Global COE Hi-Stat Discussion Paper Series 040

Research Unit for Statisticaland Empirical Analysis in Social Sciences (Hi-Stat)

Risks after Disasters:
A Note on the Effects ofPrecautionary Saving on Equity Premiums

Hi-Stat Discussion Paper

Shiba Suzuki

March 2009

Hi-Stat
Institute of Economic Research
Hitotsubashi University
2-1 Naka, Kunitatchi Tokyo, 186-8601 Japan
http://gcoe.ier.hit-u.ac.jp

Risks after disasters: A note on the effects ofprecautionary saving on equity premiums ∗Shiba Suzuki
Graduate School of Economics
Hitotsubashi University
February 25, 2009

Abstract
This paper studies the effects on equity premiums of “risks after disasters”, which are defined as a sharp rise in volatility of real per capita
GDP growth rates immediately following disasters. This paper makes threecontributions. First, we analytically demonstrate that if and only if thedegree of relative prudence is higher than 2, risks after disasters decrease
equity premiums. Second, we find that the differences between equity premiums with and without risks after disasters are quantitatively significant.
Third, equity premiums are still higher in the case of disaster than without
a disaster.

∗

The author would like to thank Makoto Saito, Shinsuke Ikeda, Yuichi Fukuta, Tokuo
Iwaisako, Tetsushi Murao, and participants at the Japanese Economic Association Annual Spring
Meeting, Osaka, 2007; the Asian FA-NFA 2008 International Conference, Kanagawa, 2008 andthe seminars at Hitotsubashi and Osaka Universities for helpful and encouraging comments. The
research has benefited from the financial supports of a grant-in-aid (the Global Center of Excellence Project on “Research Unit for Statistical and Empirical Analysis in Social Sciences,”
Hitotsubashi University) from the Ministry of Education and Science, Japan. Of course, allerrors remain my responsibility.

11

Introdctuion

Many researchers pay serious attention to the potential impacts of disasters onequity premiums. Disasters are defined as events such as wars, severe depressions
and natural disasters, which are infrequent but significantly reduce real per capita
GDP growth rates. Rietz (1988) and Barro (2006) argued that representative agent
models generate large equity premiums, if disasters are independently and identically distributed (hereafter, i.i.d.). On the other hand, Gourio (2008) examined
the fact that historically observed disasters tended to be followed by sharp rises
in real per capita GDP growth rates (hereafter, “recoveries”). Then he demonstrated that disasters with possible recoveries generated small equity premiums,
if the degree of intertemporal elasticity of substitutions (hereafter, IES) was high.
However, as discussed in a later section, historically observed disasters tended to
be followed by persistent declines in real per capita GDP growth rates as well as
by recoveries. In other words, disasters tended to be followed by sharp rises involatility of real per capita GDP growth rates (hereafter, “risks after disasters”).
This paper studies the effects of such risks after disasters on equity premiums.
Then it demonstrates that precautionary saving, rather than IES, plays an important role in determining the equity premiums. In particular, using a three-period
Lucas tree model, we analytically demonstrate that risks after disasters decrease
equity premiums if and only if the degree of relative prudence (hereafter, RP)1 is
higher than 2. This result suggests that ignoring risks after disasters causes computed equity premiums to be too large. In fact, introducing risks after disasters
into Barro’s disaster model, we find that the differences between equity premiums with and without risks after disasters are quantitatively significant. However,
we also find that the equity premiums are still higher than in the case where no
disaster occurs.
This paper is organized as follows. Section 2 constructs a three-period Lucas
tree economy and analyzes the effects of risks after disasters on equity premiums.
Section 3 introduces risks after disasters into Barro’s disaster model and explores
quantitative effects of precautionary saving on equity premiums. Section 4 offersa conclusion.

1

RP is a measure of precautionary saving proposed by Kimball (1990).

