User:Alexnally/Currently Working On/Arrow Debreu Model Setup

Model
Agent $$i$$ seeks to maximize his/her lifetime (discounted) utility across all possible (infinite) state sequence outcomes, $$s^t=(s_0,s_1,s_2,...,s_t)$$where each sequence has probability $$\pi_t(s^t|s^0)$$of occuring, given an initial state $$s^0=s_0$$.

The agent's problem is as follows

$$ max \sum_{t \mathop =0}^\infty \sum_{s^t |s^0} \beta^t \pi_t(s^t|s^0)u(c_t^i(s^t)) $$

subject to the lifetime budget constraint

$$\sum_{t \mathop =0}^\infty \sum_{s^t |s^0} q_t(s^t)c_t^i(s^t)=\sum_{t \mathop =0}^\infty \sum_{s^t |s^0}q_t(s^t)y_t^i(s^t)$$

where $$q_t(s^t)$$is the price of 1 unit of $$s^t$$consumption (i.e. of a security that pays 1 unit of consumption in $$t$$ given sequence $$s^t$$occurred) in terms of $$s^0$$consumption; $$c_t^i(s^t)$$is agent $$i$$'s consumption in time $$t$$ given sequence $$s^t$$ occurred; and $$y$$ is income.

Using Lagrande multipliers, we obtain the first order condition

$$\beta^t \pi_t(s^t|s^0)u'(c_t^i(s^t))=L^i q_t(s^t)$$

Nothing that $$q_0(s^0)=1$$ we obtain $$L^i=u'(c_0^i(s^0))$$ and our optimality condition becomes

$$q_t(s^t)=\beta^t \pi_t(s^t|s^0)\frac{u'(c_t^i(s^t))}{u'(c_0^i(s^0))}$$

which are the time-zero prices (in terms of $$s^0$$consumption) of securities that pay one unit of $$s^t$$consumption.

SUBSECTION

Market clearing conditions

SUBSECTION

Am important implication of this model is that if all agents have the same constant relative risk aversion (CRRA) utility function, of the form $$u(c)=\frac{c^{1-\sigma}}{1-\sigma}$$, then only aggregate income matters in determining securities prices. The outline of the proof is as follows:

From the first order conditions we have

$$q_t(s^t)=\beta^t \pi_t(s^t|s^0)\frac{u'(c_t^i(s^t))}{u'(c_0^i(s^0))}=\beta^t \pi_t(s^t|s^0)(\frac{c_0^i(s^0)}{c_t^i(s^t))})^\sigma=\beta^t \pi_t(s^t|s^0)(\frac{c_t^i(s^t)}{c_t^i(s^t))})^\sigma$$

and thus aggregate income matters in determining securities prices.

Example
Let u(c)=ln(c)