Spence 1984 specification, which is also used by De Bondt et al. 1992. The Levin and Reiss specification can be obtained when g = 0, and that of Sato and Suzawa
1984 results from setting g = 1, d = d 0 =0.
We assume monopolistic competition in the output market. This allows us to investigate the influence of product differentiation and the number of competitors
on technological change and welfare. We consider two kinds of goods: n differenti- ated products and one homogeneous product. The utility is quasi-linear, i.e. it adds
two kinds of sub-utility or two sectoral utility levels for the differentiated and the homogeneous goods. As in Grossman and Helpmann 1991, the utility of the
differentiated goods is a` la Dixit-Stiglitz, which means that there is linear homo- geneity among the n number of varieties. The optimal allocation of expenditures
across differentiated goods and homogeneous good yields the usual demand function for each differentiated good:
y
i
= p
i −
1
P
D a − b
X
b
7 where
4
a = 11 − r \ 1, b = 11 − z \ 1, with the following price index: P
D
=
n i = 1
p
i 1 − a
n
11 − a
8 Note that a is the elasticity of substitution across n differentiated goods, while b is
the overall price elasticity of demand. We then logically assume a \ b. The last hypothesis we need is the dynamic of exogenous 6ariables, i.e. scale of
demand, and in particular, the costs of N and K. As in Morrison and Berndt 1991, we assume that the exogenous variables increase at given constant rates X
0t
= X
00
e
− s
X
t
, c
nt
= c
n0
e
− s
n
t
, c
kt
= c
k0
e
− s
K
t
.
3. Dynamic profit maximisation and balanced growth rate
The representative firm will maximise the following intertemporal profit: Max
P
it
,N
t
t = 0
e
− rt
[y
it
p
it
, P
D
p
it
− TC
it
y
it
, A
it
− c
n
N
it
] dt 9
under the constraints of: the demand for differentiated goods, the price index, the total costs, the dynamic of the TFP and the communication externalities, as well as
the initial condition A
it = 0
= A
.
4
In monopolistic competition, marginal revenue of a firm is equal to rp Grossman and Helpmann, 1991. The condition a \ 1 is needed so that the price elasticity of demand perceived by a firm will be
larger than 1. This is required to avoid negative marginal revenue in a monopolistic situation.
The dynamic maximisation is solved by using optimal control techniques
5
. We are interested only in the symmetric Nash equilibrium, because all firms have the
same cost and price conditions, and the same spillovers, as firm 1 in Fig. 1. This yields: p
it
= p
t
and N
it
= N
t
Öi=1 . . . n. In particular, a
pt
= y
t
p
t
p
t
y
t
, the perceived price elasticity, obtained taking the price and telecommunications invest-
ment decisions of other firms as given, becomes a
P
= a − a − bn. The perceived
price elasticity is then always higher than 1. Using the growth rates of the FOC at the symmetric equilibrium see Appendix
I, we are able to calculate the steady state growth rates of the system variables. Combining them we can look at the relationship between the growth rate of the
control variable A
it
and the growth rate of the state variable we are interested in, i.e. N
it
,: A
. = d + d
0 1−g g
[cˆ
K
1 − b + dcˆ
n
− d bX
. ] = d + d
0 N. 10
where: g =
1 [1 − g − d + d
0 b−1] We easily see that the autonomous scale demand parameter has a positive impact
on the TFP growth rate, while marginal costs of capital and ITtelecommunications have a negative effect
6
. Reducing the growth rate of these marginal costs is then a measure that fosters the output growth rate. It could be argued that political
measures in favour of deregulation of the telecommunications market and stronger competition among the IT suppliers, yielding lower marginal costs, have a direct
impact on firms’ growth.
It is also interesting to look at the growth rate multiplier g. The elasticity of the own knowledge stock 1 − g has a negative effect on g: the more the own
knowledge stock is efficient, the less investment is required to increase it. The impact of ITtelecommunications on TFP growth rate can be assessed looking at
the term d + d 0 b−1. The communication technologies positively contribute to A.,
both through a direct d and an indirect effect d 0 , that stimulate further
5
We have a unique state variable A
it
, two control variables, p
it
and N
it
, and the co-state variable m
t
. Necessary conditions are also sufficient for global maximum if the Mangarasian sufficient theorem is
satisfied. This theorem contains two conditions: a the profit functions and the TFP equations are differentiable and concave in the variables N
t
, A
t
; and b in the optimal solution, m
t
] 0,
Öt[0, , if dA
t
dt is non-linear in N
t
. The model satisfies the first condition; the second condition has to be checked for each optimal solution, because dA
t
dt is non-linear in N
t
Chiang, 1992, pp. 215 – 221.
