Mathematics for Economists

Homework

(^1) Solve the problems in calculus of variations with the method of
optimal control.
(^2) Show with the method of optimal control that the problem
R 1
0
r
1 +



x

 2

dt!maxx( 0 )= 0 ,x( 1 )=1 has no solution.

(^3) Show with the method of optimal control that the problem
R 1
0 t
α
r
1 +



x

 2

dt!minx( 0 )= 0 ,x( 1 )=1 has no solution if

α> 0.

(^4) LetUbe a concave increasing utility function and letxbe a
continuously di§erentiable consumption strategy. Show that there is
no such strategyR 1 xwhich maximizes the cumulated utility
0 U(x(t))dtwithx(^0 )=0 andx(^1 )=^1 .What can we say if we
introduce the restriction
(^)
x^
^1?