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) = − u0−1 [µ (0) exp (ρs)] ds Since along any optimal path we must have limt→∞ xˆ (t) = 0, we have that Z ∞ u0−1 [µ (0) exp (ρs)] ds = 350