Price Floors and Quality Choice
Volodymyr Bilotkach 1
Department of Economics, University of California, Irvine
May 2006
Abstract
This paper studies effects of price floors in a simple model of vertical
product differentiation. We find that even non-binding price floors can
increase quality on the market, if the cost of quality is sufficiently low.
Where a binding price floor does not change the equilibrium quality, it
makes consumers worse off. There is also a possibility of overinvestment into quality as a result of the binding minimum price.
JEL Codes: D43, L13, L51
Keywords: Price Floors, Vertical Differentiation, Quality
1
Assistant Professor of Economics, University of California, Irvine. 3151 Social Science Plaza, Irvine,
CA, 92697. phone: (949)-824-5192. e-mail: [email protected]
I. Introduction
When one hears about the price floors, or legal minimum prices, the usual picture that
comes to mind is the supply-demand diagram from a principles textbook, showing that a
price floor, if set above the market-clearing price, will result in quantity supplied
exceeding the quality demanded (this is typically followed by a discussion of minimum
wages). Other uses of the minimum prices, usually not mentioned in the textbooks, relate
to preventing predatory pricing and increasing quality on the market. A good example of
the former is the existence of the sales-below-cost laws, especially in the motor fuel
industry (one recent contribution, providing survey of the relevant literature, is Skidmore
et al., 2005). In a recent study, Gruenewald et al. (2006) advocate setting minimum
prices for alcoholic beverages to decrease manufacturers’ incentives to switch to
production of lower quality product following an increase in tax rates. In fact, Russia and
Ukraine, facing significant illegal production of low quality vodka, tried imposing price
floors with the specific aim of inducing the bootleggers to increase the quality of their
product. This measure, however, generally failed: low quality vodka did not disappear,
while prices went up across the board (Kommersant, 2001).
This paper offers the theoretical analysis of effects of price floors in a model of
vertical product differentiation, where firms choose prices after they have chosen their
location in the quality space, with consumers differentiated by their willingness to pay for
quality. Our analysis yields the following results. First, introducing a price floor either
preserves the textbook maximum differentiation equilibrium (whereby one firm offers a
product of the highest possible quality, and the other one – of the lowest) or destroys it, in
which case the equilibrium involves minimum product differentiation with both firms
offering the highest possible quality. The latter case is more likely, the higher the price
floor relative to the cost of quality. Second, even non-binding price floor (i.e., minimum
price below the equilibrium price charged by the lower quality producer in the absence of
price controls) can lead to the minimum differentiation equilibrium. Third, while nonbinding price floors increase both consumer surplus and total welfare, the same cannot be
said about the binding minimum prices. Where binding minimum prices cause the
minimum product differentiation equilibrium, they can lead to socially inefficient
investment into higher quality. In case maximum product differentiation is preserved
2
despite the binding price floor, consumers are unambiguously worse off than in the
undistorted situation.
The impact of price floor on quality has not received much attention in the
literature. As far as regulation aimed at increasing quality on the market is concerned,
the researchers have devoted a lot of attention to the direct policies, which are the
minimum quality standards (e.g., Besanko et al., 1988, Boom, 1995, Ronnen, 1991).
This is understandable, since direct quality regulation is commonplace. Price regulation
in models with differentiated products has been studied by Kamien and Vincent (1991),
Ma and Burgess (1993), Bhaskar (1997) and Kemnitz and Hemmasi (2003). Kamien and
Vincent and Ma and Burgess offer models of horizontal product differentiation, allowing
the firms to increase attractiveness of their products by investing into quality (consumers
are not differentiated by the willingness to pay for quality, but uniformly prefer higher
quality to lower). In both models, convex cost of quality is an important assumption 2 .
Kamien and Vincent conclude that price floors resulting in the same quality provided by
both firms always yield over-investment into quality, which is different from the finding
of our analysis.
Ma and Burgess essentially study the social planner’s problem of
choosing the welfare-maximizing quality by setting the right two-part tariffs for the firms
to charge. Bhaskar shows that for a Hotelling duopoly with quadratic transportation costs
introducing an otherwise non-binding price floor destroys the maximum differentiation
equilibrium and decreases prices. Such a result obtains because minimum prices allow
for a less fierce price competition. Kemnitz and Hemmasi analyze effects of price
ceilings in a model of vertical product differentiation.
They find that legally set
maximum price increases quality of the product offered by the lower quality producer,
leaving that of the higher quality firm unchanged.
They do not provide welfare
comparisons of the equilibrium with and without the price controls.
