Product differentiation
with elastic demand
MAT T BIRCH
ROBERT ROSENMAN
1
Harold Hotelling, 1929
First to look formally at product differentiation.
Did so in the context of observing gradual shifting of customers between different suppliers.
Focused on geographical differentiation, but it applies equally to product characteristics.
Made the following simplifying assumptions:
•There are two firms.
•All consumers will buy one unit of the good. (Perfectly inelastic)
•Consumers are uniformly distributed.
•Consumers receive the same gross benefit from buying a good at any location or firm.
2
Harold Hotelling, 1929
0
1
0.5
𝑈𝐴 = 𝑆 − 𝑡 𝑥 − 𝐴 − 𝑝𝐴
𝐷𝐴 =
𝑈𝐴 ≥𝑈𝐵
𝑥𝑑𝑥
𝑈𝐵 = 𝑆 − 𝑡 𝑥 − 𝐵 − 𝑝𝐵
𝐷𝐵 =
𝑈𝐵 ≥𝑈𝐴
𝑥𝑑𝑥
Hotelling solution is that firms locate at the center. Minimal Differentiation.
Hotelling’s conclusion is that there would be insufficient product differentiation.
3
Maximal vs. Minimal Differentiation
Follow-up analysis came to opposite conclusion
No equilibrium when firms were too close together (d'Aspremont, Gabszewiez, &
Thisse, 1979)
Based on quadratic mismatched costs, firms will maximally differentiate. This is
the Nash equilibrium
0
0.5
1
Hotelling, 1929
d'Aspremont, Gabszewiez, & Thisse, 1979
Economides, 1986
4
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
5
Different Studies, Different Assumptions
• Consumer preference for [costly] quality yields max-min differentiation when quality costs are
low or max-max differentiation when quality costs are high (Lauga & Ofek, 2011).
• If consumers are distributed non-uniformly, there can be partial differentiation in one
dimension instead of max-min-..-min (Liu & Shuai, 2012).
•In a three firm, cube market, max-min-min does not hold. Instead there is partial differentiation
along two dimensions and minimal along the third (Feldin, 2012; see also Tabuchi, 2012).
6
What do these have in common?
Every paper mentioned so far has had an important common assumption:
Perfectly Inelastic Demand: All consumers will buy one unit of the good.
7
One Prior Paper With Elastic Demand
Economides (1984) relaxed the perfectly-inelastic demand assumption.
Back on a Hotelling line, local monopolies may form in equilibrium.
Price
0
1
8
Our Approach
•We follow Economides (1984), and use low gross surplus (i.e. elastic demand).
•We have a 2-dimensional product space, i.e. a square. Call it a Hotelling “City” with elastic
demand, if you like.
•We include salience concepts from Irmen and Thisse (1998) and Ansari et al. (1998) that allow
dimensions to differ in importance.
9
Our Results
•We show that for a wide range of consumer demand there is never maximal or minimal
differentiation in equilibrium.
