where I(ω) denotes the Poisson arrival rate of an innovation that will destroy the monopolist’s profits
in industry ω. This can be obtained assuming efficient financial markets which in equilibrium equalize
the expected return of investing in R&D to the risk-free interest rate r. In a steady state where per-
capita variables all grow at the same rate, it is easy to prove that V(ω,t) = n, and from the Euler
equation (3) we obtain r = ρ. Hence the expected value of a firm becomes
v(ω) =
q(ω) (λ (ω) - 1)
ρ+I(ω) -n
(10)
3.3 Innovation races
In each industry leaders are challenged by the innovation activity of followers that employ skilled
workers and produce a probability intensity of inventing the next version of their products. The
arrival rate of innovation in industry ω at time t is I(ω, t), and it is the aggregate summation of the
Poisson arrival rate of innovation produced by all R&D firms targeting product ω.
In each sector new ideas are introduced according to a Poisson arrival rate of innovation by use
of a CRS technology characterized by the unit cost function bwH X(ω, t), with b > 0 common in all
industries, and X(ω, t) > 0 measuring the difficulty of innovation in industry ω. Hence the production
of ideas is formally equivalent to buying a lottery ticket that confers to its owner the exclusive right to
the corresponding innovation profits, with the aggregate rate of innovation proportional to the “number
of tickets” purchased. The Poisson specification of the innovative process implies that the individual
contribution to innovation by each skilled labor unit gives an independent (additive) contribution to
the aggregate instantaneous probability of innovation: hence innovation productivity is the same if
each skilled worker undertakes her activity by working alone as when she works with others in large
firms.
The technological complexity index X(ω, t) has been introduced into endogenous growth theory
after Charles Jones’ (1995) empirical criticism of R&D based growth models that generate scale ef-
fects in the steady state per-capita growth rate. According to Segerstrom’s (1998) interpretation of
Jones’ (1995) solution to the “strong scale effect” problem (Jones 2005), X(ω, t) is increasing in the
accumulated stock of effective innovation:
X(ω, t)
X (ω,t) = μI (ω,t), (11)
with positive μ, thus formalizing the idea that early discoveries fish-out the easier inventions first,
leaving the most difficult ones for the future. This formulation implies that increasing difficulty
of innovation causes per-capita GDP growth to vanish over time unless an ever-increasing share of
resources are invested in innovation, thereby requiring a growing educated population. As it will
become clearer later, this specification of the difficulty index leads to a version of the quality ladder
model where the steady-state growth rate is proportional to the population growth rate, and policy
shocks have only temporary effects on growth14 . For this reason these frameworks are also called
14 Cozzi (2005 and 2007) has proved that in quality ladder economies the steady state growth rate could be affected
by self-fulfilling prophecies, despite the "semi-endogenous" structure. Hence sunspots may matter also in this model.
10