We develop a quality competition model to understand how price controls affect market outcomes in buyer-seller markets with discrete goods of varying quality. While competitive equilibria do not necessarily exist in such markets when price controls are imposed, we show that stable outcomes do exist and characterize the set of stable outcomes in the presence of price restrictions. In particular, we show that price controls induce non-price competition: price floors induce the trade of inefficiently high quality goods, while price ceilings induce the trade of inefficiently low quality goods

Hatfield, John William

Plott, Charles R.

Tanaka, Tomomi

Caltech Authors - Main

DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES
CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CALIFORNIA 91125
PRICE CONTROLS, NON-PRICE QUALITY COMPETITION, AND THE
NONEXISTENCE OF COMPETITIVE EQUILIBRIUM
John William Hatfield
Stanford
Charles R. Plott
California Institute of Technology
Tomomi Tanaka
Arizona State University
1 8 9 1
CA
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SOCIAL SCIENCE WORKING PAPER 1360
February 2012
Understanding Price Controls and Non-Price Competition
with Matching Theory
By JOHN WILLIAM HATFIELD, CHARLES R. PLOTT, AND TOMOMI TANAKA∗
Price controls are one of the most common
forms of market intervention by regulatory au-
thorities. Price setting was a central duty of the
Civil Aeronautics Board until the Airline Dereg-
ulation Act of 1978. The Dairy Price Support
Program sets a minimal price for milk and other
dairy products in the United States. Price caps
for pharmaceutical drugs are a pervasive form
of government intervention in Europe. And
minimum wage laws set a price floor for labor
throughout the developed world.
A great deal of empirical work suggests that
price controls induce quality-based competition
between sellers.1 However, competitive equi-
libria often fail to exist when price controls are
present, and hence theoretical analysis of the ef-
fects of price controls has proven difficult. In
particular, this lack of a theoretical foundation
undermines our ability to understand how price
controls may induce changes in product quality.
As an example, consider the case where there
are two sellers and one buyer, and two quali-
∗ Hatfield: Graduate School of Business, Stanford Uni-
versity, 655 Knight Way, Stanford CA 94305 (e-mail: hat-
field@stanford.edu). Plott: Division of the Humanities
and Social Sciences, California Institute of Technology,
1200 East California Boulevard, Pasadena CA 91125 (email:
cplott@hss.caltech.edu). Tanaka: School of Politics and Global
Studies, Arizona State University, P.O. Box 873902, Tempe,
AZ 85287 (email: tomomi.tanaka@asu.edu). We thank Marcus
Berliant, Clayton Featherstone, Scott Duke Kominers, Muriel
Niederle, Alvin E. Roth, Nilanjan Roy, William Zame and sem-
inar participants at Boston College, the 2009 Lee Center Work-
shop at Caltech, Harvard Business School, and New York Uni-
versity for helpful discussions. The financial support of the Gor-
don and Betty Moore Foundation and the support of the Caltech
Laboratory of Experimental Economics and Political Science are
gratefully acknowledged. Hatfield appreciates the hospitality of
Harvard Business School, and Tanaka appreciates the hospitality
of University of Tokyo, which hosted them during parts of this
research.
1See Plott (1965) for an analysis of non-price competition by
regulated dry cleaners, Douglas and Miller III (1974) for an anal-
ysis of non-price competition by airlines, and Joskow (1980) for
an analysis of non-price competition by hospitals. Hashimoto
(1982) and Neumark and Wascher (2001) investigate whether
minimum wage laws affect job charateristics, particularly on-
the-job training.
ties, H and L. The buyer has unit demand, and
values a high quality good at 10, while valu-
ing the low quality good at 8. Each seller has
unit supply, with the low cost seller having a
cost of production of 1 for a low quality good
and 6 for a high quality good, and the high cost
seller having a cost of production of 2 for a low
quality good and 7 for a high quality good. In
any competitive equilibrium, the low cost seller
sells to the buyer the low quality good at a price
p(L) ∈ [1, 2], with the price of the (untraded)
high quality good being in the interval p(H) ∈
[p(L) + 2, p(L) + 5]. However, if a price floor
of 5 is imposed for all qualities, then no com-
petitive equilibrium exists, as both sellers will
demand to sell since the price of the low quality
good is at least 5.
However, a stable outcome exists regard-
less of the existence of the price floor. In
the case without the price floor, we know that
any stable outcome induces the same allocation
as the competitive equilibrium (Hatfield et al.,
2012b). When the price floor of 5 is present,
our Theroem 3 demonstrates that in any stable
outcome the low cost seller sells the high qual-
ity good to the buyer at a price in the interval
[6, 7]. This example demonstrates that a price
floor may induce quality competition between
sellers, and furthermore may reduce total social
surplus.2
More generally, in this work we consider two-
sided markets with price controls as a special
case of the two-sided matching with contracts
model.3 Hatfield et al. (2012b) showed that,
when price controls are not present, the set
of competitive equilibria naturally correspond
2Indeed, in this example, all three agents are weakly worse
off.
