1
Overview
Let’s …rst look at standard preferences –additive expected utility and de…ne
some notation. Time is discrete, t = 0; 1; :::T: At each t 0, an event zt is
drawn from a set t : A history, denoted z t is a collection of events up to an
including zt , i.e., e.g., fz0 ; z1 ; :::; zt g : The set of possible histories at t is t :
Let c (z t ) denote the vector of consumption goods (possibly including
service ‡ows and leisure) consumed in period t if z t is realized:Then, the
T
standard assumption is that for any t; preferences over fc (z t ) jz t 2 t gt=0
with associated probabilities p (z t ) can be represented by the function
U
t
t
c z jz 2
t T
t=0
=
T
X
t
t=0
X
zt 2
t
p z u c z
t
=E
t
T
X
t
u c zt
t=0
where u is a smooth increasing concave function, called the per-period utility
or felicity function. Several important features of these preferences will be
relaxed during the course.
1. Curvature of u determines elasticity of substitution both between states
within a period (inverse of risk aversion) and between periods, intertemporal elasticity of substitution. Therefore, these intrinsically di¤erent
concepts cannot be disentangled theoretically.
2. No preference for time revelation –it does not matter when we get to
now our faith (unless, of course, if we can condition our actions on the
information).
3. No history e¤ect; the utility
E
T
X
t
c zt
t=s
is independent of consumption taking place before s: (However, history of course matters by a¤ecting the budget, i.e., the consumption
possibility set, usually simply captured by a state variable like wealth.
4. Time consistency; if
U c1 ; c z t
then
U
c zt
T
t=2
T
t=2
U c1 ; c0 z t
U
c0 z t
T
T
t=2
T
t=2
;
T
i.e., if the sequence of consumption fc (z t )gt=2 is preferred over fc0 (z t )gt=2
at t = 1; it is preferred also at t = 2:
1
5. Risk matters only through second order e¤ects. E.g., suppose in period
t; there is a lottery with payo¤ x~ and E x~ > 0. Then all individuals
will prefer to hold a strictly positive amount of this lottery over having
a certain consumption level c. I.e., there is a k > 0 such that for all
k 2 0; k
E (u (c + k~
x)) > u (c) :
To see this, let’s take the derivative of the left hand side w.r.t. k and
evaluate at k = 0:
@ (E (u (c + k~
x)))
jk=0 = u0 (c) E x~
@k
which is positive if E x~ > 0; regardless of riskaversion.
Empirical …ndings, introspection and lab experiments have all shown that
these implications are often invalidated.
1.1
Two important macro-puzzles.
1. Equity premium puzzle
2. Too little insurance puzzle.
Most well-known example. Mehra-Prescott, "equity premium puzzle".
Consider the following example (almost) in there original formulation and
we will return to it later. Suppose consumption grows fast with p = 1=2
and slowly otherwise. Starting from a consumption level ct , history at t + 1
is either z t+1 = zh in which case ct+1 = ct (1 + g + ) or z t+1 = zl and
ct+1 = ct (1 + g
) : We can now using the standard Euler condition for
maximizing utility to price a bond that gives 1 unit of consumption in both
cases and a share that gives 1+g + or 1+g
. The share is thought to be a
claim to the process that provides the consumption possibilities. Individuals
eat only apples. Apples are grown on a tree (or a number of identical trees(
which has exogenous but stochastic output that grows with a rate g + or
g
. A share is ownership of the tree.
The Euler equation is
u0 (ct ) = Eu0 (ct+1 ) Rt+1 :
Well known intuition, give up one unit of consumption "costs" u0 (ct ) ;
invest it and increasing consumption by Rt+1 next period gives an increase
in utility given by Eu0 (ct+1 ) Rt+1 :
We do this by de…ning state-contingent prices. In period t there is one
lottery that gives 1 unit of consumption in if growth is high and zero otherwise. There is also a lottery that gives one unit of consumption in the low
2
growth state. The price of these lotteries, ph and pl are and the returns
and
p(zl )
pl
p(zh )
ph
respectively. Therefore,
u0 (c (zh )) p (zh )
u0 (ct )
u0 (c (zl )) p (zl )
u0 (ct )
ph =
pl =
Using the standard CRRA felicity function
u (c) =
1
c
1;
recalling that the coe¢ cient of RRA is
c
u00 (c)
=
u0 (c)
Setting p (zh ) = p (zl ) =
1
2
c
(
1) c
c
2
1
=1
:
we have
ph =
pl =
(1 + g + )
2
(1 + g
)
2
1
;
1
:
A portfolio consisting of one each of the lotteries mimics perfectly the
safe one-period bond. The price of a bond is thus
pb = ph + pl
with a return,
rb =
1
:
ph + pl
Let us now compute the return on a claim to next periods dividends –a
one-period share. A portfolio consisting of (1 + g + ) of the h lottery and
(1 + g
) of the l lottery exactly such a one period share.
The price of this risky portfolio is
pr = ph ((1 + g + )) + pl (1 + g
3
)
and its expected return is therefore
rr =
=
=
1+g
ph ((1 + g + )) + pl (1 + g
1+g
(1+g+ )
2
1
(1 + g + ) +
2 (1 + g)
((1 + g + ) + (1 + g
)
(1+g
)
1
2
(1 + g
)
) )
In this simple economy, the return on a normal share, i.e., a one that
gives rights to all future dividends is the same. Why? To see this, we recall
that the price of the share with CARA utility will be proportional to current
income/consumption. The price of the share will therefore be Pr ct . and the
return
(1 + Pr ) ct+1
Pr ct+1 + ct+1
=
:
Pr c t
Pr c t
From the Euler equation,
Pr =
E
u0 (ct+1 ) (1 + Pr ) ct+1
u0 (ct ) ct
1 + Pr
Pr
1
(ct (1 + g + )) + (ct (1+g2
= (1 + Pr )
ct 1 ct
(1 + g + ) + (1 + g
)
= (1 + Pr )
:
2
2
=
((1 + g + ) + (1 + g
) )
(ct (1+g+ ))
2
))
1
(ct (1 + g
Finally, calculate the expected return on the share:
1 + Pr ct+1
E
Pr
ct
1 + pr
=
(1 + g)
pr
2 (1 + g)
=
((1 + g + ) + (1 + g
) )
;
as with the one-period share.
Using (US) data g is around 1.8% per year and is around 3.6%. Stock
market returns have averaged around 8% per year and the risk-free rate
4
))
around 1% over the last 100 years or so. Therefore
(1 + 0:018 + 0:036) 1
2
(1 + 0:018 0:036) 1
=
2
1
=
ph + pl
1 + 0:018
=
ph (1 + 0:018 + 0:036) + pl (1 + 0:018
ph =
pl
rb
rr
Can we …nd
0.94
and
0.96
0:036)
to generate the observed values of ra and rb ?
0.98
1
1.02
1.04
1.06
1.08
1.1
0
-2
-4
-6
-8
-10
Combinations of
and
such that ra = 0:08 (red) and rb = 0:01 (black)
If, for example, we set = 0:98; riskaversion. 1
should be 3.5, to
motivate 8% stock return. But then the bond return should be 7.6%, leaving
a mere 0.4% risk premium. In fact, it is di¢ cult to get the right risk premium.
Let’s look closer at the risk premium. Let’s express it as the ratio of the
price of the bond to the ratio of the price of the risky asset
pb
ph + pl
=
pr
ph (1 + g + ) + pl (1 + g
(1 + g + ) 1 + (1 + g
=
(1 + g + ) + (1 + g
5
)
)
)
1
In reality, this ratio is
1
1:01
1+g
1:08
1:05
However, by plotting
"
(1 + g + ) 1 + (1 + g
(1 + g + ) + (1 + g
against RRA = 1
)
)
1
#
g=0:018; =0:036
;
1.015
1.01
1.005
1
0.995
0.99
0.985
0
20
40
R
60
80
100
we see that it is very di¢ cult to get the right risk premium. In fact, in the
realistic case where > g; it is easy to bound the risk premium,
"
#
"
#
1
1
1
(1 + g + )
+ (1 + g
)
(1 + g
)
= lim
lim
!
1
! 1
(1 + g + ) + (1 + g
)
(1 + g
)
g<
=
1
1+g
1:0183:
The …st line comes from the fact that (1 + g
) goes to in…nity as
approach 1; while (1 + g + ) approach zero. Of course, with lower
growth and higher risk, the risk premium can get larger, but we are stuck
with data.
2. Too little risk-sharing
In a complete markets equilibrium where individuals have homothetic
preferences, e.g., CRRA, there should be full risk sharing. Consumption
6
growth should be perfectly correlated between individuals and everyone should
hold a share in a global portfolio of assets. This is not the case, obviously frictions and asymmetric information may be one explanation. But sometimes
these explanations don’t seem to su¢ ce. An example is the home bias puzzle.
All around the world local investors hold unbalanced portfolios with to much
domestic assets. It is shown in the literature that expected returns could
increase a lot, without increasing risk by having more balanced portfolios,
containing more foreign assets. The explanation cannot be that information
is superior. Then, domestic holders should sometimes have more negative
information than foreign investors, in which case they should sell moving to
foreign ones, this we don’t see. Conversely, they should sometimes go short
abroad, having an investment share above unity at home, which we don’t see
either.
1.2
Lab puzzles
Ambiguity aversion –Ellsberg Paradox.
Consider following lottery. There are two urns, each with 100 balls. In
urn 1, there are 50 red and 50 black. In urn 2, there are only red and black
balls but the proportions are unknown. The subject is given a color and can
pick one ball. If a ball with the given color comes up, the gain is 50$, if
not the gain is zero. The subject is asked to rank lotteries. Typically the
following response comes up.
1. Red from urn 1
Black from urn 1.
2. Red from urn 2
Black from urn 2.
3. Red from urn 1
Red from urn 2.
4. Black from urn 1
Black from urn 2.
This contradicts expected utility since from 2, we expect subjective probability to imply that they believe p (red) = 1=2: Then, 3 and 4 should be
with indi¤erence.
Time inconsistency.
In lab experiments, preference reversal occurs. Example.
Suppose you can choose between 10CD’s 1 year from now or 11 CD’s 1
year and a week. Often the latter is preferred. However, after a year, 10
CD’s today is preferred over 11 CD’s in a week. This is inconsistent with
standard time-additive utility with geometric discounting.
7
Other examples, people sometimes seems to pay to commit. They tend to
over-consume during the year, and, for example, ask their employer to keep
money for tax-payments at then en of the year or, say for big holidays.
First-order risk-aversion
With smooth preferences, people should as we have seen not care much
about small gambles. Big ones, on the other hand, are detrimental. In fact,
with CRRA coe¢ cient bigger than unity, su¢ ciently big losses can never
be compensated since U (ct ) is bounded from above but not from below.
