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6. (15 points) Now, we introduce taxes on net purchases into the market. Consider an economy E = (1, (Ui, Wiliel, T) with ||| agents and N commodities, where T E RN is the vector of tax rates on net purchases in the economy. All taxes are rebated as lump sums equally to all households. We impose the usual assumptions about continuity, strict concavity, monotonicity of preferences. To be clear, agent i's budget constraint is given by p.I+T-wi)+ = p.Wi+T where (v)+ indicates the vector consisting of the nonnegative entries of vector v with zeroes replacing the negative coordinates of v. i.e. agent i pays no tax on consumption of his endowment, and pays a tax Tn on each unit of good n he consumes greater than his endowment of commodity n, and thus T T. (Ii - wi)+ 11 A typical agent treats T as given - as a fixed amount like price that he could not affect. (a) Consider the following example: iel UAT, y) = + 2y, WA = (10,0) = = UB(1, y) = 2.c + y, we = (0,10) = T= (2,2) Prove (, (TA, TB)) = ((0.5, 0.5), (WA, WB)) is a competitive equilibrium. Moreover, argue that this equilibrium allocation is Pareto dominated by (TA, TB) = (0,10), (10,0)). (b) The first welfare theorem could not be applied to this economy, as the tax paid by buyers sometimes could not be received by sellers. Which step in our proof of the first welfare theorem goes wrong? 6. (15 points) Now, we introduce taxes on net purchases into the market. Consider an economy E = (1, (Ui, Wiliel, T) with ||| agents and N commodities, where T E RN is the vector of tax rates on net purchases in the economy. All taxes are rebated as lump sums equally to all households. We impose the usual assumptions about continuity, strict concavity, monotonicity of preferences. To be clear, agent i's budget constraint is given by p.I+T-wi)+ = p.Wi+T where (v)+ indicates the vector consisting of the nonnegative entries of vector v with zeroes replacing the negative coordinates of v. i.e. agent i pays no tax on consumption of his endowment, and pays a tax Tn on each unit of good n he consumes greater than his endowment of commodity n, and thus T T. (Ii - wi)+ 11 A typical agent treats T as given - as a fixed amount like price that he could not affect. (a) Consider the following example: iel UAT, y) = + 2y, WA = (10,0) = = UB(1, y) = 2.c + y, we = (0,10) = T= (2,2) Prove (, (TA, TB)) = ((0.5, 0.5), (WA, WB)) is a competitive equilibrium. Moreover, argue that this equilibrium allocation is Pareto dominated by (TA, TB) = (0,10), (10,0)). (b) The first welfare theorem could not be applied to this economy, as the tax paid by buyers sometimes could not be received by sellers. Which step in our proof of the first welfare theorem goes wrong

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