3Hatfield and Milgrom (2005) first developed the match-
ing with contracts model, building on the work of Kelso and
Crawford (1982); the possibility of such a generalization was
first noted in remarks by Crawford and Knoer (1981) and Kelso
and Crawford (1982). Klaus and Walzl (2009) and Hatfield and
Kominers (2012) extended this work to the setting of many-to
many matching with contracts.
1
2 PAPERS AND PROCEEDINGS MONTH YEAR
to the set of stable outcomes. We develop a
model of quality competition and show, using
techniques developed by Echenique and Oviedo
(2006) and Hatfield and Kominers (2012, Forth-
coming), that the set of stable outcomes is non-
empty even when price controls are imposed.
Furthermore, the model exhibits quality-based
competition induced by price controls: for a
price floor, sellers compete by offering ineffi-
ciently high quality goods, while when a price
ceiling is introduced, buyers compete by accept-
ing inefficiently low quality goods.
I. The Quality Competition Model
A. Framework
There is a finite set of unit demand buyers B,
a finite set of unit supply sellers S, and a finite
set of qualities Q, where the set Q is a finite set
of consecutive integers {qmin, . . . , qmax}. The
set of trades is defined as Ω ≡ B × S ×Q. For
each trade (b, s, q) = ω ∈ Ω, we let b(ω) ≡ b,
s(ω) ≡ s, and q(ω) ≡ q. For a set of trades
Ψ ⊆ Ω, we let Ψb ≡ {ψ ∈ Ψ : b = b(ψ)} for
each b ∈ B and Ψs ≡ {ψ ∈ Ψ : s = s(ψ)} for
each s ∈ S. A set of trades Ψ ⊆ Ω is feasible if
each agent is associated with at most one trade,
i.e., |Ψb| ≤ 1 for all b ∈ B and |Ψs| ≤ 1 for all
s ∈ S.
The set of contracts is defined as X ≡ Ω×R.
For a contract x = (ω, pω), we let b(x) ≡ b(ω),
s(x) ≡ s(ω), q(x) ≡ q(ω), and p(x) ≡ pω.
We let Yb ≡ {y ∈ Y : b = b(y)} for each
b ∈ B and Ys ≡ {y ∈ Y : s = s(y)} for
each s ∈ S. A set of contracts Y ⊆ X is an
outcome if each agent is associated with at most
one contract, i.e., |Yb| ≤ 1 for all b ∈ B and
|Ys| ≤ 1 for all s ∈ S.
The utility for a feasible set of trades for a
buyer b ∈ B is given by
ub(Ψ) ≡
{
f b + v(q(ψ)) if ∃ψ ∈ Ψb
0 otherwise.
where f b ∈ R denotes the value b obtains
from procuring a good, while v(q) is a concave
function denoting the additional utility b obtains
from procuring a good of quality q. This induces
a utility function over outcomes for b of
U b(Y ) ≡
{
f b + v(q(y))− p(y) if ∃y ∈ Yb
0 otherwise.
Similarly, the utility for a feasible set of trades
for a seller s is given by
us(Ψ) ≡
{
−cs − e(q(ψ)) if ∃ψ ∈ Ψs
0 otherwise.
where cs ∈ R denotes the cost s incurs from
producing a good, while e(q) is a convex func-
tion denoting the additional cost s incurs from
producing a good of quality q. This induces a
utility function over outcomes for s of
U s(Y ) ≡
{
p(y)− cs − v(q(y)) if ∃y ∈ Ys
0 otherwise.
Finally, for simplicity, we assume that there is
a unique quality qˆ < qmax that maximizes the
surplus v(q) − e(q); note that the assumptions
on the utility function then imply that surplus
is increasing with quality for all qualities below
qˆ, and decreasing with quality for all qualities
above qˆ.
B. Efficiency and Stability
An outcome A is efficient if it solves
A ∈ arg max
Y⊆X
∑
i∈B∪S
U i(Y ).
It is straightforward that for any efficient out-
come A, for each x ∈ A, q(x) = qˆ.
The utility function for each agent induces a
choice correspondence
Ci(Y ) ≡ arg max
Z⊆Yi
U i(Z)
for each i ∈ B ∪ S.
An outcome A ⊆ X is stable if it is
1) Individually rational: for all i ∈ B ∪ S,
Ai ∈ Ci(A).
