Critique of Hotelling
• Hotelling’s “Principle of Minimum Differentiation” was flawed
• No pure strategy exists if firms are close together. With quadratic rather than
linear transportation costs, firms optimally differentiate (DGT)
• Mixed strategies imply firms differentiate (Osborne and Pitchick)
• Firms seek differentiation to avoid unbridled price competition
• Intuitively, equilibrium, prices fall when firms get too close to each
other; it overcomes the positive impact that a unilateral move
towards the market center has on demand
• Hotelling and others ignored this strategic impact
Multiple dimensions
• Has been implicitly supposed that the one-dimension result of
maximum differentiation carries over to multiple characteristics, so a
firm that can differentiate on two (or more) dimensions (for example,
color and size) will do so on both
• Preliminary results in two dimensions (Neven and Thisse, Tabuchi)
indicate this might not be true
• Maximum differentiation in both dimensions is not an equilibrium
• Instead, strategic firms maximize differentiation in one characteristic,
minimize differentiation in the other characteristic
• So in multiple characteristic goods, in how many dimensions should
we observe product differentiation?
Analysis moved to multiple dimensions
Is it Max-Min-…-Min or Min-…-Min?
Ansari, Economides, and Steckel, 1998
Irmen and Thisse (hyperspace), 1998
Hehenkamp and Wambach (evolutionary stability ), 2010
Model set up
• Simple Hotelling type model but with several dimensions
• Quadratic transportation costs
• Represent the costs of a firm’s good not exactly matching a consumer’s
preferences. Quadratic makes it symmetric, so “too much” of some
characteristics is as bad as a like amount of “too little” of that characteristic
• Allow each characteristic to be weighted, so some characteristics matter more
than others
• Inelastic demand
Preview of the results
• With n characteristics, all weighted equally by consumers, there are n
local equilibria in which firms maximize differentiation on one
characteristic, and minimize differentiation on the others
• When there is a dominant characteristic, there is a unique equilibrium
where firms maximize differentiation on the dominant characteristic,
and minimize differentiation on the others
• Implication is that differentiation along a single dimension is sufficient
to relax price competition. With inelastic demand, profits are highest
where prices are highest, which requires the elasticity of demand to
be lowest, which means the lowest mass of marginal consumers
• Differentiation in only one dimension is sufficient to achieve this goal
• Use as an example the magazines Time and Newsweek
Basics of the model: Consumer demand
• Product characteristics are given in n with n>1.
• A’s location is described by a vector a=(a1, …,an) and likewise B’s location
is give by b=(b1,...,bn).
• Consumers are uniformly distributed over a hypecube C=[0,1]n.
Consumers get net utility from consuming 1 unit of the good according to
• S is sufficiently large that consumers always consumer 1 unit of the good
• They get a net utility that decreases in price and decreases in how far the
characteristics of the good they consumer differ from their optimum,
which is described by z=(z1,…,zn). tk weights the importance of the kth
characteristic. More salient characteristics have the highest weight.
A digression on dominance
Irmen-Thisse have tn (bn  an )  t j (b j  a j )j  n so that n is the dominant
characteristic.
Weak dominance is if  t j (b j  a j )  tn (bn  an ).
j n
Strong dominances is if tn (bn  an )   t j (b j  a j ). Strong dominance
j n
means the marginal utility loss from deviations on the dominant
characteristic is larger than the sum of marginal utility losses from other
deviations. Even if tn  ti i the salience of characteristic n is most
important. They show that this condition is always satisfied at a local
Nash equilibrium, so they focus on strong dominance.
It is this assumption that makes looking at two dimensions equivalent to
looking at n>2 dimensions
Firm demand
The demand for variant A is given by the mass of consumers for whom
the A is weakly preferred to B, that is, for whom
VA ( z )  VA ( z )
So
DA 

g (z )dz
z:VA ( z ) VA ( z )
where g (z ) is the uniform distribution.
There are only the two firms, no threat of entry. They assume a twostage game where firms first choose location and then compete on
prices. They seek a sub-game perfect Nash equilibrium
The model with n=2 (facilitates math and intuition
A consumer buying from A enjoys a utility equal to
VA ( z )  S  p A  t1 ( z1  a1 ) 2  t2 ( z2  a2 ) 2
A consumer buying from B enjoys a utility equal to
VB ( z )  S  pB  t1 ( z1  b1 ) 2  t2 ( z2  b2 ) 2
The marginal consumer, indifferent between A and B is thus located
p A  t1 ( z1  a1 ) 2  t2 ( z2  a2 ) 2  pB  t1 ( z1  b1 ) 2  t2 ( z2  b2 ) 2
Solving for z2 as a function of z1 gives
pB  p A  t1 (b12  a12 )  t2 (b22  a22 ) t1 (b1  a1 )
zˆ2 ( z1 ) 

