Lecture 7
The Solow Model and Convergence
Optimal Growth
Noah Williams
University of Wisconsin - Madison
Economics 312/702
Williams
Economics 312/702
Introducing Technological Progress
Re-introduce TFP. No limits to innovation.
Slightly simpler to introdu

Economics 702/312
Macroeconomics
Noah Williams
Problem Set 1
Due in class on February 6.
1. Some countries have introduced workweek restrictions, which we model as the requirement N N . Consider a static model where the representative agent has
preference

Econ 702: Practice Problems 5 Solution
A1.
1. The Bellman equation of this problem is
v(k) = max
cfw_log(c) + v(k 0 )
0
c,k
s.t. c + k 0 k
c, k 0 0
2. Let v (0) = 0. Then v (1) is defined as
v (1) (k) = max
log(c) + v (0) (k 0 )
0
c,k
= max
cfw_log(c)

Econ 702: Solving the social
planners problem of the
neoclassical growth model
Akio Ino
March 20th, 2015
1 / 25
Todays plan
I
I
We learned that by solving the SPP we can
compute the eqm (Negishi method).
There are two algorithm to solve the SPP.
1. Shooti

Econ 702: Practice Problems 4
Q1. Consider a version of the neoclassical growth model where household utility is given by
X
t u(ct )
t=0
where (0, 1), u0 > 0, u00 < 0, and limc0 u0 (c) = .
1. Write down the social planners problem for this model.
2. Writ

Econ 702: Practice Problems 8 Solution
A1.
1. If P A > L, then the firm demand as much labor as possible, so l = . If P A < L, then L = 0.
These solutions dont satisfy the market clearing condition. So in the equilibrium, W = AP ,
and = 0. The labor deman

Econ 702: Practice Problems 6 Solution
A1.
1. Since the Inada condition limc0 uc (c, l) = , liml1 ul (c, l) is satised, ct > 0
and lt < 1. In addition, since F (k, 0) = 0, at the solution lt > 0.
Let t and denote the Lagrange multiplier on the resource co

Econ 702: Practice Problems 3: Solution
Q1.
1. Solve the following 2 period social planners problem:
max
cfw_c0 ,c1 ,K0 ,K1 ,K2
s.t.
log(c0 ) + log(c1 )
c0 + K1 AK0
c1 + K2 AK1
0
K0 = K
c0 , c1 , k1 , k2 0
2. Using your answer in the previous problem, co

Econ 702: Practice Problems 5
Q1. Consider the social planners problem of a neoclassical growth model
max
cfw_ct ,kt t=0
X
t log(ct )
t=0
s.t. ct + kt+1 kt ,
k0 = k0
ct , kt 0,
t
t
where (0, 1) and (0, 1).
1. Write down the Bellman equation.
2. Given v (

Econ 702: Practice Problems 11
A1. Let t , t , 1 denote the Lagrange multipliers on the budget constraint, CIA constraint, and the
initial condition, respectively. Then the Lagrangian of this problem is
L=
X
t u(ct ) + (G, G, . . .) +
t=0
X
+
X
t [Pt Yt

Econ 702: Practice Problems 3
Q1.
1. Solve the following 2 period social planners problem:
max
cfw_c0 ,c1 ,K0 ,K1 ,K2
s.t.
log(c0 ) + log(c1 )
c0 + K1 AK0
c1 + K2 AK1
0
K0 = K
c0 , c1 , k1 , k2 0
2. Using your answer in the previous problem, compute the

Econ 702: Practice Problems 9: Solution
Q1. Consider the two period model we discussed in the lecture note 19:
2 periods
No labor, capital
Exogenous output Y0 , Y1
GP = cfw_G0 , G1 , 0 , 1 , B1
No initial debt: B0 = 0.
No money.
Under this setting,

Econ 702: Practice Problems 4: Answer key
A1.
1. The social planners problem for this model is
max
X
cfw_ct ,kt t=0
t u(ct )
t=0
s.t. ct + kt+1 F (kt , 1) + (1 )kt
0,
k0 = K
ct , kt 0,
t 0
t 0
2. The necessary and sufficient condition for optimality is

Econ 702: Practice Problems 7
Q1. (Programming) Consider the following social planners problem for a version of the
neoclassical growth model:
max
(ct ,kt )t=0
X
t log(ct ),
subject to
t=0
ct + kt+1 kt + (1 )kt ,
k0 = k0
ct , kt 0,
t 0
t 0
where = 0.96,