2

2

A three-period economy with risks after disasters

In order to analyze the effects of risks after disasters on equity premiums, weconstruct a three-period Lucas tree economy. In this economy, there are two
assets. One is an equity share in a Lucas tree, which produces a single perishable
consumption good as a dividend, and the other is a risk-free asset. In period 2, thenormal state (n) occurs with a probability of 1−p and the disaster state (d) occurs
with a probability of p. Once the normal state is realized in period 2, there is no
uncertainty in period 3. On the other hand, if the disaster state is realized in period
2, the dividend in period 3 is uncertain. The parameters y2n and y3n represent thedividends of the Lucas tree in periods 2 and 3, when the normal state is realized
in period 2, whereas y2d and ỹ3d represent the dividends in periods 2 and 3, when
the disaster state is realized in period 2. We assume that y2n > y2d . ỹ3d is a randomvariable, where ȳ ≡ E[ỹ3d ] and δ 2 ≡ E[(ỹ3d − ȳ)2 ]. The parameter δ represents risks
after disasters2 . E[·] denotes the mathematical expectation operator conditional
on information available in period 1. The parameter y1 represents the dividend of
the Lucas tree in period 1. We assume that there are no idiosyncratic risks.
The parameters x1 and a1 represent agents’ holdings of the equity share andthe risk-free asset in period 1. P1 and Q1 represent the prices of the equity andthe risk-free asset in period 1. xst and ast represent agents’ holdings of the equity
share and the risk-free assets in period t = 2, 3 and state s = n, d, and P2s andQs2 represent the prices of the equity and the risk-free asset in period 2 and states = n, d.
The representative agent maximizes his/her expected utility subject to budget
constraints:nonou(c1 ) + (1 − p) u(cn2 ) + u(cn3 ) + p u(cd2 ) + E[u(c̃d3 )]The parameter c1 represents consumption in period 1 and cst represents consumption in period t = 2, 3 and state s = n, d. We assume that the periodic utilityu(·) has positive first and negative second derivatives; that is, u0 > 0 and u00 < 0.
For simplicity, we ignore the subjective time preference. The budget constraints
in periods 1, 2, and 3 are as follows: c1 + P1 x2 + Q1 a2 = (P1 + y1 )x1 + a1 ,2

While we assume that the level of the dividend in period 3 after the disaster state is uncertain,ỹ d
we can interpret that the growth rate is stochastic because the dividend growth is defined as y3d .
2

3cs2 + P2s xs3 + Qs2 as3 = (P2s + y2s )x2 + a2 for s = n, d, and cn3 = y3n xn3 + an3 andc̃d3 = ỹ3d xd3 + ad3 . Then first-order conditions determine the asset prices, as follows:
h u0 (c̃d ) i
0 n
0 d
0 n
n
n u (c2 )dd u (c2 )nn u (c3 )dP1 = (1 − p)(P2 + y2 ) 0+ p(P2 + y2 ) 0, P2 = y3 0 n , P2 = E ỹ3d 0 d3 ,u (c1 )u (c1 )u (c2 )u (c2 )
h
0 d
0 n
0 d i
0 n
u (c )u (c )u (c̃ )u (c )
Q1 = (1 − p) 0 2 + p 0 2 , Qn2 = 0 3n , and Qd2 = E 0 3d .
u (c1 )u (c1 )u (c2 )u (c2 )
Hereafter, scaling u0 (·) so that, without loss of generality, u0 (y1 ) = 1.
Market clearing conditions are x1 = xst = 1, and a1 = ast = 0 for t = 2, 3 ands = n, d. Then, consumption always equals dividend: c1 = y1 , cs2 = y2s , for s = n, d,cn3 = y3n , and c̃d3 = ỹ3d . Expected equity returns and risk-free rates in period 1 are:(1−p)(P2n +y2n )+p(P2d +y2d )and Rf = Q11 . We define the expected equity premium
P1Re
as Π ≡ Rf. Then equilibrium equity premiums are defined as follows3 :

Re =

Π≡

A + αP̂
B + pP̂,(1)£
u0 (y d ) ¤where, A ≡ (1−p)2 +(1−p)p u0 (yn2 ) y3n u0 (y3n )+(1−p)2 y2n u0 (y2n )+(1−p)py2n u0 (y2d )+2u0 (y n )(1 − p)py2d u0 (y2n ) + p2 y2d u0 (y2d ), α ≡ (1 − p)p u0 (y2d ) + p2 , B ≡ (1 − p)y3n u0 (y3n ) + (1 −2p)y2n u0 (y2n ) + py2d u0 (y2d ), and P̂ ≡ E[ỹ3d u0 (ỹ3d )]. P̂ is the equity price in the disasterstate, P2d , multiplied by marginal utility in the disaster state, u0 (y2d ). However, we
hereafter refer to P̂ as the equity price in the disaster state.
Below, we analyze the effects of an increase in the risks after disasters, δ, on
the equity price, P̂ , and the equity premium, Π.
000

