asset pricing: the lucas tree model
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Asset Pricing: The Lucas Tree Model
Kaiji ChenThe University of Hong Kong
October 21, 2009
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Review of Last Class
Denition of Great Depression
A large negative deviation from trend (balanced) growth path.
On balanced growth path, capital output ratio is constant, and all per
capita variables grow at constant rate except hours per working age
person.
episode including not only sharp decline but also probably slow recovery.
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Great Depression Methodology
Growth accounting: various shocks aect aggregate output during de-
pression through three channels: eciency that inputs are combined
together for production, capital input, and labor input (from both sup-
ply and demand sides).
identify the quantitative importance of these channels through dynamic
general equilibrium model.
Simulation tells us sharp declines in TFP are important for the output
drop during U.S. Great Depression, but not the slow recovery.
Diagnose depression by measuring deviation of the models rst-order
conditions.
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Tell us at which margins the standard model misses so as to provide
directions to which types of frictions we need when constructing new
models?
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Road map of this Class
Lucas Tree Model
Applications of Lucas Tree Model to risk free assets and risky assets.
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1. LUCAS TREE MODEL
1 Lucas Tree Model
Idea
What is nancial asset? Contract that promises to deliver an amount ofgoods in some future periods and particular states.
Whats the roles of nancial asset? Agents would like to smooth consump-
tion across time and states. In a market economy, this is implemented bybuying and selling nancial assets, which transfer resources across time
and states.
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1. LUCAS TREE MODEL
How the price of an asset is determined? In a competitive equilibrium,
prices are such that all markets clear (demand meets supply).
Previously, our focus is optimal resource allocation (across time), given
resources constraint (in a social planners problem) or individual agents
budget constraints (in a competitive equilibrium).
Alternatively, given equilibrium quantities and demand function, we can
back out equilibrium prices.
In particular, take consumption process as given, solve for the equilibrium
prices of given nancial assets that transfer resource across time and dif-
ferent states.
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1. LUCAS TREE MODEL
The Economy
Consider an economy inhabited by a large number of identical agents.
Endowment: the only durable good in the economy is a set of trees, which
are equal in the number to the number of people in the economy.
Each agent starts life at time zero with one tree.
Each period, each tree yields fruit or dividends in the amount dt to its
owner at the beginning of period t.
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1. LUCAS TREE MODEL
Technology: Fruits cannot be stored. Dividends are exogenous and follow
a stochastic process.
Preference: agents in this economy consume a single good, which is fruit.
E0
1Xt=0
t
u (ct)
where u () is concave, strictly increasing and twice continuously dieren-
tiable.
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1. LUCAS TREE MODEL
Market structure: how can the agent transfer resource across time? Assume
a competitive market in trees.
Ownership of a tree at the beginning of period t entitles the owner to
receive the dividend in period t and to have the right to sell the tree at
price pt in terms of consumption good:
Since all agents are identical in terms of preference and endowment, we
can assume there is a representative agent in this economy.
In equilibrium, there is only 1 tree (supply). Our purpose is nd the
price of the tree that will make the aggregate demand of the tree equal
to 1.
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1. LUCAS TREE MODEL
Solution Strategies
Find the competitive equilibrium allocation via the social planners problem.
Calculate the FOCs for individual agents with the opportunity to buy and
sell the share of trees (assets)
Find the equilibrium prices that support the competitive equilibrium allo-
cation.
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1. LUCAS TREE MODEL
Step 1: Social Planners Problem
maxfctg
1t=0
E0
1Xt=0
tu (ct)
s:t: ct dt
Solution: ct = dt; 8t:
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1. LUCAS TREE MODEL
Step 2: Representative Agents Problem in a Competitive Market
Economy
Let st be the share of tree held at the beginning of time t and pt the price
of one share of tree at time t:
The problem of the representative agent is
maxfct;st+1g
1t=0
E0
1Xt=0
tu (ct) (1)
subject to
ct + ptst+1 = (pt + dt) st
s0 = 1 given
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1. LUCAS TREE MODEL
First-order condition
ptu0 (ct) = Etu
0 (ct+1) (pt+1 + dt+1)
The utility (shadow) cost of buying one tree today is ptu0 (ct) :
Tomorrow, each tree delivers payo pt+1 + dt+1, which is a random vari-
ables: an investor does not know exactly how much he will get from his
investment, but he can assess the probability of various possible outcomes.
The shadow value of the tree, discounted to today, is Etu0 (ct+1) (pt+1 + dt+1) :
At the margin, an investor (consumer) is indierent between buying an
additional unit of tree.
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1. LUCAS TREE MODEL
Transversality condition
limk!1
Etku0
ct+k
pt+kst+k = 0 (2)
If the expressions in equation was positive, the agent would be over-
accumulating assets so that a higher expected lifetime utility could be
achieved by increasing consumption today.
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1. LUCAS TREE MODEL
The Representative Agents Problem in Recursive Form
The states of the economy
aggregate state: d: p = p (d)
individual state: s:
Because d is time-invariant Markov process, the consumers problem is
time-invariant.
