Section Notes 7
Macroeconomic Analysis I
Economics GR5215, Fall 2021 (Miller)
October 29, 2021
Akshat Gautam and Susannah Scanlon 1
1 Consumption-Based Asset Pricing
1.1 Lucas Tree Model
The Lucas Tree model is a model for how to price assets that provide a stream of benefits that
are uncertain. In this model there are no firms and only households. There are a set of trees that
are owned by the households that produce fruit each year. Each tree is different and produces a
different amount of fruit from each other each period. Each period the households harvest the fruit
from the trees that they own and then buy and sell trees for fruit. The fruit spoils at the end of
each period so the household consumes all the fruit that they have after buying and selling trees.
To summarize, the households do the following each period in this order:
1. Households harvest fruit
2. Households trade trees for fruit between themselves
3. Households consume all their remaining fruit
In this model the amount of aggregate consumption each period is uncertain but exogenous,
it is equal to the total amount of fruit produced among all the trees. The amount of trees is also
fixed, households cannot plant any new trees. The only thing that is endogenous in this model is
the price of each tree at each date. Fully solving this model is very involved, but we can use it
solve for the prices of each tree based on the levels of expected consumption and dividends which
still gives interesting results.
To put a little bit more structure on things, let’s take the portfolio choice problem discussed in
Romer 8.5 and put a bit more structure on the problem. Suppose that there are n assets, {Ait}ni=1,
which have a price P it each period and a gross dividend D
i
t when one unit is held from t − 1 to t.
These dividends are the main source of uncertainty in this model. Note that because dividends are
uncertain, the consumption at each future point in time will be uncertain along with the interest
rate. In Micro you solve this type of problem more formally, but as noted in the previous recitation
slides the way you solve this type of problem is by looking at every single path the random variables
can take and solve for the optimal consumption path in each possible state of the world. From that
you can calculate expectations. Here we can take a few short cuts but that is gist of what we’re
doing.
1Updated October 28, 2021. Thanks to Miguel Acosta, Joe Saia & Eddie Shore for last years notes, and to those
that have pointed out errors in previous notes.
1
Now to actually solving this problem, we can write it as the following:
max
{Ct}∞t=0,{{A
i
t+1}
n
i=1}
∞
t=0
E
[
∞∑
t=0
1
(1 + ρ)t
u(Ct)
]
s.t. Ct +
n∑
i=1
P itA
i
t+1 =
n∑
i=1
DitA
i
t +
n∑
i=1
P itA
i
t
s.t. Terminal condition for each asset
Note that in the budget constraint, the right hand side is the total amount of resources available
to the agent if she sells everything she owns and the left hand side is the total amount spent. If
the agent does not trade any trees in period t then consumption is just equal to dividends. Also
note that the budget constraint does not have any expectations in it because the agent knows that
it must hold with certainty. Each period she cannot spend more than she has available to her, but
will not throw anything away. Because the budget constraint holds each period no matter what
happens in the world, it also holds in expectation. That means that the following statements are
all also true
E0
[
Ct +
n∑
i=1
P iAit+1 −
n∑
i=1
DitA
i
t
]
= 0
E1
[
Ct +
n∑
i=1
P iAit+1 −
n∑
i=1
DitA
i
t
]
= 0
Et−1
[
Ct +
n∑
i=1
P iAit+1 −
n∑
i=1
DitA
i
t
]
= 0
Et
[
Ct +
n∑
i=1
P iAit+1 −
n∑
i=1
DitA
i
t
]
= 0
We haven’t spent much time looking at infinite horizon problems, but the analysis is very similar
to finite horizon problems, except that (unless we are actually solving them) we spend less time
worrying about terminal conditions. Anyway, we can write the Lagrangian for this problem in the
following form:
L =
∞∑
t=0
E0
[
1
(1 + ρ)t
u(Ct)
]
−
∞∑
t=0
E0
[
λt
(
Ct +
n∑
i=1
P iAit+1 −
n∑
i=1
(P it +D
i
t)A
i
t
)]
To solve this we do the usual step of taking first order conditions. First we’ll take the first-order
condition for consumption in period 0
1
(1 + ρ)0
E0
[
u′(C0)
]
= E0 [λ0]
=> u′(C0) = λ0
2
Additionally, the FOC for asset i in period 0 is
E0
[
λ0P
i
0
]
= E0
[
λ1(P
i
1 +D
i
1)
]
=> λ0P
i
0 = E0
[
λ1(P
i
1 +D
i
1)
]
Now we need to do something a bit subtle. In period 1 the household is going to have more
information than in period 0, they’re going to have seen the dividends for each asset in period 1.