2) Unblocked: there does not exist a
nonempty blocking set Z ⊆ X such that
a) Z ∩A = ∅, and
VOL. VOL NO. ISSUE UNDERSTANDING PRICE CONTROLS 3
b) for all i ∈ B ∪ S, if Zi 6= ∅, then
Zi ⊆ Y i for all Y i ∈ Ci(Z ∪A).
Stability is closely related to the idea of com-
petitive equilibrium: Intuitively, for an alloca-
tionA to be unblocked, for each contract x /∈ A,
it must be the case that either b(x) or s(x) re-
jects x. If so, for each agent i we can consider
his “budget set” as the union ofAi and the set of
contracts he rejects. Hence, stability insists that
each agent chooses optimally from this “budget
set”, which is analogous to the notion of indi-
vidual maximization from competitive equilib-
rium. Furthermore, market clearing is satisfied,
as a contract x is demanded by the buyer b(x)
if and only if it is demanded by the seller s(x).
Hatfield et al. (2012b) make this notion precise,
showing that a natural correspondence between
competitive equilibria and stable outcomes ex-
ists in our setting when no price controls are im-
posed.
Furthermore, in this setting the set of stable
outcomes is equivalent to the core. This equiv-
alence non longer holds in settings with multi-
unit supply sellers and multi-unit demand buy-
ers.
C. Equilibrium
We begin by considering the case where only
contracts in X(q) ≡ {x ∈ X : q(x) = q} are
available, but no price controls are present. In
that case, A is a stable outcome if and only if
A ∈ arg max
Y⊆X(q)
∑
i∈B∪S
U i(Y );(1)
that is, A maximizes surplus given the contrac-
tual set available.
Furthermore, every contract will trade at the
same price p. The price p must be larger than
pminB (q) ≡ max
b∈Brb(Y )
f b + v(q)
so that each buyer with unmet demand does not
wish to buy a good, and
pminS (q) ≡ max
s∈s(Y )
cs + e(q)
so that each seller currently engaged in
trade wishes to do so; we let pmin(q) ≡
max{pminB (q), pminS (q)}. Analogously, the price
p must be smaller than
pmaxB (q) ≡ min
b∈b(Y )
f b + v(q)
so that each buyer currently engaged in trade
wishes to do so, and
pmaxS (q) ≡ min
s∈Srs(Y )
cs + e(q)
so that each seller not currently producing does
not wish to supply a good; we let pmax(q) ≡
min{pmaxB (q), pmaxS (q)}.
We formalize these results in Theorem 1;
proofs of this and other stated results are a
special case of results found in Hatfield et al.
(2012a).
THEOREM 1: For the contractual set X(q),
a stable outcome exists, any stable outcome A
solves (1), and each contract x ∈ A has p(x) ∈
[pmin(q), pmax(q)].
It is now straightforward to characterize the
set of equilibria for the contractual set X .
THEOREM 2: For the contractual set X , the
set of stable outcomes is the same as that given
in Theorem 1 for the quality qˆ.
Hence, in the absence of price controls, any sta-
ble outcome will be efficient. In particular, only
the efficient quality will trade. To see this, sup-
pose that y ∈ A for some outcome A and that
q(y) 6= qˆ. Now consider x such that b(x) =
b(y), s(x) = s(y), q(x) = qˆ and
p(x) = p(y) + (v(qˆ)− v(q(ψ)))− 
for some small  > 0. Then {x} is a blocking
set, as x will clearly be chosen by b(x), and also
by s(x) since
v(qˆ)− v(q(ψ))−  > c(qˆ)− c(q(ψ))
for  sufficiently small, as qˆ is the unique effi-
cient quality.
We now consider the case where a price floor
of pf is imposed, that is, the set of contracts is
given by X[pf ] ≡ {x ∈ X : p(x) ≥ pf}.
In this case, stable outcomes may no longer be
efficient, as the supply of goods of quality qˆ ex-
ceeds the demand. Hence, quality may adjust
upward to compensate. When the price floor is
just above pmax(qˆ), trade of both the efficient
4 PAPERS AND PROCEEDINGS MONTH YEAR
quality qˆ and the inefficient quality qˆ + 1 may
be observed; the efficient quality good will trade
at the price floor pf , while qˆ + 1 quality goods
will trade at pf+(v(qˆ+1)−v(qˆ)). Trades of the
inefficiently high quality good are not blocked at
this price, as any blocking contract would have
a price lower than the price floor. Finally, when
pf > p
max(qˆ + 1) − (v(qˆ + 1) − v(qˆ)), all
trade at the efficient quality will cease, as sellers
will compete away the gains from trading the ef-
ficient quality good by offering buyers more at-
tractive contracts for the inefficient quality good;
hence, only quality qˆ+1 goods will be traded in
equilibrium.
