Saving Under Uncertainty
Peter Ireland
∗
EC720.01  Math for Economists
Boston College, Department of Economics
Fall 2010
This last example presents a dynamic, stochastic optimization problem that is simple
enough to allow a relatively straightforward application of the KuhnTucker theorem. The
optimality conditions derived with the help of the Lagrangian and the KuhnTucker theorem
can then be compared with those that can be derived with the help of the Bellman equation
and dynamic programming.
1
The Problem
Consider the simplest possible dynamic, stochastic optimization problem with:
Two periods,
t
= 0 and
t
= 1
No uncertainty at
t
= 0
Two possible states at
t
= 1:
Good, or high, state
H
occurs with probability
π
Bad, or low, state
L
occurs with probability 1
−
π
Notation for a consumer’s problem:
y
0
= income at
t
= 0
c
0
= consumption at
t
= 0
s
= savings at
t
= 0, carried into
t
= 1 (
s
can be negative, that is, the consumer
is allowed to borrow)
r
= interest rate on savings
y
H
1
= income at
t
= 1 in the high state
y
L
1
= income at
t
= 1 in the low state
y
H
1
> y
L
1
makes
H
the good state and
L
the bad state
c
H
1
= consumption at
t
= 1 in the high state
∗
Copyright c
2010 by Peter Ireland. Redistribution is permitted for educational and research purposes,
so long as no changes are made. All copies must be provided free of charge and must include this copyright
notice.
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 Fall '09
 IRELAND
 Economics, Dynamic Programming, Optimization, Bellman equation, Peter Ireland∗

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