Consider the regulation of J local monopolies for different goods with inverse demand functions P_j(X_j) and cost functions C^j(X_j,e_j). Social damage depends on total pollution  and is given by a convex damage function D(E).

(a) Derive an expression for the second-best tax to jointly regulate the J local monopolists.

(b) Show that such a system is equivalent to a system of tradable permits.

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E=;