An extension to Salop’s model
Focused on variety differentiation: consumers differ on the most preferred
variety
Expands it to include quality differentiation: all consumers prefer more
quality to less, but differ in their willingness to pay for quality
The model here, each differentiated product is defined by one feature of
variety, and one feature of quality.
1. Allows exploration of substitution of variety for quality; if the
overabundance of variety from the Salop model persists
2. How firms strategic interaction define the variety-quality mix in the
market
Products are differentiated in each dimension, so within a variety,
consumers compare quality. Across varieties, they compare variety-quality
combinations (eg, computer type – Apple or IBM – and cpu speed)
Basic assumptions
Quality is independent of MC of production.
Quality comes from better design, not variable production costs.
Makes quality a fixed cost issue.
Why is this important in the Salop model?
Consumers have preferences for specific variety
At the same price, all consumers prefer higher quality to lower
quality.
They differ on their willingness to pay for higher quality
Where does quality choice fit in decisions?
Traditional Salop model has three stages of firm decision making before
consumers make their choice of whom to patronize
1. Firms decide to enter or not (long run)fore con
2. If a firm enters, it chooses location (medium run)
3. Firms in the market choose price (short run)
Adding quality gives three choices on how firms choose it
1. Quality and price are chosen simultaneously in stage 3, after location (3stage game)
2. Quality choice precedes the price stage, so is precommitted (4-stage game)
3. Quality and location are chosen simultaneously in stage 2, which is
equivalent to no precommitment in location, and has the same equilibrium
as the 4-stage game because there is no strategic value to location
Precommitment
In short run, when quality choice precedes price choice, the equilibrium
quality choice depends on the price response of neighboring firms
• Gives quality a greater strategic variable for securing revenue -- it is more useful
than when chosen simultaneously with price
• Since the strategic value is larger, less much be used to achieve strategic goals
• Result is firms use lower levels of quality when it is chosen first than when chosen
simultaneously with price – precommitment to quality lowers quality
In medium run, with fixed number of firms, precommitment brings higher
prices and profits, so there is more entry, hence more variety
Both games over-provide variety and under-provide quality, but quality
precommitment has the bigger distortion from optimum
Policy implications of the results
A key result is the inverse relationship between variety and
quality in equilibrium
This means raising quality above its equilibrium value reduces
variety
Since in equilibrium variety is over-provided and quality is
under-provided, minimum quality standards can be efficiency
improving
The model set-up
Product is defined by a pair (xj,aj) denoting its variety and quality
Variety characteristics lie on the unit circle with circumference denoted by C
Quality is chosen from the interval [0, a ] so the product space is a cylinder
of size C  [0, a ]
An alternative numeraire good is also available to all consumers
Consumers are defined by two parameters (z,)
z indicates her preferred variety, and lies on C
 denotes her relative preference for quality   [0,1] with CDF G()
Preferences are defined on C  [0,1]
Conceptualization of Salop circle with quality
We have a cylinder. The edge
of the cylinder defines the
variety, while the distance
along the tube defines
quality.
Variety
2
quality
Consumer problem
The value of one unit of product (xj,aj) sold at pj to consumer (z,) is
V ( z, , x j , a j , p j )  k   a j  p j  | z  x j |
Utility is separable in quality and variety
There is a linear penalty for variety deviating from consumer optimum
(symmetric in direction) and utility increases with quality, independent
of variety
Marginal consumer zj indifferent between (xj,aj) and (xj+1,aj+1) has
V ( z j ,  , x j , a j , p j )  V ( z j ,  , x j 1 , a j 1 , p j 1 )
 z j ( )   p j 1  p j  x j 1  x j   (a j  a j 1 )  / 2
Similarly marginal consumer zj-1 is between (xj,aj) and (xj-1,aj-1)
Location of the marginal consumers
Opening the cylinder and rotating
90o to the left shows the market
area for firm j
zj-1() zj()
The shaded area represents all
consumers who prefer to buy (xj,aj)
given current prices. Hence, the
demand for product j is
1 zj
Dj  
1
 dzdG( )   z
0 z j 1
j
 z j 1 dG ( )
0
 p j 1  p j 1  2 p j  x j 1  x j 1


/2
 E ( )(2a j  a j 1  a j 1 ) 

1
Where E ( )   dG ( ) is the expectation
0
of , ie, the average intensity of quality
preference. As this increases, the influence
of quality on demand goes up too
The profit function for firm j
With n firms and assuming zero MC of production, cost of quality is give
by C(a)=ca2/2 and fixed costs of F, firm j has a profit function
 j (p,a, x, n)  p j D j (p,a, x, n)  C (a j )  F
where
p  ( p1 ,..., pn )
a  (a1 ,..., an )
x  ( x1 ,..., xn )
are the n-tuples of strategies for prices, quality and variety
Entrylocation(quality and price): stage 3
Assume no price undercutting because it drives demand to zero
Firms choos price and quality. Thesubgame is to maximize profits wrt pj
and aj since the locations, vector x, are already made
 j
p j

p j 1
4
 pj 
p j 1
 j
a j
4

x j 1  x j 1
4
 E ( )  ca j

E ( )(2a j  a j 1  a j 1
4
 ej
 j  1,..., n
 j  1,..., n
He uses some fancy matrix algebra to show that profits for firm j are
1
1
* 2 


 j (p * (x, n), a * (x, n), x, n)  c  a j  
 F
2
 E ( ) 2c 
2
where p* and a* and p * (x, n)  ca * (x, n) / E ( ) are the vectors of
equilibrium qualities and prices. We use this for the location decision
Entrylocation(quality and price): stage 2
At stage 2, firm j chooses location to maximize
d  Lj (x, n)
 Lj (x, n)   j (p * (x, n), a * (x, n), x, n)
da*j
*
da


