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representation of reality” (Tinbergen
1933b
,p.13
3
). However, beside lag relations
other dynamic relations are possible, namely, those containing differentials and
integrals. From physics, Tinbergen knew that second-order differential equations
can generate cycles. For example, differentiating (with respect to time) an equation
containing a differential and an integral term leads to the equation of the harmonic
oscillator.
c
Z
t
a
y
_
ð
t
Þþ
by
ð
t
Þþ
y
ðτÞ
d
τ ¼
0
!
a
y
€
ð
t
Þþ
b
y
_
ð
t
Þþ
cy
ð
t
Þ¼
0
(4.7)
0
An advantage of differential equations is that differentials refer to very small
time intervals. Note that
_
y
¼
d
y
=
d
t
;
where d
t
can be approximated by a very small
difference in time
Δ
t
. So that:
y
ð
t
Þ
y
ð
t
Δ
t
Þ
y
_
(4.8)
t
Δ
Considering the shorter time many production processes need nowadays, the appearance of
only direct affective causes can be called a realistic feature in view of this. Thus, what
really matters is the question just posed: can quantities with an integral character and a
differential character, respectively, be found and do these quantities play an important role
in the business cycle? (Tinbergen
1933b
, pp. 14-15)
At a meeting of the Econometric Society in Leiden in 1933, Tinbergen raised
this question most explicitly: “Is the theory of harmonic oscillation useful in the
study of business cycles?” He proposed to start “from the mathematical nature
of harmonic oscillations and seeking among the main economic relations those
likely to fit into the harmonic pattern” (Marschak
1934
, p. 188). Accordingly, he
marshaled economic relations into two groups: (1) “differential phenomena,” that
is, functions of the rate of price change,
p
_
ð
t
Þ
, and (2) “integral phenomena,” that is,
functions of
R
p
d
t
. Statistical tests, however, showed him not to give too much
credit to most of the phenomena of group (2), because the correlations he had
hitherto found were too small (p. 188).
In his 1935 survey, Tinbergen discussed this issue again. To make “closer
approximations to reality” (p. 277), differentials,
, and integrals,
R
p
d
t
, were
added to the lag schemes. Thus, in general, the reduced form equation of a business-
cycle scheme would have the following shape:
p
ð
t
Þ
c
i
Z
tt
i
X
X
X
n
n
n
a
i
p
ð
t
t
i
Þþ
b
i
_
p
ð
t
t
i
Þþ
p
ðτÞ
d
τ ¼
0
(4.9)
1
1
1
0
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