Economic growth and economic development 680

Introduction to Modern Economic Growth
15.3.1. 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, [NL (t) , NH (t)]∞
t=0 , time paths of prices and quantities
of each machine and the net present discounted value of profits from that machine,
∞
[χL (ν, t) , xL (ν, t) , VL (ν, t)]∞
ν∈[0,NL (t)],t=0 and [χH (ν, t) , xH (ν, t) , VH (ν, t)]ν∈[0,NH (t)],t=0 ,

and time paths of factor prices, [r (t) , wL (t) , wH (t)]∞
t=0 .
An equilibrium is an allocation in which all existing research firms choose
Ô
Ê
f (ν, t) , xf (ν, t) ν∈[0,N (t)],t=0 for f = L, H to maximize profits, the evolution
f
is
determined by free entry, the time paths of factor prices,
of [NL (t) , NH (t)]∞
t=0

[r (t) , wL (t) , wH (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.
To characterize the (unique) equilibrium, let us first consider the maximization
problem of producers in the two sectors. Since machines depreciate fully after use,
these maximization problems are static and can be written as
Z NL (t)

pL (t) YL (t) − wL (t) L −
χL (ν, t) xL (ν, t) dν,
(15.11)
max
L,[xL (ν,t)]ν∈[0,N

0

max

Z

L (t)]

and
(15.12)

H,[xH (ν,t)]ν∈[0,N

H (t)]

pH (t) YH (t) − wH (t) H −

NH (t)

χH (ν, t) xH (ν, t) dν.

0

The main difference from the maximization problem facing final good producers in

Chapter 13 is the presence of prices pL (t) and pH (t), which reflect the fact that
these sectors produce intermediate goods, whereas factor and machine prices are
expressed in terms of the numeraire, the final good.
These two maximization problems immediately imply the following demand for
machines in the two sectors:
¸1/β
∙
pL (t)
L for all ν ∈ [0, NL (t)] and all t,
(15.13)
xL (ν, t) =
χL (ν, t)
and
(15.14)

∙

pH (t)
xH (ν, t) =
χH (ν, t)

¸1/β

H
666

for all ν ∈ [0, NH (t)] and all t.