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received only after his own went to the publisher, he solved the monopoly case.
More remarkably, he also found the Bertrand-Nash equilibrium for a duopoly
where the firms selling a homogeneous good are located at the opposite ends of a
finite linear market. Demand is uniformly distributed along the line and perfectly
inelastic. Each customer buys from the firm quoting the least delivered price. The
delivered price is the mill price plus transport cost which is linear in distance
(Launhardt
1885
,
28). His solution concept of “Friedensgrundlage” (base of
peace) is precisely the Nash-Equilibrium; he characterises the price mark-ups
over marginal cost chosen by the two firms as follows: “These mark-ups are the
base of peace for the competition of the two entrepreneurs. Each would, if he
reduced his price below or raised it above this level, while the competitor left his
price unchanged, lose profit.”
3
This is precisely a definition of the Nash equili-
brium. He also solves an oligopoly case with an arbitrary number of firms that,
however, all serve the spatial market from the same location.
Launhardt's innovativeness in price theory in general and spatial pricing and
market area analysis in particular is incredible, but its impact on the literature has
been limited. The Anglo-Saxon world did not notice him because Marshall did
not mention him. An English translation of his 1885 book (Launhardt
1993
) had
to wait more than a century. Hotelling was obviously not aware of Launhardt
when writing his famous
Stability in Competition
(Hotelling
1929
). Even L¨sch,
though referring to his work in two of his many footnotes (L¨sch
1940
, pp. 55 and
68), did not make use of Launhardt's deep insights but based his market areas on
Chamberlin (
1933
).
Launhardt's
third
contribution is transport cost minimization in the famous
location triangle. The task is to find—in continuous Euclidean space—a firm's
location minimizing the transport cost, if inputs have to be procured from two input
locations and products have to be delivered to an output market. Locations of input
sources and the output market, input and output quantities in tonnes, as well as
transport costs per tonne-kilometre are given. The problem is equivalent to finding
the point in a triangle minimizing the weighted sum of distances to the three
vertices. A special case of the problem with uniform weights was posed already
1643 by Pierre de Fermat and solved by Evangelista Torricelli (the inventor of the
barometer) in a ruler-and-compass construction (Martini
2001
). In a wonderful
brief and clear contribution to the journal of the German engineers' association
(Launhardt
1882
), Launhardt clearly states the generalized problem, derives the
algebraic optimum conditions, presents a graphical solution method, extends the
problem in several ways (more than three points, use of existing roads), and applies
it to a practical case. The triangle reappears in the influential work of Alfred Weber.
Weber should have referred to Launhardt, which he did not.
}
3
“Die Gewinns¨tze bilden die Friedensgrundlage f¨r den Wettkampf der beiden Unternehmer.
Jeder der beiden Unternehmer w¨rde, sobald er ¨ber oder unter diesen Einheitsgewinn ginge,
w¨hrend der Gegner seinen Preis unver¨ndert festhielte, am Gesamtgewinn verlieren” (Launhardt
1885
, p. 162).
