effect.
Y.-K. Ng / Mixed diamond goods
293
References
Yew-Kwang
values,
Ng, Diamonds
Amer.
Econom.
are a government’s
best friend:
Rev. 77 (1987) 186-191.
burden-free
taxes on goods
valued
for their