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. 1 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. 2
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