SPILLOVERS OF INNOVATION EFFECTS 29
which uses the output of sector k, and so forth. If product k is innovated, each process uses less of this product. Indirectly, each
process also requires less of any other product i because product i
is used in process k, and so forth. Propagation of the innovation effects in one sector or of coefficient changes, in general may
thus be analyzed by examining its spillovers. That is, the extent to which the production in other sectors is affected. In this paper
a surprisingly simple measure for these spillovers is developed.
The next two sections present the analytical results for process and product innovation, respectively. Using the standard Leontief
model, it turns out that the spillover measures are directly based on the elements and the column sums of the Leontief inverse.
Consequently, an alternative interpretation of the elements of the Leontief inverse in terms of innovation propagation is obtained.
As an illustration, the empirical results for the European Union in 1991 are discussed in Section 4.
2. PROCESS INNOVATION
The standard Leontief model is given by
x 5 Ax 1 f ,
1
where x is the vector of sectoral outputs and f the vector of final demands private consumption, government expenditures, invest-
ments, and exports. As usual, it is assumed that each of the n sectors produces exactly one good. A is the n 3 n matrix of input
coefficients. Its typical element a
ij
denotes the amount of product i
required, as an input in process j, for the production of one unit of output of good j. The model in Equation 1 is solved as
x 5 Lf , with L 5 I 2 A
2
1
. 2
The matrix L is the Leontief inverse, or multiplier matrix.
4
Its element l
ij
denotes the additional output of product i as required directly and indirectly per additional unit of final demand for
good j. Or, equivalently, l
ij
5 Dx
i
Df
j
. An innovation in process k is defined as
a
ik
5 1 2 aa
ik
, for i 5 1, . . . , n, 3
with 0 , a , 1. All other coefficients remain the same i.e., a
ij
5
4
It is assumed that A is nonnegative and irreducible, and has a dominant eigenvalue
smaller than one. Consequently l
ij
. 0 for all i,j 5 1, . . . , n see, e.g., Takayama, 1985,
for these and other conditions.
30 E. Dietzenbacher
a
ij
for all i 5 1, . . . , n and j ≠
k .
5
Process innovation implies that the same amounts of the inputs yield an increased amount of
output in process k. Or, in other words, per unit of output of product k the use of each input is decreased with 100a percent.
Analogous to Equations 1 and 2, the model is given by x 5
Ax 1 f , and its solution by x 5 Lf.
It should be emphasized that the innovation as defined in Equa- tion 3, is unlikely to be actually observed in exactly this form.
The common assumption that each sector produces just one good is too restrictive, in particular at the aggregation level of most
published input–output tables. In an ex-post analysis of the effects as they actually have occurred, this assumption would, therefore,
cast serious doubts on the results. The present analysis, however, examines the potential for the propagation of innovation effects,
under the hypothesis that the innovation takes the form as in Equation 3.
The effects of process innovation can now be measured as fol- lows. Per additional unit of final demand for good k, less output
needs to be produced. That is, Dx
i
Df
k
2 Dx
i
Df
k
5 l
ik
2 l
ik
, which
is negative because L L see, e.g., Takayama, 1985.
6
Using the
series expansion of the Leontief inverse i.e., L 5 I 1 A 1 A
2
1
A
3
1
. . . and writing e
k
for the kth unit vector, the vector Dx
Df
k
can be written as
DxDf
k
5 Le
k
5 e
k
1 Ae
k
1
o
∞ t
5
2
A
t
e
k
.
Similarly,
DxDf
k
5 Le
k
5 e
k
1 Ae
k
1
o
∞ t
5
2
A
t
e
k
5 e
k
1 1 2 aAe
k
1
o
∞ t
5
2
A
t
e
k
.
Thus,
DxDf
k
2 DxDf
k
5 2aAe
k
1
o
∞ t
5
2
A
t
2 A
t
e
k
. 4
Per unit final demand for good k, all outputs decrease because the innovated process k requires less of each input. This is the
direct effect as given by 2aAe
k
. The production of each of these inputs requires among other inputs the input of good k. In its
turn, the production process of good k is innovated and requires
5
See Van der Linden et al. 1993 for an application of 3 in terms of fields of influences see, e.g., Hewings et al., 1989, or Sonis and Hewings, 1989, 1992.
