INFERIOR GOOD AND GIFFEN BEHAVIOR FOR
INVESTING AND BORROWING
Felix Kubler
University of Zurich
Larry Selden
Columbia University
University of Pennsylvania
Xiao Wei
Columbia University
August 14, 2011
Abstract
It is standard in economics to assume that assets are normal goods and demand
is downward sloping in price. This view has its theoretical foundation in the classic
single period model of Arrow with one risky asset and one risk free asset, where both
are assumed to be held long, and preferences exhibit decreasing absolute risk aversion
and increasing relative risk aversion. However when short selling is allowed, we show
that the risk free asset can not only fail to be a normal good but can in fact be a Gi¤en
good even for widely popular members of the hyperbolic absolute risk aversion (HARA)
class of utility functions. Distinct regions in the price-income space are identi…ed in
which the risk free asset exhibits normal, inferior and Gi¤en behavior. An Example
is provided in which for non-HARA preferences Gi¤en b ehavior occurs over multiple
ranges of income.
We thank the anonymous referees and Edito r for their very helpful comments and suggestions.
1
Copyright © 2013 by the American Economic Association.
1 Introduction
In the classic single period model with one risky asset and one risk free asset, where both
are assumed to be hel d long, Arrow [1] shows that the risky asset is a normal good (its
demand is increasing with income or wealth) if the Arrow-Pratt [1]-[12] measure of absolute
risk aversion is decreasing. Arrow also proves that a su¢ cient condition for the income
elasticity of demand for the risk free asset to be greater than one is that relative risk aversion
is increasing. Aura, Diamond and Geanakoplos [2] point out that these two results together
imply that both assets are normal goods.
While the assumption that the risky asset be held long is relatively harmless, the same
assumption for the risk free asset is far from innocuous. Consider the case of the widely
used HARA (hyperbolic absolute risk aversion)
1
utility W (x) =
(x+a)
; where > 1,
a > 0 and x denotes wealth (or end of period consumption). Optimal holdings of th e
risk free asset will always be both positive and negative (corresponding to di¤erent income
ranges) so long as the risk preference parameter is above some minimum
critical
. Despite
the fact that this utility function satis…es the Arrow requirements of decreasing absolute
risk aversion and increasing relative risk aversion, when >
critical
the risk free asset will
always be an inferior good over some income range. And it can even be a Gi¤en good,
where corresponding to an own price increase, the asset’s positive income e¤ect swamps the
negative substitution e¤ect resulting in increased demand.
More generally, inferior good and Gi¤en behavior occur for other members of the HARA
class and other forms of utility. If Arrow’s assumption that both assets are held long is
relaxed, the only member of the HARA class for which the risk free asset and risky asset
are both always normal goods is the very special constant relative risk aversion (CRRA)
form.
2
For a number of examples, distinct regions in the price-income space are identi…ed
in which the risk free asset exhibits normal good, inferior good and Gi¤en behavior. We
show that when the risk free asset is an inferior or Gi¤en good, it can only be held short
(long) if relative risk aversion is increasing (decreasing). A non-HARA example is given
for which relative risk aversion is non-monotone and Gi¤en behavior is shown to occur over
multiple income ranges. What is particularly surprising is that in contrast to much of the
classic demand theory literature where very special forms of utility need to be constructed to
produce Gi¤en behavior, in the case of …nancial securities it arises with perfectly standard
utility functions.
3
Given that Gi¤en behavior can arise with relative ease for the commonly used HARA
1
Se e [7] for a de scription of the HARA family of utility funcitons.
2
Following Fischer [6], it is well known that the risky asset w ill be an inferior good in the case of quadratic
utility. But because this member of the HARA class exhibits increasi ng absolute risk aversion, it is rarely
assumed.
3
In their recent paper [5], Doi, Iwasa and Shimomura o bserve that the existin g demand theory literature
on Gi¤en behavior is vo id of examples based on conventional forms of utility. Indeed they too construct a
speci …c form o f utility which, although nonstandard, is argued to be well-behaved in terms of its properties.
2
utilities, it is natural to wonder what implications this behavior might have for equilibrium
asset prices.
4
By applying a not widely known certainty result of Kohli [9], one can obtain
the surprising result that in a representative agent, distribution economy, Gi¤en behavior of
the risk free asset implies that the risky asset’s equilibrium price increases with its supply.
5
In Section 2, we consider portfolios consisting of a risk free asset and a risky asset where
positive holdings of the former is not assumed. As is standard, the asset demand, or complex
securities, model is embedded in a contingent claims framework. Complete markets are
assumed.
6
We establish necessary and su¢ cient conditions for the risk free asset to be a
normal good and apply these conditions to a number of di¤erent classes of utility including
the HARA family. Section 3 examines when the risk free asset can be a Gi¤en good and
provides examples for a utility in the HARA class and for one outside the class. Section 4
considers selected extensions to a two period setting. The last section contains concluding
comments.
