Quick summary
One-dimensional vertical (quality) differentiation model is extended to
two dimensions
Use to analyze product and price competition
Two stage game
1. Firms first compete on product position
2. Then they compete on price
Firms do not tend towards maximal differentiation
When firms have equal opportunities, they end up with max-min
differentiation, just like in Irmen and Thisse
Choice of product features
Need to consider consumer preferences
In quality differentiation, all consumers agree that more of
quality is better.
The vary in their willingness to pay for quality
Need to consider strategic responses of competitors
Two forces determine the locational equilibrium (both vertical and
horizontal differentiation models)
A demand force that draws firms close together
A strategic force that causes them to differentiate
Early research on vertical differentiation maximal differentiation shows
maximal quality differentiation when demand is inelastic, but when one
option is to not buy, quality differs exist, but not maximal
Model: market and firms
Goods have more than one type of quality
product quality
service quality
Two firms, indexed by 1 and 2
choose one product to market
the product has two characteristics, x and y, defined as a point,
(xi,yi) where these define the two types of quality, and are both
bounded by (allowably different) maximum and minimum values
production has constant marginal cost, set equal to zero
regardless of product position (does not depend on quality
characteristics), and no fixed cost
Model: consumers
Consumers prefer more of each quality to less, and prefer a low price
to a higher price
Consumer reservation price, R, is sufficiently high that every consumer
in the market buys one unit from either firm 1 or firm 2
Consumer valuation is defined by
U  R  1   2  pi for i  1, 2
so consumers care about absolute rather than relative prices
The parameters 1 and 2 are uniformly distributed over the
population. We restrict the range to [0,1], where the scale is different
for (x,y) if one characteristic is more important than the other.
Characteristic competition
Asymmetric characteristics competition – each firm has a relative
advantage in one quality dimension (figure a), for example “ease of
use” and “power” (Apple versus Windows)
Dominated characteristics competition – one firm has a relative
advantage on both characteristics (Windows 10 vs Windows 7)
For both types of competition, the relative positions of the products
are described by the ratio ( x1  x2 ) / ( y1  y2 )
This ratio equals the tangent of the angle between the horizontal axis
and a line from the origin perpendicular to a line joining  x1 , x2  to  y1 , y2 
This angle is called the angle of competition
Indifference space
Panel a shows the angle of competition with
asymmetric characteristics (firm 1 dominates in x,
firm 2 dominates in y)
Consumers buy the good that maximizes their utility
– there is a set of consumers indifferent between
the two goods where
p2  p1 x1  x2
ˆ
 2 (1 ) 
y2  y1

y2  y1
1
as shown in panel b when prices are equal, in which
case the indifference line lies along the angle of
competition
Differences in prices shifts the indifference line up
or down. More skewed dominance in x or y
increases or decreases the angle of competition
Indifference space
Panel a shows the angle of competition with
dominant characteristics competition (firm 2 has
the superior product)
When prices are equal the indifference line is
outside the characteristic space – everyone buys
from firm 2. Hence, for the indifference equation to
be in the product space we need p2  p1 and we
graph ˆ ( )  p2  p1  x1  x2  as before
2
1
y2  y1
y2  y1
1
Relatively greater dominance by firm 2 in y
increases the angle of competition. Relatively
greater dominance in x decreases the angle of
competition
Price Equilibrium
Sequential game: firms first choose quality, then choose price, with
subgame perfect criterion
An equilibrium consists of product choice for firms 1 and 2 such that
 ( p , pa
) different
 ( p , p ) p  0 i,product
j  (1, 2)
neither would choose
unilaterally, for whatever
prices equilibrium that follows, using backward induction
Profit function for firm i, i=(1,2) is
*
*
(
p
,
p
A noncooperative Nash equilibrium is a pair of prices 1 2 ) such that
i
*
i
*
j
i
i
*
j
i
 i ( p1 , p2 )  pi Di ( p1 , p2 )
Asymmetric characteristics equilibrium
The indifference line is positively sloped with angle
 x1  x2 
  tan 

