1. No category

A note on proportional rationing in a PQ duopoly
Jan Zouhar1
Abstract. In a PQ oligopoly, firms pick prices and quantities simultaneously,
and unlike with the traditional Cournot and Bertrand models, market clearing
is not imposed. It is thus necessary to specify the rules of how demand is assigned to individual firms when the market does not clear, known as rationing
rules. One of the popular approaches is proportional rationing, which is justified using a notion of randomly ordered consumers coming to the market. Such
a process would render the resulting firm-specific demand a random variable,
which is ignored in the existing deterministic models. In the paper, we (a) formalize the notion of randomly-ordered consumers into a new stochastic version
of the proportional rationing scheme, (b) derive the probabilistic properties of
firm-specific demand under this new scheme, and (c) study whether the results
of the stochastic and deterministic versions are consistent.
Keywords: oligopoly, homogenous production, proportional rationing, PQ
duopoly.
JEL classification: D43
AMS classification: 91B24
1
Introduction
Since the early attempts at modelling the oligopoly market by Cournot and Bertrand, economists have
been aware of the fact that solutions to oligopoly models crucially depend on the strategic variables that
oligopolists are assumed to decide about. Even though in many real-life settings either price or quantity
can be identified as the sole strategic variable, in many other instances one has to admit that firms do
in fact decide about both prices and quantities simultaneously. This is the situation that is studied in
the framework of a PQ oligopoly, which is perhaps the most natural extension of the models studied by
Cournot and Bertrand. Just like the Cournot and Bertrand oligopolies, a PQ oligopoly is a static one-shot
non-cooperative game with perfect information, where firms sell homogenous production to price-taking
consumers and try to maximize individual profits. In a PQ oligopoly, however, firms pick their prices
and quantities simultaneously, which can lead to over-supplying or under-supplying the market. This is
a significant distinction from Cournot and Bertrand, where markets always clear.
Despite the PQ oligopoly being a straightforward extension to the traditional oligopoly models, other
concepts of incorporating both decision variables, seemingly more complicated ones, had preceded the PQ
oligopoly in the academic literature – e.g. the extensive-form vesion of the price-quantity game of Kreps
and Scheinkman [7] or the model of Davidson and Deneckere [3] that involves capacity constraints. The
reason for this, presumably, is that it is fairly difficult to obtain equilibrium results in the PQ oligopoly:
as Friedman [5] showed, the PQ oligopoly game does not have any pure-strategy equilibria under fairly
general conditions. The first characterization of the mixed-strategy Nash equilibrium (NE) in a PQ
oligopoly was discovered by Gertner [6], albeit for the duopoly case only. His work remained unpublished
and this line of research was largely abandoned by active scholars in the field in late 1980s; however, in
recent years, several researchers have taken up this research path again – see e.g. [8],[2], where a version
of Gertner’s model with discretized variables is analyzed.2
In a PQ oligopoly, as well as in any other oligopoly model that does not impose market clearance,
one has to carefully state the assumptions about rationing, which refers to the process that determines
how firm-specific demand functions are derived from the industry demand. Note that these assumptions
are a part of Cournot and Bertrand models as well. For instance, in the classical Bertrand model, the
unique lowest-priced firm covers the whole industry demand at its price, leaving nothing for the remaining
1 Univ.
2 In
of Economics, Prague, Dept. of Econometrics, nám. W. Churchilla 4, 130 67 Praha 3, [email protected]
fact, as far as I know, the term “PQ oligopoly” was coined by the author of the former of the two papers.
oligopolists, and in case of equal lowest prices, the corresponding industry demand is split evenly among
the lowest-priced firms.3 Nonetheless, rationing assumptions are especially important in case the market
does not clear: if the lowest-priced firm does not produce the entire industry demand at that price, some
of the consumers that were not served are likely to purchase the good from one of the higher-priced firms,
thus creating the spillover (or residual) demand.
