Introduction to Modern Economic Growth competitive innovations also introduce the business stealing effect discussed in Chapter 12 Consequently, the equilibrium rate of innovation and growth can now be too high or too low We now investigate this question We proceed as in the previous chapter, first deriving quantities of machines that will be used in the final good sector by the social planner In the social planner’s allocation there is no markup on machines, thus we have xS (ν, t | q) = L ψ1/β = (1 − β)−1/β L Substituting this into (14.3), we obtain Y S (t) = (1 − β)−1/β QS (t) L, where again the superscript S refers to the social planner’s allocation The net output that can be distributed between consumption and research expenditure is Y˜ S (t) ≡ Y S (t) − X S (t) −1/β = (1 − β) −1/β = (1 − β) (14.23) S Q (t) L − S βQ (t) L Z ψq (ν, t) xS (ν, t | q) dν Moreover, given the specification of the innovation possibilities frontier above, the social planner can improve the aggregate technology as follows: Q˙ S (t) = η (λ − 1) Z S (t) , since an R&D spending of Z S (t) will lead to discoveries of better vintages at the flow rate of η and each of these vintages increases average quality of machines by a proportional amount λ − Now, given this equation, the maximization problem of the social planner can be written as max Z∞ C S (t)1−θ − exp (−ρt) dt 1−θ subject to Q˙ S (t) = η (λ − 1) (1 − β)−1/β βQS (t) L − η (λ − 1) C S (t) , 620