ECON-4330 Advanced Macroeconomics I Professor Nurlan Turdaliev

Homework Assignment 1 Fall 2021

Due: Saturday, October 2, 2021, 6pm

Problem 1

Suppose the endowment is (y, 0) for all generations, and n = 4. Restrict attention to stationary

α-allocations, i.e. allocations with

c1,t = αy, c2,t = (1− α)yn, t = 1, 2, ... .

Suppose the preferences of generations 1, 2, ..., are

u(c1, c2) =

√

c1 +

√

c2,

and those of the IO are

uIO(c2) = c2.

Demonstrate that the α-allocation with α = 0.75 is not Pareto optimal.

Problem 2

Suppose the endowment is (y, 0) for all generations, and n = 1. Restrict attention to stationary

α-allocations, i.e. allocations with

c1,t = αy, c2,t = (1− α)y, t = 1, 2, ... .

Suppose the preferences of generations 1, 2, ..., are

u(c1, c2) =

√

c1 +

√

c2,

and those of the IO are

uIO(c2) = c2.

Demonstrate that the α-allocation with α = 0.2 is Pareto optimal.

Problem 3

Suppose n = 1. Suppose the endowment grows, i.e. ωt = (yt, 0) where yt = Ayt−1, and A > 1 is a

constant. Also, ω1 = (y, 0). There is a constant amount of fiat money M . Suppose the preferences

of generations 1, 2, ..., are

u(c1, c2) =

√

c1 +

√

c2,

and those of the IO are

uIO(c2) = c2.

a. Write down equations that represent the budget constraints in the first and second period

of a typical individual from generation t. Combine these constraints into a lfetime budet

constraint of this individual.

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b. Restrict attention to a stationary solution in which each generation would consume the same

fraction of its endowment when young. Write down the conditions that represents the clearing

of the money market in an arbitrary period t. Use condition to find the real rate of return of

fiat money in a monetary equilibrium. Explain the path over time of the value of fiat money.

c. Find the Golden Rule allocation for this economy.

d. Find the optimal (c∗1t, c∗2,t+1). Compare it with the Golden Rule allocation found above.

Problem 4

Consider an economy consisting of two consumers, Adam and Eve, and one good, apples. Both

consumers prefer more apples to less. There are exactly 5 apples available in this economy. Apples

are indivisible, i.e. they cannot be split into fractions of one apple. Describe all Pareto optimal

allocations in this economy. Explain.

Problem 5

Consider an economy consisting of two consumers, Adam and Eve. There are two goods, apples

and bananas. Adam likes apples and prefers more apples to less; he is indifferent to bananas. Eve,

on the other hand, likes bananas and prefers more bananas to less; she is indifferent to apples.

There are exactly 5 apples and 4 bananas available in this economy. Describe all Pareto optimal

allocations in this economy. Explain.

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