The controls: c; s0:
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1. LUCAS TREE MODEL
Bellman Equation
V (d; s) = maxc;s0
nu (c) + E
hV
d0; s0
j dio
(3)
subject to
c + ps0 = (p + d) s
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1. LUCAS TREE MODEL
First-order Conditions
u0 (c) =
p = EhVs0 d0; s0 j di
Envelop Condition
Vs (d; s) = (p + d)
Euler equation
u0 (c)p = Eh
u0
c0
p0 + d0
j di
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1. LUCAS TREE MODEL
Step 3: Equilibrium
Denition of Sequential Market Equilibrium: Given the stochastic process
fdtg1t=0 ; and the initial endowment s0 = 1; a sequential market equilib-
rium consist of allocation fct; st+1g1t=0 ; and prices fptg
1t=0 such that
Given fptg1t=0 ; fct; st+1g
1t=0 solve the representative consumers prob-
lem (1).
fptg1t=0 is such that the tree market clears: st+1 = 1; 8t: (This implies
ct
= dt)
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1. LUCAS TREE MODEL
Denition of Recursive Competitive Equilibrium: a recursive competitive
equilibrium for this economy consists of value function V (d; s) ; a set of
decision rules c (d; s) ; s0 (d; s) ; and price function p (d) such that
Given the price p (d) ; V (d; s) solve the representative consumers
problem (3) ; with the decision rules c (d; s) ; s0 (d; s) :
Market clear. p (d) is such that s0 (d; s) = 1:
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1. LUCAS TREE MODEL
Equilibrium Price
Our goal is to obtain the price that will induce the consumer to consume
the equilibrium allocation.
We impose the equilibrium allocation on the FOCs: (ct = dt)
pt = Etu0 (dt+1)
u0 (dt)(pt+1 + dt+1) (4)
The current price of a tree is equal to the expectation of the product
of the future payo on that tree with the intertemporal marginal rate of
substitution.
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1. LUCAS TREE MODEL
Equation (4) can be generalized as
pt = Et (mt+1xt+1)
where mt+1 = u0(dt+1)
u0(dt)is also called stochastic discount factor (SDF),
and xt+1 is a payo, the value of investment at time t + 1 (e.g. for a
stock, the payo is pt+1 + dt+1).
it says that one can incorporate all risk corrections by dening a single
SDFthe same one for each assetand putting it inside the expectation.
The correlation between the random components of the common dis-
count factor m and the asset-specic payo x generate asset-specic
risk corrections.
In fact, all asset pricing models amount to alternative ways of connect-
ing the stochastic discount factor to data.
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1. LUCAS TREE MODEL
Forwarding (4) by one period and use the law of iterated expectation
(EtEt+1 () = Et ()), we arrives the following expression
ptu0 (dt) = Et
1Xj=1
ju0
dt+j
dt+j + limk!1
Etku0
ct+k
pt+k (5)
No-arbitrage condition implies that the last term in (5) must be zero.
If this term is strictly positive, then the marginal utility gain of selling
share exceeds the marginal utility loss of holding the asset forever and
consuming the future streams of dividend.
Asset bubble can also be ruled out by directly referring to transversality
condition (2) ; and market clearing condition (st = 1, and ct = dt).
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1. LUCAS TREE MODEL
As a result, equation (5) becomes
pt = Et
1Xj=1
ju0
dt+j
u0 (dt)dt+j (6)
Equation (6) says that the price of a share of the tree (asset) is an ex-
pected discounted stream of future dividends, with the discount factors
intertemporal marginal rates of substitution. Here the discount factor is
time varying and stochastic, since consumption (or dividends) is time vary-
ing and stochastic.
ju0(ct+j)
u0(ct)is marginal rate of substitution between time t + j goods and
time t goods (the personal valuation (price) of one unit of time t + j
dividend in terms of time t dividend.)
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1. LUCAS TREE MODEL
Special Case 1: log utility
pt = Et
1Xj=1
jdt
dt+jdt+j
=
1 dt
Price of a share of asset only depend on it current dividend payo, not the
future payo.
In particular, assume that the fruits only take two values dt 2 fd1; d2g,
with d1 > d2: Then p1 > p2, that is the price of the tree is high in the
state when aggregate output (fruit) is high:
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1. LUCAS TREE MODEL
Intuition: when current output is high, consumer tends to transfer
resources from today to tomorrow by purchasing the tree (by saving).
As a result, the demand for the tree is high, pushing up the price of
the tree.
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1. LUCAS TREE MODEL
Intuition for the impact of an increase in future dividend dt+j on
pt
When dt+j becomes larger, this have two eects.
Income eects: an increase in dt+j increase the agents life time income.
Therefore agent would like to increases consumption at each period, in-
cluding time t:
Since dt is not changed, an increase in desirable ct induce agent to
reduce the holdings of shares of the tree (decrease in savings).
This tend to push the demand curve for asset to the left.
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1. LUCAS TREE MODEL
Substitution eects: an increase in dt+j implies a higher future return for
investing in shares of the tree. Therefore, agent would like to postpone
consumption and increase savings.