For example, consider a lottery that gives a relative loss of x, forcing a
consumption loss of xc with p = 21 and otherwise gives consumption (1 + k) c.
For di¤erent values of x; how large must k be to compensate for so that
1
U (c) = U ((1
2
1
x) c) + U ((1 + k)c)
2
Here, I plot this k as a function of x for
=
3; 4 and
6.
1.8
1.6
1.4
1.2
k1
0.8
0.6
0.4
0.2
0
0.02 0.04 0.06 0.08 0.1x 0.12 0.14 0.16 0.18 0.2
:To get people to behave like they do for small gambles, has to be so large
as to give unreasonable predictions for large gambles. In fact, sometimes no
upside can compensate for a su¢ ciently large but …nite downside. This fact
is due to that when CRRA coe¢ cient larger than 1, i.e., when < 0; utility
is bounded, since
c
< 0; 8c; < 0:
8
This means that we solving
c
1 ((1 x) c)
2
1
(1 x)
1 =
2
1
x = 1 2
=
gives the largest possible downside that could be compensated by any upside.
In the graph, we see the maximum loss occuring with 50% chance that could
be compensated by any gain as a function of the level of riskaversion.
0.8
0.6
0.4
0.2
2
4
6
8
10R
12
14
16
18
20
For = 11; x is as low as 6:1%: Would you refuse a 50/50 bet of loosing
25% of your lifetime income vs. getting the fortune of Bill Gates? If, not,
you cannot have absolute riskaversion above 3.4.
9
2
Non-additive recursive preferences
2.1
Aggregation over time
Now, disregard risk. In general, preferences can be described as a function
that associates a particular level of overall utility to any sequence of consumption levels
U (c1 ; c2 :; ; ; cT ) U fct gT0
MRS is de…ned
@U (c1 ;c2 :;;;cT )
@ct+1
@U (c1 ;c2 :;;;cT )
@ct
M RSt;t+1
We de…ne time preference as MRS along a path of constant consumption
c
(c)t;t+1 =
@U (c1 ;c2 :;;;cT )
@ct+1
j
@U (c1 ;c2 :;;;cT ) ct =c8t
@ct
noting that this may depend onf c: For the time additive utility with constant
discounting, however, we have
U=
T
X
t
u (ct )
t=s
with
(c)t;t+1 = 8c:
Koopman’s time aggregator
Assume preferences at all dates are represented by a time zero utility
function, so preferences are time consistent.
First notation,
fct ; ct+1 ; ct+2 ; ::::ct+1 g
tc
Utility at time zero is
U (0 c) = U (c0; 1 c)
Assume history independence, here marginal rate of substitution M RSt;t+1
does not depend on consumption prior to t (is this innocuous?) and if ct is
a vector, also the intra-temporal MRS between goods in t; is independent of
prior consumption. Then, but not otherwise, we can write
U (0 c) = V~ [c0 ; U1 (1 c)]
10
for an aggregator function V:and a function that gives the continuation utility
U1 (1 c) Choices over 1 c1 in particular, what maximizes U1 in some choice set,
does not depend on c0 : But the choice set can, of course, be a¤ected.
Now also assume future independence preferences over ct does not depend
on t+1 c: (Is this innocuous? Yes, clearly if c0 is a scalar, then more is just
better, but if c0 is a vector this is a restriction..One could prefer chicken
over …sh if one plans to eat a lot of …sh in the future. However, future
independence seems like a less strong assumption than history independence).
Now, we can write utility as
U (0 c) = V [u (c0 ) ; U1 (1 c)]
V aggregates utility coming from current consumption, and future consumption. It is not restricted to simply add them like standard preferences.
Finally, assume stationarity, then for all t,
U (t c) = V [u (ct ) ; U (t+1 c)] ;
and recursivity is implied
U (t c) = V [u (ct ) ; V [u (ct+1 ) ; U (t+2 c)]]
MRSt;t+1 is
@U (c1 ;c2 :;;;cT )
@ct+1
@U (c1 ;c2 :;;;cT )
@ct
=
V2 [u (ct ) ; U (t+1 c)] V1 [u (ct+1 ) ; U (t+2 c)] u0 (ct+1 )
V1 [u (ct ) ; U (t+1 c)] u0 (ct )
As we know, time preference is MRS evaluated at a constant consumption
path, where by stationarity, also u (ct ) and U (t c) is constant at u (c) and U (c)
(excuse the notation, I am here letting c denote a path of constant levels of
consumption). Then,
(u (c)) = V2 [u (c) ; U (c)]
which can depend on c unless V is a linear aggregator (standard).
The Uzawa simpli…cation is a particular example of the Koopmans aggregator.
U (t c) = u (ct ) + (u (ct )) U (t+1 c)
First order condition:
@u (ct )
(1 +
@ct;i
0
(u (ct )) U (t+1 c)) + (u (ct ))
11
@U (t+1 c)
=0
@ct;i
A bit more general, by not imposing future independence.
U (t c) = u (ct ) + (ct ) U (t+1 c)
Note that here, preference over elements in ct may depend on U (t+1 c),
which matters if ct is a vector.
First order condition:
@u (ct ) @ (ct )
@U (t+1 c)
+
U (t+1 c) + (ct )
=0
@ct;i
@ct;i
@ct;i
Examples:
Growth and …scal policy (Dolmas and Wynne (1998)). Using Usawa
max U (t c) = u (ct ) + (u (ct )) U (t+1 c)
s:t:ct = f (kt ) kt+1 gt
We can derive a Bellman equation:
J (k) = max u (f (kt )
kt+1
kt+1
gt ) + (u (f (kt )
kt+1
gt )) J (kt+1 ) :
Only non-standard is endogenous discounting.
FOC:
u0 (ct ) (1 +
0
(u (ct )) J (kt+1 )) =
(u (ct )) J 0 (kt+1 )
Envelope:
J 0 (kt ) = u0 (ct ) f 0 (kt ) + 0 (u (ct )) u0 (ct ) f 0 (kt ) J (kt+1 )
= u0 (ct ) f 0 (kt ) (1 + 0 (u (ct )) J (kt+1 )) :
Giving
J 0 (kt ) =
(u (ct )) J 0 (kt+1 ) f 0 (kt )
In a steady state
1=
(u (ct )) f 0 (kt )
Compare this to the standard case
1 = f 0 (kt )
being independent of …scal policy. In particular, an increase in g; must reduce
c one-for-one, since css = f (kss ) kss gt .
12
Now changes in g can a¤ect the steady state. To see this, consider
1=
(u (f (kss )
kss
g)) f 0 (kss )
An increase in g reduces u, suppose this makes people more patient, i.e.,
< 0: Then, the increase in g makes (u) f 0 > 1. This will lead to more
saving and a growing capital stock. Crowding out of consumption more than
one-for-one. In this case, since both 0 and f 00 are negative, there is a unique
steady state. See left panel of the …gure.
0
If, instead > 0; there may be multiple solutions to
0
1=
(u (f (kss )
g)) f 0 (kss ) :
kss
In this case, some steady states are unstable. In the right panel of the
…gure, the left steady state is unstable. A higher level of k increases u; and
more than the fall in f 0 : Therefore, > f 0 and individuals accumulates
capital. Here a reduction in g; could move the economy out of the unstable
equilibrium.
β (u ( f (k ss )− k ss − g '))
−1
β (u ( f (k ss )− k ss − g ))
−1
β (u ( f (k ss )− k ss − g '))
−1
f ' (k ss )
β (u ( f (k ss )− k ss − g ))
−1
f ' (k ss )
k ss
0
k ss
0
< 0; g increases.
> 0, g decreases.
Other examples is small open economies with a …xed interest rate, r The
steady state is
1 = (u (f (kss ) kss g)) r
With standard preferences no steady state exists generically. With
a unique one exists.
13
0
<0
2.2
Aggregation over states
As in the intro, consider states (within a period) to be z 2
associated probability measure p (z) : Utility is now
; with an
U fc (z)gz2
We can solve for the certainty equivalent consumption level
from
U fc (z)gz2 = U (f g)
of fc (z)gz2
Standard theory says
X
U (fc (z)g) =
p (z) u ((z))
z2
and since
u( ) =
X
p (z) u (c (z))
z2
we have
(fc (z)g) = u
X
1
!
p (z) u (c (z)) :
z2
For example: Suppose preferences are CRRA, then
u (c) =
1
c
So
1
=
X
1
p (z) c (z)
z2
=
X
1
p (z) c (z)
z2
!1
Note the linear homogeneity in this case. For a constant k > 0:
(k fc (z)g) = k (fc (z)g) :
Chew and Dekel generalizes this by allowing the certainty equivalent of
fc (z)g to be a more general function while maintaining …rst order conditions
that are linear in probabilities by implicitly de…ning
X
(fc (z)g) =
p (z) M [c (z) ; ] :
z2
14
As we see, this generalizes standard utility by implying that the marginal
value of consumption in state z depends on consumption in other states
through their e¤ect on : An example of this is that people might care more
(or less) about consumption in states that provide less consumption than the
certainty equivalence (disappointment aversion). Notice the relative comparison here. With concave utility, marginal utility in states with low consumption is high, but independent of consumption in other states. This is not
necessarily the case here since consumption in state z; relative to depends
on consumption in all other states since they a¤ect :
We assume that M ( ; ) = (Why?), M1 > 0; M11 < 0 (…rst order
stochastic dominance and riskaversion). Often we want to maintain the linear
homogeneity of preferences like in CRRA.
M (kc; k ) = kM (c; )
Examples:
To show that the Chew-Dekel generalizes and includes e.g., CRRA preferences: Note that if we set
M (c; ) =
1
c
(1)
+
we get
=
X
p (z)
1
c (z)
+
z2
0 =
X
z2
=
=
1
X
X
p (z) c (z)
z2
p (z) (c (z) )
z2
=
1
p (z) c (z)
X
p (z) c (z)
z2
!1
which is the CRRA certainty equivalence.
Examples:
"Weighted expected utility"
Let
c 1
c
M=
15
+
Compare to (1). Now, we have
=
X
p (z)
c (z)
c (z)
1
c (z)
c (z)
1
c (z)
+
z2
0 =
X
X
p (z)
z2
p (z) c (z)
=
z2
=
X
1
P
Here, we can interpret
c (z)
p (z) c (z) c (z)
z2
z2
p (z) c (z) c (z)
z2 p (z) c (z)
P
p (z) c (z)
z2 p (z) c (z)
P
p^ (z)
as a weighted probability. If < 0; these weights decrease in c; i.e., bad
outcomes are weighted higher than otherwise. Suppose we have two outcomes
c1 and c2 with equal probabilities 12 each. The weighted probabilities are then
p^ (z1 ) =
p^ (z2 ) =
1
c
2 1
1
c
2 1
+
1
c
2 1
1
c
2 2
1
c
2 1
+
1
c
2 1
=
c1
c1 + c1
=
c2
c1 + c1
Consequently,
=
c1
c2
c1 +
c
c1 + c1
c1 + c1 2
For any constant k; indi¤erence curves then satisfy
k=
c +
c1 +
+ 2
c1 + c2 c1 + c2
Plotting one together with standard utility we …nd these preferences produce
a higher level of riskaversion.