2 Risk Free Asset: Normal Good Behavior
2.1 Preliminaries
Throughout this Section and the next Section, we consider a single period setting in which
a consumer with a given level of income selects asset holdings so as to maximize expected
utility for end of period random consumption. In Section 4, we consider the natural
extension to a two period setting where the consumer at the beginning of period 1 chooses
a level of certain current period consumption c
1
as well as asset holdings the returns on
which fund period 2 consumption, c
2
. The notational conventions and structure of the
current Section are designed to facilitate the simplest transition to the more general two
period problem.
Consider a risky asset with payo¤
~
, where
~
is a random variable assuming the value
21
with probability
21
and
22
with the probability
22
= 1
21
. Without loss of generality,
let
21
>
22
. It is further assumed that
22
> 0: Suppose there exists a risk-free asset
with payo¤
f
> 0. Let n and n
f
denote the number of units of the risky asset and risk
free asset, respectively. Throughout this paper, we assume that
E
e
p
>
f
p
f
which can be
shown to imply that risky asset demand satis…es n > 0 for all I: In the current single
period setting, preferences are de…ned over random ec
2
and satisfy the standard expected
4
We thank one of the Re fer ees for stres sing the importance of connecting our demand theory resul ts to
their equilibrium implications.
5
For a more general analysis assuming a representative agent, exchange econo my in which the imp lications
of changing asset supplies on equilbrium asset prices and equity risk premia are examined in both one and
two period settings, see Kubler, Selden and Wei [10].
6
It should be noted that our results, Theorem (i) and (iii), extend naturally to incomplete markets.
Al though in general, The orem 1(ii) does not extend, as one might expect, it does for HAR A preference s
where markets are e¤ectively complete (see [4]).
3
utility axioms where the NM (von Neumann-Morgenstern) index W (c
2
) satis…es W
0
> 0
and W
00
< 0:
7
The expected utility function EW (ec
2
) given by
E[W (
~
n +
f
n
f
)] =
21
W (
21
n +
f
n
f
) +
22
W (
22
n +
f
n
f
) (1)
is maximized with respect to n and n
f
subject to the budget constraint
pn + p
f
n
f
= I; (2)
where p and p
f
are the prices of the risky and risk free assets and I is initial or date 1
income. De…ne the contingent claims c
21
and c
22
by
c
21
=
21
n +
f
n
f
; c
22
=
22
n +
f
n
f
: (3)
The above complex or …nancial securities problem is equivalent to a contingent claim opti-
mization where
EW (c
21
; c
22
) =
21
W (c
21
) +
22
W (c
22
) (4)
is maximized with respect to c
21
and c
22
subject to
p
21
c
21
+ p
22
c
22
= I; (5)
where
p
21
=
f
p
22
p
f
(
21
22
)
f
> 0 and p
22
=
21
p
f
f
p
(
21
22
)
f
> 0 (6)
are the contingent claims prices. The contingent claims FOC (…rst order conditions) can
be expressed as
W
0
(c
21
)
W
0
(c
22
)
=
22
21
p
21
p
22
=
def
k: (7)
Throughout we assume no arbitrage –it is easy to see that this is equivalent to
21
p
>
f
p
f
>
22
p
: (8)
It should be noted that
E
e
p
>
f
p
f
is equivalent to c
21
> c
22
or
k =
22
21
p
21
p
22
< 1: (9)
Since we do not assume an Inada condition, a minimum level of income has to be
assumed to guarantee non-negative consumption. It is easy to verify that to ensure that
c
21
; c
22
0, the minimum income level is given by
I
min
=
(
p
21
(W
0
)
1
(kW
0
(0)) (W
0
(0) 6= 1)
0 (W
0
(0) = 1)
: (10)
7
These single pe riod NM pref erences a re extended in Section 4 to the two period expected utility
EW (c
1
; ec
2
) = W
1
(c
1
) + EW
2
(ec
2
) where the consumer is choosing over both c
1
and (n; n
f
). Th e NM
utility considered here can be viewed as corresponding to the two period W
2
(c
2
):
4
0 1 2 3 4 5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
c
21
c
22
Budget
Line
Figure 1:
This condition requires I > I
min
to ensure that optimal contingent claims demand is in the
positive orthant. The application of I
min
will be illustrated for a number of speci…c utility
functions below.
Although n > 0 is ensured by the assumption
E
e
p
>
f
p
f
, the condition for n
f
> 0 is far
from free as it imposes restrictions on the consumer’s preferences. Given that
22
21
de…nes the
slope of the risky asset payo¤ ratio in the contingent claims space,
21
W
0
(c
21
)
22
W
0
(
22
21
c
21
)
measures
the slope of the tangent to the indi¤erence curve at its intersection with the n
f
= 0 ray.
The following Lemma states that for n
f
> 0; the consumer’s preferences must be such that
the MRS (marginal rate of substitution) at this ray is always less than the absolute value
of the slope of the budget line for any c
2
. See Figure 1.
