Introduction to Modern Economic Growth Given this specification, the law of motion of the technology gap in industry ν can be expressed as (14.43) ⎧ n (ν, t) + with probability ⎪ ⎪ ⎪ ⎪ ⎨ with probability n (ν, t + ∆t) = ⎪ ⎪ ⎪ ⎪ ⎩ n (ν, t) with probability 2z (ν, t) ∆t + o (∆t) (1 − z−i (ν, t) + κ) ∆t + o (∆t) − (zi (ν, t) + z−i (ν, t) + κ) ∆t − o (∆t)) Here o (∆t) again represents second-order terms, in particular, the probabilities of more than one innovations within an interval of length ∆t The terms zi (ν, t) and z−i (ν, t) are the flow rates of innovation by the leader and the follower, while κ is the flow rate at which the follower is allowed to copy the technology of the leader In the first line, the flow rate of innovation is 2z (ν, t), since the two firms are neckand-neck and undertake the same amount of research effort given by z (ν, t) (there is no conditioning on i or −i, since there is no leader and follower in this case) We next write the instantaneous “operating” profits for the leader (i.e., the profits exclusive of R&D expenditures and license fees) Profits of leader i in industry ν at time t are (14.44) Πi (ν, t) = [χi (ν, t) MCi (, t)] yi (, t) ả w (t) Y (t) w (t) − = q−i (ν, t) qi (ν, t) χi (ν, t) ³ ´ = − λ−n(ν,t) Y (t) where recall that n (ν, t) is the technology gap in industry j at time t The first line simply uses the definition of operating profits as price minus marginal cost times quantity sold The second line uses the fact that the equilibrium limit price of firm i is χi (ν, t) = w (t) /q−i (ν, t) as given by (14.37), and the final equality uses the definitions of qi (ν, t) and q−i (ν, t) from (14.41) and (14.42) The expression in (14.44) also implies that there will be zero profits in an industry that is neck-andneck, i.e., in industries with nj (t) = Followers also make zero profits, since they have no sales The Cobb-Douglas aggregate production function in (14.33) is responsible for the simple form of the profits (14.44), since it implies that profits only depend on 634