(ȳ)
Lemma 1 When the degree of relative prudence, −ȳ uu00 (ȳ)
is higher (lower) than2, P̂ is an increasing (decreasing) function of δ, for small δ.
Proof. Taking a Taylor series expansion of P̂ around ȳ gives4 :nȳ 000 o 2
00P̂ ' ȳu (ȳ) + u (ȳ) + u (ȳ) δ + o(ỹ 3 ).
20©
ª
Differencing P̂ with respect to δ yields ∂∂δP̂ = 2 u00 (ȳ) + ȳ2 u000 (ȳ) δ. Therefore:
000

(ȳ)
−ȳ uu00 (ȳ)> (<)2 implies u00 (ȳ) + ȳ2 u000 (ȳ) > (<)0, thus ∂∂δP̂ > (<)0. (Q.E.D.)

3
4

Derivations of equation (1) are described in the Appendix.
o(ỹ 3 ) refers to a higher-order term.

4Lemma 1 implies that there are two opposite effects of risks after disasters on
the equity price, P̂ . On the one hand, risks after disasters lower equity prices
because of risk aversion (u00 (ȳ)). On the other hand, risk after disasters raises
equity prices because of precautionary saving ( ȳ2 u000 (ȳ)). Thus, u00 (ȳ) + ȳ2 u000 (ȳ)
determines the total effects of risks after disasters on equity prices, P̂ . In fact, if
000

(ȳ)
RP, −ȳ uu00 (ȳ), proposed by Kimball (1990) is higher than 2, u00 (ȳ) + ȳ2 u000 (ȳ) > 0

holds.
Lemma 2 Π is a decreasing function of P̂ .
Proof. Differencing Π with respect to P̂ yields:
∂Π
∂ P̂

=

αB − pA(B + pP̂ )2.

(2)

Expanding the sign condition, αB − pA, yields5 :hn´oi³ u0 (y n )u0 (y2n )
3
n
0 n
0 d
n
αB−pA = p(1−p) (1−p) 0 d −(1−2p) u (y2 )−pu (y2 ) y3 0 n +y2 . (3)u (y2 )u (y2 )u0 (y n )Thus, u0 (y2d ) < 1 implies that the term in the bracket is negative. Now, we assume2

u0 (y n )that y2n > y2d , u0 (y2d ) < 1 always holds because of concavity of utility functions.
2<0.
(Q.E.D.)Thus, ∂∂ΠP̂

Lemma 2 argues that a rise in the equity prices in the disaster state implies
that an equity share becomes less risky and lowers the equity premiums. This
is intuitive because increases in P̂ mitigate capital losses on the occurrence ofdisasters, when consumption is low and marginal utility is high.
We thus have a key proposition of this paper.
000

(ȳ)
Proposition 1 : When the degree of relative prudence, −ȳ uu00 (ȳ), is higher (lower)

than 2, Π is a decreasing (increasing) function of δ, for small δ.
∂ P̂
= ∂∂Π
< (>)0. (Q.E.D.)
Proof. From Lemmas 1 and 2, ∂Π
∂δ
P̂ ∂δ

Proposition 1 suggests that precautionary saving plays an important role indetermining equity premiums in the context of risks after disasters. In particular,
proposition 1 suggests that when RP is higher than 2, ignoring risks after disasters
5

Derivations of equation (3) are described in the Appendix.

5

makes calibrators overestimate the equity premiums. If RP is lower than 2, theopposite occurs.
Proposition 1 holds in the case of possible recoveries, as discussed in Gourio(2008), because Proposition 1 is independent of the magnitude relationship between ȳ and y2d . In addition, we can easily show that Proposition 1 holds in thecase of government default as discussed in Barro (2006)6 .
Finally, we offer some examples of well-known utility functions.
Example 1 The quadratic utility implies that third derivatives are zero; that is,there is no precautionary saving. Thus, risks after disasters always increase equity
premiums.
Example 2 The HARA (Hyperbolic³´ Absolute Risk Aversion) utility is usuallyη