Lemma 1 The risk free asset holdings satisfy n
f
T 0 for all I i¤
W
0
(c
2
)
W
0
(
22
21
c
2
)
S k for any c
2
.
In standard demand theory, a commodity is assumed to have positive demand and is
said to be a normal good if its derivative with resp ect to income is p ositive. Given that
Lemma 1 allows for the risk free asset to be held short, we next generalize the normal good
de…nition to allow for borrowing.
5
Definition 1 The risk free asset is said to be a normal good if and only if
8
n
f
@n
f
@I
> 0: (11)
When n
f
> 0, we obtain the traditional normal goods de…nition
@n
f
@I
> 0. If n
f
< 0 the
asset will be held short, and
@n
f
@I
< 0 indicates that as income level increases, the investor
will increase the borrowing and borrowing can b e viewed as a normal good. It should be
noted that n
f
@n
f
@I
is the standard income e¤ect in the Slutsky equation. If n
f
@n
f
@I
< 0,
the risk free asset is an inferior good and the income e¤ect will become positive. This can
result in
@n
f
@p
f
being positive if the positive income e¤ect dominates the negative substitution
e¤ect.
In the analysis that follows, we will make use of the critical income level I
which serves
as the boundary along the risk free asset Engel curve dividing normal from inferior good
behavior.
Definition 2 An income level I is said to be a critical income I
if it satis…es
n
f
@n
f
@I
I=I
= 0: (12)
Throughout this paper we will require I
> I
min
to ensure that optimal consumption will
be in the positive orthant. Clearly, I
corresponds to either n
f
= 0 or
@n
f
@I
= 0: Moreover,
as we will see, there can exist multiple I
values.
2.2 Normal Good Behavior: The Canonical CRRA Case
To illustrate the role of the Lemma 1 restriction on preferences, we next consider the case
of CRRA utility.
Example 1 Suppose the NM index takes the classic CRRA form
W (c
2
) =
1
c
2
; (13)
where > 1: Will the risk free asset be held long or short? From the FOC for CRRA
preferences, using eqn. (7), the contingent claims expansion pat h is given by
c
22
= c
21
k
1=(1+)
(14)
and is linear passing through the origin with slope of k
1=(1+)
: Straightforward computation of
the condition in Lemma 1 shows that n
f
> 0, if and only if k > (
22
21
)
1+
. If we de…ne
critical
=
ln k
ln(
22
=
21
)
1; (15)
8
This same de…nition will be used for risky assets as well.
6
we have the following restriction on preferences
n
f
T 0 , T
critical
; (16)
where k is de…ned in terms of the state prices p
21
and p
22
which in turn depend on the asset
payo¤s and prices following eqn. (6). Since k < 1, the linear expansion path will rotate
clockwise as decreases because its slope k
1=(1+)
will decline. If falls below the critical value
given by the right hand side of eqn. (15), we have n
f
< 0 and the expansion path will be below
the
22
21
(or n
f
= 0) ray: Given that the Arrow-Pratt ([1]-[12]) relative risk aversion measure
R
=
def
c
2
W
00
(c
2
)
W
0
(c
2
)
= + 1; (17)
one obtains the very intuitive interpretation for eqn. (16) that if >
critical
, the consumer
is su¢ ciently risk averse that she will only hold the risk free asset long. Since for CRRA
preferences, the expansion path is linear and pass through the origin, n
f
and
@n
f
@I
always h ave
the same sign, which implies that the risk free asset is always a normal good.
2.3 Generalization of the Classic Arrow Theorems
Denoting the Arrow-Pratt ([1]-[12]) measure of absolute risk aversion by
A
(c
2
) =
def
W
00
(c
2
)
W
0
(c
2
)
; (18)
we next extend the Arrow [1] result to a contingent claims setting in which shorting the
risk free asset is allowed.
Theorem 1 For the contingent claims problem corresponding to eqns. (4) and (5), optimal
asset demands satisfy
(i)
@n
@I
T 0 if
0
A
S 0;
(ii)
@n
f
@I
T 0 i¤
A
(c
21
)
A
(c
22
)
T
22
21
;
(iii)
@(n
f
=I)
@I
T 0 if
0
R
T 0:
Remark 1 Given that
E
e
p
>
f
p
f
implies n > 0; condition (i) of Theorem 1 coincides exactly
with Arrow’s result. Condition (iii) is equivalent to Arrow’s second result relating to increasing
relative risk aversion. To see this, note that
@(n
f
=I)
@I
=
I
@n
f
@I
n
f
I
2
(19)
and assuming n
f
> 0; Arrow’s income elasticity result follows immediately from
@ (n
f
=I)
@I
> 0 ,
@n
f
=n
f
@I=I
> 1: (20)
7
Arrow’s assumption that both assets are held long clearly implies that
@n
f
@I
> 0 and the risk
free asset is a normal good as asserted by Aura, Diamond and Geanakoplos [2]. But from the
application of Lemma 1 in Example 1, we see that a ctually for n
f
> 0, one must assume that
the consumer is su¢ ciently risk averse to satisfy eqn. (16). Moreover, it follows from Example
1 that it is unnecessary to assume as in [2] that n
f
> 0 in order for both assets to be normal
goods.
