ECON209 - Spring 2024
ECON209– Problem set 4
Due Wednesday April 24
Reminders:
1. Homework must be turned in on the day it is due in through Blackboard by 10pm. Late homework will NOT be accepted unless you are sick and have a doctor’s note.
2. Homework regrading: There is a statute of limitations on regrades. If you believe a question has been incorrectly graded, please let your TA know within 2 weeks of it being returned.
3. Working in groups: You may work in groups of up to 4. BUT: You MUST put names of other group members on your homework. You MUST write up your own set of answers. Do NOT simply copy some other person’s work.
4. TYPE your work. Equations may be hand written.
Problems 1 and 2 get you to work on borrowing constraints. (Eugenio will cover very related material in his 4/12th recitation). Problem 3 concerns asset pricing. We will cover that in depth on 4/15th in class.
Problem 1: Catching up on Adam and Arthur
Remember Arthur and his son Adam from Homework 3? Suppose that both Arthur and his son Adam have preferences over consumption today (c) and consumption tomorrow (c
′
) given by √
c + 0.95√
c
′
.
Arthur’s income today (y) and tomorrow (y
′
) are y = 500, y′ = 210. Adam’s income today and tomorrow are y = 200 and y
′ = 525. Suppose there are no lump-sum taxes. Suppose that the interest rate is 5%. So far all is the same as homework 3. But now assume neither can borrow.
A) Write down Arthur’s choice problem as a savings choice problem using the numbers and functions given above. To do this substitute the budget constraints into the utility function to express the problem as one over savings. The problem should be in the form max some function of saving subject to a constraint. (Hint: what constraint implies that Arthur cannot borrow?)
B) Find Arthur’s optimal savings choice absent any borrowing constraint. If it is positive you have solved Arthur’s problem. If it is negative, impose s = 0. This is the best Arthur can do. Once you have Arthur’s saving choice, calculate his optimal consumption.
C) Write down Adam’s savings choice problem.
D) Find Adam’s optimal savings choice just as you did for Arthur. Once you have found this, find his optimal consumption choices as well.
E) Suppose the government transfers T = 100 to Arthur and T = 100 to Adam today. Recompute their optimal consumption and savings choices.
F) Calculate the responsiveness of their consumptions today to the transfer: ∆ c T (where ∆c is the change in today’s consumption induced by the transfer).
G) Based on your answers, if the government aims to encourage current consumption (per haps with a view to stimulating production), to whom (i.e. to what sort of households) should it transfer income to or cut taxes?
For each result, do NOT simply provide an answer – be sure to show your work!
Problem 2: Government Budget Constraint
Now assume that the government must balance its budget through future taxes on Adam and Arthur. So in addition to the transfer of 100 to each of them in the first period, it raises taxes by 105 dollars on each of them in the second.
A) Explain how this combination of transfers and taxes is consistent with government budget balance.
B) How will Arthur’s decisions change relative to those you found in Question 1B? How will Adam’s change relative to those you found in Question 1D? (You do not need to do more math here. You can give verbal reasoning).
Problem 3: Asset pricing
There are two possible future states, boom and recession. They occur with probability 0.5 each. Let ct and ct+1 denote an investor’s consumption in period’s t and t + 1. Assume that:
ct+1 ct = (
1 . 04 if a boom ocurs 0 .
99 if a recession occurs.
A) An investor with utility log c (and a coefficient of relative risk aversion σ = 1) and discount factor 0.9 buys a riskless asset. What is the risk-free rate of return?
B) A risky asset pays out $1.10 in the boom and $0.90 in the recession. How much would the investor pay for it? What is its risk premium? (Hint: remember that the risky return on an asset satisfies 1 + r =
D P
, where D is the risky payout and P is the price.
Use Slide 30 or 31, Lecture 14. Remember you can move variables that are certain through expectations. What variable is known by an investor when she buys an asset?)