THEOREM 3: Consider the contractrual set
X[pf ]. A stable outcome exists. There are three
cases:
1) pf < pmin(qˆ): Then any stable outcome is
as in Theorem 2.
2) pmax(qˆ) < pf < pmin(qˆ+1)− [v(qˆ+1)−
v(qˆ)]: Then in any stable outcome A, for
each x ∈ A either
• q(x) = qˆ and p(x) = pf , or
• q(x) = qˆ+1 and p(x) = pf+(v(qˆ+
1)− v(qˆ)).
and the set of buyers engaging in trade is
an element of
arg max
Bˆ⊆B
∑
b∈Bˆ
(
f b + v(qˆ)− pf
) .
3) pmax(qˆ + 1) − [v(qˆ + 1) − v(qˆ)] < pf <
pmin(qˆ+1): Then any stable outcome is as
in Theorem 1 for the contract setX(qˆ+1).
Price floors induce inefficiency in three ways:
First, there is too little trade, since some welfare
enhancing trades require a price below the price
floor to be individually rational. Second, trade
may be at an inefficient quality, since a mutually
advantageous switch to a more efficient quality
requires the price to drop below the price floor.
Finally, the wrong agents may trade, as supply
is greater than demand; a more efficient pro-
ducer can not undercut a less efficient producer
on price if the price is already at the price floor.4
4The results of this theorem can be extended to any price
floor, as well as to price ceilings. See Hatfield et al. (2012a).
II. Conclusion
The model presented here that makes specific
predictions of market equilibrium in settings
with price controls. In Hatfield et al. (2012a),
we have generalized the results here to the case
of multi-unit demand. Moreover, we report on
a series of experiments of continuous time, dou-
ble auction markets with mutiple qualities; for
each experiment, we impose a price control that
corresponds to either part 2 or part 3 of Theo-
rem 3. The experimental results are very close
to the predictions of the theory: When the price
floor falls within the regime of part 2 of Theo-
rem 3, agents trade both qualities, with the ef-
ficient quality good being traded at the price
floor and the inefficiently high quality good be-
ing traded at a price reflecting a buyer’s dif-
ference in valuation between the two qualities.
When the price floor is higher, and falls within
the regime of part 3 of Theorem 3, agents trade
only the high quality good, and do so at a price
as if that quality were the only quality available.
Furthermore, in both cases, the quantity traded
falls within the interval predicted by Theorem 3.
Analogous results are obtained when price ceil-
ings are imposed.
Stability is a robust solution concept which
generalizes the concept of competitive equilib-
rium in settings with transferrable utility. In this
work, we show that stable outcomes exist even
in settings with price controls, while competi-
tive equilibria do not. This suggests that stabil-
ity may be helpful in understanding the effects
of other market interventions. For instance, in
settings with quotas (such as import restrictions
on certain goods), competitive equilibria fail to
exist; however, stable outcomes do exist in such
settings, and hence matching theory allows us
to make sharp predictions.5 We conjecture that
there may be other settings where stability pro-
vides sharp equilibrium predictions even though
competitive equilibria fail to exist.
REFERENCES
Crawford, Vincent P. and Elsie Marie Knoer,
“Job Matching with Heterogeneous Firms and
Workers,” Econometrica, 1981, 49 (2), 437–
450.
5Using matching theory to model settings with quotas was
first suggested by Ostrovsky (2008).
VOL. VOL NO. ISSUE UNDERSTANDING PRICE CONTROLS 5
Douglas, George W. and James C. Miller III,
“Quality competition, industry equilibrium,
and efficiency in the price-constrained air-
line market,” The American Economic Re-
view, 1974, 64 (4), 657–669.
Echenique, Federico and Jorge Oviedo, “A
theory of stability in many-to-many match-
ing markets,” Theoretical Economics, 2006,
1, 233–273.
Hashimoto, Masanori, “Minimum wage ef-
fects on training on the job,” The American
Economic Review Papers and Proceedings,
1982, 72 (5), 1070–1087.
Hatfield, John William and Paul Milgrom,
“Matching with Contracts,” American Eco-
nomic Review, 2005, 95, 913–935.
and Scott Duke Kominers, “Contract De-
sign and Stability in Matching Markets,”
2012. Mimeo, Harvard Business School.
and , “Matching in Networks with Bilat-
eral Contracts,” American Economic Journal:
Microeconomics, Forthcoming.
, Charles R. Plott, and Tomomi Tanaka,
“Price controls, non-price quality competi-
tion, and stable outcomes,” 2012. Mimeo,
California Institute of Technology.
, Scott Duke Kominers, Alexandru Nichi-
for, Michael Ostrovsky, and Alexander
Westkamp, “Stability and Competitive Equi-
librium in Trading Networks,” 2012. Mimeo,
Stanford Graduate School of Business.