How should one interpret the critical
A
(c
21
)
A
(c
22
)
? It is straightforward to show that this
ratio is in fact the slope of the tangent to the contingent claims expansion path at any point
(c
21
; c
21
) along the path. Theorem 1(ii) can be viewed as requiring for n
f
to be increasing
with income that the tangent to the expansion path must have a slope steeper than that
of the n
f
= 0 ray de…ned by the risky asset payo¤
22
21
. It should also be noted that if
over an interval of income values the tangent to the expansion path has the same slope as
the c
22
=
22
21
c
21
(n
f
= 0) ray, then for that range of incomes n
f
is invariant to changes in
I. It can be shown that in the case of multiple risky assets, condition (ii) in Theorem 1
generalizes to a comparison of the angle between the tangent vector of the expansion path
in the contingent claims setting and the normal vector of the n
f
= 0 hyperplane and 90
.
Hence condition (ii) is the generic result, rather than the widely quoted Arrow condition
(i).
2.4 Canonical Inferior Good Case: HARA Preferences
The following example illustrates several important implications of Theorem 1 for a widely
used form of HARA utility.
Example 2 Preferences are de…ned by the widely used HARA form
W (c
2
) =
1
(c
2
+ a)
; (21)
where a > 0; > 1: For this utility, we have
0
A
< 0 and
0
R
> 0. Therefore, the risky asset
is always a normal good. The expansion path is given by
c
22
= k
1
1+
(c
21
+ a) a: (22)
Figure 2 illustrates expansion paths associated with di¤erent values of , where as is standard,
each p oint along an expansion path has the same price ratio but di¤erent levels of income, I.
(The expansion paths in the Figure are solid and the n = 0 and n
f
= 0 rays are dashed.) It
follows from eqn. (10) that the minimum income level I
min
to avoid bankruptcy is given by
I
min
=
ap
21
1 k
1
1+
k
1
1+
: (23)
Because
0
R
> 0; it follows from the Arrow result that the risk free asset is also a normal good
8
0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
4
c
21
c
22
Figure 2:
if n
f
> 0: However, as we show below, for the HARA utility (21) it is impossible for n
f
> 0 to
be satis…ed for all income levels. It follows from Theorem 1 that if
22
21
> k
1
1+
=
A
(c
21
)
A
(c
22
)
; (24)
which is equivalent to
<
critical
=
ln k
ln(
22
=
21
)
1; (25)
we have
@n
f
@I
< 0. Since n
f
< 0 when I = 0, the risk free asset is a normal good for all income
levels in the sense of borrowing. On the other hand if >
critical
, then we have
@n
f
@I
> 0.
Since n
f
< 0 when I = 0, the risk free asset cannot be a normal good for all income levels.
To illustrate this more explicitly, …x the parameters as follows: a = 2; p = p
f
= 1;
f
= 1;
21
= 1:2;
22
= 0:8 and
21
= 0:7: Then
critical
1:09: We plot the asset Engel curves
for = 0:5 <
critical
in Figure 3(a) and = 5 >
critical
in Figure 3(b) and indicate I
min
in
each case. It can be seen that when <
critical
, the investor will always short the risk free
asset. When the income level increases, she will borrow more, which implies that the risk free
asset is a normal good in the sense of borrowing. When >
critical
, the investor will only
short the risk free asset at the low income levels. But since
@n
f
@I
> 0 for all the income levels,
the risk free asset fails to be a normal good for the low income levels (De…nition 1). Moreover,
it is clear from Figure 3(b) that we can …nd the critical income level I
such that for I > I
,
9
0 1 2 3 4
-5
0
5
10
Income I
Demand
(a) <
critical
0 1 2 3 4
-1
-0.5
0
0.5
1
1.5
2
2.5
Income I
Demand
(b) >
critical
Figure 3:
n
f
> 0 and the risk free asset becomes a normal good. To …nd I
, note that in Figure 2 the
expansion paths corresponding to >
critical
all cross the n
f
= 0 ray. Thus for any such ;
based on the intersection point one can determine I
as follows:
I
=
a(
21
p
21
+
22
p
22
)(1 k
1
1+
)
21
k
1
1+
22
: (26)
In Figure 3(b) where = 5; we have I
min
= 0:30 and I
= 1:09. It can be veri…ed that
@I
@
< 0: Thus as the relative risk aversion parameter decreases; the critical income level I
increases. When !
critical
from above we have I
! 1 and the risk free asset becomes an
inferior good for virtually all levels of income.
Remark 2 In addition to eqn. (21) and the CRRA (13), the HARA class inclu des negative
exponential, logarithmic and quadratic utilities (e.g., [7], p.26). Each member other than the
CRRA case can generate expansion paths where the risk free asset exhibits both normal and
inferior good behavior over di¤erent income ranges.