This tend to push the demand curve for asset to the right.
When utility is log, income eect and substitution eect of an increase in
future dividend dt+j cancel out.
This leaves the demand curve for share of asset unchanged.
Hence, the price of a share of the tree is unchanged.
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1. LUCAS TREE MODEL
Special Case 2: Risk Neutrality (u0 (c) = c)
From (4) ; we have
pt = Et (pt+1 + dt+1)
Or in terms of returns, we have
Et
"pt+1 + dt+1
pt
#=
1
1 (7)
Equation (7) states that the rate of return on each asset is unpre-dictable given current information, which is taken in Finance literature
as the implication of ecient market hypothesis.
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1. LUCAS TREE MODEL
Special Case 3: A nite-state version
Assume dividend follows a time invariant Markov process dt 2n
d1;:::;dno
prob
dt+1 = di j dt = dj
= ji
The (n n) matrix with element ji is called a stochastic matrix.
The matrix satisesnX
l=1
kl = 1 for each k:
All elements in are nonnegative.
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1. LUCAS TREE MODEL
Express equation (4) as
ptu0 (dt) = Etpt+1u
0 (dt+1) + Etdt+1u0 (dt+1) (8)
Express the price at t as a function of the state p
dk
= pk:
Dene ptu0 (dt) = p
ku0
dk
vk; k = 1; :::n:
Also dene k = Etpt+1u0 (dt+1) =
n
Xl=1
dlu0 dl kl:
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1. LUCAS TREE MODEL
Then equation (8) can be expressed as
pku0
dk
= nX
l=1
plu0
dl
kl + nX
l=1
dlu0
dl
kl
or in matrix terms,
v = + v
This equation has a unique solution
v = (I )
1
(9)
The price of an asset at state k can be found from pk = vk=u0
dk
:
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2. LUCAS TREE MODEL WITH RISK-FREE BOND
2 Lucas tree Model with Risk-free Bond
Now consider the same problem , but agent can also buy and sell a risk-freeone-period discounted bond for the price R1t (Rt can be seen as the rate
of return for a one period risk free bond); which deliver one unit of goodthe next period in any state.
Assume when agents were born, they are endowed with no-risk free bondb0 = 0.
The agents budget constraint is
ct + ptst+1 + R1t bt+1 = (pt + dt) st + bt
In recursive form, R = R (d) :
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2. LUCAS TREE MODEL WITH RISK-FREE BOND
We follows the same steps in solving for the equilibrium bond
prices.
In Step 1, by solving the social planners problem, we still get ct = dt; 8t:
In Step 2, we solve for the representative agents problem and obtain
R1t = Etu0 (ct+1)
u0 (ct)
In Step 3, equilibrium condition gives bt+1 = 0: There is no one to buythese bonds or sell to, since all agents are the same in terms of preference
and endowments.
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2. LUCAS TREE MODEL WITH RISK-FREE BOND
Our goal then is to nd the equilibrium price R1t that induce agents to
choose to buy 0 bond.
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2. LUCAS TREE MODEL WITH RISK-FREE BOND
Again we impose the equilibrium allocation ct = dt to the rst-order
condition with respect to bond demand.
R1t = Etu0 (dt+1)
u0 (dt)
= 1
u0 (dt)Etu
0 (dt+1)
or
Rt =u0 (dt)
Etu0 (dt+1)
In general, we can price any asset in the same way.
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2. LUCAS TREE MODEL WITH RISK-FREE BOND
Factors aecting the price of a risk-free bond, R1 (or real interest
rate, R)
The price of risk-free bond is low when people are impatient, i.e. when
is low. If everyone wants to consume now (thus having low demand
for bond), it takes a high interest rate (a low price of risk-free bond) toconvince to save (buy the bond).
The price of risk-free bond increases with Etu0 (dt+1) and decreases with
u0 (dt), that is, decreases with consumption growth).
The more abundant is todays resource relative to tomorrows, the more
valuable is tomorrows consumption relative to todays (the larger isEtu
0(dt+1)u0(dt)
).
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2. LUCAS TREE MODEL WITH RISK-FREE BOND
The larger is the demand for this asset to transfer resource from today
to tomorrow.
Hence the higher price is risk-free bond (supply is always given).
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2. LUCAS TREE MODEL WITH RISK-FREE BOND
assume u0 (c) = c (CRRA utility):
R1t =1
Et
dt
dt+1
!
The prices of risk-free bond (real interest rate) are more sensitive to con-
sumption growth if is large.
If utility is highly curved, then investors care more about maintaining
a consumption prole that is smooth over time.
hence they are less willing to rearrangement consumption over time in
response to changes in interest rate (change of price of bond).
Thus it takes a large interest rate change to induce him to a given
consumption growth.
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2. LUCAS TREE MODEL WITH RISK-FREE BOND
Reinterpretation of the above pricing equation
Consumption growth is high when real interest rate is high (or the prices
of bond are low)
Consumption is less sensitive to real interest rate (the price of risk-free
bond ) as the desire for a smooth consumption stream, captured by ;
rise.
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