University of California, Santa Cruz
Department of Economics
Ken Kletzer
Fall 2014
Economics 205A
Advanced Macroeconomic Theory
Introduction: This first course in the year-long sequence in macroeconomic theory for PhD students
concentrates on the analytic
Economics 205A
Fall 2016
K. Kletzer
Problem Set 4
Due: Tuesday, November 22, 2016
1. In this problem, we add government to the decentralized economy. Output is produced acording
to the production function, yt = f (kt ), and capital depreciates at a consta
University of California, Santa Cruz
Department of Economics
Ken Kletzer
Fall 2016
Economics 205A
Advanced Macroeconomic Theory
Introduction: This first course in the year-long sequence in macroeconomic theory for PhD students
concentrates on the analytic
Economics 205A
Fall 2016
K. Kletzer
Notes: Consumption smoothing and tilting
Consider the household optimization problem,
[
c1 1
s
1
cfw_cs ,as+1
s=t
max
subject to the budget identity,
as+1 = (1 + r) as + ws cs
and the solvency constraint,
lim
s
given
Economics 205A
Fall 2015
K. Kletzer
Sample Answers for Problem Set 2
1. a) Ans: We have the linearized solution from Problem Set 1. The steady state, (k , c ) , is given by
1 = (1 + Af (k ) )
and
c = Af (k ) k .
The linearized dynamics are given by
ct+1 c
Economics 205A
Fall 2016
K. Kletzer
Problem Set 1: Sample answers
1. (a) Ans: Write the Lagrangian as
T
L=
t=0
t u (ct ) + t (1 ) kt + f (kt ) ct kt+1 )
and impose the restrictions that kt+1 0 and ct 0 for T t 0. The necessary conditions are
L
L
= t u (c
Economics 205A
Fall 2016
K. Kletzer
Problem Set 3: Sample Answers
1. a) Ans: The households optimization problem is
max
cfw_ct ,at+1 t0
t=0
t
c1
1
t
1
subject to the budget identity
at+1 at = rt at + wt ct
and the solvency constraint
lim
T
T
t=1
1
aT +
Economics 205A
Fall 2014
K. Kletzer
Problem Set 3
Due: Thursday, November 6
1. Consider a household that maximizes a utility over consumption given by U0 =
1
t ct 1
t=0 1 .
The
household receives a wage rate, wt , for labor which it supplies perfectly ine
Economics 205A
Fall 2012
K. Kletzer
Problem Set 2 Sample Answers
1. Ans: a) The command optimum solves
td
0
cfw_
1c te xam
1
1
tk, tc
subject to
,c k ) k( f = k
tk
0
.
t
0k
equal to
0
k
given initial
for all
b) The current-time Hamiltonian is
. )c k )
,0 = t
)1+ k(
t
.0 1+ Tk
, tc
f+1
and
) k( f +
t
) 1+tc(
0=
tk )
)1+ k(
t
.0 =
t
t
1+ T
> 0k
k
1+ Tk
L
1+t
k
1+tk
L
0 1+tk
0=
t
tk )
u = ) tc( u
1+ Tk ) Tc(
0=
u T
t T
t T
t
0=
0=
1+tk
L
0
t
t
t T
0 t 1 T
f + 1 1+t =
1+tk
L
0
c
,0 ) c( u =
L
c 0 1
Economics 205A
Fall 2008
K. Kletzer
A Primer on Intertemporal Optimization in Continuous Time
A. Discrete-time optimization
We begin with the discrete-time optimal savings problem for a representative household:
T
)tc( u
to be greater than
,
to be
tw + t
Economics 205A
Fall 2008
K. Kletzer
A Primer on Continuous-time Economic Dynamics
A. Linear Differential Equation Systems
(i) Simplest case
We begin with the simple linear first-order differential equation,
. 0x = ) 0( x
,xa = x
The general solution is
,
Economics 205A
Fall 2012
K. Kletzer
Midterm Exam Sample Answers
1. Use the discrete-time optimal growth model to answer this question. Assume production uses capital
and that representativehousehold utility is
. c gol
= 0U
N ,K( FA = Y
)
and labor under
Economics 205A
Fall 2012
K. Kletzer
Problem Set 4 Sample Answers
1. a) Ans: The household solves the problem,
0=t +
1xam cfw_
ta, tc
, tc tT tw + ta ) tr + 1( = 1+ta
0 1+ Ta
and the solvency condition,
sr + 1 1=s T
1 T mil
0a
given initial assets,
, ) tc(
Economics 205A
Fall 2012
K. Kletzer
Problem Set 3 Sample Answers
1. a) Ans: The households problem is given by
sd sc gol )ts(e
and
a
t t = t
vd vr st
,0
ar
sa, sccfw_
r
t
s
e sa mil
. For the household,
t
+
t
a
t
given initial financial wealth,
t
= U x
Economics 205A
Fall 2016
K. Kletzer
Tobins q
Consider the decentralized economy with capital stock adjustment costs. In this economy, households
supply one unit of labor perfectly inelastically, and adjustment costs are quadratic.