9
9
It should be noted that for the ne gative exponenti al case, the expan sion path will always have a slope
equal to 1. For quadratic utility, the expansio n path always h as a slope greater than 1. It follows from
Theorem 1 that
@n
f
@I
> 0 for all the income levels. Given that
E
e
p
>
f
p
f
, n
f
wi ll b e n egative at su¢ ciently
low income l evels. Therefore, the risk free asset can never be a no rmal good for all the income levels for
these two types of utility functions.
10
2.5 Risk Free Asset Engel Curve Properties: Critical Role of
0
R
We next establish an important link between
0
R
and inferior good behavior for the risk free
asset and then illustrate our conclusions with a series of examples.
Theorem 2 Assume the general NM utility (1), and complete markets with one risk free asset
and one risky asset.
(i) If
0
R
> 0, the risk free asset can become an inferior good only when n
f
< 0.
(ii) If
0
R
< 0, the risk free asset can become an inferior good only when n
f
> 0.
(iii) If the sign of
0
R
changes over its domain, the risk free asset can become an inferior good
for both n
f
< 0 and n
f
> 0.
Remark 3 In terms of Theorem 2, condition (i) is illustrated by Example 2 (above), (ii) by
Examples 3 and (iii) by Example 4.
We begin by modifying the Example 2 utility to investigate the impact of assuming
decreasing rather than increasing relative risk aversion.
Example 3 Assume
W (c
2
) =
1
(c
2
a)
; (27)
where a > 0; > 1: For this utility, we have
0
R
< 0. The same parameters are assumed as
in Example 2. Figure 4 illustrates expansion paths associated with di¤erent values of : Since
we require that c
21
; c
22
> a, I
min
= ap
21
+ ap
22
=
ap
f
f
:
10
When >
critical
, where
critical
is
de…ned by (15), the risk free asset is a normal good. (See Figure 5(b).) When <
critical
,
@n
f
@I
< 0. Since n
f
starts from
a
f
where n = 0, the risk free asset is an inferior good at low
income levels and when n
f
< 0, it becomes a normal go od. (See Figure 5(a).)
Remark 4 When one makes the reasonable assumption that n
f
can be either negative or
positive, a comparison of Examples 2 and 3 would seem to weaken Arrow’s argument for assuming
increasing rather than decreasing relative risk aversion. In Example 2 where
0
R
> 0; if >
critical
the consumer with low levels of income, I
> I > I
min
; initially shorts the risk free
asset to …nance investment in the risky asset. Whereas if one assumes exactly the same setting
except that
0
R
< 0; we see in Example 3 that for >
critical
the consumer initially holds the
risk free asset long at low levels of income and then reduces the holdings as income increases.
11
For us at least, the latter case is a priori more reasonable. The property of decreasing relative
risk aversion has received attention in empirical and experimental papers (e.g., Calvet and Sodini
10
In this case unlike the other examples, I
min
arises from the "subsistence" requirement c
2
> a rather
than from the no bankruptcy requirement c
21
; c
22
> 0.
11
It should be noted that
critical
can take on a range of values based on di¤erent assumptions of the
underlying parameters such as asset returns returns, probabilities and prices.
11
0 2 4 6 8 10
0
1
2
3
4
5
6
7
8
9
10
c
21
c
22
Figure 4:
0 2 4 6 8 10
-2
0
2
4
6
8
10
Income I
Demand
(a) <
critical
0 1 2 3 4 5 6
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Income I
Demand
(b) >
critical
Figure 5:
12
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
c
21
c
22
(a) Expansion Path
0 0.2 0.4 0.6 0.8 1
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Income I
Demand
(b) Engel Curve
Figure 6:
[3]) challenging the Arrow assumption of increasing relative risk aversion. It should also be
noted that Meyer and Meyer [11] have recently proposed employing a multiperiod version of
(27) as an alternative to the standard (internal) habit formation representation in try to resolve
the equity premium puzzle. Both the standard NM habit utility and (27) exhibit decreasing
relative risk aversion.
The next Example, based on non-HARA utility functions, illustrates that when the sign
of
0
R
varies, the sign of
@n
f
@I
can vary in both the n
f
> 0 and n
f
< 0 regions.
Example 4 Assume
W (c
2
) =
1
1
(c
2
+ a)
1
+ c
2
: (28)
where a > 0;
1
> 1: We have
0
A
< 0 and
0
R
doesn’t have a de…nite sign. An expansion
path and Engel curves are illustrated in Figure 6. It can be seen that the consumer shorts the
risk free asset at low income levels and again b eginning at intermediate income levels. But
@n
f
@I
is p ositive at low income levels and negative at high income level. At intermediate income
levels, the consumer is long the risk free asset but n
f
is not monotone in I. It can be also seen
from Figure 6(b) that there are three I
for this utility. Two correspond to n
f
= 0 and one
corresponds to
@n
f
@I
= 0.
