Arrow–Debreu Equilibrium

An Edgeworth box for consumption in each state. Two people share each other's risk here.

With no risk for the whole economy, they insure each other. Both consume exactly the same in every state. That is full insurance, the mutuality principle. When the whole economy shakes, everyone shakes together.

For example, Minsu grows 🍎 apples and Jihoon grows 🍌 bananas. If only Minsu's harvest fails, Jihoon shares. If only Jihoon's fails, Minsu shares. Both eat the same every year. But if the whole country's harvest fails, there is nothing to share.

c_{is} = \frac{\pi_s^{1/\gamma} q_s^{-1/\gamma} W_i}{\sum_t \pi_t^{1/\gamma} q_t^{1-1/\gamma}}

E_1 = E_2 \implies c_{i1} = c_{i2} \quad \text{(full insurance)}

Part of these learning paths

General Equilibrium and Welfare

Examples & Exercises

With log utility, $\pi_1 = 1/2$ , aggregate endowments $E_1 = E_2 = 10$ (no aggregate risk), and $e^1 = (7, 3)$ , verify by direct computation that the Arrow–Debreu equilibrium gives both consumers full insurance (state-independent consumption).

q_s = \frac{\pi_s (W_1 + W_2)}{E_s}, \qquad c_{is} = \frac{\pi_s W_i}{q_s}

No aggregate risk — both consume

(5, 5)

Show proof

$q_1 = 0.5 \cdot 20/10 = 1$ , $q_2 = 1$ . Wealths: $W_1 = 7 + 3 = 10$ , $W_2 = 10$ .
Demands are $c_{i1} = 0.5 \cdot 10/1 = 5$ , $c_{i2} = 5$ — both consume state-independent $(5, 5)$ . This is the mutuality principle: with no aggregate risk, risk-averse consumers fully share idiosyncratic risk.

Introduce aggregate risk into the same setup: $E_1 = 12$ , $E_2 = 8$ .

Find equilibrium consumption and show that aggregate risk cannot be insured away.

\frac{c_{11}}{c_{12}} = \frac{E_1}{E_2} = 1.5

E = (12, 8)

— consumption

(5.75, 3.83)

, tracking the aggregate.

Show proof

$q_1 : q_2 = \frac{0.5}{12} : \frac{0.5}{8} = 2:3$ . With $r = q_1/q_2 = 2/3$ , $W_1 = \frac{2}{3} \cdot 7 + 3 = \frac{23}{3}$ .
$c_{11} = 0.5 \cdot \frac{23}{3} / \frac{2}{3} = 5.75$ , $c_{12} = 0.5 \cdot \frac{23}{3} = 3.833$ . And $c_{11}/c_{12} = 1.5 = E_1/E_2$ : everyone's consumption moves in lockstep with the aggregate — aggregate risk can be shared but not eliminated.

15. General Equilibrium under Uncertainty