Economic growth and economic development 679

Introduction to Modern Economic Growth Again as in Chapter 13, we assume that all machines in both sectors are supplied by monopolists that have a fully-enforced perpetual patent on the machines We denote the prices charged by these monopolists at time t by χL (ν, t) for ν ∈ [0, NL (t)] and χH (ν, t) for ν ∈ [0, NH (t)] Once invented, each machine can be produced at the fixed marginal cost ψ in terms of the final good, which we again normalize to ψ ≡ − β This implies that total spending on machines at time t is given by ! ÃZ Z NH (t) NL (t) xL (ν, t) dν + xH (ν, t) dν X (t) = (1 − β) 0 The innovation possibilities frontier is assumed to take a form similar to the lab equipment specification in Chapter 13: (15.7) N˙ L (t) = η L ZL (t) and N˙ H (t) = η H ZH (t) , where ZL (t) is R&D expenditure directed at discovering new labor-augmenting machines at time t, while ZH (t) is R&D expenditure directed at discovering Hcomplementary machines Total R&D spending is the sum of these two, i.e., Z (t) = ZL (t) + ZH (t) The value of a monopolist that discovers one of these machines is again given by the standard formula for the present discounted value of profits: ¸ ∙ Z s Z Ê Ô 0 exp r (s ) ds χf (ν, s)xf (ν, s) − ψxf (ν, s) ds, (15.8) Vf (ν, t) = t t where f = L or H, and r (t) is the market interest rate at time t Once again, it is sometimes more convenient to work with the Hamilton-Jacobi-Bellman version of this value function, which takes the form: (15.9) r (t) Vf (ν, t) − V˙ f (ν, t) = χf (ν, t)xf (ν, t) − ψxf (ν, t) Throughout, we normalize the price of the final good at every instant to 1, which is equivalent to setting the ideal price index of the two intermediates equal to one, i.e., (15.10) Ô Ê (pL (t))1 + (1 − γ)ε (pH (t))1−ε 1−ε = for all t, where pL (t) is the price index of YL at time t and pH (t) is the price of YH We also denote the factor prices by wL (t) and wH (t) 665