Homework #3
In…nite Period Dynamic Programming: Euler Equation Method
ECONS 500, Washington State University, Fall 2020
This homework assignment is due at the beginning of lecture on September 16. Write down
the names of your group members. Please send the scanned PDF …le of your homework
answers to the email address of our TA.
1. Brock-Mirman with In…nite Horizon: Euler Equation Method
Consider the standard neoclassical growth model. For your convenience, the planning
problem is:
X
+1
t
max1 log (ct ) ;
fct ;kt+1 gt=0
t=0
given k0 > 0 and subject to the constraint that ct + kt+1 = f (kt ), where f (kt ) =
Akt + (1 ) kt . Let 0 < < 1 and = 1.
(a) Derive the Euler equation from the FOC of the Bellman Equation and the Enve-
lope Theorem.
(b) Solve the policy function from the Euler Equation through “Guess and Verify”
method. Conjecture that the policy function takes the form of
g (k) = sAk :
From the Euler Equation, verify your conjecture. Find the equation that deter-
mines s. Solve for s.
(c) Solve the policy function from the Euler Equation through “Policy Function It-
eration”method in the following steps.
In Step 0, conjecture the policy function as g0 (k) = 0.
In Step 1, plug g0 (k) into the Euler equation. Solve for the policy function
k 0 = g1 (k). From your solution, verify that g1 (k) takes the form of
g1 (k) = s1 Ak ;
where s1 is a function of model parameters.
In Step j 2, plug gj 1 (k) = sj 1 Ak from the previous step into the Euler
equation. Solve for the policy function k 0 = gj (k). From your solution, verify
that gj (k) takes the form of
gj (k) = sj Ak ;
where sj is de…ned recursively as a function of sj 1 .
1
This study source was downloaded by 100000899606070 from CourseHero.com on 09-27-2025 21:22:10 GMT -05:00
https://www.coursehero.com/file/238563331/Homework-3pdf/
In…nite Period Dynamic Programming: Euler Equation Method
ECONS 500, Washington State University, Fall 2020
This homework assignment is due at the beginning of lecture on September 16. Write down
the names of your group members. Please send the scanned PDF …le of your homework
answers to the email address of our TA.
1. Brock-Mirman with In…nite Horizon: Euler Equation Method
Consider the standard neoclassical growth model. For your convenience, the planning
problem is:
X
+1
t
max1 log (ct ) ;
fct ;kt+1 gt=0
t=0
given k0 > 0 and subject to the constraint that ct + kt+1 = f (kt ), where f (kt ) =
Akt + (1 ) kt . Let 0 < < 1 and = 1.
(a) Derive the Euler equation from the FOC of the Bellman Equation and the Enve-
lope Theorem.
(b) Solve the policy function from the Euler Equation through “Guess and Verify”
method. Conjecture that the policy function takes the form of
g (k) = sAk :
From the Euler Equation, verify your conjecture. Find the equation that deter-
mines s. Solve for s.
(c) Solve the policy function from the Euler Equation through “Policy Function It-
eration”method in the following steps.
In Step 0, conjecture the policy function as g0 (k) = 0.
In Step 1, plug g0 (k) into the Euler equation. Solve for the policy function
k 0 = g1 (k). From your solution, verify that g1 (k) takes the form of
g1 (k) = s1 Ak ;
where s1 is a function of model parameters.
In Step j 2, plug gj 1 (k) = sj 1 Ak from the previous step into the Euler
equation. Solve for the policy function k 0 = gj (k). From your solution, verify
that gj (k) takes the form of
gj (k) = sj Ak ;
where sj is de…ned recursively as a function of sj 1 .
1
This study source was downloaded by 100000899606070 from CourseHero.com on 09-27-2025 21:22:10 GMT -05:00
https://www.coursehero.com/file/238563331/Homework-3pdf/
