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CLASS 1: ASSET PRICING AND FORWARD LOOKING BEHAVIOR
- asset pricing in an infinite horizon model
- show that the price of an asset equals the ``discounted´´ present value of the
future stream of dividends
- where the discount puts a ``premium´´ on those stocks which pay dividends
sooner than later, and on those stocks which pay high dividends when consumption is
low
Asset pricing in a simple finite period model (Lucas' Cows -Trees- Model)
Assumptions:
- the economy lasts three periods, t=0,1,2
- at the beginning of period 0, the household has 1 cow,
which in period 0 (t) produces d0 (dt) dividends expressed in
terms of units of the consumption good (milk), and at the end of the period
is worth q0 (qt) units of the consumption good
- at the end of period 2 the world ends, so q2=0
- in each period the household decides how much to consume,
and how much to invest, that is, how many cows to own
let us denote st the amount of cows owned at the end of period t
(beginning of period t+1)
The household's problem
subject to:
in equilibrium all three constraints will hold with = sign, so
we can substitute the constraint in the objective function:
The first order condition (FOC) with respect to s0 is:
notice that this is a marginal condition: keeping s1 and
s-1 fixed, the household can raise its utility by
changing s0 (consuming more or less in period 0 and 1)
it can be rewritten as:
interpretation: marginal Rate of substitution (MRS) = marginal rate of transformation (MRT)
MRT=1+ capital gain + dividend payment
the second FOC (with respect to S1) is:
The two FOC's combined give:
or
In other words, we have an equation that gives us the price of the asset (cow-stock)
General equilibrium
in each period ct=dt (nothing else but cows in this economy); so the
pricing formula becomes:
in particular, if dividends are constant over time (dt=d all t):
ASSET PRICING IN AN INFINITE HORIZON MODEL
Assumptions:
- the economy lasts an infinite number of periods, t=0,1,2,...
- there is a representative household in the economy
- the aggregate supply of cows (stock) is normalized to 1, and is constant
in all periods. However the household see itself as being able to choose how much
to invest in cows (stocks)
- in each period t the cow (stock) produces dt dividends expressed in
terms of units of the consumption good (milk), and at the end of the period
is worth qt units of the consumption good
- in each period the household decides how much to consume,
and how much to invest, that is, how many cows to own
let us denote st the amount of cows owned at the end of period t
(beginning of period t+1)
subject to the budget constraint (which holds in each period t):
as before, in equilibrium all these constraints will hold with = sign, so
we can substitute the constraint in the objective function:
Let us focus on the first order condition (FOC) with respect to st
Notice that the term st appears in the sum above only in the two terms:
so the first order condition is:
which can be rewritten as:
again, this is the marginal condition (a necessary but not
sufficient condition for optimality) from period t to t+1.
let us write the same condition for period t+1:
, and substitute for
u'(ct+1)qt+1 in the previous
expression. we obtain:
following the same logic, we can write the FOC for periods
t+2,..,t+T,
and substitute again and again for
u'(ct+2) qt+2,u'(ct+2) qt+2,....
this way we obtain the following expression:
which can be rewritten as:
What happens to the right hand side of this expression as we let
?
If it is true that
,
then we obtain:
the condition
is called transversality condition. let us postpone the discussion of why the
transversality condition holds, and focus on the interpretation of the above
expression.
the above expression says that:
- the price of a stock is equal to the discounted present value of all
future dividends
- where the discount rate,
,
is equal to
the marginal rate of substitution (MRS) from time t to time t+i.
This implies that:
- other things being equal, the value of a stock that pays dividends
soon is higher that the value of a stock that pays dividends
later (
term)
- other things being equal, the value of a stock that pays dividends
when consumption is low is higher that the value of a stock that pays dividends
when consumption is high (
term)
Equilibrium
in equilibrium it must be that in each period ct=dt (nothing else but stocks
in this economy) and that st=1
the pricing formula then becomes:
in particular, if dividends are constant over time (dt=d all t):
comparative statics: in an economy where people are less impatient (higher )
the value of stocks is higher: people want to save more, but since the quantity
of stocks (trees, cows) is given, the desire to save more simply bids the
price of the stock up.
we can now discuss why the transversality condition must hold in this model:
- first, notice that is dividends dt are bounded away from zero
and infinity in all periods (
all t), the term
must also be bounded away from zero and infinity,
since ct=dt in equilibrium
- second, notice that the term
goes to zero as T goes to infinity
- this implies that, as long as qt+T does not explode towards infinity, the
transversality condition must hold
- can we have an equilibrium in which qt+T explodes towards infinity? No,
because the agents' wealth would also be shooting up to infinity, and it would not
be optimal for the agent not to consume part of that wealth
next time: pricing of durable goods
references:
- Lucas, Robert Jr., 1978, ``Asset Prices in an Exchange Economy,´´
Econometrica, vol. 42, pp. 1429-45
- Douglas T. Breeden, 1979, ``An Intertemporal Asset Pricing Model With Stochastic Consumption
And Investment Opportunities, ''Journal of Financial Economics 7, 265-296
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Marco Del Negro
2000-01-10