z1
2t2 (b2  a2 )
t2 (b2  a2 )
Without loss of generality we assume strong dominance of
characteristic 2 so
t (b  a )  t (b  a )
2
2
2
1
1
1
Firm Demand and profit
As noted, the demand for variant A is DA  z ;V ( z )V ( z ) g ( z )dz
A
B
so the middle piece of A’s demand under strong dominance is
2
2
2
2
p

p

t
(
b

a
)

t
(
b

a
A
1 1
1
2
2
2 )  t1 (b1  a1 )
DA2  B
2t2 (b2  a2 )
over the interval
p A  [ pB  t1 (b12  a12 )  t2 (b22  a22 )  2t2 (b2  a2 ),
pB  t1 (b12  a12 )  t2 (b22  a22 )  2t1 (b1  a1 )]
More on the price range
We have normalized the population to 1 so DA  1  DB . This help provide
some intuition about the price ranges
Suppose pA is at its lower limit. Then
pB  t1 (b12  a12 )  t2 (b22  a22 )  t1 (b1  a1 )  [ pB  t1 (b12  a12 )  t2 (b22  a22 )  2t2 (b2  a2 )]
DA 
2t2 (b2  a2 )
t1 (b1  a1 )  2t2 (b2  a2 )]
t1 (b1  a1 )
DA 
 1
2t2 (b2  a2 )
2t2 (b2  a2 )
Anticipating the result that a1  b1 and a2  b2 then DA  1 and A gets the
entire market. Like wise, if pA is at its upper limit it is easy to show
t1 (b1  a1 )] so if a  b and a  b then D  0 and B gets it all.
DA 
1
1
2
2
A
t2 (b2  a2 )
Since the firms are identical, you can expect that each gets ½ the market
Profit and profit maximization (check these)
Given the demand above, profit is just price times quantity, so
 pB  p A  t1 (b12  a12 )  t2 (b22  a22 )  t1 (b1  a1 ) 
 A  pA 

2
t
(
b

a
)

2
2
2

Maximizing with respect to a1, a1 and pA gives
 2t1a1  t1 

 pA 
0
a1
 2t2 (b2  a2 ) 
2
2
2
2
2



4
t
a
(
b

a
)

2
t
p

p

t
(
b

a
)

t
(
b

a

2 2
2
2
2 B
A
1 1
1
2
2
2 )  t1 (b1  a1 ) 
0
 pA 
2
2
a2
4t2 (b2  a2 )


  pB  2 p A  t1 (b12  a12 )  t2 (b22  a22 )  t1 (b1  a1 ) 

0
p A 
2t2 (b2  a2 )

If we do the same for firm 2, we get unique solutions for prices as a
function of the location parameters:
 2t2 (b2  a2 )  t1 (b1  a1 )  t2 (b22  a22 )  t1 (b12  a12 ) 
pA  

3


 4t2 (b2  a2 )  t1 (b1  a1 )  t2 (b22  a22 )  t1 (b12  a12 ) 
pB  

3


dp A
dpB
1
1

 t1 (1  2a1 )  0 for a1  and
 t1 (1  2b1 )  0 for b1 
da1
2
db1
2
Both prices rise as the products become more similar in the dominated
characteristic.
Likewise for the ranges of a2 and b2
dp A
dpB
 2t2 (1  2a2 )  0 and
 2t2 (2  b2 )  0
da2
db2
Both prices rise as the products become less similar in the dominating
characteristic.
Differentiating in the dominating characteristic lessens price
competition, while similarity in dominated increases demand. In
Hotelling the first dominated the second, so firms differentiated in the
only dimension possible.
Location equilibrium
Given the equilibrium prices the profit function for firm A is
A
2t (b


2
2
 a2 )  t1 (b1  a1 )  t2 (b  a )  t1 (b  a ) 
2
2
2
2
2
1
2
1
2
18t2 (b2  a2 )
 A t1a1  t1 2a12
1

 0  a1 
a1 18t2 (b2  a2 )
2
 A
With tedious but straightforward algebra they also show that a  0 in
2
permissible ranges for all the (a1, bi), implying the firm wants a2 as
small as possible, ie, a2=0, and likewise, it means b2=1.
If there is no dominant characteristic
With k dimensions and no strictly dominant characteristic things are a
little dicey, but if all salience characteristics are equal, there are k local
equilibria, each differentiating on one characteristic and not on the
others.
The implication of this is that what dominates can change as relative
costs of differentiating, or consumer tastes, change
So what do we have?
• Is this model much different from Hotelling? In what way yes or no?
• Does it explain the real world better than single dimension models?
• In what way?