Economic growth and economic development 296

Introduction to Modern Economic Growth
where ∇x U represents the gradient vector of U with respect to its first K arguments, and ∇y U represents its gradient with respect to the second set of K argu-

ments. Notice that (6.21) is a functional equation in the unknown function π (·) and
characterizes the optimal policy function.
These equations become even simpler and more transparent in the case where
both x and y are scalars. In this case, (6.19) becomes:
(6.22)

∂U(x, y ∗ )
+ βV 0 (y ∗ ) = 0,
∂y

where V 0 the notes the derivative of the V function with respect to it single scalar
argument.
This equation is very intuitive; it requires the sum of the marginal gain today
from increasing y and the discounted marginal gain from increasing y on the value
of all future returns to be equal to zero. For instance, as in Example 6.1, we can
think of U as decreasing in y and increasing in x; equation (6.22) would then require
the current cost of increasing y to be compensated by higher values tomorrow. In
the context of growth, this corresponds to current cost of reducing consumption
to be compensated by higher consumption tomorrow. As with (6.19), the value of
higher consumption in (6.22) is expressed in terms of the derivative of the value
function, V 0 (y ∗ ), which is one of the unknowns. To make more progress, we use the
one-dimensional version of (6.20) to find an expression for this derivative:
∂U(x, y ∗ )
.
∂x
Now in this one-dimensional case, combining (6.23) together with (6.22), we have

(6.23)

V 0 (x) =

the following very simple condition:
∂U(x, π (x))
∂U(π (x) , π (π (x)))
+β
=0
∂y
∂x
where ∂x denotes the derivative with respect to the first argument and ∂y with
respect to the second argument.
Alternatively, we could write the one-dimensional Euler equation with the time
arguments as
(6.24)

∂U(x∗ (t + 1) , x∗ (t + 2))
∂U(x (t) , x∗ (t + 1))
+β
= 0.
∂x (t + 1)
∂x (t)
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