16
3
2.8
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
We can also have Faruk Gul’s disappointment aversion. Then
(
1
(1 + ) c
+ if c (z) <
M (c; ) =
1
c
+ else
This means that all outcomes worse than the certainty equivalence get
scaled up.
=
X
X
p (z) (1 + I [c (z) < ])
z2
p (z) (1 + I [c (z) < ]) =
z2
=
1
P
The weighted probability here is
X
p (z) (1 + I [c (z) < ]) c (z)
z2
z2
p (z) (1 + I [c (z) < ]) c (z)
z2 p (z) (1 + I [c (z) < ])
P
p (z) (1 + I [c (z) < ])
z2 p (z) ((1 + I [c (z) < ]))
P
c (z)
17
1
+
For any constant k: indi¤erence curves can be written as
8
1
1
< (1+ )12 c11 + 2 c21 1 if c1 < c2
(1+ ) 2 + 2
(1+ ) 2 + 2
k=
1
c
(1+
) 12 c2
1
2
:
if c1 > c2
1
1 +
(1+ ) +
(1+ ) 1 + 1
2
2
2
2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Here, it is important to notice the kink around unity. This is …rst order risk-aversion. Individuals will be more averse to small risks than under
standard preferences.
Risk premia under Chew-Dekel Preferences.
De…ne the risk-premium R from
R = Ec (z)
(fc (z)g)
Let us consider two state case above. In the two cases of CRRA E utility
and weighted utility. Suppose c = c1 = 1
with prob 21 and c = c2 = 1 +
with prob. 12 : The shock is thus and is the standard deviation of c:
18
1.8
2
Rs tan d
(1
) + (1 + )
= 1
2
1
Rs (0) + Rs0 (0) + Rs00 (0)
2
2
1
(1 ) +(1+ )
2
6@ 1
= 6
4
@
2
(1
@2 1
16
+ 6
24
1
(1
=
2
)
1
) +(1+ )
2
Rs ( )
2
3
7
7
5
=0
1
7
7
5
2
@
3
2
2
=0
In the weighted expected utility, we have
Rwheigt =
(1
1
)
(1
+
+ (1 + )
) + (1 + )
1
Rw (0) + Rw0 (0) + Rw0 (0)
2
2
+
(1 )
+(1+ ) +
(1 ) +(1+ )
6@ 1
= 6
4
@
2
16
+ 6
24
1
=
(1
2
@
2
(1
1
)
(1
@
2 )
2
+
!
Rw ( )
2
+(1+ )
) +(1+ )
2
1
+
1
+
3
7
7
5
=0
1
3
7
7
5
2
=0
In the disappointment aversion case, the …rst derivative is not going to
19
be zero, people are not locally riskneutral. Therefore,
Rdisapp =
(1 + ) (1
1
1
Rd (0) + Rd0 (0) + Rd00 (0)
2
2
(1+ )(1 ) +(1+ )
2+
6@ 1
= 6
4
@
2
=
16
+ 6
24
2+
@
2
1
(1+ )(1
@
+ 2 (1
1
) + (1 + )
2+
Rd ( )
2
1
) +(1+ )
2+
2
1+
)
(2 + )2
!
2
3
7
7
5
=0
1
3
7
7
5
2
=0
The key here is the linear term.
2.3
Time and Risk
Let us now combine the aggregation over time and over states so we can de…ne
preferences over both states and time, allowing for rates of substitution to
di¤er depending on whether we are talking about states or time periods.
Generalize the notation so that
tc
denotes a stochastic consumption stream starting from period t; i.e., ct ; c (zt+1 ) ; :::;
and denote
U (t c) Ut
The aggregator can be written
Ut = V (ct ; (Ut+1 ))
or by imposing future independence
Ut = V (u (ct ) ; (Ut+1 ))
(2)
Note that we are now aggregating the direct utility of ct and the certainty
equivalent of the stochastic continuation payo¤ from t + 1 and onwards.
20
Thus, since at t; Ut+1 is stochastic (depending on the realization of zt+1 ), we
construct the period t + 1 certainty equivalent of this stochastic utility, i.e.,
(Ut+1 ).
If we use a standard expected utility certainty equivalent in (2), we get
what is typically called Kreps-Porteus utility (some times also Epstein-Zin,
although they often are associated with more general speci…cation of state
aggregation, a la Chew-Dekel). The key here is that we can now separate
risk-aversion, which is determined by the properties of ; from intertemporal
substitution, determined by V:
Let us use the CRRA speci…cation for ; so
(U ) = [EU ]
1
and a constant elasticity aggregator
V (u (ct ) ; (Ut+1 )) = [(1
1
)u +
(U ) ] :
If we let u be linear, u = c; so we let all curvature be taken care of by
and V; we can now interpret 1
a s the degree of risk-aversion and 1 1 as
the elasticity of intertemporal substitution. To see this simply, look at a two
period example where consumption is stochastic in the second period. In the
second period, stochastic utility is
U2 = c (z2 )
Suppose …rst second period consumption is non-stochastic at c2 : Then,
1
(U2 ) = [EU ] = c2
Then,
U1 = [(1
) c1 + c 2 ]
1
MRS is now,
@U1 @U1
=
@c2 @c1
M RSc1 ;c2
1
=
=
1
1
((1
((1
) c1 + c 2 )
) c1 + c 2 )
c2
c1
21
1
1
1
1
1
(1
c2
1
) c1
1
as usual and
c1
c2
@M RSc1 ;c2
c1
c2
@
M RS
1
= IES
c1
c2
=1
:
Note that we have imposed that U is linearly homogeneous by having the
power 1 ; which is with no consequence. Consider
U~1 = (1
) c1 + c 2
and calculate
@ U~1 @ U~1
=
=
@c2 @c1
1
c2
(1
) c1
1
=
c2
c1
1
1
It should be clear from the aggregator
(c2 (z2 )) = [Ec (z2 ) ]
1
that we can think of 1
as measuring risk aversion.
A key characteristic of Kreps-Porteus preferences is that they give rise to
preference for when risk is revealed. A basic intuition is that early revelation
implies that some risk is converted into intertemporal substitution. To see
an example of this, consider the example in the introduction, where second
period consumption is high or low (z2 = zh or zl ): Suppose that this is reveled
in the …rst period. Denote U~1 (z) …rst period utility given a realized value of
z: Then,
U~1 (zl ) = [(1
U~1 (zh ) = [(1
) c1 + c l ]
) c1 + c h ]
1
1
and
U1early =
(U1 )
=
1
((1
2
=
1
((1
2
) c1 + c h )
1
1
+
((1
2
) c1 + c h )
Compare this to the late resolution case. Then,
(U2 ) =
22
) c1 + c h )
1
1
) c1 + ch ) + ((1
2
ch + cl
2
1
1
1
and
U1late =
ch + cl
2
) c1 +
(1
!1
Comparing these, we see
U1early U1late
2
1
1
((1
) c1 + ch ) + ((1
= 4
2
2
1
) c1 + c l )
(1
) c1 +
0.0004
0.0002
0.2
0.4
0.6
0.8
1
-0.0002
-0.0004
-0.0006
Here I plotted the di¤erence for = 12 and for < ; we see that early
resolution is preferred. Note that when < , risk aversion (1
) is larger
than the inverse of intertemporal elasticity of substitution, so in a sense, it is
less costly to substitute between time periods than between states of nature.
Is this important?
With Kreps-Porteus preferences we can clearly get low interest rates if
individuals are facing little risk and high if they are facing large. But can we
get what we want when we have a representative household that prices both
risky and non-risky assets while having to bear reasonable amounts of risk
himself.
An example. Look at the case in the introduction; given c1
c2 =
ch = c1 (1 + g + ) with prob
cl = c1 (1 + g
) with prob
23
5
c1 =1
0.0006
0
ch + cl
2
!1 3
1
2
1
2
First, we note that Arrow-Debreu prices now have the property that the
price of an asset that pays out in state z = zh ; depends on consumption also
in the other state. This is clear from the expression of utility:
U1 =
(1
!1
ch + cl
2
) c1 +
where we see that the marginal contribution to utility U1 of an asset that
pays out in one of the states, say in z = zh depends on consumption also in
the other state. The simple calculation of Arrow-Debreu securities is lost.
What is now the price of a safe bond pb and a risky asset (the apple tree)
pr that pays dividends (1 + g + ) or (1 + g
) per share depending on
which state occurs? The prices have to be such that if a consumer buys one
marginal unit of the assets, utility is unchanged. Take the safe bond, it has
to satisfy
0 =
pb (1
0 =
pb (1
2
6
pb = 4
=
2 (1
@U1
@U1
+ E
@c1
@c2
0
ch +cl
2
@c1
B@
)
+ @
@c1
@ch
)
ch +cl
2
ch
2 (1
) c1
1
+ cl
1
C
A
@cl
3
7
5
ch =c1 (1+g+ );cl =c1 (1+g
)
(1 + g + ) + (1 + g
2
With the risky asset we have
0
@c
B@
0 =
pr (1
) 1+ @
@c1
) pr =
1
@
+
1
ch +cl
2
2 (1
)
ch +cl
2
)
(1 + g + )
(1 + g + )
@
+
@ch
(1 + g + ) + (1 + g
2
24
ch +cl
2
1
+ (1 + g
(1 + g
@cl
)
)
)
)
1
1
1
)C
A
((1 + g + ) + (1 + g
Note that the ratio
pb
(1 + g + ) 1 + (1 + g
=
pr
(1 + g + ) + (1 + g
)
) )
is determined by risk aversion and exactly the same as in the standard case.
Therefore it seems as we have got little help from Kreps and Porteus. Not
entirely right. We have learnt that it is di¢ cult to get the risk premia by
assuming that stock market returns are fully determined by consumption
growth. Maybe this is not the way to go.
Suppose there is some other more stable income so that the share dividend is more volatile than consumption. Say that the standard deviation of
consumption remains at while the standard deviation of dividends is x :
Then
(1 + g + ) 1 + (1 + g
pb
=
pr
(1 + g + ) 1 (1 + g + x ) + (1 + g
Setting g = 0:018;
)
)
1
1
(1 + g
x)
= 0:036 and x = 4:5; we get the following graph.
1.14
1.12
1.1
1.08
1.06
1.04
1.02
1
-50
-40
Targeting
yields
=
h
-30
(1+g+ ) 1 +(1+g
(1+g+ ) 1 (1+g+x )+(1+g
)
)
-20
1
1
(1+g
11:1; i.e., a CRRA coe¢ cient of 10.
