The Regrettable Loss of Mathematical Molding in Econometrics - Mechanism and Causality in Biology and Economics - page 62

Second, he found the empirical relationship that the C -index was a cumulative
sum of both the A and B indices:
X

t

3

1

4 A i þ 4 B i

¼ C t (4.2)

i¼ 1

Thus, according to Karsten ( 1926 , p. 417), the B -index was the “generating
force” of the three; the other two indices depended upon, and were derived from,
changes in the business index.
Equations ( 4.1 ) and ( 4.2 ) express cumulative relations of discrete processes. For
continuous processes, cumulative relations can be expressed by means of integrals, 2
Z

t

B ðτÞ d τ ¼ A ð t þ 3 Þ (4.3)

0

or by differentials:

d A ð t þ 3 Þ

¼ A

B ð t Þ¼ ð t þ 3 Þ (4.4)

d t

In classical mechanics, there is a close connection between the calculus of
variations and cause-and-effect relations. It is because of this connection that
Karsten wanted to apply the “theory of quadrature” to investigate the kind of
relations that exists between economic quantities. When a cause-and-effect relation
exists between two phenomena, then according to the quadrature theory, one
phenomenon is expected to be cumulatively affected by the other:
In the calculus such relations are familiar in the form of integrals and derivatives, and
although these functions are purely mathematical, they are useful to describe the behavior
of related forces in the physical sciences. It is the quadrature theory that economic data or
statistics betray the same relationships when similarly treated, and that when this is the
case, the economic forces or phenomena measured by statistics may be said to be in
quadrature and a real relation is strongly suggested. (Karsten 1924 , p. 14)
Tinbergen found these cumulative relations exemplary for the kind of causal
relation one could expect to find in business-cycle research. It was the application of
this connection between the calculus and causal relationships that made Karsten's
approach so appealing to Tinbergen.
Tinbergen was looking for causal explanations of business cycles, but economic
theory did not provide the appropriate mechanisms. On the one hand, business
cycles were explained by exogenous influences; on the other hand, each cycle was
examined and explained individually, or, worse still, each phase of a cycle was

2 This explains the name “quadrature theory.” Quadrature stands for the process of determining the
area of a plane geometric figure by dividing it into a collection of shapes of known area (usually
rectangles) and then finding the sum of these areas. The integral denotes this process for infinitesi-
mal rectangles.