Problem 1: Exchange economyConsider an exchange economy with two consumers, A and B, and twogoods, X and Y.Consumer A has an initial endowment TA = i >0 of good X, an initialendowment yA = 0 of good Y, and preferences over consumption bundlesthat can be represented by the Cobb-Douglas utility functionuA(TA: YA) = ",where ra is the quantity of good X, yYA is the quantity of good Y, anda € (0, 1) is a preference parameter.Consumer B has an initial endowment ig = 0 of good X, an initial endow-ment js = j > 0 of good Y, and preferences over consumption bundlesthat can be represented by the Cobb-Douglas utility functionup(rB, YB):B1-3where rB is the quantity of good X, YB is the quantity of good Y, andBE (0, 1) is a preference parameter.(a) Find the set of Pareto optimal allocations in this economy. In appro-priate diagrams, illustrate the set of Pareto optimal allocations when(i) 3 = a, (ii) 3 > a, and (iii) 3 < a.(b) Find the competitive equilibrium prices and allocations in this econ-omy. Illustrate the competitive equilibrium in an appropriate diagramfor the case when B> a.

Pareto performance, named after the Italian economist Vilfredo Pareto, is a concept used to evaluate…

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exchange economy

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

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Question

Problem 1: Exchange economy
Consider an exchange economy with two consumers, A and B, and two
goods, X and Y.
Consumer A has an initial endowment TA = i >0 of good X, an initial
endowment yA = 0 of good Y, and preferences over consumption bundles
that can be represented by the Cobb-Douglas utility function
uA(TA: YA) = ",
where ra is the quantity of good X, yYA is the quantity of good Y, and
a € (0, 1) is a preference parameter.
Consumer B has an initial endowment ig = 0 of good X, an initial endow-
ment js = j > 0 of good Y, and preferences over consumption bundles
that can be represented by the Cobb-Douglas utility function
up(rB, YB):
B1-3
where rB is the quantity of good X, YB is the quantity of good Y, and
BE (0, 1) is a preference parameter.
(a) Find the set of Pareto optimal allocations in this economy. In appro-
priate diagrams, illustrate the set of Pareto optimal allocations when
(i) 3 = a, (ii) 3 > a, and (iii) 3 < a.
(b) Find the competitive equilibrium prices and allocations in this econ-
omy. Illustrate the competitive equilibrium in an appropriate diagram
for the case when B> a.

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