1
1
j
 2c 2 a*j 

 dx
2
E
(

)
2
c

 j
0
so
and he shows dx j
which is a
dx j
result of the linear model with inelastic demand. A shift of location to the
left results in as many lost sales to the right. Equilibrium values for price
and qualities are thus
p*j (x *(n)) 
1
n
a *j (x *( n)) 
E ( )
cn
(8)
1
Price and quality both increase in the distance between firms, given by n .
As the number of firms increases, the strategic value of high quality falls
because the potential market gain falls. The level of quality and the
number of varieties are inversely related.
Entrylocation(quality and price): stage 1
Firms enter as long as profits will be positive. Hence we need to solve for
n when
 Ej (n)   Lj (x *(n), n)  0
1  E ( ) / (2c) 
n*  

F


2
1
2
which gives the solution the integer part of
Letting the superscript 3s* indicate the equilibrium from the 3-stage
game, we have
p
3 s*
j
1
 3 s*
n
a
3 s*
j
E ( )
 3 s*
cn
1  E ( ) / (2c) 


F


2
n
3 s*
1
2
x3j s*  x3j s*1 
1
n 3 s*
Since an increase in the average intensity of preference for quality gives
higher quality and lower profit when the number of firms is fixed, in the
long run, increases in the preference for quality results in fewer firms (less
variety) and more quality in long run equilibrium
Note on symmetry
Results are for a symmetric equilibrium. Prices are proportional to
quality levels, so symmetry implies equal prices, quality and profit. If
locations are asymmetric some firms have larger markets, which also
implies higher prices, quality and profits.
In such a case, entry occurs until the marginal firm makes zero profit.
All other firms will have larger market shares, meaning somewhat
fewer firms, no less variety and higher average quality, along with some
firms making profits
Entrylocationquality  price game
Now quality is announced before price is set; it is a commitment that
allows firms to reveal how aggressively they will behave in setting price.
Because it is communicated earlier, it is a more potent signal about
strategy, so firms can use “less of it” to achieve the same result.
Hence, we expect lower quality in this game than when price and
quality are a combined decision.
Working backwards, starting with price we again have
 j
p j
and

p j 1
4
 pj 
p j 1
4

x j 1  x j 1
4
 j (p **(a, x, n), a, x, n)  ( p ) 
** 2
j

c(a j ) 2
2
E ( )(2a j  a j 1  a j 1
4
F
 ej
 j  1,..., n
Price, quality and profit in the 4-stage game
Working through the stages now, like we did in the 3-stage game gives
1
p (n)  p (a **(x **, n) 
n
**
j
**
j
E ( )(b1  b2 )
a (n)  a ( x **, n) 
cn
**
j
**
j
where we know from footnote 13 that 1>b1-b2>0. Compare these to
the results from the 3-stage game for the same number of firms:
1
p (n)  p (x *(n)) 
n
**
j
*
j
E ( )(b1  b2 ) E ( )
a ( n) 

 a *j ( x *( n))
cn
cn
**
j
So for the same number of firms, price is the same, quality is lower, and
since quality is costly, profits are higher than in the 3-stage game
Entry and firms in the 4-stage game
Again looking for zero profit we find in long-run equilibrium
1
2


1

E
(

)
(
b

b
)
/
(2
c
)
1
2
n 4 s*  

F


E ( )(b1  b2 )
1
4 s*
4 s*
p j  4 s*
aj 
n
cn 4 s*
2
2
x
4 s*
j
x
4 s*
j 1
1
 4 s*
n
Since 1>b1-b2>0 we know that n4s*>n3s* so there is more variety in the
four-stage game, indicating also that price and quality are also lower
Precommitment to quality reduces quality and increases the number of
brands
The optimal number of firms
For n equally space firms all selling at the same quality and price we
can find the total society value by integrating its demand. As shown in
equation (21) this generates a total surplus value of
S (n, a)
E ( )
nca 2
1
 E ( )  nca  0  a(n) 
S (n, a )  k  E ( )a 
 nF 
.
Taking
a
nc
2
4n
Substituting this into S(n,a) and optimizing wrt n gives
1/ 4  E ( ) / (2c) 
no  

F


2
1
2
and
a o  a ( n) 
E ( )
no c
Both games give too many varieties
Comparing the optimum to the results of the two games shows
n 4 s*  n 2 s*  n o , with n 2 s* / n o  2 and so a 4 s*  a 2 s*  a o
and
S ( n 4 s* )  S ( n 3 s* )
The surplus with precommitted quality is lower than when price and
quality are chosen simultaneously
Policy implication
Quality and variety are substitutes in the market. Salop has shown that
in horizontal differentiation we get too much variety.
Adding quality exacerbates the problem.
Higher quality attracts new consumers to a market by expanding
the number of firms needed to get profit to zero.
Reducing the number of firms will improve welfare
If entry prohibition is infeasible, setting a minimum quality standard
can increase welfare by reducing the number of firms (through the
inverse relationship of quality and variety)
Surplus values with fixed
and
variable quality standards
Minimum quality standards
There is always an opportunity to
improve welfare by setting minimum
quality standards. Let A3s*  (a 3 s* , a ).
In the three stage game, any
minimum quality within this range
will improve social welfare.
Likewise in the four stage game, any
minimum quality
within A4 s*  (a 4 s* , a ) will improve
social welfare.
a 4 s*
a 3 s*
ao
a
a