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{"ops":[{"insert":"Design a government intervention in the competitive equilibrium that could restore social efficiency. In particular, a tax \u03c4 on production "},{"attributes":{"italic":true},"insert":"f "},{"insert":"("},{"attributes":{"italic":true},"insert":"n"},{"insert":") is imposed on the firms which is rebated lump-sum to the households. As a first step, set up the new firms maximization with the tax, as well as the maximization problem of the households and derive the respective first-order conditions. Show that leisure demand with taxes is given by "},{"attributes":{"italic":true},"insert":"l"},{"insert":"2"},{"attributes":{"italic":true},"insert":"nd "},{"insert":"= \u03b3(1\u2212"},{"attributes":{"italic":true},"insert":"p"},{"insert":"\u03b4) . Then, determine the optimal size (1\u2212\u03c4)+\u03b3(1\u2212"},{"attributes":{"italic":true},"insert":"p"},{"insert":"\u03b4) of the tax \u03c4 that would establish equivalence between the competitive equilibrium and Social Planner allocations.\u00a0\n\nHouseholds utility:"},{"attributes":{"italic":true},"insert":"U"},{"insert":"("},{"attributes":{"italic":true},"insert":"c"},{"insert":", "},{"attributes":{"italic":true},"insert":"l"},{"insert":") = ln("},{"attributes":{"italic":true},"insert":"c "},{"insert":"\u2212 "},{"attributes":{"italic":true},"insert":"pE"},{"insert":") + \u03b3 ln "},{"attributes":{"italic":true},"insert":"l"},{"insert":", and the budget constraint is "},{"attributes":{"italic":true},"insert":"c "},{"insert":"= "},{"attributes":{"italic":true},"insert":"w"},{"insert":"(1 \u2212 "},{"attributes":{"italic":true},"insert":"l"},{"insert":").\u00a0\nExternalities: "},{"attributes":{"italic":true},"insert":"E "},{"insert":"= \u03b4"},{"attributes":{"italic":true},"insert":"Y"},{"insert":"\nFirms produce with a linear technology using only labor input "},{"attributes":{"italic":true},"insert":"n: Y "},{"insert":"= "},{"attributes":{"italic":true},"insert":"f "},{"insert":"("},{"attributes":{"italic":true},"insert":"n"},{"insert":") = "},{"attributes":{"italic":true},"insert":"n"},{"insert":".\nAssume that 0 < \u03b4"},{"attributes":{"italic":true},"insert":"p "},{"insert":"< 1\u00a0\n\n"}]}

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