Meditations on the Politics of Limited Knowledge

The Hotelling Line: A lesson on Use and Abuse of Economic Metaphor

In Economics on April 25, 2010 at 3:02 am
Note from the editor: I am pleased to introduce our first guest blog from long-time friend and collaborator, Graham Sack. The following draws out the lessons of a bygone economic theory for a discipline in crisis. Graham’s history of the “Hotelling line” is an apt and informative follow-up to my recent lengthy post on Paul Krugman and the dangerous but seductive simplicity of reductive economic models. Another example of how insufficiently self-critical knowledge can be led astray. ~ drferris ]

In March 2009, as the markets lay in shambles and prognosticators debated the merits of the recently enacted stimulus bill, 
there quietly passed the eightieth
 birthday
 of
 the
 ‘Hotelling
 line,’
 an
 influential
 microeconomic
 model
 first
 proposed
 by
 Stanford
 professor
 Harold
 Hotelling
 in
 his
 article
 “Stability
 in
 Competition” (PDF).
 The paper
 launched
 two
 new
 sub‐disciplines
 within
 economics—the
 study
 of
 spatial
 competition
 and
 product
 differentiation—and
 is
 required
 reading
 for
 graduate
 students
 and
 aspiring
 microeconomists.
 Along
 with
 Edgeworth’s
 box and
 Nash’s
 equilibrium,
 it
 is
 one
 of
 the
 foundational
 metaphors 
of the
 discipline,
 arresting
 for 
its 
simplicity, 
clarity, 
and explanatory
 power. 
It 
is 
also
 fundamentally 
flawed.

Hotelling
 begins
 his
 article
 by
 imagining
 a
 line
 of
 fixed
 length
 (usually
 visualized 
as 
a 
boardwalk) 
along
 which
 consumers 
are
 evenly
 distributed.
There 
are
 two
 sellers
 in
 the
 market
 and
 each
 selects
 a
 location
 and
 a
 price.
 Consumers
 then
 choose 
a 
preferred 
supplier 
by 
weighing 
price 
against 
the
 disutility 
of 
traveling
 long
 distances.
 The
 sellers
 reädjust
 their
 location
 and
 price,
 trying
 to
 maximize
 profits,
 until 
an 
equilibrium
 is 
achieved.

The
 popularity
 of
 the
 model
 derives
 from
 the
 fact
 that
 it
 can
 be
 interpreted
 literally 
or 
figuratively. The 
line 
can 
represent 
geographic
 space 
(such 
as 
a
street
 or
 linear
 city)
 or
 can
 function
 as
 a
 metaphor
 for
 a
 spectrum
 of
 similar,
 substitutable
 products
 (such
 as
 ice
 cream
 flavors
 or
 automobile
 models).
 In
 the
 latter
 case,
 the
 firm’s
 location
 on
 the
 line
 represents
 its
 place
 in
 the
 product
 spectrum,
 while
 the
 distance
 to
 a
 consumer
 represents
 how
 far
 the
 firm’s
 product
 is
 from
 that
 consumer’s 
ideal.
 This 
flexibility 
of
interpretation 
enables 
the 
model 
to 
describe 
two
 apparently
 unrelated 
phenomena: 
(1) 
where 
businesses 
locate 
(spatial 
competition)
 and
 (2) 
what 
products 
they 
produce 
(product 
differentiation).

Although
 the
 metaphor
 is
 elegant,
 Hotelling
 botched
 the
 implementation. To
 simplify
 calculations,
 he
 assumed
 that
 the
 disutility
 of
 travel
 (also
 interpreted
 as
 transportation
 cost)
 was
 a
 linear
 function
 and
 arrived
 at
 the
 conclusion
 that
 both
 sellers
 will
 locate
 in
 the
 middle
 of
 the
 line
 because
 any
 deviation
 to
 the
 right
 or
 left
 would
 give
 share
 to
 the
 competitor.
 Hotelling
 concluded
 that
 sellers
 tend
 towards
 “minimal
 differentiation,” producing
 near‐identical
 products
 with
 only
 nominal
 variations.
 He
 read 
the 
phenomenon
 into 
numerous
 facets 
of 
American 
life, 
from
 the
 homogeneity
 of
 shoes
 and
 apple
 cider
 to
 the
 similarity
 of
 Democratic
 and
 Republican 
party
 platforms.

For
 fifty
 years,
 Hotelling’s
 conclusion
 was
 accepted. Then,
 in
 1979,
 a
 trio
 of
 European
 microeconomists
 led
 by
 Claude
 d’Aspremont
 (himself
 a
 Stanford
 grad)
 showed
 that
 Hotelling’s
 original
 result
 was
 incorrect.
 In
 fact,
 it
 was
 more
 than
 incorrect.
 It
 was
 the
 opposite
 of
 correct.
 The
 assumption
 of
 linear
 transportation
 costs 
in 
the
original 
paper 
led 
to 
discontinuities 
in 
the 
consumer 
utility
 function 
and
 therefore
 no
 solution
 existed:
 it
 turned
 out
 there
 was
 no
 ‘stability’
 in
 “Stability
 in Competition.” When
 d’Aspremont
 fixed
 the
 continuity
 problem
 by
 assuming
 quadratic
 transportation
 costs,
 he
 arrived
 at
 a
 result
 diametrically
 opposed
 to
 Hotelling’s: the 
two 
sellers 
will 
locate
 at 
opposite 
ends 
of 
the 
boardwalk
 rather 
than
 in
 the
 middle—there
 will
 be
 maximal
 differentiation
 rather
 than
 minimal
 differentiation. Since
 1979,
 literature
 on
 Hotelling’s
 line
 has
 proliferated:
 authors
 have
 shown
 that
 whether
 one
 arrives
 at
 minimal
 or
 maximal
 differentiation
 depends
 on
 numerous
 factors,
 including
 the
 functional
 form
 of
 the
 transportation
 costs,
 the
 type
 of
 competition
 (price
 vs.
 quantity),
 and
 whether
 search
 and
 advertising 
costs 
are
present.

