A monopolist deliberately charges the same price for differentiated products when high quality products are more likely to be allocated to low type consumers under uniform pricing. The argument can explain the use of ‘unpriced quality’ for concert tickets, sport events, and in many other situations.Price discrimination, second degree price discrimination, queuing, rationing

Pascal Courty

English

Research Papers in Economics

DEPARTMENT OF ECONOMICS
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
EUI Working Papers
ECO 2009/16
DEPARTMENT OF ECONOMICS
 
 
 
 
 
 
 
 
 
   
Unpriced Quality
 
 
 
 
 
 
 
 
 
 
 
 
   
 
Pascal Courty
  
 
EUROPEAN UNIVERSITY INSTITUTE, FLORENCE 
DEPARTMENT OF ECONOMICS
Unpriced Quality 
PASCAL COURTY
EUI Working Paper ECO 2009/16 
 
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© 2009 Pascal Courty 
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Unpriced Quality 
 
 
 
Pascal Courty
1
 
December 2008 
 
 
 
Abstract:  A monopolist deliberately charges the same price for differentiated products 
when high quality products are more likely to be allocated to low type consumers under 
uniform pricing.  The argument can explain the use of ‘unpriced quality’ for concert 
tickets, sport events, and in many other situations. 
 
 
JEL:  D42, D45. 
 
Keywords: Price discrimination, second degree price discrimination, queuing, 
rationing. 
 
 
                                                           
1 Pascal Courty, Department of Economics, European University Institute, Villa San 
Paolo, Via Della Piazzuola 43, 50133 Firenze, Italy, Pascal.Courty@IUE.it.Firms sometimes charge the same price for products of different qualities.  In the 
absence of price discrimination, high quality products, which are preferred by all 
consumers, end up being rationed.  Examples include tickets for seats of different quality 
for sports and music events.  Connolly and Krueger (2006) report that 43 percent of 
concerts for popular music in 2003 sell all seats in the house at the same price.  Consumers 
also have to wait to enjoy the most popular attractions in theme parks (Passell, 1995).  
Restaurants do not charge extra to those consumers who come during peak hours or peak 
days.  Movie theaters do the same and in addition they do not charge more for blockbusters 
(Einav and Orbach, 2007).  Finally, grocery stores often do not differentiate fresh products 
and the most appetizing qualities sell first while the last consumers pay the same price for 
inferior, old, or stale products.  In all these examples, quality is unpriced; that is, higher 
quality products are sold at the same price as lower quality ones.   
In this paper, we show that uniform pricing can dominate price discrimination even 
when there is no additional cost associated with implementing price discrimination.  We 
start with the standard model of second degree (quality) price discrimination (Mussa and 
Rosen, 1978).  To simplify the analysis, we take the distribution of goods as given.  This 
assumption is reasonable in all the examples above, and allows us to focus on revenue 
considerations alone, in studying the trade off between uniform pricing and price 
discrimination.
2   
Whether uniform pricing dominates price discrimination depends on how goods are 
allocated under uniform pricing.  Uniform pricing is more likely to be optimal under a 
reverse monotone allocation rule, which assigns high quality goods to low type consumers.  
This allocation rule is reasonable in all the applications mentioned above.  It is consistent 
with a rationing mechanism based on queues, for example, if the time value of the good 
1 decreases with consumer type (Holt and Sherman, 1982).  In addition, uniform pricing will 
dominate price discrimination when there are not too many high types and when high types 
do not value incremental units of quality too much relative to low types (so that rationing is 
not too inefficient).  
Despite the extensive economic literature on price discrimination, there is surprisingly 
little work on why firm sometimes abstain from price discriminating (Stole, 2008).   
Anderson and Dana (2008) study when the optimal product line dominates selling a unique 
product quality.  Instead, we take the product line as given and investigate whether the firm 
wants to sell differentiated products at different prices.  Miravete (2007) shows that the 
return to complex tariffs may be low and implementation costs could explain the 
prevalence of simple product lines.  We show that uniform pricing can dominate price 
discrimination even in the absence of implementation costs.       
This paper also contributes to the literature demonstrating that rationing may be more 
effective at extracting consumer surplus than market clearing (e.g. Gilbert and Klemperer, 
2000).  An important implication of our analysis is that under uniform pricing the 
monopolist will strictly favor allocation rules that are more likely to assign high quality 
goods to low types.  A queuing system may support this assignment, but more generally, 
the seller may also manipulate the release date, opening hours, payment method, or other 
features of the allocation process that differentially increase the cost that high types have to 
bare to obtain high quality products.     
 
