Introduction to Modern Economic Growth where φ < and LR (t) is labor allocated to R&D activities at time t Labor market clearing now requires LE (t) + LR (t) = L (t) , (13.34) where LE (t) is the level of employment in the production sector, and the labor market clearing condition takes into account that population is changing over time The key assumption for the model is that φ < The case where φ = is the one analyzed in the previous section, and as commented above, with population growth this would lead to an exploding path, leading to infinite utility However, the model is well behaved when φ < Aggregate output and profits are given by (13.25) and (13.26) as in the previous section An equilibrium is also defined similarly Let us focus on the BGP, where a constant fraction of workers are allocated to R&D, and the interest rate and the growth rate are constant Suppose that this BGP involves positive growth, so that the free entry condition holds as equality Then, the BGP free entry condition can be written as (see Exercise 13.16) (13.35) ηN (t)φ βLE (t) = w (t) r∗ As before, the equilibrium wage is determined by the production side, (13.13), as w (t) = βN (t) / (1 − β) Combining this with the previous equation gives the fol- lowing free entry condition ηN (t)φ−1 (1 − β) LE (t) = r∗ Now differentiating this condition with respect to time, we obtain (φ − 1) N˙ (t) L˙ E (t) + = N (t) LE (t) Since in BGP, the fraction of workers allocated to research is constant, we must have L˙ E (t) /LE (t) = n This implies that the BGP growth rate of technology is given by (13.36) ∗ ≡ gN N˙ (t) n = N (t) 1−φ 591