Introduction to Modern Economic Growth so that the price of output at every instant is normalized to Thus we impose ÃZ ! 1−ε N(t) (13.44) χ (ν, t)1−ε dν = for all t With this choice of numeraire, we obtain the consumer Euler equation as (see Exercise 13.23): C˙ (t) = r (t) − ρ C (t) With similar arguments to before, the net present discounted value of that mo- (13.45) nopolist owning the patent for product ν can be written as ¸ ∙ Z s Z ∞ 0 exp − r (s ) ds [χ(ν, s)c(ν, s) − w (s) c(ν, s)] ds, V (ν, t) = t t where w (t) c(ν, t) is the total expenditure of the firm to produce a total quantity of c(ν, t) (given the production function (13.40) and the wage rate at time t equal to w (t)), while χ(ν, t)c(ν, t) is its revenue, consistent with the demand function (13.43) The maximization of the net present discounted value again requires profit maximization at every instant Given the iso-elastic demand curve facing the firm in (13.43), this implies the following optimal monopoly price ε w (t) for all ν and t χ (ν, t) = ε−1 Since all firms charge the same price, they will all produce the same amount and employ the same amount of labor At time t, there are N (t) products, so the labor market clearing condition (13.42) implies that (13.46) c (ν, t) = l (ν, t) = L − LR (t) for all ν and t N (t) Consequently, the instantaneous profits of each monopolist at time t can be written as (13.47) π (ν, t) = χ(ν, s)c(ν, t) − w (t) c(ν, t) L − LR (t) w (t) for all ν and t = ε − N (t) Since prices, sales and profits are equal for all monopolists, we can simplify notation by letting V (t) = V (ν, t) for all ν and t 595