λcrepresented as u(c) = 1−η+ χλwith λ > 0 and χ > 0. Then, RP ≡
η
1−η
³
´−1
´−1
³
χ
χ
λc
λc
+
and
RRA
=
cλ
+
. Thus, RP > 2 implies thatcλ 2−η
1−η 1−ηλ1−ηλRRA > 2 1−ηin the case of η < 1.
2−η
Example 3 In the case of CRRA (Constant Relative Risk Aversion) utility, RP =2 implies that the degree of relative risk aversion (hereafter, RRA) equals 1, whichis the case of the log utility. That is, when a calibrator uses the CRRA utility
with a moderate degree of RRA, ignoring the risks after disasters caused computed
equity premiums to be too large.
However, Proposition 1 in this section is based on Taylor expansion arguments
for a small δ. In section 3, then, we conduct calibration exercises and evaluate
quantitatively the effects of empirically plausible degree of risks after disasters onequity premiums.

3

Calibration: Barro model with risks after disasters

To explore the quantitative effects of precautionary saving on equity premiums, we
introduce risks after disasters into Barro’s disaster model. Barro’s disaster modelis the infinitely lived representative agent model with CRRA utility.
6

Barro (2006) computed equity premiums using equity returns and defaultable government
bond returns instead of risk-free rates.

6

To conduct calibration exercises, we begin by characterizing empirically plausible degrees of risks after disasters from cross-country evidence of real per capita
GDP growth rates in disasters presented by Barro (2006, Table I, p.828-829).
Duration of disasters is distributed between one and eight years with a mode ofthree years. Then, we define three years from the beginning of the disaster as adisaster period and the succeeding three years as an aftermath period. Figure 1
shows a histogram of real per capita GDP growth rates during the disaster andthe aftermath period, where the growth rates in the disaster period are lower than
-10%7 .
From Figure 1, we find that growth rates in the disaster period are distributed
between -50% and -10% with a mode of -20%. On the other hand, growth rates in
the aftermath period are distributed between -40% and 50% with a mode of 10%.
Thus, we find that disasters tended to be followed by persistent declines in real
per capita GDP growth as well as recoveries. In other words, disasters tended to
be followed by a sharp rise in risks after disasters. In fact, the average growth ratein the aftermath period is 10.06% and the standard deviation is 21.73%.
Figure 1. Cross-countries distribution of real per-capita GDP growth20

disasteraftermath

15

10

5

0
-50

-40

-30

-20

-10

0

10

20

30

40

50

growth rates; %We introduce risks after disasters into Barro’s disaster model. Consumption
growth follows:∆ log ct = µ + (σ + δst )²t , with probability 1 − p= µ + σ²t + log(1 − b), with probability p
where the parameter ²t represents i.i.d. standard normal random variables with a
distribution of N(0,1) and b represents the size of disasters. If the previous period
7

From Maddison’s (2003) data, we limit our study to 46 events in 27 countries.

7

is a disaster, st = 1, otherwise, st = 0. That is, risks after disasters are σ + δ whilerisks in normal times are σ. If δ = 0, the model is the same as that of Barro.
Because it is difficult to obtain an analytical solution in the case of δ 6= 0, we mustuse a numerical technique to compute equity premiums.
Following Barro, we specify the calibration parameters as presented in Table
I. In our setup, since the duration of a disaster is stochastic, we do not use thehistorical distribution of b but the single value b = 43.15% in order to replicate
Barro’s equity premium if there are no risks after disasters. From Figure 1, since
standard deviation of consumption growth in the aftermath period is 21.73%, wecompute equity premiums when δ is between 3% and 23% with an increment of
5%. Table I reports results from the default considered equity premiums.
Table I. Equity-risk premiums (the case of bond default) (%)
no disasters
disasters
Total risks after disasters: δ + σ
2
5
10
15
RRA: γ = 4
0.16
3.53 3.48 3.18 2.95
RRA: γ = 1
0.04
0.37 0.37 0.39 0.41
subjective time preference ρ = 0.03, average growth µ = 2.5%,
standard deviation of consumption growth in normal times σ = 2%,
the probability of occurrence of disasters p = 1.7%,
and the default probability of government bonds in disaster q = 40%.