While price floors can indeed increase product quality, one should remember that
the price of this can be lower consumer surplus or even total welfare. With minimum
quality standards, it is also possible to make some consumers worse off than in the
‘undistorted’ equilibrium (Ronnen, 1991, however, suggests that with fixed cost of
quality and price competition it is possible to select the appropriate quality standard
2
Without the convex cost, firms will choose to supply products of infinite quality.
3
which will make all consumers unambiguously better off). An interesting topic for
further research, given that both price floors and minimum quality standards can raise
quality; will be comparing the outcomes under these two policy measures. One can argue
that even though quality standards is a more direct measure than price controls if one is to
increase quality, quality standards can in general be more costly to enforce than minimum
prices.
Therefore, if the minimum price leads to the same outcome as the quality
standard and is less costly to enforce, the regulator should definitely consider using it.
The rest of the paper is organized in a straightforward way. Section II describes
the modeling exercise and Section III concludes.
II. Model
2.1 General
Consider a textbook model of vertical product differentiation. There is a continuum of
consumers with heterogeneous preferences towards product quality. Each consumer is
assigned the location x on the [0,1] interval; the distribution of consumers along this
interval is uniform. Consumer’s location represents his/her willingness to pay for quality.
Assume two firms (A and B). The firms choose price ( p A and p B ) after they have
chosen locations (a and b) in the quality space.
Utility of consumer located at point x ∈ [0 ,1] will be defined as follows:
⎧ax − p A
Ux = ⎨
⎩bx − p B
if buys from A
if buys from B
(1)
We have vertical differentiation here: with equal prices, a higher quality good is chosen.
We also do not allow consumers to choose the outside good, so that in equilibrium
market is fully served.
Firms’ cost depends on the quality chosen as follows. Per unit cost equals cx
( c ≤ 0.5 ) for a firm located at point x ∈ [0,1].
This problem is indeed well-known and described in some IO textbooks (e.g.,
Shy, 1995 describes this model for the case of costless production). It is also known that
the location-then-price equilibrium of this game involves maximum differentiation (firms
4
choosing to locate on the opposite ends of the interval). The equilibrium prices will be
given by (assuming without loss of generality that firm A is the lower quality firm):
(1 + c )
= 23 (1 + c )
pA =
pB
1
3
(2)
The market shares are
1
3
(1 + c )
π A = 19 (1 + c )2 and π B =
1
9
(2 − c )2 , and the consumer surplus is:
CS A =
1
3
for firm A and
1
3
(2 − c )
for firm B. The profits are
(1+ c )
∫ − (1 + c )dx = − (1 + c )
1
9
1
3
2
0
∫ (x − (1 + c ))dx = (
1
CS B =
1
3
(1+ c )
2
3
2
x
2
− 23 (1 + c )x
(3)
)(
1
1
3
1+ c )
Adding all up yields the total market welfare equal to
1
18
= 16 c(c − 2 )
(c − 2)(5c − 4) .
Introducing the price floor complicates our problem as it softens the price
competition.
Fierce price competition in the classical model of vertical product
differentiation is a force that rules out minimum product differentiation and pushes the
players to locate as far from each other as possible. With the price floor firms are unable
to set their prices below the minimum specified by the regulation; therefore even the
fiercest price competition can leave marker players with positive profit, if minimum price
is above the cost. Thus, minimum price brings the possibility of an equilibrium providing
for less the maximum product differentiation.
In fact, we can easily establish one important property of the minimum product
differentiation equilibrium.
Proposition 1. If minimum product differentiation is the equilibrium with the price floor
p min > c , then in such an equilibrium firms will choose the highest possible quality, so
that a ∗ = b ∗ = 1 .
Proof: The logic is simple. If in equilibrium a ∗ = b ∗ , both firms will charge minimum
price (this is as fierce as the price competition can get), so we need only establish what
locations they will choose. Suppose both firms are located at point zero, which brings
each of them profit in the amount of half of the minimum price. This is clearly not an
equilibrium; one firm B’s response to this will be to locate marginally above zero and
5
charge minimum price, obtaining the entire market for itself. Firm A can respond by
matching firm B’s location and price, etc. Such quality competition will continue until
both firms reach maximum possible quality level, charging the established minimum
price. This description preserves the spirit of Bertrand-type undercutting, except that
firms increase quality instead of reducing prices. We can also easily establish conditions for the minimum differentiation
equilibrium to yield lower consumer surplus and total welfare as compared to the
undistorted maximum differentiation equilibrium.
