## Answered: - econ 302 macroeconomic question, only want to get the answ

econ 302 macroeconomic question, only want to get the answer of the question 7. already upload the question file.

Question 7 (from a past exam)

Assume that there are two consumers, Jerry and George. Both live for two periods, are prices takers, and have identical utility function: U(ci, cfi) = ln(ci) + ? ln(cfi) where 0 y1?(1+r)

(a) Write the optimization problem for each consumer. Use the lifetime budget constraint.

(b) Derive the Euler equation

(c) Use the Euler equation and the lifetime budget constraint to solve for Jerry?s and George?soptimal consumption plans

(i) Are the optimal consumption plans different for the two consumers? Explain why orwhy not.

(ii) Do Jerry and George engage in borrowing or lending in the first period? calculatethe amounts and explain who is borrowing and who is lending, and why.(d) Now assume that the economy changes, and borrowing is no longer allowed. (To avoidunnecessary complications, you may assume that if someone wishes to lend, there is alwaysan opportunity to do so for a rate r, e.g. to foreigners). NO CALCULATIONS

AREREQUIRED FOR THE FOLLOWING QUESTIONS, JUST EXPLAIN

.(i) Does the Euler equation that you derived in part (b) still hold for both consumers?for none? for one? explain.

(ii) Assume that the government would like give a tax rebate in order to increase currentaggregate consumption. Is it better to give the rebate to Jerry or to George? Explain.

(iii) Is Ricardian Equivalence likely to hold for the economy in this case? explain

Econ 302 ? 2015T2

NAME:

Problem Set 5

Due Thursday March 31, 2016, in class

Question 1: Data

Use FRED to download and plot the growth rates of consumptions of durables and consumption of non durables for the US economy. (These would be ?Real Personal Consumption

Expenditure: Nondurable Goods? and ?Real Personal Consumption Expenditure: Durable

Goods?; both available since 1999).

(Note 1: you don?t have to print data, just a graph; attach it to the end of the problem set of

email to the TA).

Looking at the growth rates - which series seem to be more volatile over time? Explain (brie?y)

how this is related to our discussion regarding consumption smoothing).

Solution Q1:

1

Question 2 - A short question from a previous exam

Using consumption smoothing theory, rank the following from the biggest to the smallest

consumption increase among recipients. (For simplicity, you may assume that the desired consumption plan is spending equal amounts over the life cycle). Explain your answer (unexplained

guesses = 0).

(a) an unexpected and explicitly temporary tax rebate of \$300

(b) a special dividend on a stock that the consumer owns of \$300 per shareholder, that also

lowers the value of the stock by \$300 per share (assume the consumer owns one stock).

(c) an unexpected raise of \$300 per year, e?ective immediately

Solution Q2:

2

Question 3 - From a previous ?nal exam

Around the beginning of the recent recession, U.S. in?ation forecast have risen, while short

term nominal interest rates have fallen.

(a) What does this imply for the change in the ex ante real interest rate at that time?

(b) How would this change alter the consumption plan for the current and the future periods?

[Hint: use the Euler equation, You may assume that ?(1 + r) &gt; 1 before and after the

change]

(c) During the recession we also witnessed a surprisingly slow growth of real income. How

does this surprise a?ect the optimal consumption plan ?

(d) Taking into account the e?ects of all the shocks in parts (a) and (c), what is the net e?ect

on current consumption? future consumption?

Solution Q3:

3

Solution Q3:

4

Question 4 (looking at the role of parameters using numerical examples)

Consider the standard two-periods consumption model where consumers have the utility func1

tion u(c) =

c1? ?

1

1? ?

. Furthermore, let a = 0, y = 0, and y f = 1.

(a) Calculate the optimal consumption plan, i.e. express c and cf as a function of ?, r, ?.

(Recall that you have to use both the Euler equation and the budget constraint).

(b) Set ? = 0.5, ? = 0.95 and r = 0.02. Which one is larger c or cf ? Explain why.

(c) Set ? = 2, ? = 0.95, and r = 0.02. Is the consumption pro?le more or less volatile relative

to part (b)? explain why.

Now assume that the income stream is di?erent: y = 1 and y f = 0.

(d) Set ? = 0.5, ? = 0.95 and r = 0.02. Compare c and cf to the results in part (b). Explain

the di?erences.

(e) Assume that r increases from r = 0.02 to r = 0.04. Compare the e?ect on c when ? = 0.5,

? = 2, and ? = 1. Explain.

Solution Q4:

5

Solution Q4:

6

Question 5 - Consumption Taxes (From a past exam)

At the onset of the recent recession, economists argued that a temporary reduction in consumption taxes (i.e. ?c ) should boost the demand for private consumption. Let?s examine this

argument through the logic of the Euler equation.