20
2.43
0.45

25
1.67
0.49

When RRA = 4, the equity premiums equal 0.16% in the case where no disaster occurs, 5.71% in the case of disaster, and 3.53% in the case of disaster withbond default. Risks after disasters lower equity premiums because of strong precautionary saving. In particular, the default considered equity premium is 2.43%
when σ + δ = 20% and 1.67% when σ + δ = 25%. These values indicate that thedifferences in the equity premiums with and without risks after disasters are quantitatively significant. In addition, we find that the larger risks after disasters imply
smaller equity premiums. Thus, these calibration exercises demonstrate that wemust pay serious attention to potential impacts of risks after disasters. However,
we also find that equity premiums are much higher in the case of disasters thanthe case where no disasters occur. Table I also reports the results in the case of log
utility, where RP equals 2. In this case, while risks after disasters always increase
equity premiums, magnitudes are very low.

8

4Conclusion

This paper studies the effects of risks after disasters on equity premiums. On
the one hand, we demonstrate that ignoring risks after disasters causes computed
equity premiums to be too large, if and only if RP is higher than 2. In addition,
we find that the differences in equity premiums with and without risks after disasters are quantitatively significant. In fact, many calibration exercises in asset
pricing literature tended to employ CRRA utility with RRA higher than 2, andsome empirical researchers found that RP is higher than 28 . Thus, we must pay
serious attention to potential impacts of risks after disasters. On the other hand,the equity premiums are much higher than in the case where no disaster occurs.
Thus, these results are quantitatively instructive because disaster models proposedby Rietz and Barro are potentially important. Therefore, future research should
investigate whether historically observed disasters were empirically or statisticallyconsistent with historically observed equity premiums and other asset prices.
References
Barro, R. J. (2006), “Rare Disasters and Asset Markets in The Twentieth Century,” Quarterly Journal of Economics 121(3), 823-866.
Dynan, K. E. (1993) “How prudent are consumers?”

Journal of Political

Economiy, 101(6), 1104-1113.
Gourio, F. (2008), “Disasters and Recoveries: A Note on the Barro-Rietz Explanation of the equity-risk premium Puzzle,” American Economic Review-Papers
and Proceedings, 98(2), 68-73.
Hori, M. and S. Shimiztani (2006) “Did Japanese consumers become more prudent during 1998-1999? Evidence from householde-level data,” International Economic Journal, 20(2), 197-209.
Kimball, M. (1990), “Precautionary Saving in the Small and in the Large,” Econometrica, 58(1), 53-73.
8

For example, Merrigan and Normadin (1996) showed that the estimated RP took a value ofaround 2 in U.K. Hori and Shimizutani (2006) estimated that RP is around 4 in Japan. On theother hand, Dynan (1993) found that the estimated RP is very low in U.S.

9

Maddison, A. (2003), The World Economy: Historical Statistics, (Paris: OECD).
Merrigan, P. and M. Normadin (1996) “Precautionary saving motives: an assessment from UK time series of cross-sections,” Economic Journal,, 106(438),
1193-1208.
Rietz, T. A. (1988), “The Equity Risk Premium: A Solution,” Journal of Monetary Economics 22, 117-131.
Appendix
Derivation of the equation (1) We describe the derivation of the equation(1) in detail. Using the definition of asset prices and the first order conditions, theequity premium is written as:ReRf(1 − p)(P2n + y2n ) + p(P2n + y2n )
Q1
=
P1

Π ≡

³
=

n 0 n
o
n h 0 d io´³´u (y )u (ỹ )
u0 (cd )u0 (cn )(1 − p) y3n u0 (y3n ) + y2n + p E ỹ3d u0 (y3d ) + y2d(1 − p) u0 (c21 ) + p u0 (c21 )2n
o 0 n 2 n h 0 d i
o 0 d
0 n
u (ỹ )u (y )n u (y3 )n u (y2 )(1 − p) y3 u0 (yn ) + y2 u0 (y1 ) + p E ỹ3d u0 (y3d ) + y2d u0 (y21 )2

2

As discussed above, we assume that u0 (y1 ) = 1. Then, the denominator is writtenas:n u0 (y n )oon h u0 (ỹ d ) i
3
3
n
d
n
0 n
d
(1 − p) y3 0 n + y2 u (y2 ) + p E ỹ3 0 d + y2 u0 (y2d )u (y2 )u (y2 )hi
= (1 − p)y3n u0 (y3n ) + (1 − p)y2n u0 (y2n ) + pE ỹ3d u0 (ỹ3d ) + py2d u0 (y2d )Thus, the denominator is written as B + pP̂ , whereB ≡ (1 − p)y3n u0 (y3n ) + (1 − p)y2n u0 (y2n ) + py2d u0 (y2d )P̂ ≡ E[ỹ3d u0 (ỹ3d )].

