Introduction to Modern Economic Growth value can be written as: Z (13.7) V (ν, t) = t ∞ ¸ ∙ Z s 0 exp − r (s ) ds [χ(ν, s)x(ν, s) − ψx(ν, s)] ds t where r (t) is the market interest rate at time t Alternatively, assuming that the value function is differentiable in time, this could be written in the form of HamiltonJacobi-Bellman equations as in Theorem 7.10 in Chapter 7: (13.8) r (t) V (ν, t) − V˙ (ν, t) = χ(ν, t)x(ν, t) − ψx(ν, t), where x(ν, t) and χ(ν, t) are the profit-maximizing choices for the monopolist Exercise 13.1 asks you to provide a different derivation of this equation than in Theorem 7.10 13.1.2 Characterization of Equilibrium An allocation in this economy is defined by the following objects: time paths of consumption levels, aggregate spending on machines, and aggregate R&D expenditure [C (t) , X (t) , Z (t)]∞ t=0 , time paths of available machine types, [N (t)]∞ t=0 , time paths of prices and quantities of each machine and the net present discounted value of profits from that machine, [χ (ν, t) , x (ν, t) , V (ν, t)]∞ ν∈N(t),t=0 , and time paths of interest rates and wage rates, [r (t) , w (t)]∞ t=0 An equilibrium is an allocation in which all existing research firms choose ∞ [χ (ν, t) , x (ν, t)]∞ ν∈[0,N(t)],t=0 to maximize profits, the evolution of [N (t)]t=0 is deter- mined by free entry, the time paths of interest rates and wage rates, [r (t) , w (t)]∞ t=0 , are consistent with market clearing, and the time paths of [C (t) , X (t) , Z (t)]∞ t=0 are consistent with consumer optimization We now characterize the unique equilibrium of this economy Let us start with the firm side Since (13.6) defines isoelastic demands, the solution to the maximization problem of any monopolist ν ∈ [0, N (t)] involves setting the same price in every period (see Exercise 13.2): (13.9) χ(ν, t) = ψ for all ν and t 1−β That is, all monopolists charge a constant rental rate, equal to a mark-up over the marginal cost Without loss of generality, let us normalize the marginal cost of 575