Assume that consumers live for three periods and that the objective is to maximize the discounted sum of lifetime utility. Consumers choose c0 , c1 .c2 (for periods t = 0, 1, 2), subject to

a lifetime budget constraint. Denote income in the three periods by y0 , y1 .y2 . Assume that

a = 0(no initial assets), 0 &lt; ? &lt; 1 is the discount factor, u(c) is a standard utility function

with u (c) &gt; 0 and u (c) &lt; 0. Finally, assume that consumption is taxed at a rate ?t , a rate

that may vary over time.

As a result, the consumer problem can be written as:

max {u(c0 ) + ?u(c1 ) + ? 2 u(c2 )}

c0 ,c1 ,c2

subject to:

(1 + ?0 )c0 +

(1 + ?1 )c1 (1 + ?2 )c2

y1

y2

+

= y0 +

+

2

1+r

(1 + r)

1 + r (1 + r)2

(a) Denote the Lagrange multiplier on the budget constraint by ?. Write the Lagrange function

for this problem.

(b) Derive the ?rst order conditions for consumption in periods 0, 1, 2. Use this set of ?rst

order conditions to derive the two Euler equations for this problem.

(c) Show that if the consumption tax ? is constant over time, then it does not a?ect the

optimal allocation of consumption over time. (Hint: how does the Euler equation for this

case compares to an Euler equation for a model without taxes?)

(d) Using the Euler equation, explain what should happen to the consumption plan if consumption tax is decreased in period 1. (i.e ?1 is lower than expected).

Solution Q5:

7

Solution Q5:

8

Solution Q5:

9

Question 6 (In?nite Horizon (mostly for practicing derivatives))

In class we derived the Euler equation for a two period model, and claimed that it generalizes

to in?nite horizon. Let?s show this step by step.

Assume that consumers live forever and maximize the discounted sum of lifetime utility. Consumers choose c0 , c1 , c2 , .... subject to a lifetime budget constraint. Assume that a = 0(no initial

assets), 0 &lt; ? &lt; 1 is the discount factor, u(c) is a standard utility function with u (c) &gt; 0 and

u (c) &lt; 0.

The consumer problem can be written as:

?

? t u(ct )

max

c0 ,...,c?

t=0

?

s.t.:

t=0

yt

ct

=

t

(1 + r)

(1 + r)t

Note that this can be written in a longer form (but easier to work with):

max {u(c0 ) + ?u(c1 ) + ? 2 u(c2 ) + ... + ? t u(ct ) + ? t+1 u(ct+1 ) + ...}

c0 ,...,c?

s.t.:

c0 +

c1

c2

ct

ct+1

+

+ ... +

+

+ ... =

2

t

1 + r (1 + r)

(1 + r)

(1 + r)t+1

y2

yt

yt+1

y1

+

+ ... +

+

+ ...

y0 +

2

t

1 + r (1 + r)

(1 + r)

(1 + r)t+1

(a) Denote the Lagrange multiplier on the budget constraint by ?. Write the Lagrange function

for this problem. (Hint: not magic here... just write down the constrained maximization

problem).

(b) Derive the ?rst order conditions with respect to consumption in (the arbitrary) periods t

and t + 1.

(c) Use the two ?rst order conditions to derive the Euler equation.

(d) Provide a brief interpretation for the Euler equation - in what sense is the Euler equation

an optimality condition?

Solution Q6:

10

Solution Q6:

11

Solution Q6:

12

Question 7 (from a past exam)

Assume that there are two consumers, Jerry and George. Both live for two periods, are prices

takers, and have identical utility function: U (ci , cf ) = ln(ci ) + ? ln(cf ) where 0 &lt; ? &lt; 1, and i

i

i

can be J for Jerry, or G for George.

Both consumers have initial wealth a = 0. The endowment (or income) process is di?erent:

Jerry:

George:

where we assume that y0 &gt;

f

yJ = y1

yJ = y0

yG =

y1

1+r

f

yG = (1 + r)y0

y1

?(1+r)

(a) Write the optimization problem for each consumer. Use the lifetime budget constraint.

(b) Derive the Euler equation

(c) Use the Euler equation and the lifetime budget constraint to solve for Jerry?s and George?s

optimal consumption plans

(i) Are the optimal consumption plans di?erent for the two consumers? Explain why or

why not.

(ii) Do Jerry and George engage in borrowing or lending in the ?rst period? calculate

the amounts and explain who is borrowing and who is lending, and why.

(d) Now assume that the economy changes, and borrowing is no longer allowed. (To avoid

unnecessary complications, you may assume that if someone wishes to lend, there is always

an opportunity to do so for a rate r, e.g. to foreigners). NO CALCULATIONS ARE

REQUIRED FOR THE FOLLOWING QUESTIONS, JUST EXPLAIN.

(i) Does the Euler equation that you derived in part (b) still hold for both consumers?

for none? for one? explain.

(ii) Assume that the government would like give a tax rebate in order to increase current

aggregate consumption. Is it better to give the rebate to Jerry or to George? Explain.

(iii) Is Ricardian Equivalence likely to hold for the economy in this case? explain

Solution Q7:

13

Solution Q7:

14

Solution Q7:

15

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