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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?

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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.

It is perfectly normal to make trades for receiving something in a contingency that might never materialize. Indeed, that is how insurance works, and risk sharing is the major reason why we cannot just trade after the contingency is known.

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  • $\begingroup$ I see what you mean. Is it that each contingency can be recognized as a lottery and the price reflects its value including its chance? And I think my worry about breaking the budget constraint is actually somehow resolved by the market clearing at each period. Now I can roughly get the idea but the whole maximization setting is still not crystal clear to me. I would appreciate it if you could elaborate a little bit more or give a very simple but intuitive example. $\endgroup$
    – Alalalalaki
    Commented May 15, 2022 at 14:54
  • $\begingroup$ 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. $\endgroup$
    – Michael Greinecker
    Commented May 15, 2022 at 18:24

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