•We analyze different levels of demand and explain how the structure of product differentiation
evolves as demand levels change.
10
The Consumer
Consumers are distributed 𝑔(𝑥) which we assume to be uniform over the square.
Consumer x located at (𝑥1 , 𝑥2 ) who buys from firm A located at (𝑎1 , 𝑎2 ) for a price 𝑝𝐴 has utility
𝑈𝐴 = 𝑆 − 𝑡1 𝑥1 − 𝑎1
Dimension 2 is dominant over dimension 1:
2
− 𝑡2 𝑥2 − 𝑎2
2
− 𝑝𝐴
𝑡2 ≥ 𝑡1 ≥ 0
Reservation utility is 0.
11
A little example… Cars
𝑥22
Consumer 2
Firm B
Size
Size
𝑏1
𝑥21
Consumer 1
𝑥11
“Off-Road” Capabilities
𝑎2
𝑥12
Firm A
𝑏1
𝑎1
“Off-Road” Capabilities
12
Demand
Demand at firm A is given as
𝐷𝐴 =
𝑈𝐴 ≥𝑈𝐵 ,0
𝑔(𝑥)𝑑𝑥
When the firms are far enough from each other and the edges of the square, the plot of the
marginal consumers forms an ellipse:
𝑆 − 𝑝𝐴 = 𝑡1 𝑥1 − 𝑎1
2
+ 𝑡2 𝑥2 − 𝑎2
2
What do we mean by “low” or “small” demand?
• We mean that under optimal pricing, both demand ellipses can fit entirely within the square.
13
Cars continued…
Consumer 2
Firm B Demand Ellipse
Size
Size
Firm B
Consumer 1
Firm A
“Off-Road” Capabilities
Firm A Demand Ellipse
“Off-Road” Capabilities
14
Constrained Demand
Scenario 1: Firm A is unconstrained.
Firm B is constrained.
Scenario 2: Both firms are
constrained.
Firm B
Firm
A A
Firm Firm
A
B
15
Mathematical Notation for Small Enough
We denote the highest level of 𝑆 that permits two unconstrained demand
ellipses to coexist as 𝑆.
16
Elastic Demand: What do we mean?
What do we mean by “low” or “small” demand?
We focus on 𝑆 ≤ 𝑆
𝑆>𝑆
𝑆=𝑆
𝑆<𝑆
17
A Little Geometry Review
𝑆 − 𝑝 = 𝑡1 𝑥1 − 𝑎1
2
+ 𝑡2 𝑥2 − 𝑎2
2
𝑟2
𝑟1
The semi-major axis, 𝑟1 =
𝑆−𝑝
.
𝑡1
The semi-minor axis, 𝑟 2 =
𝑆−𝑝
.
𝑡2
Area 𝐷 = 𝜋𝑟1 𝑟 2
If 𝑡1 = 𝑡2 this is a circle with radius 𝑆 − 𝑝.
18
General Form Profit Maximization
We seek a Nash Equilibrium in price and location. Firm A’s problem is
max 𝑝𝐴 𝐷𝐴 𝑎1 , 𝑎2 , 𝑝𝐴 ; 𝑏1 , 𝑏2 , 𝑝𝐵
𝑎1 ,𝑎2 ,𝑝𝐴
𝑠. 𝑡. (𝑎1 , 𝑎2 ) ∈ 0,1 × 0,1
The Kuhn-Tucker conditions:
𝜕𝐷
𝑝𝐴 𝜕𝑎𝐴 + 𝜆𝐴1 − 𝜇𝐴1 = 0
1
𝜕𝐷
𝑝𝐴 𝜕𝑎𝐴 + 𝜆𝐴2 − 𝜇𝐴2 = 0
2
𝜕𝐷
𝑝𝐴 𝜕𝑝 𝐴 + 𝐷𝐴 = 0
𝐴
𝜆𝐴1 𝑎1 = 𝜆𝐴2 𝑎2 = 𝜇𝐴1 1 − 𝑎1 = 𝜇𝐴2 1 − 𝑎2 = 0
19
Corner [Edge] Solutions?
For each binding location constraint, the firms cut demand in half.
No corner solutions in Nash equilibrium.
20
Interior Solution FOCs
The location conditions say that for a given price we maximize the area of the ellipse within the
square.
𝑝𝐴
𝜕𝐷𝐴
𝜕𝑎1
=0
𝑝𝐴
𝜕𝐷𝐴
𝜕𝑎2
=0
The price condition indicates that we choose price such that marginal revenue is zero.
𝑝𝐴
𝜕𝐷𝐴
𝜕𝑝𝐴
+ 𝐷𝐴 = 0
21
Far enough apart.
In equilibrium:
𝜕𝐷𝐴
𝜕𝑎1
=
𝜕𝐷𝐴
𝜕𝑎2
=
𝜕𝐷𝐵
𝜕𝑎1
=
𝜕𝐷𝐵
𝜕𝑎2
=0
This means that for 𝑆 ≤ 𝑆 firms locate so that demand ellipses do not overlap with each other
(FC) or with the square boundaries (BC).
Arrows depict violations
of optimality conditions.
