Economic growth and economic development 300

Introduction to Modern Economic Growth
Thus using the notation y = π (x) and combining these two equations, we have
α [π (x)]α−1
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
1
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
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