eco 512
ECO 512
Spring 2019
Locay
Final Exam - Optimal Control Problems
1. Carlos' lifetime utility is given by:
0
( )
T
t e c t dt
He begins life with assets equal to 0
s . His assets evolve over time according to the following:
( ) ( ) ( )s t rs t c t m
where r is the interest rate and m is a positive medical cost that Carlos must pay every instant
he is alive. Carlos' problem is to choose a consumption path, ( )c t , and a lifespan, T, so as to
maximize lifetime utility:
0
max ( )
( ),
T
t e c t dt
c t T
Assume that r . Find Carlos' optimal * ( )c t and
* T .
2. United Way, UW, has $1 billion dollars to allocate to a continuum of charities. Charities are
denoted by the fraction of their revenue they use for administration, [0,1] . is uniformly
distributed over the interval [0,1]. This means that the density of is simply ( ) 1f . Let
( )x be the amount UW gives to charity . Then the amount actually received by the
recipients of charity is (1 ) ( )x . UW aims to choose ( )x so as to maximize:
1 1
1/ 2 1/ 2
0 0
(1 ) ( ) ( ) (1 )x f d x d
subject to the budget constraint:
1 1
0 0
( ) ( ) 1x f d xd
Recall that for this integral constraint you create a state variable, ( )z , where z x , and
(1) 1z and (0) 0z . Find the optimal * ( )x .
3. Ivette retires at time with savings ( )s s . She lives to time T . She wishes to choose a
consumption path, ( )c t , that maximize her utility over her retirement:
ln ( )
T
t e c t dt
Her assets evolve according to ( ) ( ) ( )s t rs t c t . Ivette can choose how long she lives, as
long as T . She is also free to chose ( )s T as long as it is non-negative. Assume that
r , and that rs e . Find the optimal
* * * ( ), ( ), ,c t s t T and
* * ( )s T . Find the maximized
value of utility, ( , )V s .