y

y
 2 1
1
With product positions fixed the indifference line is shifted up or down
by changes in p2  p1.
Look at the problem from firm 1’s perspective (so p2 is fixed).
We see in the next figure there are four boundaries
p1u is the lowest price at which no
consumers buy from firm 1
p1l is the lowest price at which no
consumers buy from firm 1
These are the upper and lower
bounds of the price the firm will
charge given p̂2
The two remaining key prices are
those where the indifference line
n
p
passes through (0,0), at price 1 ,
and when it passes through (1,1), at
price p1m
Price equations
Replacing 1 and  2 in the boundary equations with the boundary point
coordinates gives
where, for example, in (3) we set 1  1,  2  0.
These are all increasing in p̂2 , and, if it is in the equation, increasing in
x1  x2 and decreasing in y2  y1.
The greater firm 1’s relative advantage over 2, the higher the price it can
charge to generate a similar demand
Characteristic i dominance, i=x,y
If α45o we have x1  x2  y2  y1 and p1  p1  p1  p1 which is
characteristics x dominance, while if x1  x2  y2  y1 then 45o  α we
have characteristic y dominance and p1l  p1m  p1n  p1u
l
1
𝑝1𝑙
𝑝1𝑙
1
𝑝1𝑚
n
m
𝑝1𝑛
u
𝑝1𝑚
𝑝1𝑛
𝑝1𝑢
𝑝1𝑢
2
45o α, characteristic y dominance
2
α45o characteristic x dominance
Characteristic x dominance
Three regions
In R1x demand for firm 1 increases at an
increasing rate
In R 2 demand for firm 1 increases at an
x
constant rate
In R3x demand for firm 1 increases at an
increasing rate
To see the rate of change
R1x . As price falls
Suppose
we
are
in
u
m
p
p
from 1 to 1 the area of demand grows
at an increasing rate
We can easily see that z1 is given by
and demand by
Total demand under characteristic x dominance
It is easy to see that
Note D2=1-D1 and we
have three regions
that correspond to
the regions in the
above graph. In
region 1 D1 is convex,
in region 2 it is linear
and in region 3 it is
convex. Firm 2’s
demand follows the
opposite pattern
R
3
x
R2x
R1x
Equilibrium prices
2
p
D
Start in region 2. Total profit is price times quantity, so x 1 .
Maximizing this with respect to px gives a unique solution
n
*
m
p

p

p
if in region 2 so 1
1
1 but that is only satisfied if ( x1  x2 )  ( y2  y1 )
(the condition for characteristic x dominance which we started with). A
similar condition holds that puts firm 2 in the same region.
m
*
n
p

p

p
A similar analysis under characteristic y dominance results in 1
1
1
so whichever characteristics is dominant in the asymmetric dominance
situation we end up in the middle region
Dominated characteristic competition
In this case one firm (2) has higher quality in both characteristics. With
characteristic x dominance we have p1m  p1u  p1l  p1n
1
p1u
l
1
p
m
1
p
R1
R2
p1n
R3
2
With the same type of tortured
logic they show that there are
price equilibrium in all three
regions. This is different from
the asymmetric characteristic
competition.
There is a similar picture if y is
dominant, but the ordering of
m
l
u
n
prices becomes p1  p1  p1  p1
because the blue lines are flatter.
Product equilibrium strategy
Product positions are dependent on the equilibrium prices
Process
1. Establish which regions need to be considered
2. Calculate the firms profit function within each relevant region
3. Find the FOC for profit along with the demand restrictions that
define the region indicate the maximum profit location in a region
4. Compare the maximum profit in each region to find the optimal
location choice
Location decision under asymmetric dominance
Need only look in region 2 for each firm. From the demand for each
firm, and the equilibrium prices we find
Now look at table 1
Implications about produce choice
Maximum quality in both directions yields the highest profit, therefore
both firms want to choose that location
Both can’t be there, and it is not clear which will, so we can arbitrarily
pick one
Given that one firm is in the supreme position of highest quality in both
characteristics, it is the strategy of the lower quality firm that
determines how much differentiation there is
One solution is max-min (panel a on next page), in which both firms
have maximum quality in one characteristic, one has maximum quality
in the first characteristic, and the other has minimal quality in the
second characteristic
Other solutions are max-max, and max partial
Max-min
Max-max
Max-partial
What is interesting about these three pictures?
If the range in quality in one direction is approximately equal to that of
the other, the result is max-min
As the ratio of the ranges grows, we move to max-max
But as it gets very large, we get max-partial. Moreover, as the range of
x quality relative to the range of y quality gets bigger and bigger, the
partial differentiation gets further away from max-max (but not
necessarily closer to max-min
So why? (they do a lousy job of explaining why these results make
sense)
Both firms want to have maximum quality. When quality dimensions
are approximately equal, but because of strategic force to avoid price
competition, only one locates there. The other firm, to maintain
demand as much as possible keeps quality high in one dimension, but
differentiates in the other dimension to reduces price competition.
When quality are very different maintaining high quality in the lesser
dimension does not bring enough demand to counter the price
competition effect. So the second firm relinquishes demand (by
lowering the quality on the second dimension) but gains in price.
But the strategic tradeoff doesn’t always dominate. If the ranges of
quality get sufficiently large, relative quality would get too low (they
would lose too much demand for any diminishment of the price
competition to compensate, so chooses an intermediate level in the
second quality dimension.
So why is this important?
Depending on the range of quality differences, we go, as the range of x
relative to the range of y increases, from max-min to max-max to maxpartial in differentiation along the two quality dimensions
This is different from everything we’ve seen before, because within one
model we are seeing all possibilities
Also shows that the way quality differentiates and how consumers
value different quality components are important in determining the
equilibrium (different consumer values would show up as different
scales, increasing or decreasing the ranges of x and y)
Helps explain the wide variety of quality we see among differentiated
products