Introduction to Modern Economic Growth where (14.9) Q (t) = Z q(ν, t)dν is the average total quality of machines This expression closely parallels the derived production function (13.12) in the previous chapter, except that instead of the number of machine varieties, N (t), labor productivity is determined by the average quality of the machines, Q (t) This expression also clarifies the reasoning for the particular functional form assumptions above In particular, the reader can verify that it is the linearity of the aggregate production function of the final good, (14.3) in the quality of machines that makes labor productivity depend on average qualities With alternative assumptions, a similar expression to (14.8) would still obtain, but with a more complicated aggregator of machine qualities than the simple average (see, for example, Section 14.3) As a byproduct, we also obtain that aggregate spending on machines is (14.10) X (t) = (1 − β) Q (t) L Similar to the previous chapter, labor, which is only used in the final good sector, receives an equilibrium wage rate of β Q (t) L 1−β We next specify the value function for the monopolist of variety ν of quality (14.11) w (t) = q (ν, t) at time t As in the previous two chapters, despite the fact that each firm generates a stochastic stream of revenues, the presence of many firms with independent risks implies that each should maximize expected profits The net present value of expected profits can be written in the Hamilton-Jacobi-Bellman form as follows (14.12) r (t) V (ν, t | q) − V˙ (ν, t | q) = π(ν, t | q) − z(ν, t | q)V (ν, t | q), where z(ν, t | q) is the rate at which new innovations occur in sector ν at time t, while π(ν, t | q) is the flow of profits This value function is somewhat different from the ones in the previous chapter (e.g., (13.8)), because of the last term on the right-hand side, which captures the essence of competitive innovations When a new innovation occurs, the existing monopolist loses its monopoly position and is replaced by the 615