25
i
0.98
0
-10
x) g=0:018; =0:036;x=4:5
= 1:05
To also get the riskfree rate right, we need
pb =
1
1:01
with a reasonable value of : Setting
= :99 )
1
and using
"
2 (1
)
=
= 0:497 49
11:1; we solve
(1 + g + ) + (1 + g
2
)
(1 + g + )
1
+ (1 + g
)
1
#
g=0:018; =0:036; =
1
=
we …nd = 0:34; so the intertemporal elasticity of substitution is 1 0:34
1:515: This is quite far away from the EU case where we would have IES= 1 ( 111:1) =
0:08: Producing an interest rate of
0"
#
(1
+
g
+
)
+
(1
+
g
)
1
1
@
(1 + g + )
+ (1 + g
)
2 (1
)
2
which equals 13:8%:
Recent paper by Bansal and Yaron use Kreps-Porteus speci…cation with
CRRA=10 and IES 1.5, i.e., a large deviation from standard EU. They use
data to show that dividends are more volatile than consumption, 4.5 times
higher. As you can see, these assumptions seems able to account for the
risk-premia.
2.4
Closed form value function in partial equilibrium
Consider an individual with no labor income who has access to a market
~ His budget
for investments with an exogeneous stochastic i.i.d. return R:
constraint is thus
~ t+1
At+1 = (At ct ) R
We can now write a Bellman equation
W (At ) = max V u (ct ) ;
ct
W (At
where as above
u (c) = c
(W ) = [EW ]
26
1
~ t+1
ct ) R
g=0:018; =0:036;
and
V (u; ) = [(1
1
)u +
(3)
] :
Conjecture that
W (A) =
1A
and
c=
for yet undetermined coe¢ cients
Using this, we …nd that
2A
and
1
2:
~ t+1 =
ct ) R
W (At+1 ) = W (At
1
~ t+1 =
ct ) R
(At
1
~
(1
2 ) At Rt+1
1
(W (At+1 )) =
E
1
~
2 ) At Rt+1
(1
1
=
1
(1
~ t+1
E R
2 ) At
Using this in the Bellman equation;
1
1
1 At
=
(1
)(
2 At )
+
1
(1
2 ) At
~ t+1
E R
1
=
(1
)(
2)
+
(1
1
2)
1
~ t+1
E R
At
which is satis…ed, provided
1
1
=
(1
)(
2)
+
1
(1
2)
~ t+1
E R
1
:
The …rst-order condition for ct is the same as choosing
2
(4)
optimally,
1
@
(1
)(
2) +
~
2 ) E Rt+1
1 (1
@
=0
(5)
2
Solving these equations gives us the solution to the problem. Assuming
~ makes it possible to calculate
a speci…c form for the distribution of R;
1
~
~
E R
: If, for example, if ln R
r is normal with mean m and
standard deviation ; i.e., ln R s n (R; m; ) we have
~
E R
=
= e
1
2 2
2
m+
1
~
E R
Z1
1
p
2
= em+
2
2
27
;
(er ) e
(r m)2
2 2
dr
Note here that increasing has a direct e¤ect on the expected return, so
it is not a mean preserving spread, i.e.,
2
~ = em+ 2 :
E R
We can de…ne a mean-preserving spread as increasing
2
m
; then
2
~ =e
E R
2
2
m
+
2
2
by letting m =
:
2
That is, if for any , the mean of r is m 2 ; and the standard deviation ;
an increase in ; is a mean preserving spread. In this case,
1
~
E R
= em
2
2
+ 2
2
2
(1
2
= em
)
Clearly, we here see that an increase in ; while keeping the mean of the
return constant, reduces the certainty equivalent, since
1 and more so
the smaller is :
1
~
; the choice of 2 solves (5), implying that 2
Denoting R
E R
is a root of
2
1
1
2
1
1
2
(1
(1
)+ (
)+ (
1
1
(1
(1
2 ) R)
= 0:
0
2 ) R)
= 0
In the simplest case of unitary intertemporal elasticity of substitution,
i.e., = 0; the solution is 2 = 1
:
We should then substitute our optimized value of 2 back into the Bellman equation and …nd 2 in (4). Note, however, that the formulation in (3)
is not valid if = 0:We can however, look at the limit as ! 0;
V (u; ) = lim ((1
!0
)u +
)
1
= u1
Using this is formalution plus ut = ct = 2 At and (W (At+1 )) =
in the Bellman equation gives
1 (1
2 ) At R with 2 = 1
1 At
1
=
=
=
=
=
u1t
t
((1
) At )1 ( 1 At R)
At ((1
))1 ( 1 R)
At exp ((1
) ln (1
) + ln
1
))
( R)
1 ((1
= (1
) ( R) 1
28
> 0:
1
R)
~ is log-normal
In which case we conclude that if R
W (A) = (1
2
em+
)
1
2
A
and
c = (1
) A:
In this case, high riskaversion or higher risk reduces welfare by reducing
the certainty equivalent return (high riskaversion means low ) but saving is
una¤ected. What happens if > 0? First we note that higher risk reduces
R and therefore tends to reduce 1 ; then, in (??) a lower R and lower 1
reduces (increases) the term ( 1 (1
> (<) 0. In optimum,
2 ) R) if
this term must equal
1
(1
)
2 1
2
which is downward sloping since
d
2
1
1
2
d
(1
)
<1
=
(1
)
1
1
+
2
2
< 0:
2
1.2
1
0.8
0.6
0.4
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Therefore, an increase in risk leads to a higher (lower) 2 if > (<) 0.
I.e., consumption increases and savings decreases if the certainty equivalent
return decreases i¤ the intertemporal elasticity of substitution is higher than
unity.
29
3
Ambiguity
Aversion to unknown odds, as is demonstrated in labs, e.g., the Ellsberg
paradox, is given axiomatic foundations by Gilboa and Schmeidler. They
show that reasonable axioms capturing such ambiguity averse behavior can
be represented by a sort of max min preferences. Suppose it is known
that there is a set of possible realizations of the state, z 2 with associated
consumption levels c (z) : Then, assume that the individual does not know
the probability distribution over these states, but he knows that this probability distribution belongs to a set : Call the elements of this set p being
particular possible probability distributions. An element p is thus a vector
of probabilities p (z) : Then, preferences are given by
X
U (fc (z)g) = min
p (z) c (z) = min Ep c (z) :
p2
p2
z2
As with standard preferences, it is not necessary to take this literally in
the sense the individuals actually maximize these minimized preferences. It
may, for example, not be possible to ask individuals to tell us directly about
:
A simple example, consider again the Ellsberg paradox.
There are two urns each with 100 balls. In urn 1, there are 50 red and 50
black. In urn 2, there are only red and black balls but the proportions are
unknown. The subject is given a color and can pick one ball. If a ball with
the given color comes up, the gain is 50$, if not the gain is zero. The subject
is asked to rank lotteries. Typically the following response comes up.
1. Red from urn 1
Black from urn 1.
2. Red from urn 2
Black from urn 2.
3. Red from urn 1
Red from urn 2.
4. Black from urn 1
Black from urn 2.
Suppose we now ask individuals about the "ambiguity premium", i.e., we
…nd value r; such that if we let urn 1 contain 50 r red balls and 50 + r black
ones.
Red from urn 1 Red from urn 2.
and hopefully if we let urn 1 contain 50 + r red balls and 50
ones
Black from urn 1
Black from urn 2.
30
r black
In this case, the relevant p0 s are the shares of red balls in urn 2. The
:
value of r; pins down ;being the interval 50100r ; 50+r
100
Given this, the value of urn 2 in a bet on red is
min
r 50+r
p2[ 50
; 100 ]
100
=
pU (50) + (1
p) U (0)
50 + r
50 r
U (50) +
U (0)
100
100
with the same value in a bet on black since in this case, we have
min
r 50+r
p2[ 50
; 100 ]
100
pU (0) + (1
p) U (50)
50 + r
50 r
U (0) +
U (0):
100
100
Suppose now that there are two individuals, one owns an asset that gives
50 dollars with a probability p he knows is 50% but that he cannot verify this
knowledge to the other person. Otherwise it pays zero. The other person is
in the same position, he owns an asset that gives 50 dollars with probability
of p = :5 that he can not credibly verify. Suppose individuals consume 1+the
payo¤ from the asset and have log utility.
Under standard assumption including that both individuals assign a 50%
probability to the other’s urn. Individuals should share the risk and get a
payo¤ of
1
1
1
ln (51) + ln (26) + ln 1 = 2:612
4
2
4
If instead individuals have ambiguity aversion the probability of winning in the "foreign" asset is p0 2 : How big share ! should he choose
to invest in the other persons asset. Given a p 2 ; the four states of the
world, fhigh; highg ; flow; highg ; fhigh; lowg ; flow; lowg happen with probabilities, 12 p; 21 p; 12 (1 p) ;and 12 (1 p) : Given the symmetric nature of the
game, we focus on the case when the price of the "foreign" asset in terms
of the "domestic" is unity. Consumption, given ! is then 51; :1 + !50; 1 +
(1 !) 50; 1: The utility of the optimal portfolio is therefore
min max
p2P
!
1
1
1
p ln (1 + 50) + p ln (1 + !50) + (1
2
2
2
p) ln (1 + (1
!) 50) +
The …rst-order condition for the maximization problem is
d
1
p ln (1
2
+ 51) + 12 p ln (1 + !50) + 21 (1
d!
31
p) ln (1 + (1
!) 50)
1
(1
2
p) ln 1
52p 1
50
Of course, this is increasing in p: Note also that for p = 1=52; ! = 0 and
for p < 1=52; ! < 0: For example, if p = 0:01; !
0:01;i.e., a short position
in the foreign asset. The short position implies that if the foreign asset pays
50, a payment from hom to abroad takes place.
Now, we have to pick p:To do this, we consider the maximized value, i.e.,
!=
1
1
1
p ln (1 + 51) + p ln (1 + !50) + (1
2
2
2
max
!
=
1
(p ln (51) + ln (52) + p ln (p) + (1
2
p) ln (1
p) ln (1 + (1
!) 50)
p))
Let’s plot this, what we see at the this is increasing in p for p close to 0.5.
But, it is not monotone.
2.6
2.5
2.4
2.3
2.2
2.1
2
0
0.1
0.2
0.3
p
32
0.4
0.5
2.01
2
1.99
1.98
1.97
0
0.02
0.04
p
0.06
0.08
0.1
.
In fact,
arg min
p
1
(p ln (51) + ln (52) + p ln (p) + (1
2
p) ln (1
p)) =
1
52
which corresponds to a share ! = 0: Why is this? The answer is that if the
individual would be short in the foreign asset, what is a bad realization has
‡ipped. It is now the state when the foreign asset pays 50, happening with
probability p: Therefore, in the range where ! < 0; a higher p means lower
utility.
