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 <math>u(c)=ln(c)