Proposition 2. In the minimum differentiation equilibrium as described by Proposition 1,
consumer surplus will be less than without the price controls if p min > 181 (11 − c 2 + 10c ) ;
total welfare will be lower than in the undistorted equilibrium if c > 0.2
Proof: In the minimum differentiation equilibrium, prices will be equal to the minimum
price, firms’ total profit will be p min − c and total consumer surplus is
1
2
− pmin ,
calculated easily as:
1
CS = ∫ ( x − p min )dx = 12 − p min
(4)
0
Comparing (4) to the sum of consumer surpluses implied by (3) we obtain that for
(
)
p min > 181 11 − c 2 + 10c the consumer surplus in the minimum differentiation equilibrium
will be less than that without price controls.
Total welfare in the minimum differentiation equilibrium is equal to
1
2
− c ; that is,
it does not depend on the exact level of minimum price. Subtracting this from the total
welfare we obtained without price controls, we obtain the condition for the minimum
differentiation equilibrium to yield lower total welfare than the maximum differentiation
one without price controls (despite the fact that the former yields higher average quality).
This will be the case if:
1
18
(1 + c )(5c − 1) > 0
(5)
Or if c > 0.2 That is, if cost of quality is sufficiently high, the minimum differentiation
equilibrium will results in the socially inefficient investment into high quality. We still,
however, need to show that minimum differentiation can indeed be the equilibrium here.
6
The analysis will differ somewhat for the cases of binding and non-binding
minimum price. That is, we technically need to consider three cases. If price is not
binding for either the lower quality (firm A) or the higher quality (firm B) player, then
p min ≤
1
3
1
3
(1 + c ) ;
(1 + c ) < p min
p min >
2
3
≤
(1 + c ) .
if the price floor is binding for firm A, but not firm B, then
2
3
(1 + c ) ;
finally, if the minimum price is binding for both firms, then
It will become evident, however, that there are only two cases to
consider here: the analysis only changes when price becomes binding for firm A.
2.2 Non-Binding Minimum Price
Characterization of the equilibria with the non-binding price floor is given by the
following Proposition.
(1 + c ) , then minimum product differentiation as described
2
in Proposition 1 is the equilibrium if p min ≥ c + 92 (1 + c ) . Otherwise, maximum product
Proposition 3. If c < p min ≤
1
3
differentiation with prices given by (2) is the equilibrium.
Proof: As before, firms will choose price after they have chosen their location in the
quality space. Introducing the price floor expands the set of strategies available to the
players. Namely, at the second stage either firm can now choose to charge the regulated
minimum price. As firm A is still considered the ‘lower quality’ firm (so that a ∗ ≤ b ∗ ), it
will also be true that p ∗A ≤ p B∗ ; so that if firm B chooses to charge the minimum price at
the second stage of the game, so will firm A. Also, if firm A chooses the highest quality
at the first stage, so will firm B. The equilibrium in the case where neither firm chooses
tot charge the minimum price has been worked out before. Both firms choosing the price
floor at the second stage implies (by Proposition 1) selecting maximum product quality
and minimum product differentiation equilibrium. Firm A will choose to locate at the
upper end of the interval and charge the minimum price if by doing so it is able to obtain
higher profit than in the textbook minimum differentiation equilibrium. Namely, this will
happen if:
1
2
( pmin − c ) ≥ 19 (1 + c )2
(6)
7
Combining (6) with the requirement that the price floor be non-binding, we obtain the
following condition for the minimum product differentiation equilibrium to exist:
1
3
(1 + c ) ≥ pmin ≥ c + 92 (1 + c )2
(7)
The set of minimum prices for which inequality (7) can be satisfied will be non-empty for
c≤
1
2
(3
)
3 − 5 (slightly less than 0.1). Moreover, with the non-binding price floor firm
A will always combine charging the minimum price with choosing the highest available
quality. To show that this will be so, assume the opposite is true: firm A chooses p min at
the second stage and a ∗ < b ∗ at the first. Firm B’s second-stage best-response function
is:
pˆ B =
1
2
[b(1 + c ) − a + p min ]
(8)
This in turn implies the following first-stage profit function for firm A:
⎡ b − a + c − p min ⎤
⎥
2(b − a )
⎣
⎦
π A = ( p min − ca )⎢
(9)
If a < b , we obtain (after some algebra):
(
)
∂π A − c(b − a ) + (bc − p min )
=
< 0`
2
∂a
2(b − a )
2
2
(10)
This implies that firm A will have incentive to move towards the lower end of the quality
interval. Yet, locating at the lower end, firm A will at the second stage set the price
above the price floor.
