Introduction to Modern Economic Growth Thus using the notation y = π (x) and combining these two equations, we have α [π (x)]α−1 =β for all x, xα − π (x) [π (x)]α − π (π (x)) which is a functional equation in a single function, π (x) There are no straightforward ways of solving functional equations, but in most cases guess-and-verify type methods are most fruitful For example in this case, let us conjecture that (6.27) π (x) = axα Substituting for this in the previous expression, we obtain αaα−1 xα(α−1) = β , xα − axα aα xα2 − a1+α xα2 β α = , α a x − axα which implies that, with the policy function (6.28), a = βα satisfies this equation Recall from Corollary 6.1 that, under the assumptions here, there is a unique policy function Since we have established that the function π (x) = βαxα satisfies the necessary and sufficient conditions (Theorem 6.10), it must be the unique policy function This implies that the law of motion of the capital stock is (6.28) k (t + 1) = βα [k (t)]α and the optimal consumption level is c (t) = [1 − βa] [k (t)]α Exercise 6.7 continues with some of the details of this example, and also shows how the optimal growth equilibrium involves a sequence of capital-labor ratios converging to a unique steady state Finally, we now have a brief look at the intertemporal utility maximization problem of a consumer facing a certain income sequence Example 6.5 Consider the problem of an infinitely-lived consumer with instantaneous utility function defined over consumption u (c), where u : R+ → R is strictly increasing, continuously differentiable and strictly concave The individual discounts 286