6
The marginal cost of capital and ITtelecommunications has a negative influence on the balanced growth rate because it is inversely related to the steady state growth rate of ITtelecommunications
investment. This effect could seem counter intuitive, as higher input prices imply that the potential benefits of telecommunications investment per unit of output increase. But in a situation with a price
elastic demand curve a, b \ 1, higher input prices also imply that the price level will increase, which decreases demand more than proportionally.
telecommunications investments by increasing the productivity of these efforts. Moreover, the increased productivity induced by the communication technologies
decreases total production costs and then the price level cost reducing effect. It follows that the firm’s revenue is increased by b − 1 times the productivity change.
The externality parameter e does not appear in the TFP growth rate. However, it plays an important role in the steady state values of the TFP and the telecommu-
nications investment level, as we will see in the following Sections 4 – 6. Manipulating the FOC, we also acquire the two differential equations dAdt =
fA, N, dNdt = gA, N, which are necessary to study the system dynamics. However, these differential equations depend on time: costs, price and the market
scale factor grow in time at an exogenous rate. In order to apply the phase diagram technique, it is a prerequisite that the variable t does not enter into differential
equations as a separate argument the system has to be autonomous; otherwise, each point in a phase space can imply different directions of the system over time.
When this is the case, it is not possible to make qualitative statements about the characteristics of a possible equilibrium Chiang, 1984. In order to apply the phase
diagram analysis to the inter-temporal maximisation with exponentially growing prices and costs, we have to remove the time component from the dynamic
problem.
In the Appendix, we use a time elimination method to make the system autonomous: the endogenous variables will be deflated with their steady state
growth rates; the appendix also describes the dynamic properties of the system. The two dynamic constraints in terms of the redefined or deflated variables are:
dA
d
dt =
A
d g
N
d d
1
1 + en − 1 1 − e
2
n − 1
n
d
− s
a
A
d
11 dN
d
dt = l
1
N
d d
1
l
2
N
d 1 − d
1
− l
3
A
d b + g − 2
12 where:
A
d
= A
t
e
d + d 0 [cˆ
K
1 − g + cˆ
n
− bX
. ]gt
= A
t
e
− s
a
t
N
d
= N
t
e
1 − g[cˆ
K
1 − b + bX .
− c
n
]gt
= N
t
e
− s
N
t
g = 1
[1 − g − d + d 0 b−1]
d + d 0 =d
1
l
1
= 1
1 − d
1
, l
2
= r + s
a
1 − g l
3
= h
X
00 b
c
n0
1 + en − 1 1 − e
2
n − 1
n
d
d + d
1 + en − 1
n
c
K0 1 − b
a
p
− 1
a
p b
n
a − b1 − a
13
Note that the subscript d means that the steady state values are obtained after the time elimination method has been applied, and the depreciation factor s
a
is the steady state growth rate of A. The initial values of the exogenous variables also
appear X
00
, c
n0
, c
k0
.
Equating the dynamic equations to zero, we obtain the steady state value of A
d
and N
d 7
: A
d
= X
00 b
c
K0 1 − b
c
n0
r + s
a
a
P
− 1
a
P b
n
a − b1 − a
S
N
S ´
N
n
11 − g + d
1
b − 1
s
a d
1
− 11 − g + d
1
b − 1
14 N
d
= X
00 b
c
K0 1 − b
r + s
a
a
P
− 1
a
P
n
a − b1 − a
S ´
N
c
n0
n
11 − g + d
1
b − 1d
1
s
a 2 − g − b1 − g + d
1
b − 1
S
N d
1
b − g − 21 − g1 − g + d
1
b − 1d
1
15 where the externalities obtained by using a communication network are as follows:
S
N
= 1 + en − 1
1 − e
2
n − 1
n
d
S ´
N
= d +
d 1 + en − 1
n
16
4. Analysis of the steady state TFP and ITcommunication levels
