Solutions toProblem Set # 4
Macro 453
October 18, 2003
1 Problem 1: Incomplete Markets and Asset
Prices
This problem investigates the effects of market incompleteness on asset pric
ing following Constantinides and Duffie (1996). There is a continuum of
individuals with identical CRRA preferences: u (c) = c1−γ / (1 − γ) and sub
jective discount factor is β.
1.1
Part (a)
Write down the Euler equation for individual i and asset j. This equation
should relate the gross return Rtj+1 to the growth rate of consumption for
i
individual i: cit+1 /c
.
t
The standard Euler equation for each individual i and asset j will be
"
¡ i ¢#
0
u
c
1 = βEt Rtj+1 0 t+1
u (c
it )
and with our preferences specification
"
µ i ¶−γ #
c
1 = βEt Rtj+1 t+1
c
it
1.2
Part (b)
Now assume that the growth rate of individual consumption satisfies,
cit+1
ct+1 i
=
ε ,
i
ct
ct t+1
(1)
where ct+1 /ct is the growth rate of aggregate consumption. Here εit+1 rep
resents the idiosyncratic component of consumption growth. Conditional on
σ 2t+1 , ct+1 /ct and Rtj+1 , assume εit+1 is independent across individuals and
log εit+1 is distributed N(− 12 σ 2t+1 , σ 2t+1 ). The cross sectional variance parame
ter, σ 2t+1 , is a random variable from the point of view of time t, it becomes
known at time t + 1. Note that σ 2t+1 is not assumed to be independent of
j
(i.e. the three variables may be correlated with each other).
ct+1 /ct and Rt+1
Use the individual Euler equation from (a) to show that:
"
#
¶
µ
¡ 2 ¢ ct+1 −γ j
(2)
1 = βEt Φ σ t+1
Rt+1
ct
where
h¡
¡ 2 ¢
¢−γ 2 i
i
Φ σt+1 ≡ E εt+1
|σ t+1 = exp
½
¾
γ (1 + γ) 2
σ t+1 .
2
From part (a)
1 = βEt
= βEt
"
Rtj+1
"
Rtj+1
µ
µ
cit+1
cit
ct+1
ct
¶−γ #
¶−γ
=
¡ i ¢−γ
εt+1
#
¡
¢
Recall that we know log εit+1 | X ∼ N(− 12 σ 2t+1 , σ 2t+1 ) where
³
´
j
R
X 0 = σ2t+1 ct+1
t+1
ct
We can rewrite
1 = βEt
"
j
Rt+1
µ
ct+1
ct
¶−γ
#
n h¡
¢−γ io
i
exp log εt+1
and using the law of iterated expectations
"
#
µ
¶−γ h
n h¡
i
¢−γ io
ct+1
j
i
1 = βEt Rt+1
Et exp log εt+1
|X =
ct
"
#
µ
¶−γ
£
©
ª
¤
ct+1
j
i
Et exp −γ log εt+1 | X
= βEt Rt+1
ct
From the known properties of the normal distribution we know that if z ∼
N(µ, σ 2 ) and c a constant, then
¶
µ
1 2 2
E [exp cz] = exp cµ + c σ
2
It follows
"
µ
ct+1
ct
¶−γ
½
1 2
1
γσ t+1 + γ 2 σ 2t+1
1 = βEt Rtj+1
exp
2
2
"
µ
¶−γ
¾#
½
c
γ
(1
+
γ)
t+1
j
σ 2t+1
exp
= βEt Rt+1
ct
2
¾#
=
which is what we wanted to prove!
1.3
Part (c)
Specialize the above by assuming that,
σ 2t+1
= A − B log
Show that,
ˆ t
1 = βE
"µ
ct+1
ct
µ
ct+1
ct
¶−γ̂
¶
Rtj+1
.
#
holds with,
1
γ̂ = γ + γ (1 + γ) B
2½
¾
1
β̂ = β exp
γ (1 + γ) A
2
(3)
From part (b) we have
"
µ
¶−γ
µ
¶¸¾#
½
·
ct+1
ct+1
γ (1 + γ)
j
A − B log
=
exp
1 = βEt Rt+1
ct
2
ct
"
µ
¶−γ
¶¾#
½
¾
½
µ
ct+1
γ (1 + γ)
γ (1 + γ)
ct+1
j
=
exp
A exp −
B log
= βEt Rt+1
ct
2
2
ct
"
µ
¶−γ µ
¶− γ(1+γ)
¾#
½
B
2
c
c
γ
(1
+
γ)
t+1
t+1
j
A
=
exp
= βEt Rt+1
ct
ct
2
#
½
¾ "
µ
¶− γ+ γ(1+γ)
B]
2
γ (1 + γ)
ct+1 [
j
= β exp
A Et Rt+1
2
ct
which is what we wanted to prove (note we are assuming that A and B are
constants).