To find the competitive
Economics 205A
Fall 2016
K. Kletzer
Problem Set 2
Due: Tuesday, October 18, 2016
1. This problem examines how a temporary productivity shock aects the optimal growth solution.
The production function is given by yt = At f (kt ) for f (k) strictly concave.
Economics 205A
Fall 2014
K. Kletzer
Problem Set 2
Due: Monday, October 27, 2014
1. This problem examines how a temporary productivity shock affects the optimal growth solution.
The production function is given by yt = At f (kt ) for f (k) strictly concave
Economics 205A
Fall 2014
K. Kletzer
Problem Set 4
Additional Study Problems for Midterm
1. In this problem, we compare the command optimum to the decentralized equilibrium for a representative agent economy. A representative household seeks to maximize th
Economics 205A
Fall 2014
K. Kletzer
Problem Set 5
Due: December 1, 2014
1. Is this problem, we add uncertainty to the optimal consumption problem. Assume that wt = w (st ) is
i.i.d. with distribution over the finite space, cfw_w (1) , ., w (N ). That is,
Economics 205A
Fall 2014
K. Kletzer
Problem Set 6
Due: December 10, 2014
1. In this problem, we compare complete and incomplete markets in the two household endowment
i
economy. The endowment of each household, yt , is stochastic but the total endowment o
Economics 205A
Fall 2014
K. Kletzer
Problem Set 1
Due: Friday, October 17, 2014
1. Consider the simple command economy with capital in which the planner seeks to maximize the utility
function
T
t u (ct )
U0 =
t=0
given the resource identity
kt+1 kt = f (
Economics 205A
Fall 2014
K. Kletzer
Notes on the Linear Growth Model
Start with the equation of motion for the capital stock,
kt+1 = f (kt ) + (1 ) kt ct
and let f (kt ) = Akt for a positive constant A. This is a production function that displays constant
Economics 205A
Fall 2014
K. Kletzer
Midterm Exam
Monday, November 17, 2014
1. Use the basic representative agent economy to answer this question. The households utility function
0=t
= 0U
) tk(
f = ty
the production function is
, ) tc( u t
is
and the reso
Economics 205A
Fall 2014
K. Kletzer
Final Exam
Thursday, Decmber 18, 2014
Part I (20 points each):
1. Consider a representative household economy with production and a government. The households
utility is U0 =
t
t=0 u (ct ).
The representative firms prod
Economics 205A
Fall 2014
K. Kletzer
Notes: Consumption smoothing and tilting
Consider the household optimization problem,
max
cfw_cs ,as+1
s=t
c1 1
s
1
subject to the budget identity,
as+1 = (1 + r) as + ws cs
and the solvency constraint,
lim
s
1
1+r
st
Economics 205A
Fall 2014
K. Kletzer
A Primer on Intertemporal Optimization in Continuous Time
A. Discrete-time optimization
We begin with the discrete-time optimal savings problem for a representative household:
T
t u (ct )
max
cfw_ct ,at
(1)
t=0
subjec
Economics 205A
Fall 2016
K. Kletzer
Problem Set 3
Due: Tuesday, November 2
1. Consider a household that maximizes a utility over consumption given by U0 =
P
t=0
t
c1
1
t
1 .
The
household receives a wage rate, wt , for labor which it supplies perfectly in
Economics 205A
Fall 2016
K. Kletzer
Notes on linearization of the optimal growth model
I. Begin with our discrete-time system derived by maximizing the planners utility subject to its resource
constraint:
u (ct ) = 1 + f (kt+1 ) u (ct+1 )
and
kt+1 kt = f
Economics 205A
Fall 2016
Ken Kletzer
Problem Set 1
Due: Tuesday, October 4, 2016
1. Consider the simple command economy with capital in which the planner seeks to maximize the
utility function
T
U0 =
t u (ct )
t=0
given the resource identity
kt+1 kt = f