  The validity of the renormalization group approach for large $N$ is clarified
by using the vector model as an example. An exact difference equation is
obtained which relates free energies for neighboring values of $N$. The
reparametrization freedom in field space provides infinitely many identities
which reduce the infinite dimensional coupling constant space to that of finite
dimensions. The effective beta functions give exact values for the fixed points
and the susceptibility exponents.
