Consider a one-period exchange economy with two agents \(i \in\{1,2\}\), two possible states of the world \(\left\{\omega_{1}, \omega_{2}\right\}\) and two consumption goods. Suppose that \(\omega_{1}\) and \(\omega_{2}\) have the same probability of occurrence, i.e., \(\mathbb{P}\left(\omega_{1}\right)=\mathbb{P}\left(\omega_{2}\right)=1 / 2\). The endowments of the agents, expressed in terms of units of the two goods, are given by \(e^{1}=(1,0)\) and \(e^{2}=(0,1)\), in correspondence of both states of the world. In aggregate terms, there is one unit of each of the two goods in both states of the world. The preferences of the two agents are characterized by the state dependent utility function

\(u^{i}\left(\omega, x_{1}^{i}(\omega), x_{2}^{i}(\omega)\right)=\beta(\omega) \sqrt{x_{1}^{i}(\omega)}+(1-\beta(\omega)) \sqrt{x_{2}^{i}(\omega)}, \quad\) for \(\omega \in\left\{\omega_{1}, \omega_{2}\right\}\), where \(\beta\left(\omega_{1}\right)=1\) and \(\beta\left(\omega_{2}\right)=0\), with \(x_{n}^{i}(\omega)\) denoting the demand by agent \(i\) of good \(n\) in correspondence of state \(\omega\). This specification means that agents do not receive any utility from the second good in correspondence of the state of the world \(\omega_{1}\).

(i) In this economy, there are no security markets, but only a spot market (i.e., agents trade immediately before the realization of the state of nature). Suppose first that there is no private information. Show that in equilibrium each agent consumes \(1 / 2\) of each good and has an expected utility equal to \(\sqrt{1 / 2}\).

(ii) Suppose now that the state of the world is perfectly revealed to the two agents before the opening of the market. Show that in equilibrium each agent only consumes his initial endowment, getting an utility of 1 in one state of the world and 0 in the other state of the world.

(iii) Deduce that the revelation of the state of the world before the opening of the market decreases the social welfare ( Hirshleifer effect, see Sect. 8.2).