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2. Consider the optimal commodity taxation problem in a perfectly competitive economy with two consumers. Consumer 1' has income M l and spends his income on two (private) goods, 1:1 and $2. The before-tax prices of the two goods are normalized to be 1. The after-tax price of good 3' is g; = 1 +tj, where tj is the tax on private good 5] and j = 11 2. Consumer i's preferences are given by the following utility function: U\"i = a'h1(:c'i) + (1 ai)h1($): 0