In period 1 they are going to use this information to revaluate their expected path of consumption
by resolving their Lagrangian with the new information. This is what we mean by the household
solves the consumption decision for every possible path of the world. Doing this is will give them
the following first order conditions for period 1
1
(1 + ρ)1
E1
[
u′(C1)
]
= E1 [λ1]
=>
1
1 + ρ
u′(C1) = λ1
And for each asset:
E1
[
λ1P
i
1
]
= E1
[
λ2(P
i
2 +D
i
2)
]
=> λ1P
i
1 = E1
[
λ2(P
i
2 +D
i
2)
]
While we don’t learn much from the FOC for the assets we do learn something important from
the FOC for consumption, that is that discounted marginal utility in period 1 will be equal to the
Lagrange multiplier no matter what the dividends are in period 1. The household knows this in
period 0 and will incorporate it into their decision making. Combining all this information we get
the following first order conditions
u′(C0) = λ0
u′(C1) = λ1
λ0P
i
0 = E0
[
λ1(P
i
1 +D
i
1)
]
Combining yields
u′(C0)P
i
0 =
1
1 + ρ
E0
[
u′(C1)(P
i
1 +D
i
1)
]
More generally, for all i and t:
u′(Ct)P
i
t =
1
1 + ρ
Et
[
u′(Ct+1)(P
i
t+1 +D
i
t+1
]
) (1)
Note that while at time t, λt+1 and Ct+1 are unknown and so cannot be taken out of the
expectations operator, it is known that they will be equal to one another so we can substitute one
for the other, even within the expectations operator.
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Let us pause here and ponder this equation for a minute. First let’s rearrange it slightly:
P it = Et
11 + ρ · u
′(Ct+1)
u′(Ct)︸ ︷︷ ︸
SDF
· (P it+1 +D
i
t+1)
This relates the price of asset i to it’s expected dividend and the consumer’s time t stochastic
discount factor. First, the SDF captures “discounting” in the sense of the underlying patience
of the consumer. A patient consumer has a low discount rate or, equivalently, a high 1
1+ρ
. The
more patient they are, then, the more they are willing to pay for the asset today to get the return
Dit+1 tomorrow. An impatient consumer, on the other hand, would rather spend the money on
the consumption good today, so the asset would look less valuable to them. What distinguishes
the SDF from the normal discount factor, however, is that it is “utility-weighted” in a very special
way. What matters for consumers when deciding between consuming today vs. tomorrow are
the relative marginal utilities, and the SDF contains this ratio. Suppose, for example, that the
consumer expects to be very rich tomorrow, so that in expectation u′(Ct+1) is low (remember—
diminishing marginal utility (or, u′′ < 0)). For a fixed level of consumption today, this implies a
low SDF—the consumer knows they will be rich tomorrow, so there is no use in delaying today’s
consumption by buying an asset. Put differently, the consumer does not much value this asset or
its return, so they are not willing to spend much on it.
Now, let’s return to equation (1). If we define 1 + rt+1 ≡
P it+1+D
i
t+1
P it
, then (1) becomes
u′(Ct) =
1
1 + ρ
Et
[
u′(Ct+1)(1 + r
i
t+1)
]
Suppose that utility is quadratic. Then this is
1− aCt =
1
1 + ρ
Et
[
(1− aCt+1)(1 + rit+1)
]
=
1
1 + ρ
Et
[
1 + rit+1
]
−
a
1 + ρ
Et
[
Ct+1(1 + r
i
t+1)
]
(2)
Now, remember that for any two random variables X and Y , E[XY ] = E[X] · E[Y ] + Cov(X,Y ),
so that we can rewrite equation (2) as
1− aCt =
1
1 + ρ
Et
[
1 + rit+1
]
−
a
1 + ρ
(
Et[Ct+1]Et[1 + rit+1] + Covt(Ct+1, 1 + r
i
t+1)
)
Rearranging, this is
Et[1 + rit+1](1− aEt[Ct+1]) = (1 + ρ)(1− aCt) + aCovt(Ct+1, 1 + r
i
t+1)
and dividing we get
Et[1 + rit+1] =
(1 + ρ)(1− aCt) + aCovt(Ct+1, 1 + rit+1)
1− aEt[Ct+1]
.
Now the important part is the following: suppose that there is risk-free asset, for which we will
use the subscript F . That is, the consumer knows DFt+1 at time t. Since she also knows P
F
t , this
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means that she knows rFt+1 at time t; that is, from the perspective of time t information, r
F
t+1 is a
constant. The covariance of any random variable with a constant is zero. Thus,
Et
[
1 + rFt+1
]
=
(1 + ρ)(1− aCt)
1− aEt[Ct+1]
.
So if we subtract the risk free pricing condition from the asset i condition, we get
Et[rit+1]− r
F
t+1 =
aCovt(Ct+1, 1 + r
i
t+1)
1− Et[Ct+1]
On the left hand side, we have what we call the “expected return premium” relative to the risk-free
rate—that is, how much larger is the return on this asset than that of the risk-free asset. Or, how
much does the consumer need to be compensated to take on this extra risk. The answer to all
of these questions rests importantly on the covariance term. If the asset has a high return when
consumption is high (i.e., Covt(Ct+1, 1 + r
i
t+1) > 0), then it had better offer a very high return on
average. This is capturing risk not in a pure volatility sense but, instead, along the margin that
is important to the consumer—the volatility of their consumption. An asset that only pays off
in a boom had better pay off a lot, because when the recession comes (and the asset isn’t paying
anything) the consumer is going to need some money to smooth their consumption as they’d like.
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Consumption-Based Asset Pricing
Lucas Tree Model