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Let $x_i(p, w_i)$ be consumer $i$'s (Marshallian) demand and $y_j(p)$ be firm $j$'s profit-maximizing supply at price $p$. A competitive equilibrium is a price $p$ such that $$\sum_{i=1}^I x_i(p, w_i(p, e_i)) = \sum_{j=1}^J y_j(p).$$ By strict concavity of $u_i$'s, consumer demands are functions, rather than correspondences. Assume also that $y_j(p)$ are ...