22
Profit Maximization
Profit function:
Π=𝑝
Optimal price condition:
𝑆−𝑝
𝑡1
𝑆−𝑝
𝑡2
𝜋 𝑆−𝑝
𝜋𝑝
−
=0
𝑡1 𝑡2
𝑡1 𝑡2
Optimal price:
𝑝𝐴 = 𝑝𝐵 =
Equilibrium Profit:
Π𝐴 = Π𝐵 =
𝑆
2
𝑆2
2 2𝑡1 𝑡2
23
Location Constraints (BC) and (FC)
Firms locate sufficiently within the square that they are not constrained by the edges. (BC)
𝑎1 , 𝑎2 ∈
𝑆−𝑝𝐴
,1
𝑡1
−
𝑆−𝑝𝐴
𝑡1
×
𝑆−𝑝𝐴
,1
𝑡2
−
𝑆−𝑝𝐴
𝑡2
𝑏1 , 𝑏2 ∈
𝑆−𝑝𝐵
,1
𝑡1
−
𝑆−𝑝𝐵
𝑡1
×
𝑆−𝑝𝐵
,1
𝑡2
−
𝑆−𝑝𝐵
𝑡2
Firms Locate far enough apart that demand ellipses do not overlap. (FC)
2𝑆
𝑏2 −𝑎2 2 +1
𝑡2 𝑏2 −𝑎2 2 +𝑡1
≤
𝑏1 − 𝑎1
2
+ 𝑏2 − 𝑎2
2
24
More about cars…
Why isn’t consumer 2 served when S is low?
2
2
Firm B Demand Ellipse
Size
Size
Firm B Demand Ellipse
The BC:
That’s why!
1
1
Firm A Demand Ellipse
“Off-Road” Capabilities
Firm A Demand Ellipse
“Off-Road” Capabilities
25
Location Constraints
26
Central Result
We can distinguish three cases of low-S product differentiation.
𝑡1
8
𝑡1 𝑡2
,
8 8
𝑡2
,𝑆
8
• 𝑆 ∈ 0,
• 𝑆∈
• 𝑆∈
⇒
1
4
𝑟2
⇒ 𝑟2 ≤ <
⇒
1
4
𝑟1
𝑟 2 ≤ 𝑟1 ≤
1
4
<
≤ 𝑟1
We still are interested in 𝑆 > 𝑆 but there significant are analytical challenges associated using calculus and
constrained ellipses in Nash Equilibrium.
27
Product Differentiation 1
When demand is low, such that 𝑆 ∈ 0,
dimensions.
𝑡1
8
, we can have differentiation on either or both
𝑎1 ≠ 𝑏1
𝑎2 ≠ 𝑏2
𝑎1 = 𝑏1
𝑎2 ≠ 𝑏2
Firm A
𝑎1 ≠ 𝑏1
𝑎2 = 𝑏2
28
Product Differentiation 2
𝑡
𝑡
When 𝑆 ∈ 1 , 2 the firms will differentiate along the dominant dimension and may also
8 8
differentiate along the dominated dimension.
𝑎1 = 𝑏1
𝑎2 ≠ 𝑏2
Firm A
𝑎1 ≠ 𝑏1
𝑎2 ≠ 𝑏2
Firm A
Never
𝑎2 = 𝑏2
Firm A
Error!
29
Product Differentiation 3
𝑡
When 𝑆 ∈ 2 , 𝑆 the firms will differentiate on both dimensions, with greater differentiation
8
along the dominant dimension.
30
So… what?
Us
Partial-Partial
Them
Max-Min
Max-Max
Min-Min
31
Conclusion [of this part]
In common with literature:
• Differentiation is more pronounced on dominant dimension to relieve price competition.
• Always differentiation on at least one dimension.**
In contrast with literature:
•
•
•
•
Never maximal or minimal differentiation for wide range of demand.
Dimensions of differentiation and degree of differentiation are dependent on demand.
For some range there must be differentiation on both dimensions.
For low demand there is an infinite number of equilibria.
Allows for much more realistic outcomes.
32
Intermediate S
•Demand is high enough that optimally priced firms cannot have unconstrained demand ellipses.
•Our objective: trace the optimal location path as the disc expands.
•ADDITIONAL ASSUMPTION (1):
𝑡1 = 𝑡2 = 1
•Demand ellipses are now circular discs.
𝑟=
𝑆−𝑝
33
Special case: S=S and t1=t2=1
Circles just fit along the diagonal, so S 
 1 2
1 
a
,
a

