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 equilibrium prices and the value of λ so that the two problems have the same solution.) Can you relate this to the two welfare theorems? Consider a two good, two individual endowment economy. Suppose that individual A's endowment is givenby (e^, e^) and individual B's endowment is given by (e̱e, e³). Also assume individual A's preferences aregiven 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 twoindividuals. ¿ 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 thereverse is true if 0; is small. The social planner's problem is to maximize society's welfare by allocatingconsumption among the two individuals. The solution will be a Pareto optimum. We can obtain all possiblePareto optima by varying the weights of the problem.

Planner's Problem:The planner's problem is to maximize social welfare, given by the social welfare…

Answered: Consider a two good, two individual…

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Question

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 (x^p_A; y^p_A) and (x^p_B; y^p_B).

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, p_x, and p_y 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 equilibrium prices and the value of λ so that the two problems have the same solution.)

Can you relate this to the two welfare theorems?

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.

Definition Video Definition Point where the demand and supply curves intersect. At the equilibrium price the quantity supplied of a good is equal to the quantity demanded. When the market is not in equilibrium, market surplus or market shortage occurs. The region that is above the price equilibrium indicates market surplus and the region below the price equilibrium indicates market shortage. Video

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