Most researchers use one of the two popular rationing schemes known as parallel rationing and proportional rationing. As the title suggests, this paper focuses on the latter. For two firms with different
prices, proportional rationing works as follows: if the low-priced firm supplies only a fraction φ of the
industry demand at the low price, the high-priced firm faces a demand of (1 − φ) times the industry
demand at its price. Gertner [6, p. 62] and Binmore [1, p. 308] mention that this is consistent with a
demand function that represents a large number of consumers coming to the market in a random order,
each only interested in buying a single unit of the good and willing to pay anything less than or equal
to their acceptance threshold. It is worth noting that this process automatically renders firm-specific
demand a random variable – a fact that has been ignored in the existing research into NE outcomes of
PQ duopolies so far.
The aim of this paper is to (a) formalize the notion of randomly-ordered consumers into a new version
of the proportional rationing scheme, (b) derive the probabilistic properties of firm-specific demand under
this new scheme, and (c) analyze whether the explicit incorporation of randomness in the model changes
the current results obtained from the deterministic version of proportionate rationing.
2
The model
We consider a static one-shot non-cooperative game in a duopoly with homogeneous production. The
duopolists, firms 1 and 2, simultaneously pick prices and quantities, and both are trying to maximize
their individual profit. In the sequel, we use i, j to denote either 1, 2 or 2, 1. The profit of firm i can be
expressed as
πi = pi min{qi , di } − γi (qi ),
(1)
where pi , qi are the price and production quantity picked by firm i, di is the quantity demanded of firm
i’s goods, and γi is firm i’s cost function. Note that total cost depends solely on qi , implying that free
disposal of unsold goods is assumed. A strategy consists in specifying both price and production quantity.
We allow for mixed strategies – thus, formally, a strategy of firm i is a random vector4 (Pi , Qi ). Firmspecific demand, eventually realized as di , depends on the following: (a) both firms’ prices and quantities,
(b) the industry demand function, and (c) the rationing scheme.
Industry demand. Industry demand is defined as follows. We assume there are n consumers, each
willing to buy 1 unit of production in case the price does not exceed his/her acceptance threshold, denoted
t. Therefore,
δ(p) = the number of consumers with t at or above p =
n
X
1[p,∞) (tc ),
(2)
c=1
where 1 is the indicator function. Depending on the distribution of acceptance thresholds in the population of consumers, the industry function defined in this way can take on all sorts of shapes. For instance,
if the distribution is uniform over a certain interval, the resulting industry demand is approximately
linear; if it is positively skewed, industry demand resembles a convex function.5
Rationing scheme. The rationing scheme defines the process of how possible demand spillovers are
created. The n consumers come to the market in a random order. Each consumer c decides whether and
from whom to demand the good according to the following rules:
(i) if no price is less than or equal to c’s acceptance threshold, c leaves without purchase;
3 The inelegant property of this rationing scheme is the inherent discontinuity of demand in points where prices are equal.
This motivates the use of models of differentiated oligopoly, originally conceived by Dixit [4] and later used in several papers
related to the PQ oligopoly, such as [3] and [5].
4 Throughout the text, we use capitals to denote random variables, and their lower-case counterparts to indicate specific
values.
5 Obviously, the industry demand from (2) is a step function; as such, it can be neither linear nor convex. However, its
smoothed version may have these properties.
(ii) c demands the good from the firm with the lowest price; if there are multiple lowest prices, he/she
picks among them at random with equal probabilities,
(iii) if the selected firm has nothing left to sell, c leaves this firm out of further consideration and proceeds
again from step (i).
After consumer c leaves, either with or without a purchased good, the next consumer turns up and
proceeds in an analogous fashion. Once all n consumers have left, the quantity demanded of firm i, di , is
determined as the total number of consumers that have come and demanded the firm’s goods (no matter