Joskow, Paul L., “The effects of competition
and regulation on hospital bed supply and the
reservation quality of the hospital,” The Bell
Journal of Economics, 1980, 11 (2), 421–447.
Kelso, Alexander S. and Vincent P. Craw-
ford, “Job Matching, Coalition Formation,
and Gross Substitutes,” Econometrica, 1982,
50, 1483–1504.
Klaus, Bettina and Markus Walzl, “Stable
Many-to-Many Matchings with Contracts,”
Journal of Mathematical Economics, 2009,
45 (7-8), 422–434.
Neumark, David and William Wascher,
“Minimum wages and training revisited,”
Journal of Labor Economics, 2001, 19 (3),
563–595.
Ostrovsky, Michael, “Stability in supply
chain networks,” American Economic Re-
view, 2008, 98, 897–923.
Plott, Charles R., “Occupational self-
regulation: A case study of the Oklahoma
dry cleaners,” The Journal of Law and
Economics, 1965, 8, 195–222.


Understanding Price Controls and Non-Price Competition with Matching Theory

371
American Economic Review: Papers & Proceedings 2012, 102(3): 371–375
http://dx.doi.org/10.1257/aer.102.3.371
Price controls are one of the most common 
forms of market intervention by regulatory 
authorities. Price setting was a central duty of 
the Civil Aeronautics Board until the Airline 
Deregulation Act of 1978. The Dairy Price 
Support Program sets a minimum price for milk 
and other dairy products in the United States. 
Price caps for pharmaceutical drugs are a perva-
sive form of government intervention in Europe. 
And minimum wage laws set a price floor for 
labor throughout the developed world.
A great deal of empirical work suggests that 
price controls induce quality-based competition 
between sellers.1 However, competitive equi-
libria often fail to exist when price controls are 
present, and hence theoretical analysis of the 
effects of price controls has proven difficult. In 
particular, this lack of a theoretical foundation 
1 See Plott (1965) for an analysis of nonprice competition 
by regulated dry cleaners, Douglas and Miller III (1974) for 
an analysis of nonprice competition by airlines, and Joskow 
(1980) for an analysis of nonprice competition by hospitals. 
Hashimoto (1982) and Neumark and Wascher (2001) inves-
tigate whether minimum wage laws affect job characteris-
tics, particularly on-the-job training. 
Understanding Price Controls and Nonprice Competition 
with Matching Theory †
By John William Hatfield, Charles R. Plott, and Tomomi Tanaka*
* Hatfield: Graduate School of Business, Stanford 
University, 655 Knight Way, Stanford, CA 94305 (e-mail: 
hatfield@stanford.edu); Plott: Division of the Humanities 
and Social Sciences, California Institute of Technology, 1200 
East California Boulevard, Pasadena, CA 91125 (e-mail: 
cplott@hss.caltech.edu); Tanaka: School of Politics and 
Global Studies, Arizona State University, P.O. Box 873902, 
Tempe, AZ 85287 (e-mail: tomomi.tanaka@asu.edu). We 
thank Marcus Berliant, Clayton Featherstone, Scott Duke 
Kominers, Muriel Niederle, Alvin E. Roth, Nilanjan Roy, 
William Zame, and seminar participants at Boston College, 
the 2009 Lee Center Workshop at Caltech, Harvard Business 
School, and New York University for helpful discussions. 
The financial support of the Gordon and Betty Moore 
Foundation and the support of the Caltech Laboratory of 
Experimental Economics and Political Science are grate-
fully acknowledged. Hatfield appreciates the hospitality of 
University of Tokyo, which hosted them during parts of this 
research.
† To view additional materials, visit the article page at 
http://dx.doi.org/10.1257/aer.102.3.371.
undermines our ability to understand how price 
controls may induce changes in product quality.
As an example, consider the case where there 
are two sellers and one buyer, and two quali-
ties, H and L. The buyer has unit demand, and 
values a high quality good at 11, while valu-
ing the low quality good at 8. Each seller has 
unit supply, with the low cost seller having a 
cost of production of 1 for a low quality good 
and 6 for a high quality good, and the high cost 
seller having a cost of production of 2 for a 
low quality good and 7 for a high quality good. 
In any competitive equilibrium, the low cost 
seller sells to the buyer the low quality good 
at a price p(L) ∈ [1, 2], with the price of the (untraded) high quality good being in the inter-
val p(H ) ∈ [ p(L) + 3, p(L) + 5]. However, if a 
price floor of 5 is imposed for all qualities, then 
no competitive equilibrium exists, as both sell-
ers will demand to sell since the price of the low 
quality good is at least 5.