,
 1 2  

2

2
2

2


 1
1 2 
,
 b1 , b2   

 2 2 2 2 
1
3 2 2
and r 
1
and there are two solutions
2 2

1
1 
,

 2 2 2 2 
 1 2 1 2 
,
 b1 , b2   

 2 2 2 2 
 a1 , a2   
34
S a little bigger than 𝑆
•Characterizing demand. 𝑆 is “a little bigger” than 𝑆.
•Firm A’s demand is 𝐷𝐴 𝑎1 , 𝑎2 , 𝑝𝐴 ; 𝑏1 , 𝑏2 , 𝑝𝐵 = 𝜋 𝑆 − 𝑝𝐴 − 𝐴1 − 𝐴2 − 𝐴3
•Firm B’s demand is 𝐷𝐵 𝑏1 , 𝑏2 , 𝑝𝐵 ; 𝑎1 , 𝑎2 , 𝑝𝐴 = 𝜋 𝑆 − 𝑝𝐴 − 𝐵1 − 𝐵2 − 𝐵3
35
The easy part: Price
Interpreting general results:
• Price will be chosen such that marginal revenue is zero.
• Then 𝑝𝐴 = 𝑝𝐵 ⇒ 𝑟𝐴 = 𝑟𝐵 = 𝑟 = 𝑆 − 𝑝
• At any given price, the firm chooses location to maximize demand.
• For this section we treat the optimal radius as exogenous, and track optimal locations as the radius
increases.
1
1
• Assume 𝑎1 , 𝑎2 ≤ 2 and 𝑏1 , 𝑏2 ≥ 2.
36
Because 𝑝𝐴 = 𝑝𝐵 :
•Cross-section from center of circle to firm overlap
r
C
𝑟2
r
−
𝑐2
r
𝑟2 − 𝑐2
C
r
•Firms with same radii share overlapped demand equally
•This is better news than you can possibly imagine!
37
The constrained disc:
We assume
𝑡1 = 𝑡2 = 1
𝐷𝐴 = 𝜋𝑟 2 −
3
𝑖=1
𝑟 2 sin−1
1
𝑎3 = 2
𝑏𝑖 ≡
𝑟
− 𝑎𝑖 𝑟 2 − 𝑎𝑖2
𝑎1 = min 𝑎1 , 1 − 𝑎1
𝑎2 = min 𝑎2 , 1 − 𝑎2
𝑎𝑖 ≡
𝐷𝐵 = 𝜋𝑟 2 −
𝑟 2 −𝑎𝑖2
3
𝑖=1
𝑏1 − 𝑎1
𝑟 2 sin−1
2
+ 𝑏2 − 𝑎2
𝑟 2 −𝑏𝑖2
𝑟
1
We assume 𝑎1 , 𝑎2 ≤ 2
2
− 𝑎𝑖 𝑟 2 − 𝑏𝑖2
1
𝑏1 = min 𝑏1 , 1 − 𝑏1
𝑏2 = min 𝑏2 , 1 − 𝑏2
1
𝑏3 = 2
𝑏1 − 𝑎1
2
+ 𝑏2 − 𝑎2
We assume 𝑏1 , 𝑏2 ≥ 2
2
38
FOC
𝜕𝐷𝐴
𝜕𝑎1
=0⇒
𝑏1 −𝑎1
2
𝜕𝐷𝐴
𝜕𝑎2
𝑟2−
1
4
−
2
𝑎
1− 21
𝑟
𝑟2
−
𝑎12
𝑏1 −𝑎1 2 + 𝑏2 −𝑎2 2
𝑏1 −𝑎1 2 + 𝑏2 −𝑎2 2
=0⇒
𝑏2 −𝑎2
2
𝑟
𝑟2−
1
4
𝑟
−
2
𝑎
1− 22
𝑟
𝑟2
−
𝑎22
𝑏1 −𝑎1 2 + 𝑏2 −𝑎2 2
𝑏1 −𝑎1 2 + 𝑏2 −𝑎2 2
+
−
+
−
𝑎12
𝑟 2 −𝑎12
𝑏1 −𝑎1
8
𝑎22
𝑟 2 −𝑎22
𝑏2 −𝑎2
8
−
1
4
𝑏1 −𝑎1
𝑟2−
1
4
𝑏1 −𝑎1 2 + 𝑏2 −𝑎2 2
𝑏1 −𝑎1 2 + 𝑏2 −𝑎2 2
1
𝑟2−
4
𝑏1 −𝑎1
2+
𝑏2 −𝑎2
1
𝑟2 − 𝑏1 −𝑎1 2 + 𝑏2 −𝑎2 2
4
1−
𝑟4
=0
2
1
𝑏2 −𝑎2
−4
𝑟2−
1
4
𝑏1 −𝑎1 2 + 𝑏2 −𝑎2 2
𝑏1 −𝑎1 2 + 𝑏2 −𝑎2 2
1
𝑟2−
4
+
1
𝑟2 − 𝑏1 −𝑎1 2 + 𝑏2 −𝑎2 2
4
1−
𝑟4
+
=0
𝑏1 −𝑎1 2 + 𝑏2 −𝑎2 2
39
Amazingly these reduce to
𝜕𝐷𝐴
𝜕𝑎1
𝜕𝐷𝐴
𝜕𝑎2
𝜕𝐷𝐵
𝜕𝑏1
𝜕𝐷𝐵
𝜕𝑏2
=2
=2
𝑟2
𝑟2
=2
𝑟2
=2
𝑟2
𝑏1 −𝑎1
𝑑
𝑟2
𝑏2 −𝑎2
𝑑
𝑟2
−
𝑏12
𝑎1 −𝑏1
−
𝑑
𝑟2
−
𝑏22
𝑎2 −𝑏2
−
𝑑
𝑟2
−
−
𝑎12
𝑎22
−
−
1 2
𝑑
4
= 0 (1)
1 2
𝑑
4
= 0 (2)
−
1 2
𝑑
4
= 0 (3)
−
1 2
𝑑
4
= 0 (4)
−
−
𝑑=
𝑏1 − 𝑎1
2
+ 𝑏2 − 𝑎2
1
Assume 𝑎𝑖 ∈ 0, 2 , 𝑖 = 1,2
Let 𝑏𝑖 = min 𝑏𝑖 , 1 − 𝑏𝑖
40
2
About the derivatives
Firm A and Firm B have identical optimality conditions given 𝑎1 , 𝑎2 ≤
1
2
1
2
and 𝑏1 , 𝑏2 ≥ .
Two big IF’s:
•One solution requires 𝑎𝑖 = 1 − 𝑏𝑖 . Are there non-firm-symmetric location paths
to worry about? No, more on that below.
•One solution requires 𝑎1 = 𝑎2 . Are there off-diagonal location paths to worry
about? Probably
41
Nash equilibrium requires firm symmetry
Firm symmetry: ai=1-bi
Proof structure:
assume interior solutions
assume firm A is in the lower left quadrant
show that the FOC require ai=1-bi
does not solve for the Nash equilibrium or prove its existence, just
establishes a requirement for one to exist
42
Symmetry requirement for Nash
b1  a1 2 d 4
2
2
r  a1 
r 
2d
4
(1)
4
b

