Consider a pure exchange economy with two goods. (a) Show that the gross substitute property implies z(p) = 0 and z(p') #0 implies p.z(p') > 0, (1) so that a unique Walrasian equilibrium is globally stable. (b) Prove that if (1) holds, the set of equilibrium price vectors {p = R² : z(p) = 0} is a convex set. (c) Suppose that all the consumers in the economy have quasilinear utility functions of the form u² (x₁, x₂) v(x₁) + x₂, where v() is strictly concave. Show that the gross substitute property is then satisfied, so that a Walrasian equilibrium is unique and (globally) stable. =

Consider a pure exchange economy with two goods. (a) Show that the gross substitute property implies z(p) = 0 and z(p') #0 implies p.z(p') > 0, (1) so that a unique Walrasian equilibrium is globally stable. (b) Prove that if (1) holds, the set of equilibrium price vectors {p = R² : z(p) = 0} is a convex set. (c) Suppose that all the consumers in the economy have quasilinear utility functions of the form u² (x₁, x₂) v(x₁) + x₂, where v() is strictly concave. Show that the gross substitute property is then satisfied, so that a Walrasian equilibrium is unique and (globally) stable. =

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

Please, answer only (c).

Uniqueness and Stability
Consider a pure exchange economy with two goods.
(a) Show that the gross substitute property implies
z(p) = 0 and z(p') #0 implies p-z(p') > 0,
(1)
so that a unique Walrasian equilibrium is globally stable.
(b) Prove that if (1) holds, the set of equilibrium price vectors {p = R² : z(p) = 0} is a convex
set.
(c) Suppose that all the consumers in the economy have quasilinear utility functions of the form
u²(x1, x2)
=
v(x₁) + x2,
where v(.) is strictly concave. Show that the gross substitute property is then satisfied, so
that a Walrasian equilibrium is unique and (globally) stable.

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Step 1: Define general equilibrium in microeconomics

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Step 3: Prove that if (1) holds, the set of equilibrium price vectors is a convex set

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