1-Example 
Assume there are two types of consumer, two types of good, and each consumer can 
consume at most one good.  Consumer t=L,H values v
t
s a good of quality s=l,h such that 
                                                                                                                                                                      
2 Since we ignore cost issues, selling different goods at different prices corresponds to 
2 v
t
h>v
t
l, v
Η
s>v
L
s, and v
H
h-v
H
l>v
L
h−v
L
l.  All consumers value the high quality good more, the 
high type values any quality more than the low type, and the high type values an increment 
in quality more than the low type.  In addition, we also assume that v
H
l>v
L
h.  There are 
φ∈[1/2,1] high type consumers and 1-φ low types.  There is a unit continuum of goods.  To 
simplify, we assume that the fraction of high quality goods is equal to φ and we show later 
that the results generalize.  Under price discrimination, the monopolist fully extracts the 
surplus of the low type consumers, pl= v
L
l, binds the incentive compatibility constraint of 
the high types, ph= v
L
l+(v
H
h-v
H
l), and earns revenue  
R
d=v
L
l+φ(v
H
h-v
H
l). 
Under uniform pricing, we assume that the goods are allocated according to an inverse 
monotone allocation rule: high quality goods are first allocated to low types.  Under 
uniform pricing, the monopolist charges v
L
h and earns profits R
u= v
L
h (there are enough 
high quality goods for the low types and high types buy since v
H
l>v
L
h).
3   The gains from 
using uniform pricing instead of price discrimination, ΔR=R
u-R
d, can be expressed as ΔR = 
v
L
h−v
L
l−φ(v
H
h-v
H
l).  The monopolist uses uniform pricing when this expression is positive, 
that is, when 
(v
H
h-v
H
l)/(v
L
h−v
L
l) <1/φ. (1) 
This condition is more likely to hold for low fractions of high types and when high types 
do not value quality much more than the low types, so that rationing is not too inefficient.  
The intuition is that the reverse monotone allocation rule increases the willingness to pay 
of the low types more than what is lost from the high types under price discrimination.  
                                                                                                                                                                      
price discrimination (Clerides, 2004).   
3 Because rationing is deterministic, the monopolist could earn more by charging 
more for low quality goods but this odd prediction does not carry through when 
rationing is random (see next section). 
3 Next, we generalize the analysis to a continuum of consumers and goods and to arbitrary 
rationing rules.   
 
2-Analysis 
There is a unit mass of consumer with type distribution F(t), t∈[t
L,t
H].  The monopolist 
has to price a given continuum of goods of quality distributed according to G(q), q∈[ql,qh].  
Consumer t gets utility U
t(q)-p from buying a good of quality q at price p, where U
t
q>0, 
U
t
qq<0, and U
t
tq>0.  The single crossing condition implies that it is efficient to allocate 
higher quality goods to higher types.  Denote the efficient allocation t
e(q) defined by 
F(t
e(q))=G(q).  We assume that under price discrimination there is full market coverage.
4   
Under uniform pricing, consumers buy a lottery over quality.  The probability density 
that type t receives a good of quality q is π
t(q) with associated distribution Π
t(q).  The 
lotteries π
t() are given.  The rationing rule is such that market clearing takes place, 
∫
tH
tLπ
t(q)dt=g(q), for all q.  In addition, we assume  
∫ ∫ ≥
h
l
h
l
q
q
L L q
q
t t dq q U q dq q U q ) ( ) ( ) ( ) ( π π   for all t  (A1) 
This condition guaranties that under uniform pricing all types participate if the lowest type 
does.
5
Under price discrimination, denote the pricing rule p(q) and the profit maximizing 
allocation rule t
pd(q).  The participation constraint of consumer t
L, p(ql)= U
L(ql), together 
with the consumers’ first order conditions, U
t
q(q)=pq(q), define the pricing rule as a 
function of the allocation rule 
                                                           
