Economic growth and economic development 364

Introduction to Modern Economic Growth
Theorem 7.14 implies the following necessary condition for an interior continuously
differentiable solution (ˆ
x (t) , yˆ (t)) to this problem. There should exist a continuously differentiable function µ (t) such that
y (t)) = µ (t) ,
u0 (ˆ
and
µ˙ (t) = ρµ (t) .
The second condition follows since neither the constraint nor the objective function
depend on x (t). This is the famous Hotelling rule for the exploitation of exhaustible
resources. It charts a path for the shadow value of the exhaustible resource. In
particular, integrating both sides of this equation and using the boundary condition,
we obtain that
µ (t) = µ (0) exp (ρt) .
Now combining this with the first-order condition for y (t), we obtain
yˆ (t) = u0−1 [µ (0) exp (ρt)] ,
where u0−1 [·] is the inverse function of u0 , which exists and is strictly decreasing
by virtue of the fact that u is strictly concave. This equation immediately implies
that the amount of the resource consumed is monotonically decreasing over time.
This is economically intuitive: because of discounting, there is preference for early
consumption, whereas delayed consumption has no return (there is no production or
interest payments on the stock). Nevertheless, the entire resource is not consumed
immediately, since there is also a preference for smooth consumption arising from
the fact that u (·) is strictly concave.
Combining the previous equation with the resource constraint gives
x˙ (t) = −u0−1 [µ (0) exp (ρt)] .
Integrating this equation and using the boundary condition that x (0) = 1, we obtain
Z t
xˆ (t) = 1 −
u0−1 [µ (0) exp (ρs)] ds.
0

Since along any optimal path we must have limt→∞ xˆ (t) = 0, we have that
Z ∞
u0−1 [µ (0) exp (ρs)] ds = 1.
0

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