Hence, the non-binding minimum price will result in either
minimum differentiation equilibrium with both firms choosing the highest quality (if
inequality (7) is satisfied) or the textbook maximum product differentiation outcome
otherwise. We have established the following. First, even the non-binding minimum price
can lead to an increase in quality in equilibrium, but only if quality is not costly. Second,
since the upper limit of c for which inequality (7) can be satisfied is not more than 0.2, by
Proposition 2 the non-binding price floor will necessarily lead to higher total and
consumer welfare.
Also, firm B will clearly be worse off under the minimum
differentiation equilibrium than in the undistorted one, as it both has to charge lower
price and loses market share.
8
2.3 Binding Minimum Price
Now, suppose p min >
1
3
(1 + c ) ,
so that the price floor is at least above the undistorted
equilibrium price of the low quality firm A. This effectively means that the lower quality
firm A will have to charge the price floor in equilibrium. The following proposition
describes equilibria with the binding price floor.
Proposition 4. If p min > 13 (1 + c ) , minimum product differentiation with both firms
offering the highest quality product and charging the regulated minimum price will be the
equilibrium if p min ≥ 12 c + c(4 + c ) or p min ≥ 1 + c . Otherwise, maximum product
(
)
differentiation is the equilibrium with p ∗A = p min and p B∗ =
1
2
(1 + c + p min ) .
Proof: The case of the binding price floor is easier since this is the price firm A will
charge at the second stage of the game. Further, we have established while proving
Proposition 3 that if firm A chooses at the first stage to locate at a ∗ < b ∗ , it will locate at
the lower end of the quality spectrum. Suppose this is where firm A locates. Firm B’s
best response to such an action will be to locate at the opposite end and set the price
according to:
pˆ B = max{p min , 12 (1 + c + p min )}
(11)
It is easy to see that pˆ B = p min (so that the price floor technically becomes binding for
firm B) if p min ≥ 1 + c . If firm B charges the minimum price, then firm A will match it
and locate at the upper end of the interval. Therefore, if p min ≥ 1 + c , minimum product
differentiation will be the equilibrium.
When choosing its location at the first stage of the game, firm A will compare the
profit from locating at the upper end of the interval (which will be location of firm B) to
that from choosing the lowest possible quality. If firm A locates at the lower end of the
quality spectrum and charges the minimum price; and firm B sets its price above the price
floor according to (11), firm A’s profit will be:
π A = 12 p min (1 + c − p min )
(12)
Then, firm A will choose to offer the highest quality if:
1
2
( p min − c ) ≥ 12 p min (1 + c − p min )
(13)
9
Solving this inequality, we conclude that for the minimum product differentiation to be
the equilibrium in case 1 + c > p min >
p min ≥
1
2
(c +
c(4 + c )
1
3
(1 + c ) the following inequality must be satisfied:
)
(14)
Otherwise, maximum product differentiation will be the equilibrium with p ∗A = p min and
p B∗ =
1
2
(1 + c + p min ) . We have thus established that binding price floor will not necessarily lead to
higher equilibrium quality. For the maximum product differentiation to survive the
imposition of the binding price floor, the price floor needs to be low relative to the cost of
producing the high quality product, which is an intuitive result.
It is also quite
understandable that consumer and total welfare comparisons become more complicated
under the binding minimum price, as discussed in the next sub-section.
At this point, we can offer a graphical generalization of equilibria in presence of
price floors (see Figure 1 below).
Figure 1 Equilibria with Price Floors
10
As can be seen from figure 1, in most cases the non-binding price floor does not change
the textbook maximum differentiation equilibrium. In general, minimum prices are more
likely to lead to increase in equilibrium quality for lower cost of producing the higher
quality good, which is understandable.
2.4 Welfare Analysis
We have quite easily established that the non-binding minimum price yields net welfare
gains as compared to the undistorted equilibrium, if minimum differentiation equilibrium
obtains as a result. The reason for this outcome is simple. Non-binding price floors can
result in the minimum product differentiation equilibrium only where the cost of quality
is sufficiently low; yet, for the minimum differentiation to yield lower total welfare we
require high cost of quality.
Consumers are necessarily better off with minimum
differentiation resulting from the non-binding price floor since they obtain higher quality
products at a lower price.
With the binding price floor, consumers with low valuation of quality can be
worse off in either the minimum or the maximum differentiation equilibrium. In the
former case, they are paying higher price for higher quality, which they do not value a
lot; in the latter they will be paying more for the same quality. When price floor
preserves the maximum product differentiation, all consumers seem to be worse off, since
prices for both the high and the low quality goods increase. Yet, the market share of the
lower quality good is also shrinking with the higher price floor, meaning that more
consumers are getting the high quality good than before. Yet, the price is also higher,
meaning the net outcome is uncertain.