1.4
Part (d)
How can these results help explain the equity premium puzzle?
Recall that the equity premium puzzle is a quantitative puzzle, not a
qualitative one. From the Euler equation (3) for a risky asset j
"µ
"
¶−γ̂ #
¶−γ̂ #
µ
£ j ¤
ct+1
ct+1
j
ˆ t R
ˆ
1 = βE
+ βCov
Rt+1
(4)
,
t+1 Et
ct
ct
and for a risk free rate asset
ˆ RF Et
1 = βR
t+1
"µ
ct+1
ct
¶−ˆγ #
(5)
Note that what we need to explain the equity premium is a negative covari
ance term in equation (4) which is consistent with the data: in booms both
stock returns and consumption increase and viceversa in recessions. The issue
turns out to be quantitative in the sense that to match the equity premium
that emerges form the data we need a γ much higher than how it seems to be
from empirical studies. This comes from the fact that the covariance term is
bounded by the variance of consumption. The standard model implies that
what matters is the variance of aggregate consumption, but if we introduce
idiosyncratic shocks as in this exercise we can take into account the fact that
consumption is much more volatile. Note that from part (c) we can express
our Euler Equation as observationally equivalent to the standard one, where
the new parameters are γ̂ and β̂. And note that for B > 0 and A > 0 we have
that γ < γ̂ and β < β̂, which both make the premium larger! In particular,
the usual Mehra and Prescott calculation now implies that what is required
to be very high to match the data is now γ̂ which allows for some B to have
a much slower γ since
1
γ̂ = γ + γ (1 + γ) B
2
2
Problems 2
There is an handout for this in my mail box in front of the office.
3
3.1
Problem 3
Part (a)
Let us define the expected profits of the firm if it undertakes the investment
in period 1 Π (1) and the expected profits of the firm if it does not undertake
the investment Π (0). Then we have
Π (1) = π 1 + E (π 2 ) − I
and
Π (0) = 0
It is then clear that the firm will prefer to undertake investment at period 1
than not to undertake it iff Π (1) > Π (0), i.e.
π 1 + E (π 2 ) > I
3.2
Part (b)
Let us define the expected profits of the firm if it undertakes the investment
in period 2 Π (2). Then we can derive that
Π (2) = Pr (π 2 > I) E (π 2 − I | π 2 > I)
Solutions PS4 Macro III
1 Two-Sided Lack of Commitment: Stationary
Allocations
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sÜÜ àìÇëâîN
2
Risk Free rate Puzzle
HÇIsÜÜ àìÇëâî ÇïïÇÑâÜÖN
tâÜìÜ ÇìÜ ïòê áêìÑÜî ÑêñèïÜìÇÑïäèà âÜìÜN fêì âäàâ çÜóÜçî êá γ[ ïâÜ ÜçÇîïäÑäïö
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ïê ÑêèóäèÑÜ ïâÜ äèÖäóäÖñÇç ïê îÇóÜ Üèêñàâ ïê ÇÑâäÜóÜ ïâÜ ìÜíñäìÜÖ àìêòïâ ìÇïÜ
äè ÑêèîñéëïäêèN êè ïâÜ êïâÜì âÇèÖL âäàâ γ éÜÇèî âäàâ ìäîå ÇóÜìîäêè ÇèÖ ïâñî
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Ç çêòÜì êèÜ ïê ÇÑâäÜóÜ ïâÜ ÖÜîäìÜÖ àìêòïâ ìÇïÜ äè ÑêèîñéëïäêèN tê îÜÜ ïâÇï
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HÉI tâäî àìÇëâ ìÜ-ÜÑïî âêò ïâÜ îÜÑêèÖ ÜffÜÑï ÖêéäèÇïÜî äè ïâäî ÑÇîÜ áêì γ Éäà
ÜèêñàâN nêïäÑÜ ïâÇï òÜ èÜÜÖ γ ^ SP ïê ÉÜ ÇÉçÜ ïê ÜôëçÇäè ïâÜ óÇçñÜ êá ïâÜ ìäîå
áìÜÜ ìÇïÜN
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çêòÜìäèà ïâÜ óÇìäÇèÑÜ êá ÑêèîñéëïäêèN
1 These parameters imply a standard deviation for log-output of .29 and a first-order autocorrelation of .5, matching findings by Heaton and Lucas (1996) using the PSID.
2 Hint: Make sure you first check for perfect risk sharing. If full risk sharing is available
take that allocation. Otherwise compute the allocation that satisfies the requirements in part
(e), which may imply autarky or some insurance (watch out: do not compute an allocation
with more variability than autarky!).
T