Mathematics for Economists

Production saving model

Consider the terms withkt+ 1 witht<T.It is in two terms of the sum

βtU(f(kt)kt+ 1 )+βt+^1 U(f(kt+ 1 )kt+ 2 )

Di§erentiating with respect tokt+ 1 and assuming that there is an internal

solution

βtU^0 (f(kt)kt+ 1 )( 1 )=βt+^1 U^0 (f(kt+ 1 )kt+ 2 )f^0 (kt+ 1 )= 0

DeÖnition

The second order di§erence equation

U^0 (f(kt)kt+ 1 ) = βU^0 (f(kt+ 1 )kt+ 2 )f^0 (kt+ 1 )

k 0 = a,kT+ 1 = 0

is the Euler equation for the production saving model.