# advanced macroeconomics problems till 8th March

4 Problems has to be finish till 8th March

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Problem Set

2

– Consumption & Government
Due

1

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1:59 PM Saturday 9th March 201

3

Problem 1
Consider the following dynamic infinite time horizon household optimization problem:

max
{ct}∞t=0

{
U =

∞∑
t=0

βt ln (ct)

}

s.t.

at+1 = (1 + r) (at − ct + yt)

where ct is consumption at time t and at denotes the stock of assets at time t. The household
has an initial stock of assets a0 (a one-period real pure discount bond) and income yt = y for all
t. The interest rate on assets r is constant over time.

a) Solve the above problem using dynamic programming, find the first order conditions and
provide their economic interpretation. Determine the household’s optimal consumption in
each period and show that consumption grows at the constant rate.

b) Suppose that β = 0.98 and determine the effect of a 1% increase in period 0 income on period
0 consumption. How does this result compare to the permanent income hypothesis?

Problem 2
Suppose you had the following model:

max
{ct}∞t=0
{
U =
∞∑
t=0

βtc0.5t

}
s.t.

i) ct + kt+1 = yt

ii) yt = kαt

a) Find the first order condition for the choice of kt+1.

b) Find the steady-state values of k, c, and y from the first order conditions.

c) Find the second order Taylor’s approximation of u (ct) = (kαt −kt+1)
0.5.

1

Problem 3
In a two period model suppose the government provides good G to a household that must be
paid for with taxes T on this household. Assume that the household receives this government
good only in the first period of their lives. The government must collect taxes in the first period
to finance the provision of the government good. The household chooses the optimal amount of
consumption in each period taking as given the government good. The household solves thus the
following utility maximization problem:

max
Ct,Ct+1

{U = ln Ct + β ln Ct+1 + a ln Gt}

subject to the following constraint:

Ct +
Ct+1
1 + r

= Yt −Tt

where a,β > 0.

a) Find the optimal values of Ct,Ct+1.

b) Substitute the optimal values you obtained in a) back into the utility function to obtain an
expression for utility that depends on G and T .

c) Now have the government choose the optimal level of T that maximizes the utility expression
in b) subject to the constraint that Gt = Tt (balanced budget).

d) Given the solution to T obtained in c) substitute it back to the expressions for Ct and Ct+1
in a).

e) If β = 1 and r = 0 is C∗t+1 > C∗t ? Why or why not?

Problem 4
Consider an economy with N individuals which lasts for T < ∞. All agents have identical preferences given by:

T∑
t=1

βt−1 ln
(
c
j
t

)
The individuals are endowed with income yjt . In period t = 1 they are able to trade a complete

set of real pure discount bonds and current consumption. The aggregate endowment in t is:

Yt =
N∑
t=1

y
j
t

There is a government in this economy able to make lump-sum taxes and transfers across
individuals, and that borrows and lends over time one-period real pure discount bonds. Let
bt+1 be the quantity of bonds issued by the government at time t. Let the net interest rate to
be paid on debt issued at t be rt+1. The government can run deficit or surplus, however, it
must pay back any loan. The government begins period 1 with b1 = 0. Before trade begins
the government commits to a plan specifying individual-specific and time specific taxes τt and
transfers gt, t = 1, 2, . . . ,T . The government takes into account any possible effects of its plans
on equilibrium prices.

2

a) Write down the government budget constraint in each period t.

b) Write down the intertemporal government budget constraint.

c) Suppose now that the government is able to borrow and lend freely from foreign banks at
a fixed interest rate r = (1/β) − 1. The consumers cannot do so, however. What is the
intertemporal budget constraint now?

d) Set up and solve the social planner’s problem for aggregate consumption. What should the
government do in order to maximize the overall level of utility derived from consumption in
this economy? Will the government be able to smooth the consumption path of individuals?
Why or why not?

3

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