his is about an exchange economy with uncertainty. There are \( I \) consumers, one good, and \( S \) states of the world. Consumption occurs at period 1. Let \( e_i \in \mathbb{R}^S \) denote the initial endowment of consumer \( i \), and assume that \( e = \sum_{i=1}^I e_i \) is not constant across states (there is aggregate uncertainty in the economy). Each consumer's utility function is \[ u_i(x_i) = \sum_{s=1}^S \pi_s \log (\alpha_i + x_{is}) \] where \( 0 < \pi_s > 1 \) is the probability of state \( s \) (obviously \( \sum_{s=1}^S \pi_s = 1 \)), and \( x_i = (x_{i1}, ..., x_{iS}) \) is a state-dependent consumption plan. Note that consumers' utility functions only differ in \( \alpha_i \). A consumption plan is feasible for agent \( i \) if it belongs to the set \( \{x_i \in \mathbb{R}^S | x_{is} > 0 \text{ for each } s = 1, ..., S\} \). An allocation is denoted as \( x = (x_1, ..., x_I) \in \mathbb{R}^{S \times I} \). Given a price vector \( p = (p_1, ..., p_S) \in \mathbb{R}_+^S \) (normalized so that \( \sum_{s=1}^S p_s = 1 \)), each consumer chooses \( x_i \) to maximize her utility subject to her budget constraint. Let \( x^*, p^* \) denote a competitive equilibrium of this economy. \begin{enumerate} \item[a.] Given an economic interpretation of the parameter \( \alpha_i \)