However, a stable outcome exists regardless 
of the existence of the price floor. In the case 
without the price floor, we know that any stable 
outcome induces the same allocation as the com-
petitive equilibrium (Hatfield et al. 2012). When 
the price floor of 5 is present, our Theorem 3 
demonstrates that in any stable outcome the 
low cost seller sells the high quality good to the 
buyer at a price in the interval [6, 7]. This exam-
ple demonstrates that a price floor may induce 
quality competition between sellers, and further-
more may reduce total social surplus.2
More generally, in this work we consider two-
sided markets with price controls as a special 
case of the two-sided matching with contracts 
model.3 Hatfield et al. (2012) showed that, when 
2 Indeed, in this example, all three agents are weakly 
worse off. 
3 Hatfield and Milgrom (2005) first developed the match-
ing with contracts model, building on the work of Kelso and 
Crawford (1982); the possibility of such a  generalization 
Contents
Understanding Price Controls and Nonprice Competition 
with Matching Theory † 371
I. The Quality Competition Model 372
A. Framework 372
B. Efficiency and Equilibrium 372
C. Equilibria of the Quality  
Competition Model 373
II. Conclusion 374
REFERENCES 375
MAY 2012372 AEA PAPERS AND PROCEEDINGS
price controls are not present, the set of com-
petitive equilibria naturally correspond to the 
set of stable outcomes. We develop a model of 
quality competition and show, using techniques 
developed by Echenique and Oviedo (2006) and 
Hatfield and Kominers (2012a, b), that the set of 
stable outcomes is nonempty even when price 
controls are imposed. Furthermore, the model 
exhibits quality-based competition induced by 
price controls: when a price floor is introduced, 
sellers compete by offering inefficiently high 
quality goods, while when a price ceiling is 
introduced, buyers compete by accepting inef-
ficiently low quality goods.
I. The Quality Competition Model
A. Framework
There is a finite set of unit demand buyers B, 
a finite set of unit supply sellers S, and a finite 
set of qualities Q, where the set Q is a finite set 
of consecutive integers { q min  , … ,  q max  }. The set 
of trades is defined as Ω ≡ B × S × Q. For 
each trade (b, s, q) = ω ∈ Ω, we let b(ω) ≡ b, 
s(ω) ≡ s, and q(ω) ≡ q. For a set of trades Ψ ⊆ Ω, we let  Ψ b ≡ {ψ ∈ Ψ : b = b(ψ)} for 
each b ∈ B and  Ψ s ≡ {ψ ∈ Ψ : s = s(ψ)} for 
each s ∈ S. A set of trades Ψ ⊆ Ω is feasible if 
each agent is associated with at most one trade, 
i.e., |  Ψ b | ≤ 1 for all b ∈ B and |  Ψ s | ≤ 1 for all 
s ∈ S.
The set of contracts is defined as X ≡ Ω × ℝ. 
For a contract x = (ω,  p ω ), we let b(x) ≡ b(ω), 
s(x) ≡ s(ω), q(x) ≡ q(ω), and p(x) ≡  p ω . We 
let  Y b ≡ { y ∈ Y : b = b(y)} for each b ∈ B and 
Y s ≡ { y ∈ Y : s = s(y)} for each s ∈ S. A set of 
contracts Y ⊆ X is an outcome if each agent 
is associated with at most one contract, i.e., |  Y b | ≤ 1 for all b ∈ B and |  Y s | ≤ 1 for all s ∈ S.
The utility for a feasible set of trades for a 
buyer b ∈ B is given by
 u b (Ψ) ≡ {    f b + v(q(ψ))   0    if ∃ ψ ∈  Ψ b otherwise,  
where  f b ∈ ℝ denotes the value b obtains from 
procuring a good, while v(q) is a  concave 
was first noted in remarks by Crawford and Knoer (1981) 
and Kelso and Crawford (1982). Klaus and Walzl (2009) 
and Hatfield and Kominers (2012) extended this work to the 
setting of many-to many matching with contracts. 
 function denoting the additional utility b 
obtains from procuring a good of quality q. 
This induces a utility function over feasible 
outcomes for b of
 U b (Y ) ≡ {    f b + v(q(y)) − p(y)  0    if ∃y ∈  Y b  otherwise. 
Similarly, the utility for a feasible set of trades 
for a seller s is given by
 u s (Ψ) ≡ {  − c s − e(q(ψ))   0    if ∃ψ ∈  Ψ s otherwise,  
where  c s ∈ ℝ denotes the cost s incurs from pro-
ducing a good, while e(q) is a convex function 
denoting the additional cost s incurs from pro-
ducing a good of quality q. This induces a utility 
function over feasible outcomes for s of
 U s (Y ) ≡ {  p(y) −  c s − e(q(y))   0    if ∃y ∈  Y s  otherwise. 
