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 (d_t) dividends expressed in terms of units of the consumption good (milk), and at the end of the period is worth q₀ (q_t) 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 s_t the amount of cows owned at the end of period t (beginning of period t+1)

The household's problem

$$max_{c_0,c_1,c_2,s_0,s_1} u(c_0)+\beta u(c_1)+ \beta^2 u(c_2)$$

subject to:

$$q_0 s_0+c_0 \leq (q_0+d_0)1$$

$$q_1 s_1+c_1 \leq (q_1+d_1)s_0$$

$$c_2 \leq d_2 s_1$$

in equilibrium all three constraints will hold with = sign, so we can substitute the constraint in the objective function:
$\begin{array}{l} max_{s_0,s_1}\\ u((q_0+d_0)1-q_0 s_0)+\beta u((q_1+d_1)s_0-q_1 s_1)+ \beta^2 u(d_2 s_1) \end{array}$

The first order condition (FOC) with respect to s₀ is:

$$u'(c_0)q_0=\beta u'(c_1) (q_1+d_1)$$

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:

$$\frac{u'(c_0)}{\beta u'(c_1)}=\frac{q_1+d_1}{q_0}=1+\frac{q_1-q_0}{q_0}+\frac{d_1}{q_0}$$

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:

$$u'(c_1)q_1=\beta u'(c_2) d_2$$

The two FOC's combined give:

$$u'(c_0)q_0=\beta u'(c_1) d_1+\beta^2 u'(c_2) d_2$$

or

$$q_0=\frac{\beta u'(c_1)}{u'(c_0)} d_1+\frac{\beta^2 u'(c_2)}{u'(c_0)}d_2$$

In other words, we have an equation that gives us the price of the asset (cow-stock)

General equilibrium

in each period c_t=d_t (nothing else but cows in this economy); so the pricing formula becomes:

$$q_0=\frac{\beta u'(d_1)}{u'(d_0)} d_1+\frac{\beta^2 u'(d_2)}{u'(d_0)}d_2$$

in particular, if dividends are constant over time (d_t=d all t):

$$q_0=\beta d+\beta^2 d$$

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 d_t dividends expressed in terms of units of the consumption good (milk), and at the end of the period is worth q_t 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 s_t the amount of cows owned at the end of period t (beginning of period t+1)

$$max_{{c_t}_{t=0}^\infty} \sum_{t=0}^\infty \beta^t u(c_t)$$

subject to the budget constraint (which holds in each period t):

$$q_t s_t+c_t \leq (q_t+d_t)s_{t-1}$$

as before, in equilibrium all these constraints will hold with = sign, so we can substitute the constraint in the objective function:

$$max_{{s_t}_{t=0}^\infty} \sum_{t=0}^\infty \beta^t u((q_t+d_t)s_{t-1}-q_t s_t)$$

Let us focus on the first order condition (FOC) with respect to s_t
Notice that the term s_t appears in the sum above only in the two terms:

$$\beta^t u((q_t+d_t)s_{t-1}-q_t s_t) + \beta^{t+1} u((q_{t+1}+d_{t+1})s_t-q_{t+1} s_{t+1})$$

so the first order condition is:

$$\beta^t u'(c_t)q_t=\beta^{t+1} u'(c_{t+1}) (q_{t+1}+d_{t+1})$$

which can be rewritten as:

$$u'(c_t)q_t=\beta u'(c_{t+1}) (q_{t+1}+d_{t+1})$$

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:

$$u'(c_{t+1})q_{t+1}=\beta u'(c_{t+2}) (q_{t+2}+d_{t+2})$$

, and substitute for u'(c_t+1)q_t+1 in the previous expression. we obtain:
$\begin{array}{lll} u'(c_t)q_t & = &\beta u'(c_{t+1}) d_{t+1} +\beta^2 u'(c_{t+... ...um_{i=1}^2 \beta^i u'(c_{t+i}) d_{t+i} +\beta^2 u'(c_{t+2}) q_{t+2} \end{array}$

following the same logic, we can write the FOC for periods t+2,..,t+T, and substitute again and again for u'(c_t+2) q_t+2,u'(c_t+2) q_t+2,....
this way we obtain the following expression:

$$u'(c_t)q_t= \sum_{i=1}^T \beta^i u'(c_{t+i}) d_{t+i} +\beta^T u'(c_{t+T}) q_{t+T}$$

which can be rewritten as:

$$q_t= \sum_{i=1}^T \beta^i \frac{u'(c_{t+i})}{u'(c_t)} d_{t+i} +\beta^T \frac{u'(c_{t+T})}{u'(c_t)} q_{t+T}$$

What happens to the right hand side of this expression as we let $T \rightarrow \infty$?

If it is true that $lim_{T \rightarrow \infty} \beta^T \frac{u'(c_{t+T})}{u'(c_t)} q_{t+T}=0$, then we obtain:

$$q_t=\sum_{i=1}^{\infty} \beta^i \frac{u'(c_{t+i})}{u'(c_t)} d_{t+i}$$

the condition

$$lim_{T \rightarrow \infty} \beta^T \frac{u'(c_{t+T})}{u'(c_t)} q_{t+T}=0$$

is called transversality condition. let us postpone the discussion of why the transversality condition holds, and focus on the interpretation of the above expression.

$$q_t=\sum_{i=1}^{\infty} \beta^i \frac{u'(c_{t+i})}{u'(c_t)} d_{t+i}$$

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, $\beta^i \frac{u'(c_{t+i})}{u'(c_t)}$, 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 ($\beta^i$ 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 ( $\frac{u'(c_{t+i})}{u'(c_t)}$ term)

Equilibrium

in equilibrium it must be that in each period c_t=d_t (nothing else but stocks in this economy) and that s_t=1

the pricing formula then becomes:

$$q_t=\sum_{i=1}^{\infty} \beta^i \frac{u'(d_{t+i})}{u'(d_t)} d_{t+i}$$

in particular, if dividends are constant over time (d_t=d all t):

$$q_t=\frac{\beta}{1-\beta} d$$

comparative statics: in an economy where people are less impatient (higher $\beta$) 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:

$$lim_{T \rightarrow \infty} \beta^T \frac{u'(c_{t+T})}{u'(c_t)} q_{t+T}=0$$

• first, notice that is dividends d_t are bounded away from zero and infinity in all periods ( $0<a<d_t<A<\infty$ all t), the term $\frac{u'(c_{t+T})}{u'(c_t)}$ must also be bounded away from zero and infinity, since c_t=d_t in equilibrium
• second, notice that the term $\beta^T$ goes to zero as T goes to infinity
• this implies that, as long as q_t+T does not explode towards infinity, the transversality condition must hold
• can we have an equilibrium in which q_t+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