Mathematics for Economists

Calculus of variations

Zb

a

F



t,x,x



dt! maxmin

x(a)=xa,x(b)=xb

Letybe an arbitrary function, called variation, withy(a)=y(b)= 0 .Let

ψ(λ)$

Zb

a

F



t,x(t)+λy(t),x(t)+λy(t)



dt

Ifxis an optimal function then by Fermatís principleψ^0 ( 0 )= 0.