In general, since trade can only increase utility, the worst p; is the one
that implies autarky.
The situation would be quite di¤erent under asymmetric information, in
which case we would expect to sometimes see agents with more information
(domestic) go short in home securities.
Let’s take another example, suppose production in each country is 2 or 1,
so the state space is f1; 2; 3; 4g ; implying output fy1 ; y2 g is f2; 2g ; f2; 1g ; f1; 2g
or f1; 1g : Utility is
X
Ui = min
p (z) ln ci (z)
p2
Suppose also individual the knows that the probability of there own production being high is 0.5, while the set of possible probabilities for the other
countries production being high is 0:5 2 ; for 2 [ a; a] and that these
events are independent.
The agent decides how much to invest abroad !: Due to symmetry, the
relative price of the two assets, foreign and domestic production should be
33
one. The budget constraint is therefore for agent 1.
c1 (1) = 2; c1 (2) = (1
!) 2 + !; c1 (3) = (1
!) + 2!; c1 (4) = 1
Given ; the maximization problem is
X
max
p (z) ln ci (z)
!
1 1
1 1
2 ln 2 +
+ 2 ln ((1 !) 2 + !)
! 2
2
2 2
1 1
1 1
+
2 ln ((1 !) + 2!) +
+ 2 ln 1
2 2
2 2
= max
The …rst order condition is
d
1
2
1
2
+2
ln ((1
1
2
!) 2 + !) +
d!
1
2
2
ln ((1
!) + 2!)
= 0
1
2
! =
Substituting this into the utility function, yields
X
max
p (z) ln c
6
!
=
=
1
2
1
2
2
ln 2 +
1
1
ln 2 + ln 3
4
2
1
2
1
+2
2
ln ((1
1
+
4
(ln 2) +
!) 2 + !) +
ln (1 + 4 ) +
1
2
1
2
1
4
2
ln ((1
!= 12 6
ln (1
4 ):
Now, this the …rst order condition for minimizing this is
d
1
4
ln 2 + 12 ln 3
(ln 2) +
1
4
1
giving min = 12
: This means that
in particular that
! min =
1
2
6
+
d
ln (1 + 4 ) +
1
4
ln (1
will never be chosen larger than
min
:
34
=
1
2
6
= 0:
12
!) + 2!)
4 )
1
12
and
0.65
0.6
0.55
0.5
0.45
0.4
0.35
-0.2
-0.1
0
0.1
0.2
This is implies the important result that shortsales cannot occur. This
is in line with empirical evidence and is not the prediction of models with
asymmetric information.
Why is this? Note …rst that if ! = 0; the probability of success has no
e¤ect on utility. If ! is negative, a reduction in p actually increases welfare.
Why, the stream of payment from the foreign asset is
! (2p + (1
p)) = !(1 + p);
which decreases in p if ! is negative. In this case, high production abroad is
the bad state!
35
4
Time-inconsistency and temptation
Lab evidence discussed in the introduction shows preference reversal, quicker
discounting in for close time periods than for distant. Also preference for
commitment. People sometimes prefer to restrict their future behavior –
force themselves to save, hide the jar of cookies, not bring to much money
to the bar, and so on.
Evidence that hyperbolic discount factor represents time preference better
than geometric.
Two approaches:
Quasi geometric preferences.
Preferences over sets (menus), allows modelling of temptation, cost of
employing self control and welfare analysis.
4.1
Quasi-geometric preferences
Between current and next period, an "extra" discount factor ; is introduced
(the
model).
Self 0
U0 (0 c) = u (c0 ) +
Self 1
Self 2
u (c1 ) +
2
u (c2 ) +
3
2
u (c3 ) +
U (1 c) = u (c1 ) +
u (c2 ) + u (c3 ) +
U (2 c) = u (c2 ) + ( u (c3 ) +
3
u (c3 ) +
)
This implies that preferences are changing over time.
2 0
u0 (c2 )
u (c2 )
=
u0 (c1 )
u0 (c1 )
u0 (c2 )
u0 (c1 )
Self 0 M RS1;2 =
Self 1 M RS1;2 =
Self 1 cares relatively less of period 2 utility than self 1.
Assumptions about behavior:
The consumer cannot commit to future actions.
The consumer is “sophisticated”: he realizes that his preferences will
change and makes the current decision taking this into account.
The decision-making process is viewed as a dynamic game, with the
agent’s current and future selves as players. (Alternative: “naive” behavior. The agent thinks that he will not change preferences.)
36
Focus is on Markov equilibria, but other equilibria with trigger strategies also exists. For example, a self "behaves" and does not overconsume as long as previous selves has behaved well.
Markov equilibria can be strange or non-existent. Standard existence
theorems not applicable.
Example: Consumption and savings problem.
Suppose the agent has log period utility
Ut (t c) = ln (ct ) +
1
X
s
ln (ct+s )
s=1
and face a constant return r. The budget constraint is
at+1 = r (at
ct )
The state variable is at and let us implicitly de…ne a continuation value
from
J (at ) = ln ct + J (r (at ct )) ;
(6)
for some value ct :
If = 1; J is also the value function if ct is the argmax to the RHS
(the whole equation is then the Bellman equation). Under quasi geometric
discounting, ct is NOT arg maxc ln ct + J (r (at ct )) instead
ct = arg max ln ct +
ct
J (r (at
ct ))
J (r (at
ct ))
and the utility of self t is
W (at ) = max ln ct +
ct
Note that the value of giving assets to the next self is depreciated by the
fact that self 0 knows that self 1 is going to overconsume in the eyes of self
0.
Now, we can guess that J has the form
J (at ) = A + b ln at
Given this, ct is the solution to the …rst order condition
0 =
=
1
ct
1
ct
J 0 (r (at
b
r (at
at
) ct =
1+ b
37
ct )
ct )) r
r
Substituting this in (6) gives
A + b ln at = ln
at
+
1+ b
at
1+ b
A + b ln r at
= (1 + b) ln at + A + b ln
r b
1+ b
ln (1 +
b)
This is satis…ed for all at iif
(1 + b) = b
1
) b=
1
implying
ct =
at
=
1
1+
1
1
1
(1
+ ln
1
)
at
and A is
A=
ln 1
1
r
(1
)
1
(1
)
:
1
As we see, if < 1; consumption is higher and savings are lower than in
the time-consistent case, when the consumption rate is 1
:
Let’s now …nd the commitment solution if self 0 determines all consumption values. In this case, we …rst calculate Jc (at ) ;which is the continuation value when everything is determined by self 0. Note, however, that
future selves will agree with self zero on this. The di¤erence between the nocommitment case is that now the continuation value maximizes the standard
Bellman equation without any : Thus, Jc must satisfy,
Jc (at ) = max ln ct + Jc (r (at
ct
ct ))
If we don’t remember the solution to this, we do as usual. We take the …rst
order condition
1
= rJ 0 (r (at ct )) :
ct
Guessing
Jc (at ) = Ac + bc ln at
implies
1
=
ct
ct
bc r
r (at
at
=
:
1 + bc
38
1
ct )
Substituting,
1
at
+ Ac + bc ln rat 1
1 + bc
1 + bc
1
= ln at + ln
+ Ac + bc ln at + bc ln r
1 + bc
Ac + bc ln at = ln
Ac + bc ln at
bc
1 + bc
Giving
) bc = (1 + bc ) ; bc =
ct =
at
= (1
1 + bc
Ac = ln
1
1+
= ln (1
) Ac =
1
1
1
) at
+ Ac +
ln (1
)+
1
1+
1
1
!
ln r
1
1
ln r
1
1
) + Ac +
1
1
ln r
1
As we see, the coe¢ cient on ln at+1 is the same in both cases, commitment
and no commitment. The di¤erence between the constants under no commitment and commitment is negative, I think. The fact that the coe¢ cient
on ln at+1 is the same in Jc and J; implies that the marginal value of leaving
assets to self 1 is the same in both cases. Thus, consumption in period 0 is
independent of whether there is commitment or not. Note that two forces
here are a¤ecting the results. On the one hand, giving assets to self 1 under
no commitment has a lower value since he consumers too much in the eyes
of self 0. This reduces the incentive to save for self 0. On the other hand,
this leaves self 2, 3,... with too little consumption and the way only way self
0 can increase consumption of self 2,3,... is to save. This increases the value
of saving. Apparently, this two e¤ects cancel in the log utility case.
From period 1 and onwards, savings is higher under commitment.
>
1
(1
)
Commitment would of course increase welfare for self 0; it can never
reduce it. What about later selves?
39
In the no commitment case, self 1 gets
Wnc = ln ct+1 + J r at+1 ct+1
1
ln
at+1
1
(1
)
1
ln rat+1
+
A+
1
1
(1
)
1
(1
)
1
=
ln at+1 + ln
+
1
1
(1
)
A+
1
1
ln r
1
With commitment, the continuation value is di¤erent since now self 0
determines everything.
Self 1 gets under commitment
ln (1
= ln (1
1
=
) at+1 +
) at+1 +
(1
)
1
Jc (r at+1 )
bc ln (r at+1 ) +
ln at+1 + ln (1
Ac
)+
1
ln r +
Ac
Commitment gives extra value which is good also for self 1, but she cannot
control her consumption which reduces the value. For self 1, commitment
therefore be better than no commitment, also if it is done by self 1. For later
individuals, it may be even better with previous commitment since asset
levels are higher.
4.2
Preferences over choice sets, Gul and Pesendorfer
An alternative approach. Does not assume multiple selves, no game thus
no multiplicity. Also allows resistance to temptation and to model costs of
resisting temptation.
Two subperiods.
Second subperiod preferences de…ned over ordered pairs (A; x), where
A is a choice set and x 2 A is a choice (consumed).
De…nition: y tempts x if (fxg; x) is preferred to (fx; yg; x). That is,
individuals are better of getting x without having y in the choice set.
Assumptions:
1. Eliminating temptations cannot make the consumer worse o¤.
40
(1
)
2. If y tempts x, then x does not tempt y.
3. The utility of a …xed choice is a¤ected by the choice set only
through its most tempting element.
Second-period preferences induce …rst-period preferences over choice
sets themselves: A
B if and only if there is an x 2 A such that
(A; x) is preferred to (B; y) for all y 2 B.
The above assumptions imply what is labelled set betweenness:
A
B)A
A[B
B:
Choice sets cannot be compared simply by looking at their "best" or chosen elements. Instead, the utility of a …xed choice depends on the choice
set (through its most “tempting”element). Note that this violates one
of the axioms in standard theory. Removing a non-choosen element
from a choice set cannot change utility or behavior (independence of
irrelevant alternatives).
Set betweenness allows three possibilities:
1. Standard decision maker: A
A[B
B.