a
d
r 2  a22  2 2 r 2 
2d
4
(2)
4
a

b
d
r 2  b12  1 1 r 2 
2d
4
(3)
a2  b2 2 d 4
r b 
r 
2d
4
(4)
2
2
2
Square both side of all equations.
Then (1) and (3) imply
a1  b1  (1  b1 )
and (2) and (4) imply
a2  b2  (1  b2 )
the last equalities coming from the FOC
that b1  min{b1 , (1  b1 )}
43
One possible solution
From (1) and (2) and symmetry requirement
r 2  a12
r 2  a22

1  2a1
1  2a2
One possible solution to this is
a1  a2
44
Next Steps
1. We have shown that a Nash equilibrium requires 𝑎𝑖 = 1 − 𝑏𝑖 .
2. Have also shown that for any r, one possible solution to the individual firm
problem, given that a Nash Equilibrium exists, is a1=a2. There is another solution
with a1 a somewhat complicated function of a2 and r.
3. Conditional on 1 and 2 we can find the relationship between a1 and r so that
MR=0. This relationship depends on S, so S will characterize our equilibria.
4. Show when 𝑎𝑖 = 1 − 𝑏𝑖 and a1=a2 we have a Nash equilibrium for any S (may
not be unique).
45
Still limited
•This constrained disc analysis applies to a range of S from 𝑆 to some 𝑆.
•At some point we may switch structures: The third case is Irman and Thisse
46
Still limited
•This constrained disc analysis applies to a range of S from 𝑆 to some 𝑆.
•We conjecture that the equilibrium approaches max-min differentiation as S gets large.
47
Conclusion
Even without capturing the complicated intermediate 𝑆 equilibria, we have still added to our
understanding of product differentiation.
•First multi-dimensional setting with elastic demand.
•Never max or min on any dimension.
•Always differentiation in one dimension or both.
•For some values of S there must be differentiation in both dimensions.
•Differentiation is stronger on dominant dimension.
48