14.02 Principles of Macroeconomics: Problem Set 5 - MIT ...

14.02 Principles of Macroeconomics
Problem Set #5
Due: November 20, 2009 at 4pm
1 Intergenerational altruism and Ricardian Equivalence (50
points)
The country of Bequestia has only two consumers, Joe Senior and Joe Junior. At time t = 0; Joe
Senior is young, and Joe Junior has not been born yet. At time t = 1; Joe Senior is old and Joe
Junior is born, and at t = 2 Joe Junior is old and Joe senior is dead. They are both PIH consumers
with preferences given by:
U(c young ; c old ) = ln c young + ln c old
where 2 [0; 1]: Both Joes have an income of y young = 10 when young and y old = 10 when old. The
interest rate is r > 0:
1. (5 points) Write both consumers’intertemporal budget constraints.
2. (5 points) Using the optimality condition c old = (1 + r)c young and the budget constraint, …nd
each consumers’consumptions as a function of r and :
3. (6 points) Find the aggregate consumption C t and savings S t of the economy at times t = 0; 1; 2:
4. (7 points) Now assume that the government of Bequestia decides to spend G = 5 at t = 0 and
give it as gift to Joe Senior. The gift will be …nanced by lump-sum taxes T at t = 2. Since the
government must satisfy its intertemporal budget constraint,
G +
T
(1 + r) 2 = 0:
Find the level of consumption and saving for the economy at t = 0; 1; 2 as a function of r and
:
5. (6 points) Does Ricardian equivalence hold in this economy? Why?
From now on, assume Joe Senior cares about Joe Junior’s future, and his utility function is
now
U(c young;sr ; c old;sr ; c young;jr ; c old;jr ) = ln c young;sr + ln c old;sr + ln c young;jr + 2 ln c old;jr
where c young;sr is Senior’s consumption when young, c young;jr is junior’s consumption when
young and so on. Junior does not solve a consumption problem, he simply follows the consumption
path that Senior picks for him. Income of Senior and Junior remain the same as
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efore. Government spending and taxes are zero in all periods. Also assume from now on that
r = 0 and = 1:
6. (5 points) Find the consumption of Joe Senior and his bequest to (consumption of) Joe Junior.
[Hint: with = 1; r = 0 you need to do very little math. Why?]
7. (5 points) How does Senior’s consumption pattern in part 6 compare to that in part 2?
8. (5 points) Now the government implements the same spending pattern as in part 4, with G = 5
at t = 0 …nanced by taxes at time t = 2: Find Senior and Junior’s consumption pattern.
9. (6 points) Does Ricardian equivalence hold in this economy? Why?
10. BONUS (10 points) Would Ricardian equivalence hold if Senior’s utility function is
What about
a) U(c young;sr ; c old;sr ; c young;jr ; c old;jr ) = ln c young;sr + ln c old;sr + 2 ln c old;jr ?
b) U(c young;sr ; c old;sr ; c young;jr ; c old;jr ) = ln c young;sr + ln c old;sr + ln c young;jr ?
2 Rules of thumb and Ricardian Equivalence (50 points)
Consider an economy where savings follow the rule of thumb that they are a constant fraction s of
income, i.e. S t = sY t with s 2 (0; 1): The production function is Y t = A t L 1=2 ; and the labor force
is …xed at L = 1: The growth rate of productivity is g; i.e. A t+1 = (1 + g)A t ; with A 0 = 1: The
government spends G t each period and collects a lump-sum tax of T t : The economy exists for three
periods t = 0; 1; 2:
1. (8 points) Find the path for private savings S pvt
t = Y t T t C t and public savings S gov
t = T G
if G t = T t = 0 for all t:
2. (8 points) Find the path for private savings if government spending is …xed at a fraction
p 2 (0; 1) of GDP and the government is running a balanced budget every period.
3. (8 points) Now consider the case where the government spends G 0 = pY 0 and zero in all other
periods, and collects taxes T 2 = pY 2 in period 2: Find private and public savings. Find the
path for consumption.
4. (8 points) In order to implement the policy in part 3, the government borrows in period 0
and re-pays its debt in period 2 using the money collected from taxes. Find the maximum
per-period interest rate r that the government is willing to pay.
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5. (8 points) Is there Ricardian equivalence in this economy? Explain. [Hint: Ricardian equivalence
has to do with aggregate savings and consumption].
6. (10 points) Now assume that aggregate savings S is allowed to depend on G (as well as Y as
before.) Find a dependence that S must have on G so that Ricardian equivalence holds in this
economy.
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14.02 Principles of Macroeconomics
Fall 2009
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