# Arrow-Debreu Market with Uncertainty

I am reading some lecture note on stochastic macro model. Say an endowment economy with agent $$i=1,2$$, who receive the random endowment each period $$e_{t}^{i}\left(s^{t}\right)$$ where $$s^{t}=\left(s_{0}, s_{1}, \ldots s_{t}\right)$$ is the event history with probability $$\pi_{t}\left(s^{t}\right)$$.

Then a competitive Arrow-Debreu equilibrium is defined as prices prices $$\left\{\hat{p}_{t}\left(s^{t}\right)\right\}_{t=0, s^{t} \in S^{t}}^{\infty}$$ and allocations $$\left(\left\{\hat{c}_{t}^{i}\left(s^{t}\right)\right\}_{t=0, s^{t} \in S^{t}}^{\infty}\right)_{i=1,2}$$ such that

• 1 Given $$\left\{\hat{p}_{t}\left(s^{t}\right)\right\}_{t=0, s^{t} \in S^{t}}^{\infty}$$, for $$i=1,2,\left\{\hat{c}_{t}^{i}\left(s^{t}\right)\right\}_{t=0, s^{t} \in S^{t}}^{\infty}$$ solves \begin{aligned} \max _{\left\{c_{t}^{i}\left(s^{t}\right)\right\}_{t=0, s^{t} \in S^{t}}^{\infty}} \sum_{t=0}^{\infty} \sum_{s^{t} \in S^{t}} \beta^{t} \pi_{t}\left(s^{t}\right) U\left(c_{t}^{i}\left(s^{t}\right)\right) \text{s.t.}\\ \sum_{t=0}^{\infty} \sum_{s^{t} \in S^{t}} \hat{p}_{t}\left(s^{t}\right) c_{t}^{i}\left(s^{t}\right) \leq & \sum_{t=0}^{\infty} \sum_{s^{t} \in S^{t}} \hat{p}_{t}\left(s^{t}\right) e_{t}^{i}\left(s^{t}\right) \\ c_{t}^{i}\left(s^{t}\right) \geq & 0 \text { for all } t, \text { all } s^{t} \in S^{t} \end{aligned}
• 2 $$\hat{c}_{t}^{1}\left(s^{t}\right)+\hat{c}_{t}^{2}\left(s^{t}\right)=e_{t}^{1}\left(s^{t}\right)+e_{t}^{2}\left(s^{t}\right) \text { for all } t, \text { all } s^{t} \in S^{t}$$

The utility maximization is intuitive, i.e. maximizing the expectation of consumption over all possibilities. But I don't understand the budget constraint, which is summed over the histories with no probabilities. It seems to me this is a budget constraints over all counterfactual worlds but in reality we only have one realization?

There are several ways to "decentralize" trades, but it is easiest to think that all trades are made before any uncertainty is revealed. Then $$\hat{p}_{t}\left(s^{t}\right)$$ is how much an agent has to pay initially to receive one unit of the consumption good in period $$t$$ in state $$s_t$$, whether the state later realizes or not.
• Consider a good available at a single period in one of two states. You could also embed this in an infinite horizon economy in which the endowments in all other periods are zero. The two agents have an endowment of $(0,1)$ and $(1,0)$ respectively. Both have utility function of the form $u(c_1,c_2)=\sqrt c_1+\sqrt c_2$. There is a unique equilibrium in which the good has the same price in both states and both consume $(0.5,0.5)$; there is perfect risk sharing. May 15, 2022 at 18:24