2. An economy consists of two consumers with labels i=1,2. They exchange two goods labelled j=1,2. Suppose there is a fixed total endowment ej of each good to be distributed between two consumers. Let c denote i's consumption of good j. Suppose that each consumer i has preference represented by the utility function Uici, c) = a Inc + aznc = = = = where the parameters aj are positive and independent of i, with Q1 + A2 1. Suppose the goods are to be distributed in order to maximize social welfare in the form of the weighted sum W = B_U1 + B2U2 where are positive and B1 + B2 = 1 2-A) Formulate the welfare maximization problem with one equality constraint for each of the goods. 2-B) Write down the Lagrangian where ); denotes the Lagrange multiplier associated with the constraint for good j. Find the welfare maximizing distribution of the goods. 2-C) Verify that hj aw* = where W* denotes the maximum value of W. 2 2. An economy consists of two consumers with labels i=1,2. They exchange two goods labelled j=1,2. Suppose there is a fixed total endowment ej of each good to be distributed between two consumers. Let c denote i's consumption of good j. Suppose that each consumer i has preference represented by the utility function Uici, c) = a Inc + aznc = = = = where the parameters aj are positive and independent of i, with Q1 + A2 1. Suppose the goods are to be distributed in order to maximize social welfare in the form of the weighted sum W = B_U1 + B2U2 where are positive and B1 + B2 = 1 2-A) Formulate the welfare maximization problem with one equality constraint for each of the goods. 2-B) Write down the Lagrangian where ); denotes the Lagrange multiplier associated with the constraint for good j. Find the welfare maximizing distribution of the goods. 2-C) Verify that hj aw* = where W* denotes the maximum value of W. 2