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Homework 9 Advanced Micro 1. From the Jehle and Reny text do the following problems: 5.19, 5.25, 5.26, 5.27, 5.31, and 5.33. 2. Consider an economy

2. Consider an economy consisting of two people, A and B who have utility functions and
endowments as follows: please see attached file!

a) Find the core of this economy and illustrate it in an Edgeworth Box diagram.
b) What is the Walrasian equilibrium of this economy?
c) Suppose we add two people to the above economy, C and D such that C and A have
identical preferences and endowments as do B and D. Can you find one allocation
in the core of the original economy of two individuals which is not in the core of
the economy consisting of all four of them?

3. Consider an economy of two people and two goods. Total initial resources are e1+e2 >>0.
Individuals have identical homothetic utility functions, ui(xi) = u(xi), where u is a
continuous, strongly increasing and strictly concave function.

a) Show that the Walrasian equilibrium price vector is independent of the initial
distribution of endowments.
b) Show that the set of Walrasian equilibrium allocations coincides with the diagonal
of the Edgeworth box.

Hint: recall that a homothetic utility function has the property that the MRS is constant
along rays through the origin. You might also find it easier to do part (b) before part (a).

4. Consider an economy of two people, A and B who have utility functions and
endowments as follows: see file
a) Illustrate the contract curve and the core in an Edgeworth box diagram.
b) Define the relative price: see file. Use the Kuhn-Tucker conditions to show that P=1 is a
general equilibrium price ratio.
c) Show that P =1.5 is also a GE price ratio.
d) What would you guess are all of the GE price ratios in this economy?
Homework 9 Advanced Micro 1. From the Jehle and Reny text do the following problems: 5.19, 5.25, 5.26, 5.27, 5.31, and 5.33. 2. Consider an economy consisting of two people, A and B who have utility functions and endowments as follows; !" 1 2 1 2 2 1 2 1 2 ( , ) 2ln 2ln (8,2) ( , ) (2,8) A A A A A A B B B B B B U x x x x e U x x x x e # $ # # $ # a) Find the core of this economy and illustrate it in an Edgeworth Box diagram. b) What is the Walrasian equilibrium of this economy? c) Suppose we add two people to the above economy, C and D such that C and A have identical preferences and endowments as do B and D. Can you find one allocation in the core of the original economy of two individuals which is not in the core of the economy consisting of all four of them? 3. Consider an economy of two people and two goods. Total initial resources are e 1 +e 2 >>0. Individuals have identical homothetic utility functions, u i (x i ) = u(x i ), where u is a continuous, strongly increasing and strictly concave function. a) Show that the Walrasian equilibrium price vector is independent of the initial distribution of endowments. b) Show that the set of Walrasian equilibrium allocations coincides with the diagonal of the Edgeworth box. Hint: recall that a homothetic utility function has the property that the MRS is constant along rays through the origin. You might also find it easier to do part (b) before part (a). 4. Consider an economy of two people, A and B who have utility functions and endowments as follows: 1 2 1 2 1 2 1 2 ( , ) 2 and e (10,6) ( , ) 2 and e (6,10) AA BB U x x x x U x x x x # $ # # $ # a) Illustrate the contract curve and the core in an Edgeworth box diagram. b) Define the relative price x y P P P # . Use the Kuhn-Tucker conditions to show that P=1 is a general equilibrium price ratio. c) Show that P =1.5 is also a GE price ratio. d) What would you guess are all of the GE price ratios in this economy?
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