Economic growth and economic development 589

Introduction to Modern Economic Growth
value can be written as:
Z
(13.7)
V (ν, t) =

t

∞

¸
∙ Z s
0
0
exp −
r (s ) ds [χ(ν, s)x(ν, s) − ψx(ν, s)] ds
t

where r (t) is the market interest rate at time t. Alternatively, assuming that the
value function is differentiable in time, this could be written in the form of HamiltonJacobi-Bellman equations as in Theorem 7.10 in Chapter 7:
(13.8)

r (t) V (ν, t) − V˙ (ν, t) = χ(ν, t)x(ν, t) − ψx(ν, t),

where x(ν, t) and χ(ν, t) are the profit-maximizing choices for the monopolist. Exercise 13.1 asks you to provide a different derivation of this equation than in Theorem
7.10.
13.1.2. Characterization of Equilibrium. An allocation in this economy
is defined by the following objects: time paths of consumption levels, aggregate
spending on machines, and aggregate R&D expenditure [C (t) , X (t) , Z (t)]∞
t=0 , time
paths of available machine types, [N (t)]∞

t=0 , time paths of prices and quantities of
each machine and the net present discounted value of profits from that machine,
[χ (ν, t) , x (ν, t) , V (ν, t)]∞
ν∈N(t),t=0 , and time paths of interest rates and wage rates,
[r (t) , w (t)]∞
t=0 .
An equilibrium is an allocation in which all existing research firms choose
∞
[χ (ν, t) , x (ν, t)]∞
ν∈[0,N(t)],t=0 to maximize profits, the evolution of [N (t)]t=0 is deter-

mined by free entry, the time paths of interest rates and wage rates, [r (t) , w (t)]∞
t=0 ,
are consistent with market clearing, and the time paths of [C (t) , X (t) , Z (t)]∞
t=0 are
consistent with consumer optimization. We now characterize the unique equilibrium
of this economy.
Let us start with the firm side. Since (13.6) defines isoelastic demands, the
solution to the maximization problem of any monopolist ν ∈ [0, N (t)] involves
setting the same price in every period (see Exercise 13.2):
(13.9)

χ(ν, t) =

ψ
for all ν and t.
1−β

That is, all monopolists charge a constant rental rate, equal to a mark-up over the
marginal cost. Without loss of generality, let us normalize the marginal cost of

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