2. Preference for commitment and self-control: A A [ B B.
Interpretation: there is an element in B that tempts me. Nevertheless, I choose the same element in A and A [ B; but if faced
with only A; I don’t have to take the e¤ort of controlling myself.
Thus A
A [ B: Furthermore, A [ B
B since the choice in
A [ B provides higher utility,than the tempting choice.
3. Preference for commitment and succumbing to temptation: A
A [ B B.
Interpretation: there is an element in B that tempts me. A
A [ B since it provides higher utility. Faced with the tempting
choice, however. I cannot resist. I choose the same element in
A [ B and B and there is no cost of controlling myself:Thus,
A[B B
4.3
The representation theorem
The assumptions implies that preference over sets in the …rst period can be
written
W (A) = maxx2A fU (x) + V (x)g maxx~2A V (~
x)
41
Second period, preference are represented by
W (A; x) = fU (x) + V (x)g
maxx~2A V (~
x)
Interpretation:
U determines the commitment ranking (i.e., the utility of singleton sets,
no temptation).
V determines the temptation ranking (i.e., V gives higher values to
more tempting elements).
arg maxx~2A V (~
x) is the most tempting element in A.
The second-period choice chooses x by solving
maxfU (x) + V (x)g
x
maxx~2A V (~
x)
If a person is given x 2 A, without anything else to choose from, there
is no cost of self control. The utility is
U (x) + V (x)
V (x) = U (x) :
If a person chooses x 2 A; the disutility of self-control is V (x)
maxx~2A V (~
x) 0;so utility is
U (x) + V (x)
maxx~2A V (~
x):
If a person chooses x~ = arg maxx~2A V (~
x), he gives in to temptation,
there is no cost of self-control, and the utility is
U (~
x) + maxx~2A V (x)
4.4
maxx~2A V (~
x) = U (~
x):
A 2-period consumption-savings model
Consumption today and tomorrow.
Neoclassical production.
Standard budget set (borrowing and lending at r).
General equilibrium.
42
With U (c1 ; c2 ) playing the role of U and V (c1 ; c2 ) the role of V , let the
temptation function V have a stronger preference for present consumption. For example, let
U (c1 ; c2 ) = u(c1 ) + u(c2 )
and
V (c1 ; c2 ) =
with ;
(u(c1 ) +
u(c2 )) ;
< 1.
Aggregate resource constraint given by
c1 + k2 = f (k1 )
c2 = f (k2 )
Strength of temptation determined by : Standard model when
As ! 1; Laibson model.
= 0:
In equilibrium choices are made to maximize
U (c1 ; c2 ) + V (c1 ; c2 )
In competitive general equilibrium individuals take prices (here interest
rate) as given, provides a linear budget set for the individual.
43
c2
Competitive equilibrium
U(c1,c2)
U(c1,c2)+V(c1,c2)
V(c1,c2)
c1
Temptation
Best without temptation
Actual choice
Policy implications. Command optimum can achieve arg max U , without
any temptation cost. Nothing is better than this. A subsidy to investments
(tax on …rst period consumption) can improve upon laissez faire. Does so
by reducing temptation.
For example, let u (x) = ln (x). Then, choices are governed by solving
max (ln c1 ) + ln(c2 )) + (ln(c1 ) +
s.t. c1 +
c2
1+r
ln(c2 ))
= w
c1 = arg max ((1 + ) ln c1 + (1 +
1+
= w
1 + + (1 + )
) ln((1 + r) (w
which increases in :
The maximum temptation is
ct1 = arg max (ln(c1 ) +
w
=
1+
44
ln((1 + r) (w
c1 )))
c1 )))
Interesting implication
Compare autarky, i.e., each individual runs his own machine. Then the
interest rate is not exogenous.
Autarky
c2
U(c1,c2)+V(c1,c2)
V(c1,c2)
Actual choice
Temptation in comp. eq.
Temptation under autarky
c1
Result: Autarky delivers the same allocation, but at higher welfare.
Why?
The choice sets shrinks, the temptation to overconsume is reduced and
the cost of resisting temptation falls.
4.5
Macroeconomic applications
Krusell, Kuruşçu, Smith "Temptation and Taxation"
Consider long horizons: the limit of the …nite-horizon problems.
Study competitive equilibrium under two kinds of parametric restrictions:
1. Logarithmic utility, Cobb-Douglas production, and full depreciation: full analytical solution of recursive competitive equilibria.
2. Iso-elastic utility and no restrictions on technology: analytical
characterization of steady state and computational analysis of dynamics.
45
Analysis: Vary and
interest rate constant.
Results:
, while adjusting
to keep the steady-state
Almost observationally equivalence (like in Barro "Laibson meets Ramsey", when the utility is log, the speed of adjustment to the steady state
does not depend on (as in Barro).
With more (less) curvature in utility, the speed of adjustment is decreasing (increasing) in .
The e¤ects of on the speed of adjustment is quantitatively small:
observational equivalence found in Barro “almost”carries over.
Savings should be subsidized. But not much, and the welfare gains are
small (little reduction in temptation costs).
46
5
Habits
6
Topic 4. Habits
We have previously assumed history independence, meant to mean that the
marginal rate of substitution, evaluated at t between:
goods consumed at t and t + 1; and
di¤erent goods consumed in the same time is independent of the consumption history prior to t:
This is not necessarily a good assumption.
There are, in particular two, cases in which we might want to relax history
independence.
1. The case when previous consumption leads to higher aspiration. To
live on a small budget may be easier if you are used to it than if your
are used to the good life.
2. When relative consumption matters for utility. "Catching up with the
Joneses or "Poverty is more easily accepted if it is shared by everyone"
Ernst Wigfors, Social Democratic Finance Minister 1932-49.
3. The relative taste for some goods is a¤ected by previous consumption
of them, e.g., food, culture goods and drugs.
In the literature, the …rst case have been used to try to explain asset
market puzzles, i.e., why standard models have great problems explaining
the co-movements of prices and consumption. The third case is used to
explain, for example, cultural diversity.
To simplify, in particular in order to be able to specify recursive preferences, utility is assumed to be
Ut =
1
X
j
u (ct+j ;
t+j )
j=0
where
t
= v (~
ct 1 ; c~t 2 ; :::~
ct
n) ;
is denoted the habit. A couple of things to note,
1. We maintain time additivity here, although this should be straightforward to generalize to the constant elasticity aggregator.
47
2. c~t s can denote the own previous consumption of the household, the
consumption of some reference group or some combination. If c~t s =
ct s (own consumption), we have what is called "internal habits" while
if it is the consumption of some reference group, it is called "external"
habit. Abel uses a geometric average
t
1
cD
t 1 Ct
D
1
;
where Ct 1 is aggregate consumption. Here = 0 gives the standard
model, 6= 0; and D = 1;gives the internal habit case and D = 0 the
external case.
3. History matters only though the habit function, i.e., through
v (~
ct 1 ; c~t 2 ; :::~
ct
n)
4. We assume either a …nite value of n or at least that
@v (~
ct 1 ; c~t 2 ; :::~
ct n )
lim
= 0:
n!1
@~
ct n
so that we can hope to …nd stationary decision rules.
6.1
Optimal consumption under external vs. internal
habits.
Suppose the representative agent solves
max
1
X
t
u (ct ;
t)
t=0
s.t. At+1 = (At ct ) r
1 D
vt = cD
t 1 Ct 1
and a no-Ponzi condition.
Let’s look at the Bellman equation. Note that now, vt is a state variable.
Therefore
V (At ; vt ) = max u (ct ;
ct
s.t. At+1 = (At
t)
+ V (At+1 ; vt+1 )
1
ct ) r; vt = cD
t 1 Ct
D
1
The …rst order condition is
u1 (ct ; vt ) =
V1 (At+1 ; vt+1 ) r
=
V1 (At+1 ; vt+1 ) r
48
V2 (At+1 ; vt+1 )
D
@vt+1
@ct
vt+1
V2 (At+1 ; vt+1 )
ct
Clearly, the …rst order condition is not a¤ected if D = 0: Suppose instead
D > 0: Then an increase in today’s consumption increases the habit. Suppose this reduces utility, then this implies that there is a negative dynamic
e¤ect of consumption which will show up in a negative V2 : Therefore, the
V2 (At+1 ; vt+1 ) is positive and marginal utility of consumpterm
D vt+1
ct
tion should be set higher in period t:
Consider the case of external habits, D = 0 . We have seen that in this
case, the FOC is the same as under no habits. Suppose …rst for simplicity
that
u (ct ; t ) = ln ct ln vt
Recall the solution strategy in the case when we expect that there can be
an analytical solution.
1. Write Bellman equation.
2. Guess a functional form of the value function with unknown parameters. From the Bellman equation we see that it has to be of the same
functional form as the per-period utility function.
3. Solve for the choice variable that maximizes the RHS of the Bellman
equation given our guess on the value function.
4. Substitute your optimal choice variable into the RHS of the Bellman
equation to express the maximized RHS.
5. Verify that the Bellman equation is satis…ed for all values of the state
variables by …nding the unknown parameters.
If step 5 fails you have made an incorrect guess and must start with
another. However, most problems do not admit closed form solutions for the
value function in which case this approach is useless.
Now, we guess that the value function is
V (At ; vt ) = B1 ln At + B2 ln vt + B3
for the unknown coe¢ cients B1 ; B2 and B3 :
49
The FOC is
1
ct
1
ct
=
=
B1 r
At+1
B1
At c t
) ct =
1
1 + B1
= (At ct ) r
B1
=
rAt
1 + B1
At+1
At ;
Which we recognize well.
We also have
1
ct+1 =
At+1
1 + B1
1
B1
=
1 + B1 1 + B1
1
B1
=
1 + B1 1 + B1
B1
=
rct
1 + B1
rAt
r (1 + B1 ) ct
Substituting this into our guess gives
At
1 + B1
At
= ln
1 + B1
B1 ln At + B2 ln vt + B3 = ln
ln vt + (B1 ln At+1 + B2 ln vt+1 + B3 )
ln vt +
B1 ln
B1 r
At + B2 ln vt+1 + B3
1 + B1
This cannot work unless we get rid of vt+1 in the RHS. To do this we note
that
vt+1 = Ct :
and in general equilibrium
Ct =
Ct
=
vt+1 =
B1
rCt 1
1 + B1
B1
r Ct 1
1 + B1
B1
r vt
1 + B1
50
Therefore,
B1 ln At +B2 ln vt +B3 = ln
At
ln vt +
1 + B1
B1 ln
B1 r
At + B2 ln
1 + B1
r B1
vt + B3
1 + B1
We solve this by equalizing the coe¢ cients on the di¤erent terms
B1 = (1 + B1 )
B2 =
1 + B2
B3 =
(B1 + B2 ) ln B1 r
(1 + B1 +
B2 ) ln (1 + B1 ) + B3
1
B1 =
1
1
B2 =
1
Again using the FOC, we get
ct =
At
= (1
1 + 11
) At
As we see, consumption not a¤ected by the habit. This is due to the
log utility. Marginal utility of consumption is c1t regardless of vt since u is
separable in c and v:
Let us therefore consider a generalization. Instead of solving the full
problem, we can at least characterize consumption dynamics.