6
For vectors and matrices we adopt the following expressions and notations. Positive,
x 0, if x
i
. 0 for all i. Nonnegative, x 0, if x
i
0 for all i. Semipositive, x . 0, if x 0 and x
≠ 0.
SPILLOVERS OF INNOVATION EFFECTS 31
less inputs, and so forth. All these effects are captured by the term
o
∞ t
5
2
A
t
2 A
t
e
k
, the indirect effects. In the same way, the effects of process innovation can be de-
scribed per additional unit final demand of good j ≠
k . From
DxDf
j
5 e
j
1 Ae
j
1
o
∞ t
5
2
A
t
e
j
5 e
j
1 Ae
j
1
o
∞ t
5
2
A
t
e
j
,
it immediately follows that
DxDf
j
2 DxDf
j
5
o
∞ t
5
2
A
t
2 A
t
e
j
. 5
Because j ≠
k , the direct effect is absent, implying that all effects
are indirect. The terms Dx
i
Df
k
2 Dx
i
Df
k
for i 5 1, . . . , n express the effects of process innovation as embodied in an additional unit final
demand for good k. The innovation in process k has propagated and its effects are also present in the other final demands. The
terms Dx
i
Df
j
2 Dx
i
Df
j
with j ≠
k indicate the effects as embodied
in an additional unit final demand for good j. In constructing a measure for the amount of propagation, the
change in the output of sector i is related to the change in sector k
, i.e., the sector where the innovation has taken place.
Dx
i
Df
j
2 Dx
i
Df
j
Dx
k
Df
j
2 Dx
k
Df
j
5 l
ij
2 l
ij
l
kj
2 l
kj
6
Next, it is shown that expression 6 equals l
ik
l
kk
2 1. Note that
this ratio does not depend on j. Therefore, Equation 6 measures the propagation of the innovation in process k, irrespective of the
specific structure of final demands.
Observe that A 5 A 2 aAe
k
e9
k
, where a prime is used to denote transposition. Using well-known formulae for the inverse
of a sum of matrices see, e.g., Henderson and Searle, 1981, for an overview yields
L 2 L 5 2aLAe
k
e9
k
L [1 1 ae9
k
LAe
k
].
Using LA 5 L 2 I, it follows that
L 2 L 5 2 a L 2 Ie
k
e9
k
L [1 1 ae9
k
L 2 Ie
k
.
The denominator is a scalar which shall be denoted by h
k
,
h
k
5 1 1 al
kk
2 1.
Notice that L 2 L is a matrix of rank one. Its typical element l
ij
2 l
ij
is obtained as e9
i
L 2 Le
j
. Hence,
32 E. Dietzenbacher
l
ij
2 l
ij
5 2 al
ik
l
kj
h
k
, for i ? k, l
kj
2 l
kj
5 2 a l
kk
2 1l
kj
h
k
. 7
Substituting Equation 7 into 6 yields
Dx
i
Df
j
2 Dx
i
Df
j
Dx
k
Df
j
2 Dx
k
Df
j
5 l
ik
l
kk
2 1
, 8
which holds for all j 5 1, . . . , n. Note that the ratio in Equation 8 is independent of both a and
the index j. Obviously, the magnitude of the total effects does depend on a and the specific vector f. Compare, for example, the
case after innovation x 5 Ax 1 f with the case before innovation x 5 Ax 1 f
. Then Equation 7 implies
x
i
2 x
i
5 2al
ik
o
j
l
kj
f
j
h
k
x
k
2 x
k
5 2a l
kk
2 1
o
j
l
kj
f
j
h
k
.
Although the propagation of innovation effects clearly depends upon a and f, its distribution over the sectors is always the same.
This is a consequence of the matrix L 2 L having rank one.
The spillover of innovation effects is defined as the percentage of the total ouput change that occurs in sectors i other than the
innovated sector k. Define the kth column sum of the Leontief inverse as c
k
, that is,
c
k
5
o
i
l
ik
.
Then, the spillover of the effects of process innovation in sector k
is given as s
k
5 100[
o
i?k
x
i
2 x
i
][
o
i
x
i
2 x
i
], which yields
s
k
5 100c
k
2 l
kk
c
k
2 1.
Alternative measures that are related to s
k
are the average sectoral spillover defined as s
k
n 2 1 and the relative spillover 100c
k
2 l
kk
l
kk
2 1, which relates the output changes in sectors
i to the change in the innovated sector k.
3. PRODUCT INNOVATION
