Homework #5
Mathematical Methods II
Spring 2010
1. This problem examines the labor supply decision of a consumer who derives utility from the
consumption of both market-produced and home-produced goods. Let c1t be the amount of
market-produced goods consu
Homework #4
Mathematical Methods II
Spring 2010
1. Consider the dynamic program
P (x) = max xT Qx + aT Ra + 2aT W x + E P x |x
a
where Q, R, and W are matrices such that the return function is jointly strictly concave in
(x, a) and
x
= Ax + Ba +
N (0, )
Homework #3
Mathematical Methods II
Spring 2010
1. Consider the following variant of the standard deterministic growth model.
The social planner can only change the capital stock if she pays a xed
cost . Preferences are logarithmic: u (c) = log (c). Produ
Homework #2
Mathematical Methods II
Spring 2010
1. Suppose that we have the quadratic dynamic programming problem
xT P x = max xT Qx + aT Ra + xT W a + aT Sx + xT AT + aT B T P (Ax + Ba)
a
where x is an n 1 vector (with 1 as the rst element) and a is a k
Homework #1
Mathematical Methods II
Spring 2010
1. Consider the dynamic program
v (x) = max Qx2 + Ra2 + 2W ax + v (x )
a
with law of motion
x = Ax + Ba.
Assume that (Q, R, W ) are such that the objective function is strictly
increasing and strictly concav
Final Exam
Mathematical Methods
ECON 180.616
1. Consider an individual who faces the dynamic programming problem
V (k ) = max (1 + ) u (c) + V k
k ,c
max cfw_u (c)
k ,c
subject to the budget constraints
c+k
f (k )
c+k
f (k )
and capital and consumption
1
Dynamic Programming
These notes are intended to be a very brief introduction to the tools of dynamic
programming. Several mathematical theorems the Contraction Mapping Theorem (also called the Banach Fixed Point Theorem), the Theorem of the Maximum (or
1
Finite State Space Dynamic Programming
Assume that the state space is nite: k ,k cfw_k1 , ., kn . The value function
then takes on only nitely-many values:
v (ki ) =
max
k cfw_k1 ,.,kn
cfw_u (f (ki ) + (1 ) ki k ) + v (k ) .
Straightforward grid search