whether they were served or not). It is worth noting that the rationing process defined above is not
restricted to a duopoly model; it is applicable to oligopolies with an arbitrary number of firms.
3
Probabilistic properties of firm-specific demand
Due to the random order of consumers in the rationing scheme, the eventual firm-specific demand di is
an outcome of a random variable, Di . If prices are different and the low-priced firm i covers the entire
industry demand at its price, then Di = δ(pi ) and Dj = 0 with probability 1. In case the low-priced
firm produces less then the entire industry demand, it sells out all its production, and with a non-zero
probability some of the unserved customers will recalculate their plans and demand the good from the
high-priced firm. In other words, the spillover effect is likely to set in, and the high-priced firm faces
residual demand that is positive with a non-zero probability. The actual probability distribution of this
residual demand is given in Proposition 1.
Proposition 1. If pi > pj and qj < δ(pj ), residual demand for firm i’s goods follows the hypergeometric
distribution:
Di ∼ Hypergeometric(δ(pj ), δ(pi ), δ(pj ) − qj ),
(3)
meaning that
P{Di = di } =
δ(pi )
di
δ(pj )−δ(pi ) δ(pj )−qj −di
,
δ(pj ) δ(pj )−qj
and
EDi =
δ(pj ) − qj
δ(pi ).
δ(pj )
(4)
Proof. At price pj , there are δ(pj ) consumers willing to buy from firm j (denote this set of consumers as
A), and δ(pi ) of them are willing to accept price pi as well (set B). Consumers are coming at random
order, first of qj them buying from firm j. The residual demand is formed by those out of the remaining
δ(pj )−qj consumers who belong to B. As all orderings of consumers are equally likely, the situation is akin
to making δ(pj ) − qj random draws without replacement from population A [of size δ(pj )] containing a
subset of successes B [of size δ(pi )]. The number of successes has thus the distribution posited above.
In case prices are equal, the probability distribution of Di is more complicated to derive. The reason
is that both kinds of spillover effects can be present, from firm i to firm j and the other way around,
depending on the results of the random selection process of individual buyers [see rule (ii) above]. In
Proposition 2, we derive the characterization of the support of the joint distribution of Di and Dj , given
the prices, production quantities and industry demand. (It is worth noting that the results in Proposition
2 do not hinge on that the buyers pick evenly-priced firms with equal probabilities; the result holds as
long as the probabilities are non-zero for both firms, see the proof of the proposition below.)
Proposition 2. If pi = pj = p, the support of the distribution of (Di , Dj ) is made up by all integral
points lying in set S defined as the set of all 2-tuples (di , dj ) ∈ [0, δ(p)]2 that satisfy the inequalities
di ≥ δ(p) − qj ,
(5)
dj ≥ δ(p) − qi ,
(6)
di + dj ≥ δ(p),
(7)
with the further requirement that at least one of these inequalities is tight in each point of S.
Proof. Firstly, note that if there are δ(p) buyers willing to buy at price p, firm-specific demand has to
be an integer between 0 and δ(p). Next, if qj < δ(p) and more than qj customers will first come to buy
from firm j, then all coming after the qj th one will be turned down, and will subsequently demand the
good from firm i as well. Thus, (5) has to hold. Swapping i and j in the previous argument gives (6).
The requirement in (7) is straightforward – there are δ(p) buyers, and each will demand the good from
i, j or both.
Finally, we have to prove that at least one of the inequalities has to be tight in any (di , dj ) having a
positive probability. If di + dj > δ(p), some buyers have demanded the good from both firms, i.e. one
or both firms must have run out of supplies. Moreover, one of them must have been the first to run
out supplies; if this was firm i, then necessarily dj = δ(p) − qi , because all consumers but the qi that
purchased the good from firm i demanded the good from firm j (either directly or through the spillover
effect). Analogously, if firm j is the first to run out of supplies, di = δ(p) − qj . In other words, if (7) is
strict, one of (5),(6) has to be tight, which completes the proof.
In Proposition 3, we derive the actual probability mass function of the joint distribution of Di , Dj .
Proposition 3. If pi = pj = p, the joint distribution of Di , Dj is given by a probability mass function
that can be expressed as
P{(Di , Dj ) = (di , dj )} =