12
12
As suggested by this Example, the utili ty (28) may o¤er interesting potential in providing a partial
respons e to Hart’s [8] query as to what assumptions are required to get meaningful comparative statics
…ndings for the risk free asset when the util ity function does not exhibit the portfolio separation proper ty
implied by the HARA class.
13
3 Gi¤en Good Behavior
3.1 Law of Demand Violations and Risk Free Asset Gi¤en Behavior
Given our …ndings that the risk free asset can become an inferior good, it is natural to
wonder if it can also be a Gi¤en good, i.e., the risk free asset can violate the (own good)
LOD (Law of Demand)
@n
f
@p
f
> 0. To see that shorting is allowed by this de…nition, note …rst
that if n
f
< 0, when the price p
f
increases, the e¤ective cost of borrowing
f
p
f
will decrease.
And if the consumer responds to the decreased cost of borrowing by the increasing borrowing
when n
f
< 0; then borrowing satis…es the LOD. On the other hand, if the consumer reduces
the borrowing, when the cost decreases the risk free asset becomes a Gi¤en good.
It is well known that if one of the goods is a Gi¤en good, then the LOD is violated.
However, violation of the LOD is only a necessary and not a su¢ cient condition for Gi¤en
good behavior. Moreover, Quah ([13], Proposition 1) has shown that the LOD is violated
in the …nancial securities setting, if and only if it is also violated in the contingent claims
setting. Since in the contingent claims setting, both contingent claims commodities are
normal goods, one may wonder whether Gi¤en good behavior can ever occur in the complex
security setting. In the next Subsection, we provide two examples illustrating that the risk
free asset can indeed be a Gi¤en good.
13
3.2 Gi¤en Behavior: Income and Price Regions
Next we show for both the HARA utility used in Example 3 and the non-HARA utility
in Example 4 that by choosing the appropriate parameter values, the risk free asset can
become a Gi¤en good. Also for the former, we characterize regions in the (p
f
; I) parameter
space corresponding to normal, inferior and Gi¤en behavior.
First it should be noted that for Example 5 since the demand function is linear in income
n
f
= (p
f
) + (p
f
)I; (29)
one can compute the break-even income for Gi¤en behavior I
G
by solving the equation
@n
f
@p
f
= 0 as follows
I
G
=
0
(p
f
)
0
(p
f
)
: (30)
Example 5 Assume the following specialization of the Example 3 HARA utility
W (c
2
) =
(c
2
a)
; (31)
13
Due to the equivalence of the LOD between the contingent claims setting and compl ex security settin g,
these examples show that the LOD can be vio lated even when both goods are nor mal.
14
0.95 1 1.05 1.1 1.15 1.2 1.25
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
p
f
Income
Normal
Good
Region
Inferior Good Region
Giffen Good Region
Bankruptcy Region
Normal Good Region
Figure 7:
where a > 0.
14
Assume the parameters a = 2; p = 1;
f
= 1;
21
= 1:2;
22
= 0:8 and
21
= 0:7: When <
critical
, the risk free asset will exhibit regions of normal good, inferior
good and Gi¤en good. See Figure 7. We have I
min
=
ap
f
f
. It is clear from Figure 7 that there
are two separate normal good regions. To understand why, note that
@p
21
@p
f
< 0 and
@p
22
@p
f
> 0 )
@k
@p
f
< 0: (32)
Therefore, for
critical
=
ln k
ln(
22
=
21
)
1, we have
@
critical
@p
f
=
1
k ln(
22
=
21
)
@k
@p
f
> 0: (33)
When p
f
is small,
critical
< and the risk free asset is always a normal good. When p
f
is
large enough such that
critical
> ; I
becomes positive, implying inferior good behavior when
I < I
and normal good behavior when I > I
: In terms of the Figure, as p
f
approaches 0:98
from above,
critical
! and I
! 1:
Given that in Example 5 the contingent claims are normal goods, it is natural to ask
why the risk free asset can be a Gi¤en good. As in the typical potato Gi¤en story, the
consumer’s income is only slightly above I
min
and c
22
is close to the subsistence level a. It
is clear from Figure 5(a) that most of her income is invested in the risk free rather than
14
It can be shown that when a < 0, the risk fre e asset can exhibit Gi¤en good behavior for appropriate
parameters.
15
the risky asset. In this case, safety from starvation is provided by the risk free asset rather
than potatoes. Now if p
f
increases, the return on the asset
f
p
f
falls and the substitution
e¤ect tends to drive the consumer to reduce her holdings of the risk free asset. But if she
does, since
22
p
<
f
p
f
, c
22
will decline
15
and she will face a greater risk of starvation and
hence the associated income e¤ect outweighs the substitution e¤ect leading her to actually
increases her demand for risk free asset. In other cases, the risk free asset can be a Gi¤en
good at income levels not necessarily close to I
min
and the impact of the price change on
the consumer’s risk aversion and desire for safety are the keys to explaining the behavior.
