m each consumer '5' has strictly monotone preferences f,- and initial endowment wi >> 0. Given an allocation :3 = (:L'l,~- ,sz), say that z' envys j in :1: if ZL'j >7 sci. An allocation a," is called envyfree if there exists no pair of consumers i and j such that i envys j in a. (a) Is every competitive equilibrium allocation envyfree? Either prove that this is true or briey explain why this is not true. (b) An allocation 1!: contains an envy cycle if there exists a set of consumers {211, - -- ,ik} such that 2'1 = ilk, and min\" >7\" mi\" in :1: for each n = 1, - -- ,k. Can a competitive equilibrium allocation in this economy contain an envy cycle? Either prove that this is impossible or briey explain why this is possible. (0) Consider a social planner whose objective is to ensure that all allocations are envyfree. Suppose that the planner can intervene by changing consumers' initial endowment, subject to the xed social endowment. Can the planner nd an intervention such that, after the intervention, every competitive equilibrium application is envyfree? Either prove that this is true, or briey explain why this is not true. (d) Now let m = L = 2. Consumer 1's endowment is (0,1) where 0 is con sumer 1's endowment of goods 1 and 1 his endowment of goods 2. Con sumer 2's endowment is (1, 0). Consumer 1's utility function is U1($%, 33%) = ln(a:}) + 2ln(:1:), where 35% > U is consumer 1's consumption of goods 1 and an; > 0 his consumption of goods 2. Consumer 2's utility function has the form U 2($,$) = ($)0'5(x)0'5. Find the competitive equilibrium in this economy (normalise the price of goods 1 by 1.) (e) Let V1($11,:c) = 9(U"'(mi,$g)), where g() is some strictly increasing func tion. If consumer i's utility function is V1113?\" 1:3), would the equilibrium you get in ((1) change? explain briey