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.