Consider a two good, two individual endowment economy. Suppose that individual A's endowment is given by (e^, e^) and individual B's endowment is given by (e̱e, e³). Also assume individual A's preferences are given by: and individual B's preferences are given by: UA (xA, YA) = a log(xд) + (1 − a) log(YA) UB (XB,YB) = log(xB) + (1 − y) log(YB). Consider the following social welfare function: O AUA (xA, YA) + BUB (XB,YB) · We can interpret this as a social planner that wishes to allocate the economy's endowment between the two individuals. ¿ is descriptive of how important individual i is to the social planner, i = {A, B}. Therefore, a large value of 0; indicates to us that the social planner gives a lot of weight to individual i, while the reverse is true if 0; is small. The social planner's problem is to maximize society's welfare by allocating consumption among the two individuals. The solution will be a Pareto optimum. We can obtain all possible Pareto optima by varying the weights of the problem.
Write down the Planner’s problem, as well as the Lagrangian. (Hint: There should be two constraints because aggregate consumption of each good cannot exceed the economy’s endowment.) Label the multipliers ϕ1 and ϕ2.
What are the first order conditions of the Planner’s problem? (Hint: There should be 6.).
Find the solution to the Planner’s problem and label the quantities (xpA; ypA) and (xpB; ypB).
If we compare the first order conditions of the Planner’s problem to the first order conditions of a corresponding competitive equilibrium (CE), we can obtain a relationship between ϕx, ϕy, λ, px, and py so that the CE and the Planner’s problem give us the same solution. This is called decentralizing the Planner’s problem.
Decentralize the Planner’s problem. (find the
Can you relate this to the two welfare theorems?
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