Suppose
1
u (ct ; vt ) =
ct
vt
1
1
u1 (ct ; vt ) =
ct
ct
vt
1
1
The Euler condition under purely external habits is the usual
u1 (ct+1 ; vt+1 )
r
u1 (ct ; vt )
1=
In general equilibrium vt = ct 1 , giving
1
1=
ct+1
1
ct
ct+1
vt+1
ct
vt
1
1
1
r=
ct+1
1
ct
1
ct+1
ct
1
ct
ct
1
51
r = ct+1 ct
(
1)+
ct
(1
1
)
r
Taking logs
(ln ct+1
0 = ln r
ln ct ) = ln r
In the case
constant at
ln ct+1 + (
(1
) (ln ct
(1
)) ln ct + (1
ln ct 1 )
) ln ct
= 0;the standard case, the growth rate of consumption is
ln ct+1
ln ct =
ln r
;
as we should now from standard models.
We have also seen that with = 1;
ln ct+1
ln ct = ln r
Dynamics becomes interesting now under
ln ct+1
ln ct
6= 1 and
> 0: De…ne
gt+1
Then,
gt+1 =
ln r
(1
)
(7)
gt
When riskaversion is low (IES high), that is a > 0; we get oscillations! A
low growth rate is followed by a high and vice versa. The oscillations may
even be unstable if (1 ) > 1: If instead < 0;we get a monotone path,
(1
)
that is stable if
< 1:
Note that we have assumed a constant interest rate r; this is quite easy
to relax. With a varying interest rate, for example if we include capital
accumulation, we still have
gt+1 =
6.2
ln rt+1
(1
)
gt :
Adding stochastics
With stochastics, we have
Et
1
ct+1
Et ct+1 ct
Et
1
ct+1
ct
rt+1 =
(
1)
ct+1
ct
gt+1
Et (e
ct
1
(
1)
1
(1
ct
ct
gt
rt+1 = (e )
52
1
ct
rt+1 = ct ct
rt+1 =
)
1
ct
1
(1
)
)
:
1
: : :
This can be analyzed by linearization. Suppose
rt+1 = ezt+1 r
where r = 1; giving a steady state of the economy.
Approximating around g = 0; z = 0,
(egt+1 )
ezt+1 r
(egt )
(1
r
)
gt+1 r + rzt+1
1 + (1
) gt
Giving
Et ( r
gt+1 r + rzt+1 ) = 1 + (1
(1
Et gt+1 =
) gt
)
Et zt+1
gt +
;
which can provide interesting dynamics.
To understand the reason for the oscillatory behavior, it may be of help to
note that individuals with habits might prefer variations over time. Clearly,
when = 0; individuals are risk averse for > 0 and also averse to variations
over time. Consider instead the case when = 1: Assume that (individual
and aggregate) consumption is
c (1 + ") if t odd
c (1 ") else.
ct =
In this case, utility in even periods (multiplied by (1
is
ut =
1+"
1 "
and in odd
ut+1 =
1 "
1+"
) for convenience)
1
1
we have
ut + ut+1
=
2
=
1 " 1
1+"
1 " 1
1+"
2
6d
1 + "6
4
= 1 + 2 (1
+ 1+"
1 "
2
+ 1+"
1 "
2
1
( 11+"" )
1
1
1
+( 11+"" )
2
d"
3
7
7
5
"=0
2
) "
2
53
2
1
2
d
"2 6
6
+ 4
2
( 11+"" )
1
+( 11+"" )
2
d"2
3
7
7
5
"=0
which is increasing in "2 :
6.3
Asset market implications.
Abel has shown that habits also may have an ability to explain asset market
puzzles. Let us …nally go over this.
Using
1
u (ct ; vt ) =
1
1 ct 1
u1 (ct ; vt ) =
ct vt
ct 1
vt
utility in period t can be written
Ut =
s.t. At+1
At+2
1
ct 1
1 ct+1 1
+
vt
1
vt+1
D
ct ) rt+1 ; vt = ct 1 Ct1 1D
ct+1 ) rt+2
1
= (At
= (At+1
+ V (At+2 ; vt+2 )
Therefore
@Ut
1
=
@ct
ct
ct
1
=
ct
ct
1
=
ct
1
ct+1
D
t
t+1
1
D
t
ct
!
1
ct+1 t
ct t+1
1
ct+1
ct
1
1
1
D
t
t
Now, de…ne gross output growth
yt+1
yt
xt+1
and since the economy is closed
xt+1 =
and thus
t+1
vt
=
ct+1
Ct+1
=
ct
Ct
1
cD
t Ct
D
1
cD
t 1 Ct
D
1
54
!
1
ct
= xt ;
1
t
t+1
!
implying
@Ut
1
=
@ct
ct
1
ct
1
D
t
1
= ct
t
= ct
t
1
1
ct+1
ct
xt+1
xt
D
1
1
1
t
t+1
!
!
(8)
Ht+1
where
1
xt+1
xt
Clearly the marginal utility of consumption in t is increasing in Ht+1 :The
latter, in turn, is high when xt+1 is low relative to xt if > 0, and vice-versa
otherwise. So if growth is expected to be low between t and t + 1; and IES
is large ( > 0), this boosts the marginal utility of consumption. In other
words, an expectation of low growth has a negative e¤ect on savings. Note
that this goes against the standard smoothing results that if you expect low
income in the future, this strengthens the savings motive. Furthermore, in
asset market equilibrium, this e¤ect tends to reduce the price of assets, i.e.,
increasing the expected return.
Let us now …rst consider the case when D = 0: We will see that we can
get some interesting results for bond and asset returns.
t
= ct t 1 :The Euler equation implies
When D = 0; Ht = 18t and @U
@ct
as usual
@Ut
@Ut+1
= Et rt+1
@ct
@ct+1
1
c
1 = Et rt+1 t+1 t+11
ct t
Ht+1
1
D
Now consider the endowment economy, where ct = yt and we de…ne
yt+1
xt+1
:
yt
Then, we have
c
1 = Et rt+1 t+1
ct
1
t+1
1
t
= Et rt+1 xt+1 xt
(
1)
Now, consider a risky share that pays yt as dividend (the apple trees)
with price pr;t : The price of this asset must satisfy
rr;t+1 =
pr;t+1 + yt+1
:
pr;t
55
Denoting the price-dividend ratio
pr;t
yt
wt
we can write
wt+1 yt+1 + yt+1
;
wt yt
1 + wt+1
xt+1 :
=
wt
rr;t+1 =
Now, using this last expression for the return on stocks into the Euler
equation yields
(
1)
1 = Et rt+1 xt+1 xt
1 + wt+1
(
1)
= Et
xt+1 xt+1 xt
wt
(
1)
wt = xt
Et (1 + wt+1 ) x1t+1 ;
since wt is known in t:
When growth rates are i.i.d., this can be calculated quite easily. In particular, we will show that the expression
1
(1 + wt+1 ) xt+1
Et
is constant at some value A, so that we can write
wt = Axt
(
1)
for some A and verify that this satis…es the pricing equation (11).
Using our "guess", we have
Axt
(
1)
= xt
(
A = Et
1)
Et
(
1 + Axt+1
(
1 + Axt+1
1)
1)
1
xt+1
;
x1t+1
(
=
Et x1t+1 + Et Axt+1
=
1
Et xt+1
+ Et Axt+1
(1
1)+(1
)(1
)
)
Clearly, the RHS is a constant if growth rates are i.i.d. and we have
established that
(
1)
wt = Axt
56
where
Ex1
A =
A =
1
+ EAx(1
Ex1
Ex(1 )(1 )
)(1
)
To calculate the expected stock market return, we use
rr;t+1 =
=
1 + wt+1
xt+1
wt
(
1)
1 + Axt+1
Axt
(
1)
xt+1
and
Et rr;t+1 =
=
1 + AEx
(
Axt
1 + AEx
Axt
(
(
1)
Ex
1)
(
1)
1)
Ex
As we see, the expected return is time dependent, despite the i.i.d. assumption, provided 6= 0: Why?
Furthermore, if > 0; and < 1; the denominator decreases in xt and
yt
the expected return is thus higher when xt
is high. If > 1; the
yt 1
opposite is true.
We can easily calculate the unconditional return, i.e., the average return
over time
1 + AEx ( 1)
Ex :
Err = E
Ax ( 1)
In a similar fashion, the unconditional return on bonds is
Erb =
Ex (
(Ex
1)
)
Consider the special case when output growth x is lognormal, with mean
g and standard deviation :
Recall that then
1 2
Ex = eg+ 2 :
Furthermore, if ln x is normal, ln x = ln x is normal with mean g and
standard deviation :Thus,
Ex = e
57
g+
2 2
2
:
Now, let ln x be normally distributed with mean g and standard deviation
. Then,
Ex1
A =
+ AEx(1
Ex1
Ex(1
1
(
)
)
e(1
=
e(1
1
1) 1 + AEx
AEx (
(
ln Err = ln Ex
= ln e
)(1
)(1
1)g+( (
1))
2
(
)2 2
2
(1
((1
)g+
)(1
2
))2 2
1)
1)
2
2
)(1
)g+
eg+
Ex
2
2
+ Ae(1+
(
1))g+(1+ (
1))2
A
2
2
!
and
ln ERB
Ex (
(Ex
rB = ln
= ln
e
(
1)g+
e
= (
(
1)
)
( (
g+
1))2 2
2
2 2
2
1)) g +
!
( (
1))2
2
2
2
ln
Setting, = 0; = 0:036; g = 0:018; = :99 and plotting against rs and
rB against 1
; we have the no habit case.
58
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
2
4 Relative Riskaversion
6
8
10
12
Average stockmarket return rS (solid line) and safe return rB against risk
aversion ( ): Standard utility ( = 0):
Keeping the other parameters, but now introducing external habits by
setting = 1; the returns are given in the second …gure.
59
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
2
4 Relative riskaversion
6
8
10
12
Average stockmarket return rS (solid line) and safe return rB against risk
aversion ( ): Extrenal Habit ( = 1):
As we see, the stock market return quickly becomes very high as we reduce
:
6.4
Appendix: The case when D > 0:
The stochastic Euler equation
Et
Note that is
ator.