The 
salient 
question,
 of 
course,
 is 
how
 fifty 
years
 could 
go 
by 
without 
anyone
 catching
 Hotelling’s
 mistake. The
 math
 is
 not
 difficult
 (for
 academic
 economists,
 anyway),
 requiring
 only
 algebra
 and
 basic
 calculus.
 Why
 did
 no
 one
 see
 the
 error?
 One
 answer
 is
 that
 the
 model’s
 power
 as
 a
 metaphor
 overwhelmed
 and
 obscured
 inconsistencies
 in
 its
 logic
 and
 predictions. Hotelling’s
 mapping
 of
 the
 abstract
 idea
 of
 “difference”
 onto
 concrete,
 easily
 visualizable
 space
 is,
 along
 with
 Marshall’s
 supply
 and
 demand
 curves,
 one
 of
 the
 most
 elegant
 acts
 of
 representation
 in
 all
 of
 economics,
 and
 economists,
 like
 poets,
 were
 persuaded
 by
 symmetry,
 beauty,
 and
 concision.
The 
artistry 
of 
Hotelling’s
 model 
temporarily 
eclipsed 
the 
incongruities 
of
 minimal
 differentiation.

Hotelling’s
 article
 was
 published
 in
 Econometrica in
 March
 of
 1929,
 a
 mere
 six
 months
 before
 the
 Great
 Stock
 Market
 Crash.
 Unlike
 macroeconomics,
 faulty
 microeconomics
 doesn’t
 blow
 up
 markets
 or
 create
 depressions—at
 least
 not
 directly—and
 Hotelling’s
 error
 is
 negligible
 compared
 to
 those
 made
 by
 central
 bankers
 and
 policymakers
 of
 the
 time.
 But
 the
 difference
 is
 one
 of
 degree
 rather
 than
 of
 kind. Hotelling
 made
 a
 basic
 conceptual
 error
 in
 his
 model
 of
 economic
 behavior
 and 
applied 
his 
results 
too 
broadly. 
His 
model 
was accepted 
uncritically 
by
 others 
and 
reproduced
 for 
half 
a 
century.

The
 relevance
 to
 contemporary
 economics
 should
 be
 apparent.
 In
 the
 aftermath 
of 
the 
financial 
crisis 
and 
a 
year 
of
 chastening 
public 
rebukes, 
economists
 are
 under
 pressure
 to
 re‐examine
 their
 methods
 of
 representation
 and
 argumentation
 and
 to
 consider
 the
 factors
 that
 persuade
 them
 to
 believe
 in
 one
 model
 over
 another.
 The
 story
 of
 the
 Hotelling
 line
 is
 an
 example
 of
 how
 powerful
 metaphors
 can
 illuminate,
 but
 also
 blind. A
 greater
 attention
 to
 metaphor
 may
 help
 contemporary
 economists
 to
 develop
 more
 realistic
 and
 descriptively
 powerful
 models
 and
 avoid
 multi‐decade
 errors
 such
 as
 the
 theory
 of
 “minimal
 differentiation.”

~ Graham Alexander Sack

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  1. Great article, Graham. Thanks for the contribution. You are so right to draw attention to the power of metaphor and its use and abuse. More to come on that…

    With respect to product differentiation, check James Surowiecki’s recent “Financial Page” in the New Yorker, “Soft in the Middle,” on Apple’s iPad, General Motors, and the shrinking middle of the consumer market.

    Surowiecki notes Apple’s fundamental gamble with the iPad (as with their other products) that people will pay for quality. With respect to the Hotelling line, their gamble is to locate at one end of the line — in this case, the high end of the consumer market — high quality, high price.

    For Apple, which has enjoyed enormous success in recent years, “build it and they will pay” is business as usual. But it’s not a universal business truth. On the contrary, companies like Ikea, H. & M., and the makers of the Flip video camera are flourishing not by selling products or services that are “far better” than anyone else’s but by selling things that aren’t bad and cost a lot less. These products are much better than the cheap stuff you used to buy at Woolworth, and they tend to be appealingly styled, but, unlike Apple, the companies aren’t trying to build the best mousetrap out there. Instead, they’re engaged in what Wired recently christened the “good-enough revolution.” For them, the key to success isn’t excellence. It’s well-priced adequacy.

    These two strategies may look completely different, but they have one crucial thing in common: they don’t target the amorphous blob of consumers who make up the middle of the market. Paradoxically, ignoring these people has turned out to be a great way of getting lots of customers, because, in many businesses, high- and low-end producers are taking more and more of the market….

    While the high and low ends are thriving, the middle of the market is in trouble. Previously, successful companies tended to gravitate toward what historians of retail have called the Big Middle, because that’s where most of the customers were. These days, the Big Middle is looking more like “the mushy middle”… The products made by midrange companies are neither exceptional enough to justify premium prices nor cheap enough to win over value-conscious consumers….

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