Econ 702: Practice problem 12
Q1. Consider the following real business cycle model.
Production function: F (Kt , At Lt ) = (Kt ) (At Lt )1 where (0, 1).
Productivity is uncertain. Let ht (A0 , A1 , . . . , At ) denote the history of shocks up until the

Econ 702: Practice Problems 10: Solution
A1.
1. Let t denote the Lagrange multiplier on the GBC in period t. Then the first order conditions are
G0 : 0 (G0 ) = 0
(1)
G1 : 0 (G1 ) = 1
1 : 0 = 1 1
B
(2)
0l
1l
(3)
(1 0l ) = 0 [(1 0l ) 0l (1 0l )1 ]
(1 1l ) =

Econ 312/702 - Practice Problems Solution
Professor: Noah Williams, TA: Junhyong Kim & Tomoaki Kotera
1
(a) The rm prot maximization problem is for any t:
max zt kt rt kt = (zt rt )kt .
kt
The optimal decision is
kt
0
=
[0, ]
if
rt > zt
if
rt = zt .
if
r

k
=
k = syt
c = (1
n
s n
1
s)
=n
s n
s n
kt
N
kt = syt
Nt
1
kt
k ) k = 0
1
1
n)kt
nyt kt = (s
k =
.
kt .
s n
1
1
, y =
s n
1
,
pop
.
k
y
n
@pop
@n
=
@k
@n
n
s
=
1
(1 )
(s
n
)1
< 0.
k
c .
n
1
+
ns
n
2
=
1s
n
2
s > 0.
(s
n)k
k
(s
n)k
n
k = (s
k = k
n)

Econ 312/702 - Problem Set 3 Solution
Professor: Noah Williams, TA: Junhyong Kim & Tomoaki Kotera
Q1
The problem is:
0
max
u(c)
+
u(c
)
where: u(c) = ec
0
c,c ,s
subjected to: c +
c0
=y
1+r
Note that:
Inada condition does not hold, i.e., limc0 u0 (c) =

Noah Williams
Department of Economics
University of Wisconsin
Economics 312
Macroeconomics
Spring 2014
Final Examination Solutions1
FOR GRADUATE STUDENTS ONLY
Instructions: This is a 120 minute examination worth 100 total points. Each question is
worth 25

Econ 312/702 - Handout Last Week
Professor: Noah Williams, TA: Junhyong Kim & Tomoaki Kotera
Exercise 1: Dynamic model revisited (2016 Final)
See the solution of previous exam posted on webside.
Exercise 2: Simple Labor Search Model of McCall in infinite

Leisure Luxuries and the Labor Supply of Young Men
Mark Aguiar
Mark Bils
Kerwin Charles
Erik Hurst
September 15, 2016
Preliminary and Incomplete
Abstract
We explore the declining market hours of men in the last fifteen years, with a particular focus on le

Government Spending Multipliers in Good Times
and in Bad: Evidence from U.S. Historical Data
Valerie A. Ramey
University of California, San Diego and NBER
Sarah Zubairy
Texas A&M University
Abstract
We investigate whether U.S. government spending multipli

Econ 312/702 - Handout Week 14
Professor: Noah Williams, TA: Junhyong Kim & Tomoaki Kotera
Exercise 1: Sticky prices and efficiency wages revisited
Consider a generalized efficiency wage model where effort also depends on the outside options
available to

Econ 312/702 - Handout Week 14
Professor: Noah Williams, TA: Junhyong Kim & Tomoaki Kotera
Exercise 1: Sticky prices and efficiency wages revisited
Consider a generalized efficiency wage model where effort also depends on the outside options
available to

.
Econ 702/312 Review Session
.
Junhyong Kim & Tomoaki Kotera
March 27th
Junhyong Kim & Tomoaki Kotera
Econ 702/312 Review Session
March 27th
1 / 23
A Couple of Information
Midterm 2: This Wednesday in class time.
Extra OH:
Today
Tomorrow
Tomo
16:30-18:30

Economics 702/312
Macroeconomics
Noah Williams
Problem Set 3
Due in class on March 8.
1. Consider a simple two period model where an individual receives income y the first
period only (nothing in the second) and can borrow and lend freely at a constant
in