@Ut
@ct
@Ut
@Ut+1
= Et+1 rt+1
@ct
@ct+1
is actually not realized at t; thus the expectations oper-
Et
@Ut
=
@ct
1 =
Et rt+1
@Ut+1
@ct+1
t+1
Et rt+1 @U
@ct+1
t
Et @U
@ct
60
:
(9)
Now, it is convenient to …nd an expression for
@Ut+1
@ct+1
Et
@Ut
@ct
Shifting (8) forward, yields
@Ut+1
= Ht+2
@ct+1
Using,
t+1
t
= xt and
@Ut+1
@ct+1
Et
@Ut
@ct
ct+1
ct
1
t+1 ct+1 :
= xt+1 ;yields
Ht+2 t+11 ct+1
=
Et Ht+1 t 1 ct
Ht+2
=
Et Ht+1
Ht+2
(
=
xt
Et Ht+1
1
t+1
t
1)
ct+1
ct
xt+1 :
Now, consider a risky share that pays yt as dividend (the apple trees)
with price pr;t : The price of this asset must satisfy
rr;t+1 =
pr;t+1 + yt+1
:
pr;t
Denoting the price-dividend ratio
pr;t
yt
wt
we can write
wt+1 yt+1 + yt+1
;
wt yt
1 + wt+1
xt+1 :
=
wt
rr;t+1 =
Now, using this last expression for the return on stocks into the Euler
equation yields
1 =
wt =
t+1
Et rr;t+1 @U
@ct+1
t
Et @U
@ct
;
t+1
Et (1 + wt+1 ) xt+1 @U
@ct+1
t
Et @U
@ct
61
;
(10)
since wt is known in t:
From (10), we have
t+1
Et (1 + wt+1 ) xt+1 @U
@ct+1
wt =
Et
@Ut
@ct
t+1
Et (1 + wt+1 ) xt+1 @U
@ct+1
=
Et
@Ut
@ct
Et (1 + wt+1 ) xt+1 ct+1 t+11 Ht+2
Et ct t 1 Ht+1
=
Et (1 + wt+1 ) xt+1
=
1
ct+1
ct
t+1
t
Ht+2
Et (Ht+1 )
1
Et (1 + wt+1 ) xt+1 xt+1 (xt )
=
Ht+2
Et (Ht+1 )
Et (1 + wt+1 ) xt
=
(
1)
Ht+2 x1t+1
Et Ht+1
xt
=
(
1)
1
Et (1 + wt+1 ) Ht+2 xt+1
:
Et Ht+1
Using the law of iterated expectations
1
Et 1 + wt+1 )Ht+2 x1t+1 = Et (1 + wt+1 ) xt+1
Et+1 (Ht+2 )
De…ne
Jt
Et (Ht+1 ) = 1
Dxt
(
1)
1
Et xt+1
:
Then we have
wt =
xt
(
1)
1
Et (1 + wt+1 ) Jt+1 xt+1
:
Jt
(11)
We now need to …nd wt as a function of state variables (which are they?)
that satis…es (11).
When growth rates are i.i.d., this can be calculated quite easily. In particular, we will show that the expression
Et (1 + wt+1 ) Jt+1 x1t+1
62
is constant at some value A, so that we can write
(
1)
Axt
wt =
Jt
for some A and verify that this satis…es the pricing equation (11).
Using our "guess", we have
!
!
(
1)
(
1)
(
1)
Ax
Axt
xt
t+1
=
Et
1+
Jt+1 x1t+1
Jt
Jt
Jt+1
!
!
(
1)
Axt+1
1
Jt+1 xt+1
A = Et
1+
Jt+1
(
=
1
Et Jt+1 xt+1
+ Et Axt+1
=
Et Jt+1 x1t+1 + Et Axt+1
(1
1)+(1
)(1
)
)
So
A 1
(1
Et xt+1
)(1
)
= Et Jt+1 x1t+1
(12)
Now, under the assumption of i.i.d. output (consumption) shocks,
Et x1t+1 = Et x1t+2 = Ex1
we have
Jt = Et Ht+1
= 1
Dxt
(
1)
Et x1t+1
= 1
Dxt
(
1)
Ex1
(
1)
so
Jt+1 = 1
Jt+1 x1t+1
1
= xt+1
Dxt+1
1
Ex1
(
Dxt+1
1)
Ex1
Take conditional expectation at t
Et Jt+1 x1t+1
= Et x1t+1 1
Dxt+1 Ex1
= Et x1t+1
DEx1
= Ex1
DEx1
= Ex1
1
63
(
1
Et xt+1
xt+1
E x(1
DE x(1
)(1
)(1
)
)
1)
Finally, use this in (12), and use the i.i.d. assumption to replace conditional expectations
A 1
(1
Et xt+1
)(1
)
=
A =
Ex1
1
Ex1
1
(1
DE x(1
)(1
)
DE x(1 )(1
Ex(1 )(1 ) )
)
which is clearly a constant under the i.i.d. assumption.
To calculate the expected stock market return, we use
pSt+1 + yt+1
pSt+
(1 + wt+1 ) xt+1
=
wt
S
Rt+1
=
and
(
Et
S
Et Rt+1
1+
=
=
Axt+1
Jt+1
1)
xt+1
wt
Ex + Et
1+ (
Axt+1
Jt+1
(
Axt
Jt
Which we at least can simulate.
64
1)
1)
;
7
Loss-Aversion
Substantial amounts of lab-evidence suggests that individuals behave like if
the formed reference levels for consumption. Preferences over actual consumption then depend in a particular way of consumption relative to this
reference level. Speci…cally, preferences are consistent with utility maximization if
1. utility is concave in consumption if consumption is above the reference
level,
2. utility is convex in consumption if consumption is below the reference
level, and
3. marginal utility is discretely larger below the reference level than what
it is above.
The utility function can then be depicted as follows:
Two important implications of this is that
1. Individuals have a strict distaste also for abitrarily small gambles (because of the discontinuity).
2. Individuals are risk-lovers for losses. This means that they may prefer
a 50/50 bet of loosing x or nothing over a sure loss of x=2:
Kahneman & Tversky proposes
u (c
(c r) if c r
( (c r)) else
r) =
65
and in lab-experiments …nds that
following graph1
=
= 0:88; and
= 2:25as in the
1
0.5
0
-0.5
-1
-1.5
-2
-0.8 -0.6 -0.4 -0.2
0x
0.2
0.4
0.6
0.8
1
In the paper by Bowman et al., this implication is expanded into a dynamic setting. One implication is then that consumption may respond asymmetrically to positive and negative news about future income. To get the intuition, consider a two period setting and suppose that income at the outset
is expected to be w in both periods. Suppose also that the reference point
for consumption is r = w: For simplicty suppose that the interest rate equals
the subjective discount rate. Clearly, the optimal consumption is now c = w
in both periods.
Consider now a positive but uncertain signal about period 2 income. Say
that income is w + 2x with probability 1=2 and w with probability 1=2: Expected lifetime income is then 2w + x and unless there is some precautionary
savings, consumption in period 1 will be w + 12 x: This is the permanent income hypothesis. In any case, consumption will certainly increase when this
positive signal comes. Behavior is "standard" for gains.
Consider now instead a negative signal saying that second period income
is w 2x with probability 12 and w otherwise. Now, it may very well pay for
the household to continue to consume w in period 1 and then with probability
1=2 consume w also in the …nal period and with probability 21 consume w 2x,
rather than consuming the permanent income w 12 x in the …rst period, in
which case second period consumption is either w + 12 x or w 32 x: Why?
1
A better formulation, allowing ;
< 0; is u (c
66
r) =
(
(c r)
if c
( (c r))
r
else
Let r denote the reference level for consumption. Then, the expected
utility of the …rst strategy is
1
1
u (w r) + u (w r) + u (w 2x r)
2
2
1
0+0
(2x)
2
and for the second strategy is it
u w
r
1
x
2
=
1
1
1
x + u w+ x
2
2
2
+
1
2
1
x
2
1
r + u w
2
3
x
2
3
x
2
The di¤erence is
1
x
2
1
(2x)
2
=
1
(2) +
2
1
2
r
3
2
+
1 1
+
x
2 2
!
x
3
x
2
!
1+
1
2
x
Under the assumption a = ; this is positive if
!
1+
3
1
1
1
+
> 0
(2) +
2
2
2
2
>
2
2 +2
1 1+
2
1
+
2
2
3
2
2
1 1
;
4 2
Due to the convexity of utility in losses, its better to take a chance that
consumption might not need to be reduced below the reference point. In
other words, there is a tendency that consumption does not fall "unless it is
clear that it has to". Bowman et al documents such an asymmetry in U.S.
consumption data.
In a dynamic setting, a key issue is how reference points are formed.
Unfortunately, not much empirical evidence is collected regarding this issue.
Bowman et al assume r1 ; the reference point for period 1, is given and that
r2 = (1
) r1 + c 1
If = 0; we have the case discussed above –static reference points. For
= 1; next periods reference points is completely determined by the previous
periods consumption. Here, we could think both of the case when the agent
internalize the e¤ect her consumption has on the reference point and the case
when she doesn’t (due to naivite or external reference points).
67
7.1
Loss-aversion as commitment (Hassler&Rodriguez
Mora)
Two types of rational and non-altruistic individuals, (poor) workers
and entrepreneurs, living in a two period OLG-setup.
The workers make no private choices, having a …xed wage normalized
to zero, consuming in the second period of life, having high marginal
utility since they are poor.
i2
Young entrepreneurs in t choose investments it at a utility cost 2t ,
returning it (1
t+1 ) in second period of life when consumption takes
place and the capital fully depreciates.
Old workers get a transfer Gt , …nanced by taxes on installed capital.
Taxes, t 2 [0; 1], are determined without commitment by probabilistic
voting with equal weight on all living individuals. Alternative interpretation, a benevolent planner that cares equally of all living individuals.
Without commitment, the only Markov equilibrium is one with 100
percent taxation since temptation to tax installed capital is too high.
As Köszegi and Rabin, we consider the case when reference points for
consumption are forward-looking. We can refrase reference points for
cosumption in terms of the corresponding tax-level, r :
We require
r
t+1
to be in the set of equilibrium tax rates for t + 1.
We allow politicians to a¤ect reference points by making "promises"
about the future. But remember that the promise is empty – the
politician does not remain in o¢ ce nor runs again and he has no formal
commitment power.
The promise can a¤ect the future if it is believed, in which case it
becomes the the reference point.
It is believed if it is done by the winning candidate and is in the set of
equilibria for next period. If the promise is not an equilibrium, rt+1 is
some element of the set of equilibrium tax rates.
As a variation, we consider the opposite case of history dependence.
Reference tax-levels are backward-looking, rt+1 = t
68
7.1.1
Results
Under both backward and forward-looking reference points, there is a Markov
equilibrium with limited amounts of taxation. Dynamics di¤er between the
two cases.
The level of taxation in equilibrium depends inversely on on the degree
of loss-aversion.
Intuition: If people have reference points, implying that they feel "entitled" to some return on their investments – if becomes politically costly to
go against this. If the entitlements are not too large, they will be satis…ed in
equilibrium.
69