Finally, for simplicity, we assume that there 
is a unique quality   q <  q max  that maximizes 
the surplus v(q) − e(q); note that the assump-
tions on the utility function then imply that 
surplus is increasing with quality for all quali-
ties below   q, and decreasing with quality for all 
qualities above   q.
B. Efficiency and Equilibrium
An outcome A is efficient if it solves
A ∈  arg max 
Y⊆X
   ∑ 
i∈B∪S
 
 
   U i (Y ).
It is straightforward that for any efficient out-
come A, for each x ∈ A, q(x) =   q.
The utility function for each agent induces a 
choice correspondence
 C i (Y ) ≡  arg max 
Z⊆ Y i 
  U i (Z )
for each i ∈ B ∪ S.
We define an equilibrium as a stable outcome. 
An outcome A ⊆ X is stable if it is
 (i) Individually rational: for all i ∈ B ∪ S, 
A i ∈  C i (A).
VOL. 102 NO. 3 373UNDERStANDING PRICE CONtROLS AND NON-PRICE COMPEtItION wItH MAtCHING tHEORY
 (ii) Unblocked: there does not exist a non-
empty blocking set Z ⊆ X such that
  a)  Z ∩ A = ∅, and
  b)  for all i ∈ B ∪ S, if  Z i ≠ ∅, then 
 Z i ⊆  Y i for all  Y i ∈  C i (Z ∪ A).
Stability is closely related to the idea of com-
petitive equilibrium: Intuitively, for an alloca-
tion A to be unblocked, for each contract x ∉ A, 
it must be the case that either b(x) or s(x) rejects 
x. If so, for each agent i we can consider his 
“budget set” as the union of  A i and the set of 
contracts he rejects. Hence, stability insists that 
each agent chooses optimally from this “budget 
set,” which is analogous to the notion of individ-
ual maximization from competitive equilibrium. 
Furthermore, market clearing is satisfied, as a 
contract x is demanded by the buyer b(x) if and 
only if it is demanded by the seller s(x). Hatfield 
et al. (2012) make this notion precise, showing 
that a natural correspondence between competi-
tive equilibria and stable outcomes exists in our 
setting when no price controls are imposed.
Furthermore, in this setting the set of stable 
outcomes is equivalent to the core. This equiva-
lence no longer holds in settings with multiunit 
supply sellers and multiunit demand buyers.
C. Equilibria of the Quality  
Competition Model
We begin by considering the case where only 
contracts in X(q) ≡ {x ∈ X : q(x) = q} are avail-
able, but no price controls are present. In that 
case, A is a stable outcome if and only if
(1) A ∈  arg max 
Y⊆X(q)
   ∑ 
i∈B∪S
 
 
   U i (Y );
that is, A maximizes surplus given the contrac-
tual set available.
Furthermore, every contract will trade at the 
same price p. The price p must be no less than
  p B min  (q) ≡  max  
b∈B∖b(Y )  f b + v(q)
so that each buyer with unmet demand does not 
wish to buy a good, and
  p S min  (q) ≡  max 
s∈s(Y )  c s + e(q)
so that each seller currently engaged 
in trade wishes to do so; we let  p min  (q) ≡ max{  p B min  (q),  p S min  (q)}. Analogously, the price 
p must be no greater than
  p B max  (q) ≡  min 
b∈b(Y )  f b + v(q)
so that each buyer currently engaged in trade 
wishes to do so, and
  p S max  (q) ≡  min  
s∈S∖s(Y )  c s + e(q)
so that each seller not currently producing 
does not wish to supply a good; we let  p max  (q) ≡ min{  p B max  (q),  p S max  (q)}.
We formalize these results in Theorem 1; 
proofs of this and other stated results are a 
special case of results found in Hatfield et al. (2011).
THEOREM 1: For the contractual set X(q), 
there exists at least one stable outcome A; for 
any such A, (1) holds and moreover for each 
x ∈ A, p(x) ∈ [  p min (q),  p max (q)].
It is now straightforward to characterize the 
set of equilibria for the contractual set X.
THEOREM 2: For the contractual set X, the set 
of stable outcomes is the same as that given in 
theorem 1 for the quality   q.
Hence, in the absence of price controls, any 
stable outcome will be efficient. In particular, 
only the efficient quality will trade. To see this, 
suppose that y ∈ A for some outcome A and that 
q(y) ≠   q. Now consider x such that b(x) = b(y), 
s(x) = s(y), q(x) =   q, and
 p(x) = p(y) + (v(  q) − v(q( y ))) − ϵ
for some small ϵ > 0. Then {x} is a blocking set, 
as x will clearly be chosen by b(x), and also by 
s(x) since
 v(  q) − v(q(y )) − ϵ > c(  q) − c(q(y))
for ϵ sufficiently small, as   q is the unique effi-
cient quality.