Quite surprisingly, as we show next for the Example 5 (and 3) HARA utility, it is
possible to …nd a region in the price, income space where the risk free asset exhibits Gi¤en
behavior for any value of the risk aversion parameter and for any distribution of the risky
asset’s returns.
Proposition 1 Assume the NM HARA utility in eqn. (31). Then for any a > 0; >
1;
21
;
22
and
21
;
22
> 0, there exists an income level I and a speci…c range of p
f
such
that the risk free asset becomes a Gi¤en good:
Remark 5 The intuition for this result can be seen in terms of Figure 7. When
critical
= ;
which corresponds to the vertical at p
f
0:98; the I
G
curve and the I
min
line will always
intersect at a point on the vertical. At this point, the slope of the I
G
curve is greater than the
I
min
line. Hence, there will always be a Gi¤en region as shown in the Figure to the right of the
intersection point on the vertical.
We conclude this Section by showing that for non-HARA preferences, Gi¤en good be-
havior can occur over multiple regions of income.
Example 6 Assume
W (c
2
) =
(c
2
+ 0:2)
9
9
+ c
2
; (34)
Figure 8 illustrates that there exist two regions of income where
@n
f
@p
f
> 0 . Gi¤en behavior for
the intermediate income range 3:93 < I < 4:94 is clear from Figure 8(a). The lower range is
shown in the magni…ed view in Figure 8(b):
4 Two Period Setting
In this Section, we will extend our analysis to a two period setting. Consider maximizing
EW (c
1
; ec
2
) = W
1
(c
1
) +
21
W
2
(c
21
) +
22
W
2
(c
22
) (35)
15
Noticing that
M c
22
=
22
M n +
f
M n
f
= p
f
M n
f
f
p
f
22
p
;
if M n
f
< 0, we have M c
22
< 0.
16
0 5 10 15
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
Income I
dn
f
/dp
f
(a)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
-20
-10
0
x 10
-5
Income I
dn
f
/dp
f
Bankruptcy Region
(b)
Figure 8:
with respect to c
1
; c
21
and c
22
subject to
p
1
c
1
+ p
21
c
21
+ p
22
c
22
= I: (36)
Because W (c
1
; c
2
) is additively separable, it is clear that conditional on a …xed c
1
one can
optimize
21
W
2
(c
21
) +
22
W
2
(c
22
) for c
21
and c
22
independent of W
1
(c
1
) resulting in the
…rst order condition for the conditional optimization
W
0
2
(c
21
)
W
0
2
(c
22
)
= k; (37)
which is identical to that obtained in the single period case. Given the conditionally
optimal c
21
and c
22
demands, (35)-(36) can then be solved for optimal c
1
: It should be
stressed that if we go from the single pe riod W (c
2
) in Section 2 to the current two period
W (c
1
; c
2
) = W
1
(c
1
) + W
2
(c
2
) then all of the Lemm as, Theorems and Corollaries in Section
2 continue to hold except that the condition for c
1
> 0 must be added to the no bankruptcy
restriction. The reason is as follows. The comparative statics results in Section 2 are based
on a comparison of @c
21
=@I and @c
22
=@I, which can be obtained from di¤erentiation of the
…rst order condition and the budget constraint. As argued above, the …rst order condition
remains the same in the two period setting. Although the budget constraint changes, this
change does not a¤ect our comparison results.
Although we can …nd the Gi¤en good behavior for the risk free asset in the two period
setting when choose the appropriate parameters, it is important to note that the above
argument does not imply that the risk free asset is a Gi¤en good in the two period case
17
whenever it is in the one period case. To see this, note that the price of the risk free asset
a¤ects demand both directly and indirectly
dn
f
dp
f
=
@n
f
(p
f
; c
1
)
@p
f
+
@n
f
(p
f
; c
1
)
@c
1
@c
1
@p
f
: (38)
The …rst term on the right hand side is the direct e¤ect of p
f
on n
f
through the conditional
portfolio optimization while the second term is the impact through optimal period one con-
sumption. Suppose that for the income after period one consumption, I p
1
c
1
; the risk free
asset is a Gi¤en good, then
@n
f
(p
f
;c
1
)
@p
f
> 0: This Gi¤en behavior is reinforced (diminished)
depending on whether the second term is positive (negative). It is straightforward to show
that
@n
f
(p
f
;c
1
)
@c
1
has the same sign as
@n
f
(p
f
;c
1
)
@I
which is positive if the risk free asset is a
Gi¤en good. But since
@c
1
@p
f
can be positive or negative, the sign of
dn
f
dp
f
is uncertain.
5 Concluding Comments
In this paper, Arrow’s seminal single period results on the relation between asset demand,
risk aversion and income are extended by dropping his restrictive assumption that the risk
free asset is held long. When shorting is allowed, even for well-behaved utility functions
satisfying Arrow’s assumptions that
0
A
< 0 and
0
R
> 0, the risk free asset can not only
become an inferior good but also a Gi¤en good. The sign of
0
R
plays a critical role in
determining whether inferior and Gi¤en behavior occurs when the risk free asset is held
long or short. We investigate when Gi¤en good behavior occurs and the relation between
its occurrence in one and two period settings. In addition to providing general results, we
illustrate them with numerous examples based on HARA and non-HARA utility functions.