Proposition 2 defines the condition for higher consumer welfare in the minimum
product differentiation equilibrium as compared to the case without the price distortions.
In the case of the binding price floor preserving the maximum product differentiation, the
consumer surplus will be:
Cˆ S =
1
2
(1+ c − pmin )
∫p
0
=
1
8
min
1
dx +
[(c − p
1
2
min
∫ [x − (1 + c + p )]dx =
(1+ c − pmin )
1
2
min
)2 + 1 − 2c − 6 p min ]
11
(15)
Subtracting from (15) the consumer surplus that obtains without the price controls, we
get:
Cˆ S − (CS A + CS B ) =
1
72
(1 + c − 3 pmin )(17 + 5c − 3 pmin )
(16)
Expression (16) will be negative (meaning price controls make consumers worse off) if
1
3
(1 + c ) < p min
<
1
3
(17 + 5c ) ; which comfortably embraces the region where the binding
price floors preserve the maximum product differentiation equilibrium, while increasing
prices. Thus, such consumers will be unambiguously worse off in this situation. Effects
of the binding minimum price on consumer surplus are summarized in the following
figure.
Figure 2 Binding Price Floors and Consumer Surplus
Note: all comparisons are relative to the undistorted equilibrium
While there is a good chance that binding price floors can make consumers worse off, it
is also true that firms’ profit increases as a result of this policy. Again, Proposition 2
describes the total welfare comparison for the minimum product differentiation
equilibrium. This comparison depends only on the cost of quality. The total welfare for
12
the case of maximum product differentiation equilibrium with the binding price floor is
given by:
TW =
1
8
(1 + p min − c )(3 − 3c − p min )
(17)
Subtracting the total welfare without the price controls from (17), gives the expression
1
72
(1 + c − 3 p min )(7c − 5 + 3 p min ) ; which will be non-negative (implying the non-binding
price floor leads to the net welfare gain despite the fact it makes consumers worse off) for
1
3
(1 + c ) ≤ p min
≤
1
3
(5 − 7c )
and negative otherwise. This means that where the binding
minimum price preserves the maximum product differentiation and makes consumers
worse off than in the undistorted equilibrium, it can still lead to total welfare gains, due to
higher profit that firms obtain. Figure 3 gives the graphical representation of effects of
the binding minimum price on total welfare in our model.
Figure 3 Binding Price Floors and Total Welfare
Note: all comparisons are relative to the undistorted equilibrium
13
III. Concluding Comments
Price floors are used by the regulators as an indirect policy instrument aimed at
increasing quality of products offered on the market. Since setting a minimum quality
standard is a more direct way to achieve the goal of higher quality, the literature has paid
much more attention to quality standards than to price regulation. Yet, if it is less costly
to enforce the regulated price than the quality standard, and if price regulation can
achieve the goal, then the regulator should seriously consider using this instrument. We
however do not know what will happen to the classical model of vertical product
differentiation if we plug the minimum price into it.
We know, however, (due to
Kemnitz and Hemmasi, 2003) that a price ceiling in a vertically differentiated duopoly
will increase quality on the market. We also know (Kamien and Vincent, 1991; Ma and
Burgess, 1993) that in the horizontal product differentiation models, where firms are
allowed to invest into quality and consumers have identical preferences towards the
quality but not towards the firms, price regulation can indeed achieve higher quality, but
overinvestment is also possible.
The goal of this paper is to learn about the effects of the price floors in a vertically
differentiated duopoly model. We find that the price floor either preserves the textbook
maximum differentiation equilibrium or destroys it, replacing it with the minimum
product differentiation with both firms offering the highest possible quality. The latter
case is more likely, the higher the price floor relative to the cost of quality. Second, even
non-binding price floor can lead to the minimum differentiation equilibrium. Third,
while non-binding price floors increase both consumer surplus and total welfare, the
same cannot be said about the binding minimum prices. Where binding minimum prices
cause the minimum product differentiation equilibrium, they can lead to socially
inefficient investment into higher quality. In case maximum product differentiation is
preserved despite the binding price floor, consumers are unambiguously worse off than in
the undistorted situation.
We have re-discovered the trade-off between higher quality and higher cost of
providing it. We have shown that price floors can indeed increase the quality and lead to
net welfare gains. So, for a non-empty set of combinations of parameter values, price
floors can accomplish essentially the same goal as minimum quality standards. Future
14
research can look at whether minimum quality standards uniformly accomplish this goal
better than price floors do, or if price regulation can in fact outperform direct quality
regulation in terms of its effect on equilibrium quality, prices, consumer surplus, and total
welfare.
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