In steady state the growth rate of L(t) and H(t) is equal to n.
3.2 Manufacturing
Firms can hire unskilled workers to produce any consumption good ω ∈ [0, 1] of the second best quality
under a constant returns to scale (CRS) technology with one worker producing one unit of product.
However, in each industry the top-quality product can be manufactured only by the firm that has
discovered it, whose rights are protected by a perfectly enforceable patent law.
As usual in Schumpeterian models with vertical innovation (see e.g. Grossman and Helpman, 1991,
and Aghion and Howitt, 1992) the next best-quality of a given good is invented by means of innovation
activity performed by challenger firms in order to earn monopoly profits that will be destroyed by
the next innovator. During each temporary monopoly the patentholder can sell the product at prices
higher than the unit cost. We assume that the patent expires when further innovation occurs in
the industry. Hence monopolist rents are destroyed not only by obsolescence but also because a
competitive fringe can copy the product using the same CRS technology.
The unit elastic demand structure13 encourages the monopolist to set the highest possible price
to maximize profits, but the existence of a competitive fringe sets a ceiling to it equal to the lowest
unit cost of the previous quality product. This allows us to conclude that the price p (jmax (ω, t), ω, t)
of every top quality good is:
p (jmax(ω, t), ω, t) = λ (ω) , for all ω ∈ [0, 1] and t ≥ 0.
(7)
Our fiscal policy tool will be sector specific per-capita public spending G(ω, t) ≥ 0, for all ω ∈ [0, 1]
and t ≥ 0. The government uses tax revenues to finance public spending in different sectors and
we assume that the government budget is balanced at every date: N(t)T(t) = N(t) J01 G(ω,t)dω .
Moreover, we will assume N (t)T (t) < γN(t), in order to guarantee that public expenditure is feasible.
Since we are interested in steady states, in which per-capita variables are constant, from now on we
will drop time indexes from per-capita taxation and per-capita expenditures.
From the static consumer demand (2) we can immediately conclude that the demand for each
product ω is:
N(t)J^01 cθdθ + N(t)G (ω) ≡ cN(t) + N(t)G (ω) =
λ (ω) + λ (ω) ≡ λ (ω) + λ (ω) q( ),
(8)
where c = J01 cθdθ is average per-capita consumption. Sectorial market-clearing conditions imply that
demand equals production of every consumption good by the firm that monopolizes it, q (ω). It follows
that the stream of profits accruing to the monopolist which produces a state-of-the-art quality product
will be equal to:
π(ω) = q(ω) (λ (ω) - 1) = (cN(t) + G(ω)N(t)) (1 - ɪ)) .
(9)
A firm that produces good ω has an expected discounted value of
π ч π(ω)
q(ω) (λ (ω) - 1)
r + I(ω) - vjω't; ,
v(ω,t)
v(ω) =---------г—-
r + I(ω) - v(t)
v(ω,t)
1 3 Any CES utility index with elasticity of substitution not greater than one would imply this result.