4 A sufficient condition for full market coverage is U
L(ql)>(1-F(t))U
t(q(t)) for all t 
where q(.)=(t
e)
-1(.).  This condition implies that serving [t
L,t
H] dominates serving only 
[t,t
H] for any t. 
5 This condition holds under a reverse monotone allocation if U
t(q’(t))> U
L(ql) for all t 
where q’ is defined by G(q’(t)=1-F(t). 
4 ∫ + =
q
q
q t
q l
L
l
pd
dq q U q U q p ) ( ) ( ) (
) ( . 
Taking full derivatives in the consumer first order condition with respect to t implies that 
t
pd(q) is increasing, which together with full market coverage, implies t
pd(q)=t
e(q).  After 
integration, we obtain the revenue under price discrimination  
∫∫⎟
⎠
⎞ ⎜
⎝
⎛ + =
h
ll
e q
q
q
q
x t
q l
L pd dq q g dx x U q U R ) ( ) ( ) (
) (  
and after integration by parts, we get 
∫ − + =
h
l
e q
q
q t
q l
L pd dq q G q U q U R )) ( 1 )( ( ) (
) ( . 
Under assumption A1, the optimal uniform price is   with profits   ∫
h
l
q
q
L L dq q U q ) ( ) ( π
∫ =
h
l
q
q
L L u dq q U q R ) ( ) ( π  
Uniform pricing weakly dominates price discrimination if and only if R
U≥ R
PD
( ) () ( ) 0 ) ( ) ( 1 ) ( ) ( 1
) ( ≥ − − Π − ∫
h
l
e q
q
q t
q
L
q
L dq q U q G q U q .   (2) 
This establishes our main result which we now discuss.  Under a reverse monotone 
allocation rule the lowest type gets the highest good for sure, Π
L(q)=0 for all q<qh, and 
condition (2) becomes  
() () 0 ) ( ) ( 1 ) (
) ( ≥ − − ∫
h
l
e q
q
q t
q
L
q dq q U q G q U  
which is the continuous version of (1).  A sufficient condition for uniform pricing to be 
optimal is 
) (
) (
) ( 1
) ( 1
) (
q U
q U
q G
q
L
q
q t
q
L
e
≥
−
Π −
    for all q∈[ql,qh].    (3) 
Again, this condition is equivalent to condition (1) in the two type case with an inverse 
monotone allocation rule.  In general, it is less likely to hold if there is a large fraction of 
high types (first-order-stochastic-dominance shift in G(.)), if the allocation rule is closer to 
5 reverse monotone (first-order-stochastic-dominance shift in Π
L(.)), and if higher types are 
not willing to pay much more than the lowest type for incremental units of quality so that 
rationing inefficiencies are not too high (the ratio on the right hand side of inequality (3) is 
small).  Any allocation rule away from reverse monotone reduces the chance that uniform 
pricing is optimal.  For example, under a random allocation rule, Π
t(q)=G(q), condition (2) 
is violated, and price discrimination is preferred.    
In a market with no consumer heterogeneity at all, price discrimination and uniform 
pricing are equivalent.  Price discrimination dominates uniform pricing when there much 
consumer heterogeneity (condition (2) is violated).  Therefore, uniform pricing can be 
strictly optimal when there is some consumer heterogeneity but not too much. 
From a welfare point of view, three points deserve mention.  When uniform pricing is 
optimal, the allocation of goods is inefficient.  In fact, mandating the firm to use price 
discrimination would implement the first-best allocation rule.  In addition, uniform pricing 
may generate queuing inefficiencies (which are not captured in the current model).  
Finally, total consumer welfare decreases since overall welfare decreases and firm revenue 
increases.  Some consumers, however, may be better-off.
6
 
3-Conclusions 
The price discrimination literature has overlooked the possibility to deliberately sell 
vertically differentiated products at the same price.  Our analysis proposes a rationale for 
observing ‘unpriced quality’ in environments where price discrimination does not entail 
any additional implementation costs and should, according to standard theory, dominate 
uniform pricing.   
                                                           
6 This will be the case if   where q(.)=(t () )) ( ( )) ( ( ) ( ) ( ) ( ) ( t q p t q U dq q U q q U q
t q
q
L L t t h
l
− > − ∫ π π
e)
-1(.). 
6 We assumed that the set of goods was given and focused on the monopoly revenue 
maximization problem.  The analysis could be extended to endogenous product qualities.  
Clearly, the results hold for cost functions that sufficiently constrain the monopolist’s 
choice of product line, so that the profit maximizing product line satisfies condition (2).  
Whether the analysis extends to general cost functions is an interesting question for future 
research. 
7 References 
Anderson, Eric and James Dana.  (2008). “Integrating Models of Price 
Discrimination.” Mimeo, Northwestern. 
Connolly, Marie and Alan B. Kruger (2006), “Rockonomics: The Economics of 
Popular Music.” Handbook of the Economics of Art and Culture. 
Clerides, Sofronis. (2004). “Price Discrimination with Differentiated Products: 
Definition and Identification.”  Economic Inquiry 42(3), 402-412.  
Einav, Liran and Orbach, Barak. (2007). “Uniform Prices for Differentiated Goods: 
The Case of the Movie-Theater Industry.”  International Review of Law and Economics, 
27(2), 129-153. 
Gilbert, Richard and Paul Klemperer. (2000).  “An equilibrium theory of rationing.”  
RAND Journal of Economics.  31, 1-21. 
Holt, Charles, and Roger Sherman. (1982).  “Waiting-Line Auctions.” Journal of 
Political Economy. 90, 280-94. 
Miravete, Eugenio. (2007).  “The Limited Gains from Complex Tariffs.” CEPR 
Discussion Paper No. 4235.    
Mussa, M., and S. Rosen (1978) “Monopoly and Product Quality,” Journal of 
Economic Theory 18: 301-317.   
Passell, Peter.  “Economic Scene; Disneyland and the old U.S.S.R.: Sharing something 
in common.”  New York Time, April 27, 1995. 
Stole, Lars.  “Price Discrimination and Competition.”  In M. Armstrong and R. Porter: 
Handbook of Industrial Organization, Volume 3. Amsterdam: Elsevier, 2007. 2221-99  
 
8 

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