二、Let us consider a pure exchange economy with two agents, a and b, and two goods, 1 and 2. Let the initial endowments for the two agents be, respectively, $ea = [1 \quad 0]$, $eb = [0 \quad 1]$. The two agents would like to exchange part of the goods they own, at some price which we want to determine. Let us assume a Cobb-Douglas utility function for the two agents: $ua(x{1a}, x{2a}) = x{1a}^\alpha x{2a}^{1-\alpha}$, $ub(x{1b}, x{2b}) = x{1b}^\beta x{2b}^{1-\beta}$, where $x_{ij}$ is the consumption of good $i = 1, 2$ by agent $j = a, b$, and $\alpha, \beta \in (0,1)$ are parameters specifying the preferences of the two agents.

Let $p1$ and $p2$ be the prices of the two goods, 1 and 2, respectively.