Problem Set 10
EC720.01  Math for Economists
Peter Ireland
Boston College, Department of Economics
Fall 2009
Due Tuesday, November 24
1. LinearQuadratic Dynamic Programming
This problem will give you more practice with dynamic programming under certainty; it
presents another case in which an explicit solution for the value function can be found using
the guessandverify method. The problem is to choose sequences
{
z
t
}
∞
t
=0
for a flow variable
and
{
y
t
}
∞
t
=1
for a stock variable to maximize the objective function
∞
X
t
=0
β
t
(
Ry
2
t
+
Qz
2
t
)
,
subject to the constraints
y
0
given and
Ay
t
+
Bz
t
≥
y
t
+1
for all
t
= 0
,
1
,
2
, ...
, where
β
,
R
,
Q
,
A
, and
B
are constant, known parameters.
This
problem can be described as being “linearquadratic,” because the constraint is linear and
the objective function is quadratic. The discount factor lies between zero and one, 0
< β <
1,
and to make the objective function concave, it is helpful to assume that
R <
0 and
Q <
0
as well.
a. Write down the Bellman equation for this problem.
b. Now guess that the value function also takes the quadratic, timeinvariant form
v
(
y
t
;
t
) =
v
(
y
t
) =
Py
2
t
,
where
P
is an unknown constant. Using this guess, derive the firstorder condition for
z
t
and the envelope condition for
y
t
.
c. Use your results from above to show that the unknown
P
must satisfy
P
=
R
+
βA
2
QP
Q
+
βB
2
P
.
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 Fall '09
 IRELAND
 Economics, Quadratic Formula, Equations, Optimization, The Land, objective function, Bellman equation, Jacopo Francesco Riccati

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