Introduction to Modern Economic Growth the technology gap of the industry and aggregate output This will simplify the analysis below by making the technology gap in each industry the only industryspecific payoff-relevant state variable The objective function of each firm is to maximize the net present discounted value of net profits (operating profits minus R&D expenditures and plus or minus patent fees) In doing this, each firm will take the sequence of interest rates, [r (t)]∞ t=0 , ∞ the sequence of aggregate output levels, [Y (t)]∞ t=0 , the sequence of wages, [w (t)]t=0 , the R&D decisions of all other firms and policies as given Note that as in the baseline model of Schumpeterian growth in Section 14.1, even though technology and output in each sector are stochastic, total output, Y (t), given by (14.33) is nonstochastic denote the distribution of indus14.3.2 Equilibrium Let µ (t) ≡ {µn (t)}∞ P∞n=0 tries over different technology gaps, with n=0 µn (t) = For example, µ0 (t) denotes the fraction of industries in which the firms are neck-and-neck at time t Throughout, we focus on Markov Perfect Equilibria (MPE), where strategies are only functions of the payoff-relevant state variables MPE is a natural equilibrium concept in this context, since it does not allow for implicit collusive agreements between the follower and the leader While such collusive agreements may be likely when there are only two firms in the industry, in most industries there are many more firms and also many potential entrants, making collusion more difficult Throughout, we assume that there are only two firms to keep the model tractable (see below for references on MPE) The focus on MPE allows us to drop the dependence on industry ν, thus we refer to R&D decisions by zn for the technological leader that is n steps ahead and by z−n for a follower that is n steps behind Let us denote the list of decisions by the leader and follower with technology gap n at time t by ξ n (t) ≡ hzn (t) , χi (ν, t) , yi (ν, t)i and ξ −n (t) ≡ z−n (t) Throughout, ξ will indicate the whole sequence of decisions at every state, ξ (t) ≡ {ξ n (t)}∞ n=−∞ 2There are two sources of abuse of notation here First, pricing and output decisions, given by (14.37) and (14.38), depend on the aggregate level of output Y (t) as well However, profits, as given by (14.44), and other choices not depend on Y (t), and we suppress this dependence without any ∞ ∗ affect on the analysis Second, the sequences [χ∗i (ν, t)]∞ t=0 and [yi (ν, t)]t=0 are stochastic, while the rest of the objects specified above are not Since the stochastic nature of these sequences has no effect on the analysis, we suppress this feature as well 635