Problem 5. We want to divide one divisible resource among two agents. Think, for exam- ple,...

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Problem 5. We want to divide one divisible resource among two agents. Think, for exam- ple, about dividing an amount of money between the two agents, or allocating computing power between two tasks that need access to a powerful server. To make it simple, we take the amount of resource to equal 1. So there are two agents, A = {1,2} and we need to choose (x1, x2) 0 with x1+x2 1. Suppose that agent i derives a utility u(xi) = x; from an amount of resource xi. Represent all the feasible pairs (u1(x1), u2(x2)) in R with the given resources. Find all the Pareto optimal allocation of resources, and show that an allocation (x1, x2) is Pareto optimal if and only if there are 0, i = 1,2, with 1 + 2 > 0, so that x solves the problem max{1u1 (1) + A2u2(Y2) Y1, Y2 0, Y1 + Y2 1}. Provide an interpretation of (A1, A2). Problem 5. We want to divide one divisible resource among two agents. Think, for exam- ple, about dividing an amount of money between the two agents, or allocating computing power between two tasks that need access to a powerful server. To make it simple, we take the amount of resource to equal 1. So there are two agents, A = {1,2} and we need to choose (x1, x2) 0 with x1+x2 1. Suppose that agent i derives a utility u(xi) = x; from an amount of resource xi. Represent all the feasible pairs (u1(x1), u2(x2)) in R with the given resources. Find all the Pareto optimal allocation of resources, and show that an allocation (x1, x2) is Pareto optimal if and only if there are 0, i = 1,2, with 1 + 2 > 0, so that x solves the problem max{1u1 (1) + A2u2(Y2) Y1, Y2 0, Y1 + Y2 1}. Provide an interpretation of (A1, A2).