We now consider the case where a price floor 
of  p f is imposed; that is, the set of contracts is 
MAY 2012374 AEA PAPERS AND PROCEEDINGS
given by X[  p f ] ≡ {x ∈ X : p(x) ≥  p f }. In this 
case, stable outcomes may no longer be efficient, 
as the supply of goods of quality   q exceeds the 
demand. Hence, quality may adjust upward to 
compensate. When the price floor is just above 
p max  (  q), trade of both the efficient quality   q and 
the inefficient quality   q + 1 may be observed; 
the efficient quality good will trade at the price 
floor  p f , while   q + 1 quality goods will trade 
at  p f + (v(  q + 1) − v(  q)). Trades of the inef-
ficiently high quality good are not blocked 
at this price, as any blocking contract would 
have a price lower than the price floor. Finally, 
as the price floor  p f rises above  p max  (  q + 1) − (v(  q + 1) − v(  q)), we reach another tipping 
point, as all trade at the efficient quality ceases. 
At such high price floors, sellers compete away 
the gains from trading the efficient quality good 
by offering buyers more attractive contracts for 
the inefficient quality good; hence, only qual-
ity   q + 1 goods will be traded in equilibrium.
THEOREM 3: Consider the contractual set 
X[  p f ]. A stable outcome exists. there are three 
cases:
 (i)  p f <  p min  (  q) : then any stable outcome 
is as in theorem 2.
 (ii)  p max  (  q) <  p f <  p min  (  q + 1) − [v(  q + 
1) − v(  q)] : then in any stable outcome 
A, for each x ∈ A either
  • q(x) =   q and p(x) =  p f , or
  •   q(x) =   q + 1 and p(x) =   p f + (v(  q + 1) − v(  q)),
  and the set of buyers engaging in trade is 
an element of
  arg max 
  B ⊆B
 {  ∑ 
b∈  B 
 
  (   f b + v(  q) −  p f )} .
 (iii)  p max  (  q + 1) − [v(  q + 1) − v(  q)] <  p f <  p min  (  q + 1) : Then any stable out-
come is as in Theorem 1 for the contract 
set X(  q + 1).
Price floors induce inefficiency in three ways: 
First, there is too little trade, since some welfare 
enhancing trades require a price below the price 
floor to be individually rational. Second, trade 
may be at an inefficient quality, since a mutually 
advantageous switch to a more efficient quality 
requires the price to drop below the price floor. 
Finally, the wrong agents may trade, as supply is 
greater than demand; a more efficient producer 
can not undercut a less efficient producer on 
price if the price is already at the price floor.4
II. Conclusion
The model presented here makes specific 
predictions of market equilibrium in settings 
with price controls. The results of our earlier 
work Hatfield, Plott, and Tanaka (2011) gen-
eralize the results here to the case of multiunit 
demand. Moreover, in that work, we report on 
a series of experiments of continuous time, 
double auction markets with multiple qualities; 
for each experiment, we impose a price control 
that corresponds to either part (ii) or part (iii) 
of Theorem 3. The experimental results are very 
close to the predictions of the theory: When the 
price floor falls within the regime of part (ii) of 
Theorem 3, agents trade both qualities, with the 
efficient quality good being traded at the price 
floor and the inefficiently high quality good 
being traded at a price reflecting a buyer’s dif-
ference in valuation between the two qualities. 
When the price floor is higher, and falls within 
the regime of part (iii) of Theorem 3, agents 
trade only the high quality good and do so at 
a price as if that quality were the only quality 
available. Furthermore, in both cases, the quan-
tity traded falls within the interval predicted 
by Theorem 3. Analogous results are obtained 
when price ceilings are imposed.
Stability is a robust solution concept which 
generalizes the concept of competitive equilib-
rium in settings with transferable utility. In this 
work, we show that stable outcomes exist even 
in settings with price controls, while competi-
tive equilibria do not. This suggests that stabil-
ity may be helpful in understanding the effects 
of other market interventions. For instance, in 
settings with quotas (such as import restrictions 
on certain goods), competitive equilibria fail to 
exist; however, stable outcomes do exist in such 
settings, and, hence, matching theory allows us 
4 The results of this theorem can be extended to any price 
floor, as well as to price ceilings. See Hatfield, Plott, and 
Tanaka (2011). 
VOL. 102 NO. 3 375UNDERStANDING PRICE CONtROLS AND NON-PRICE COMPEtItION wItH MAtCHING tHEORY
to make sharp predictions.5 We conjecture that 
there may be other settings where stability pro-
vides sharp equilibrium predictions even though 
competitive equilibria fail to exist.
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Understanding Price Controls and Non-Price Competition with Matching Theory

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