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   .