January 24, 2001
Eric Rasmusen notes, erasmuse@indiana.edu
Http://Php.Indiana.edu/~erasmuse
61. Harold Hotelling, ``Stability in Competition,'' Economic
Journal, 39: 41-57 (March 1929).
p. 347. In both the original article and the 1952 reprint in Stigler
and Boulding's AEA Readings in Price Theory, there is a typo: the
equations are
f(q_1+q_2) + q_1 f(q_1+q_2) =0
and
f(q_1+q_2) + q_2 f(q_1+q_2) =0,
lacking the subscripts in the second term of each equation to indicate
derivatives.
p. 350. Figure 2 should have the lowest diagonal line hitting the P_1
axis at 30. In the original and in the reader, it looks like it hits at
about 35. We haven't corrected that, but the 1952 reprint in Stigler
and Boulding's AEA Readings in Price Theory did.
p. 351. The line where dpi_1/dp_1=0 is the reaction curve for player 1.
It is the line that connects RTE in Figure 2.
Footnote 8. This footnote is confusing because it redraws Figure 2 in
words, and so looking at Figure 2 to understand it is not very
helpful. Hotelling ought to have given us a fresh figure to illustrate
this case.
He is talking about the situation where the two firms are located too
close to each other for the equations he just described to be
applicable. In his example here, l=20, a=11, b=8, and c=1. Then, the
two boundary lines for solutions are
p_1-p_2 = (+_) (1)(20-11-8) = (+_)1,
which means that the two firms' prices cannot differ by more than 1, or
one firm will get all the other's customers, and, in fact, get the
entire line of length 20. The calculus method used in the paper was
valid only under the assumption that the firms each held onto their own
extreme customers and split the ones in between them. In the example,
the calculus solutions yield
p_1 = (1) (20+ (11-8)/3)) = 21
and
p_2 = (1) (20- (11-8)/3)) = 19.
That is a difference of 2, not 1, so all of the customers would prefer
Firm 2.
The essential problem is that when the firms are located close
together, the model is a lot closer to pure, undifferentiated Bertrand
competition. If a=b, for example, so the firms were on top of each
other, it is clear that Hotelling's price equations wouldn't work, but
they would yield definite numbers nonetheless-- identical prices of 20
if l=20, c=1, and a=b, for example. Hotelling seems to have noticed
this kind of problem when he thought about optimal location of an
entrant-- see footnote 9 on page 354.
Do not trust his supposed solution in footnotes 8 and 9.
p. 354. Hotelling suggests that someone setting up a subdivision, a
new tract of homes and businesses, would want to locate two businesses
so as to maximize the businesses's profits, which would allow him to
sell the land to those businesses at higher prices. He does not note
that so doing reduces the value of the homes in the subdivision even
more, because the businesses' price gains are the homeowners' losses,
plus the homeowners would have higher transportation costs if the
business locations were inefficient.