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

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 702: Practice Problems 8
Q1. Consider the simplified version of the model we discussed in the lecture. In that model, given GP =
(G, ), an equilibrium is a sequence (c , l , L , P , W , ) such that
a. Household optimization: (c , l ) solves
max log(c

Econ 702: Practice Problems 6
Q1. Consider a two period neoclassical growth model where household utility is given by
log(c0 ) + log(1 l0 ) + [log(c1 ) + log(1 l1 )]. The social planners problem is
max
c0 ,c1 ,l0 ,l1 ,k0 ,k1 ,k2
log(c0 ) + log(1 l0 ) + [l

Econ 702: Practice Problems 11
Q1. In order to understand how the price level is determined in the neoclassical growth model with money,
consider the following model.
Output is exogenous and constant at Y .
P t
Utility:
t=0 u(ct ) + (G0 , G1 , . . .).

Econ 702: Practice Problems 9
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, given GP

Econ 702: Practice Problems 1: S
A1. Profit is defined as
t = F (Kt , Lt ) rt Kt wt Lt
We will prove that if firms maximize profit taking prices as given and the production function exhibits
constant return to scale (CRS), then t = 0.
First, CRS implies t

Programming in python
Akio Ino
February 13th, 2015
1 / 17
Todays plan
I
In the last lecture, we learned
1. how to open python
2. how to run a python code
3. how to write a python code
I
Today we will learn
1. convenient ways to use python
2. how to use Nu

Econ 702: Practice Problems 1
Q1. Suppose that the production function F (K, L) exihibits constant return to scale and differentiable.
Prove that the profit of firms is zero when firms maximize their profit.
Q2. Suppose that the production function is Cob

Econ 702: Practice Problems 2
Q1. Suppose that the production function F (K, L) exihibits constant return to scale and differentiable.
Prove that the marginal product of capital and labor is homogeneous of degree zero.
Q2. Modify the sample code in my web

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 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 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 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,