Appendix
A Proof of Lemma 1
From the de…nition of n
f
, we have
n
f
> 0 ,
c
22
c
21
>
22
21
: (39)
Using the …rst order condition, we will obtain
n
f
> 0 , W
0
(c
21
) < kW
0
(
22
21
c
21
) (40)
for any c
21
> 0. Since a similar argument can be applied to the other cases, we can conclude
n
f
T 0 , W
0
(c
2
) S kW
0
(
22
21
c
2
) for any c
2
> 0: (41)
18
B Proof of Theorem 1
Di¤erentiating the FOC with respect to the income I yields W
00
(c
21
)
@c
21
@I
= kW
00
(c
22
)
@c
22
@I
:
Di¤erentiating the budget constraint with respect to the income I, we obtain p
21
@c
21
@I
+
p
22
@c
22
@I
= 1: Combining the two equations above yields
@c
21
@I
=
1
p
21
+ p
22
A
(c
21
)
A
(c
22
)
and
@c
22
@I
=
A
(c
21
)
A
(c
22
)
p
21
+ p
22
A
(c
21
)
A
(c
22
)
; (42)
where we have used
W
00
(c
21
)
kW
00
(c
22
)
=
A
(c
21
)
A
(c
22
)
: Noticing that
@n
@I
=
@c
21
@I
@c
22
@I
21
22
=
p
21
+ p
22
A
(c
21
)
A
(c
22
)
1
(
21
22
)
f
1
A
(c
21
)
A
(c
22
)
(43)
and c
21
> c
22
, we have
0
A
T 0 ,
@n
@I
S 0: (44)
Also notice that
@n
f
@I
=
21
@c
22
@I
22
@c
21
@I
(
21
22
)
f
=
p
21
+ p
22
A
(c
21
)
A
(c
22
)
1
(
21
22
)
f
21
22
A
(c
22
)
A
(c
21
)
: (45)
Therefore, we have
21
A
(c
21
) T
22
A
(c
22
) ,
@n
f
@I
T 0: (46)
Finally, since
n
f
I
=
1
p
n
n
f
+ p
f
; (47)
we have
@(n
f
=I)
@I
T 0 ,
@(n
f
=n)
@I
T 0 , n
@n
f
@I
n
f
@n
@I
T 0: (48)
Noticing that
n
@n
f
@I
n
f
@n
@I
=
p
21
+ p
22
A
(c
21
)
A
(c
22
)
1
c
21
(
21
22
)
f
1
R
(c
22
)
R
(c
21
)
(49)
and c
21
> c
22
> 0, we have
0
R
T 0 ,
@s
@I
T 0 ,
@(n
f
=I)
@I
T 0: (50)
19
C Proof of Theorem 2
If
0
R
> 0, it follows from Theorem 1(iii) that
@(n
f
=I)
@I
> 0 ,
@n
f
@I
>
n
f
I
: (51)
If n
f
> 0, then the risk free asset is a normal good. Therefore, the risk free asset can
become an inferior good only when n
f
< 0. The other cases can be proved similarly.
D Proof of Proposition 1
It can be veri…ed that the optimal demand for the risk free asset is given by
n
f
= (p
f
) + (p
f
)I; (52)
where
(p
f
) =
a
f
ap
f
f
(p
f
) and (p
f
) =
21
k
1
1+
22
p
21
+ p
22
k
1
1+
(
21
22
)
f
: (53)
Since
0
(p
f
) =
a (p
f
)
f
+
ap
f
0
(p
f
)
f
; (54)
It follows from eqn. (30) that
I
G
=
0
(p
f
)
0
(p
f
)
=
a (p
f
)
f
0
(p
f
)
+
ap
f
f
: (55)
And we can also calculate that
@I
G
@p
f
=
a
0
(p
f
)
f
0
(p
f
)
+
a (p
f
)
f
@
1
0
(
p
f
)
@p
f
+
a
f
=
2a
f
+
a (p
f
)
f
@
1
0
(
p
f
)
@p
f
: (56)
It follows from eqn. (15) that when =
critical
,
21
k
1
1+
22
= 0 ) (p
f
) = 0: (57)
Therefore, we have
I
G
=
ap
f
f
= I
min
and
@I
G
@p
f
=
2a
f
>
a
f
=
@I
min
@p
f
: (58)
Denoting the critical p
f
corresponding to =
critical
as p
c
f
, due to the continuity of I
G
and
I
min
, we can conclude that there exists an > 0 such that if p
f
2
p
c
f
; p
c
f
+
, we have
I
G
> I
min
; implying that the risk free asset is a Gi¤en good.
20
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21
