Introduction to Modern Economic Growth where the constraint equation uses net output, (14.23), and the resource constraint, (14.1) In this problem, QS (t) is the state variable, and C S (t) is the control variable It can be verified that this problem satisfies all the assumptions of Theorem The current-value Hamiltonian for this problem can be written as h i ¡ S S S ¢ C S (t)1−θ − S −1/β S S ˆ βQ (t) L − η (λ − 1) C (t) +µ (t) η (λ − 1) (1 − β) H Q ,C ,µ = 1−θ The necessary conditions for a maximum are ¡ ¢ ˆ C QS , C S , µS = C S (t)−θ = µS (t) η (λ − 1) = H ¡ ¢ ˆ Q QS , C S , µS = µS (t) η (λ − 1) (1 − β)−1/β βL = ρµS (t) − µ˙ S (t) H Ô Ê lim exp (t) àS (t) QS (t) = t→∞ Moreover, it is straightforward to verify that the current-value Hamiltonian is concave in C and Q, so any solution to these necessary conditions is an optimal plan Combining these conditions, we obtain the following growth rate for consumption in the social planner’s allocation (see Exercise 14.5): (14.24) ´ 1³ C˙ S (t) −1/β S = g η (λ − 1) (1 − β) ≡ βL − ρ C S (t) θ Clearly, total output and average quality will also grow at the rate gS in this allocation Comparing g S to g ∗ in (14.21), we can see that either could be greater In particular, when λ is very large, gS > g ∗ , and there is insufficient growth in the equilibrium We can see this as follows: as λ → ∞, gS /g ∗ → (1 − β)−1/β > In contrast, to obtain an example in which there is excessive growth in the equilibrium, suppose that θ = 1, β = 1/2, λ = 1.0355, η = 1, L = and ρ = 0.071 In this case, it can be verified that gS ≈ 0, while g∗ ≈ 0.015 This illustrates the counteracting influences of the appropriability and business stealing effects discussed above The following proposition summarizes this result: Proposition 14.3 In the model of competitive innovations described above, the decentralized equilibrium is generally Pareto suboptimal, and may have a higher or lower rate of innovation and growth than the Pareto optimal allocation 621