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
Product innovation in sector k is defined as
a
kj
5 1 2 aa
kj
, for j 5 1, . . . , n 9
with 0 , a , 1. All other coefficients remain the same. In matrix notation, Equation 9 is written as A 5 A 2 ae
k
e9
k
A .
SPILLOVERS OF INNOVATION EFFECTS 33
Let the final demand for an arbitrary good, say j, be increased. This affects the production in sector k. Directly, because process
j uses less of product k as an input. And indirectly, because process
j uses all products as an input while each of these products requires
less inputs from sector k. This yields
Dx
k
Df
j
2 Dx
k
Df
j
5 l
kj
2 l
kj
5 e9
k
L 2 Le
j
5 2aa
kj
1
o
∞ t
5
2
e9
k
A
t
2 A
t
e
j
. 10
The first term indicates the direct effect, the second term the indirect effects. An additional final demand of product j also affects
the production in sector i ≠
k . In this case, however, the decrease
in production is due only to indirect effects;
Dx
i
Df
j
2 Dx
i
Df
j
5 l
ij
2 l
ij
5
o
∞ t
5
2
e9
i
A
t
2 A
t
e
j
. 11
The measure for propagation is based again on the ratio in
Equation 6. Using A 5 A 2 ae
k
e9
k
A and substituting AL 5
L 2 I
in the inverse of a sum of matrices yields
L 2 L 5 2aLe
k
e9
k
L 2 I[1 1 ae9
k
L 2 Ie
k
].
The denominator equals h
k
5 1 1 al
kk
2 1 again, which gives
l
ij
2 l
ij
5 2 al
ik
l
kj
h
k
, for j ? k, l
ik
2 l
ik
5 2 al
ik
l
kk
2 1h
k
.
Substituting these expressions in Equation 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
, 12
which holds for all j 5 1, . . . , n. Note that this ratio is again independent of both the specific final demand f
j
and a. The spillover of the product innovation effects is measured as the
percentage of the total ouput change that occurs in the sectors i ≠
k . This yields
s˜
k
5 100c
k
2 l
kk
c
k
.
Observe that the spillover s˜
k
for product innovation is substantially smaller than s
k
for process innovation. As Equation 10 for product innovation shows, Dx
k
Df
j
2 Dx
k
Df
j
involves a direct effect, while Dx
i
f
j
2 Dx
i
Df
j
in Equation 11 includes only indirect effects. For process innovation, however, Equations 4 and 5 indicate that either
all output changes include a direct effect i.e., when j 5 k or no output change includes a direct effect i.e., when j
≠ k
.
34 E. Dietzenbacher
The terms l
ik
l
kk
2 1 in Equation 8 and l
ik
l
kk
in Equation 12 provide an alternative interpretation of the elements of the Leon-
tief inverse. For example, l
ik
l
kk
measures the output gain in sector i
relative to the output gain in sector k, induced by an arbitrary increase of the final demands, when product innovation takes
place in sector k. That is to say, the propagation to sector i of a product innovation in sector k.
4. EMPIRICAL RESULTS