10

On the other hand, the numerator is written as:³´
n u0 (y n )on h u0 (ỹ d ) i
o´³
3
3
n
0 d
n
d
d
0 n
(1 − p) y3 0 n + y2 + p E ỹ3 0 d + y2
(1 − p)u (y2 ) + pu (y2 )u (y2 )u (y2 )on h u0 (ỹ d ) i
o
n
3
d
n 0 n
n 0 n
d 0 n
0 n
= (1 − p) y3 u (y3 ) + y2 u (y2 ) + p(1 − p) E ỹ3 0 d u (y2 ) + y2 u (y2 )u (y2 )
n u0 (y d )
o
n £
o
¤
+(1 − p)p 0 2n y3n u0 (y3n ) + y2n u0 (cd2 ) + p2 E ỹ3d u0 (ỹ3d ) + y2d u0 (y2d )u (y2 )2=nu0 (y d ) o(1 − p)2 + (1 − p)p 0 2n y3n u0 (y3n ) + (1 − p)2 y2n u0 (y2n ) + p(1 − p)y2d u0 (y2n )u (y2 )inu0 (y n ) o h+(1 − p)py2n u0 (y2d ) + p2 y2d u0 (y2d ) + p2 + p(1 − p) 0 2d E ỹ3d u0 (ỹ3d )u (y2 )

Thus, the numerator is written as A + αP̂ , wherenu0 (y d ) o(1 − p)2 + (1 − p)p 0 2n y3n u0 (y3n ) + (1 − p)2 y2n u0 (y2n )u (y2 )
n 0 d+(1 − p)py2 u (y2 ) + (1 − p)py2d u0 (y2n ) + p2 y2d u0 (y2d )u0 (y n )α ≡ (1 − p)p 0 2d + p2 .
u (y2 )A ≡P̂As a result, we derive equation (1), that is, Π ≡ A+α.
B+pP̂

Derivation of the equation (3) We describe the derivation of the equation(3) in detail.
αB − pAononu0 (y2n )
d 0 d
n 0 n
2
n 0 n
=
(1 − p)p 0 d + p(1 − p)y3 u (y3 ) + (1 − p)y2 u (y2 ) + py2 u (y2 )u (y2 )hn
u0 (y d ) o− p (1 − p)2 + (1 − p)p 0 2n y3n u0 (y3n ) + (1 − p)2 y2n u0 (y2n )u (y2 )i+(1 − p)py2n u0 (y2d ) + (1 − p)py2d u0 (y2n ) + p2 y2d u0 (y2d )

11

=

n
u0 (y n )u0 (y d ) o(1 − p)2 p 0 2d + p2 (1 − p) − p(1 − p)2 − (1 − p)p2 0 2n y3n u0 (y3n )u (y2 )u (y2 )no
0 nu (y )+ (1 − p)2 p 0 2d + p2 (1 − p) − p(1 − p)2 y2n u0 (y2n ) − (1 − p)p2 y2n u0 (y2d )u (y2 )
no u0 (y n )u0 (y n )= (1 − p)p (1 − p) 0 2d − (1 − 2p)u0 (y2n ) − pu0 (y2d ) y3n 0 3d
u (y2 )u (y2 )on
0 n
u (y )+(1 − p)p (1 − p) 0 2d − (1 − 2p)u0 (y2n ) − pu0 (y2d ) y2nu (y2 )

Thus, the sign condition is written as equation (3),hnoi³ u0 (y n )´
u0 (y2n )
3
0 n
0 d
n
n
αB − pA = p(1 − p) (1 − p) 0 d − (1 − 2p) u (y2 ) − pu (y2 ) y3 0 n + y2 .
u (y2 )u (y2 )
When the term in the bracket is negative,onu0 (y2n )(1 − p) 0 d − (1 − 2p) u0 (y2n ) − pu0 (y2d ) < 0
u (y2 )
³ u0 (y n ) ´2
u0 (y n )− (1 − 2p) 0 2d − p < 0⇔ (1 − p) 0 2d
u (y2 )u (y2 )u0 (y n )Therefore, if u0 (y2d ) < 1, the term in bracket is negative, which implies that the2sign condition is negative.

12