−δ(p) δ(p)

2



di





δ(p)−di 

X
k + qi − 1 min{qj , δ(p) − qi )} − k

−
min{δ(p),q
+q
}

i
j
2

k
δ(p) − di − k
k=0
=



δ(p)−dj 
X

k + qj − 1 min{qi , δ(p) − qj )} − k

− min{δ(p),qi +qj }

2



k
δ(p) − dj − k

k=0





0
if di + dj = δ(p),
if dj = δ(p) − qi ,
(8)
if di = δ(p) − qj ,
otherwise.
for any (di , dj ) ∈ {0, 1, . . . , δ(p)}2 .
Proof. First note that conditions in the first three cases of (8) correspond to (5)–(7); it is thus easily
verified that (8) gives a non-zero probability for all points in the support (derived in Proposition 2), and
zero otherwise.
At price p, there are δ(p) consumers coming and demanding the goods from firms i and j. It is useful
to define
(
1 if cth consumer decides to go to firm i first,
σc =
0 if cth consumer decides to go to firm j first,
and σ = (σ1 , . . . , σδ(p) ). As the inital decisions made by individual consumers are independent, an
arbitrary sequence σ happens with a probability of 2−δ(p) . Next, we verify the correctness of the formulas
in (8) for the different cases:
Case di + dj = δ(p). This condition requires that no consumers are turned down by either firm, i.e. there
are no spillover effects. Therefore, any sequence σ with di 1s and [δ(p) − di ] 0s produces the same
outcome, and there are δ(p)
such sequences.
di
Case dj = δ(p) − qi . This condition implies that i was the first firm to have run out of supplies (or,
more precisely, no consumers purchased from i through the spillover effect). The corresponding σ
sequences have no more than qj 0s preceding the qi th 1. The number of different sequences that have
k 0s before the qi th 1 is given by
k + qi − 1 δ(p)−qi −k
.
2
k
| {z }
the last δ(p) − qi − k elements are irrelevant
However, we also require that quantity demanded from i equals di ; this includes both the direct and
the spillover demand. If di = δ(p) − x, it means that there are x consumers who both (a) decided
to go to firm j first and (b) were not turned down (i.e. they indeed purchased the good). In case
qi + qj ≥ δ(p), this corresponds with sequences having x 0s in total; if qi + qj < δ(p), this corresponds
with x 0s in the first qi + qj elements of the sequence, since after qi + qj consumers have come in,
firm j will have sold out all its production. Therefore, the number of suitable sequences that have k
0s before the qi th 1 and produce the firm-specific demands (di , dj ) is
x − k 0s in the first qi + qj elements and after qi th 1
k + qi − 1
k
{z
}
|
z
k 0s before the qi th 1
}|
{
min{δ(p), qi + qj } − qi − k max{0,δ(p)−qi −qj }
2
.
x−k
|
{z
}
the last δ(p) − qi − qj elements are irrelevant
Summation across all k ≤ x and a slight rearrangement gives the formula in (8).
Case dj = δ(p) − qi . The result is obtained easily from symmetry; it suffices to swap the i and j subscripts in the previous argument.
4
Does randomness matter?
In this section, we demonstrate that the traditional deterministic model of proportional rationing is not
fully consistent, in terms of the expected equilibrium outcomes, with the random-ordering model – despite
the notion of randomly ordered customers being used as the justification for the former. As a benchmark,
we use Gertner’s [6] model. Gertner defines firm-specific demand as follows:

δ(pi )




 δ(p ) − min{q , d }
j
j
j
δ(pi )
di =

δ(pj )




max {δ(pi )/2, δ(pi ) − qj }
if pi < pj ,
if pi > pj ,
(9)
if pi = pj .
Gertner’s findings for his model are as follows. There is no equilibrium in pure strategies. A mixedstrategy NE exists, and its characterization differs with varying returns to scale. With constant or
decreasing marginal cost, firms produce the entire industry demand at their price, the price has a continuous distribution on a certain interval, and expected profits are zero. With increasing marginal cost,
the equilibrium distribution of prices has a mass point at the upper support, where the firm produces
half the industry demand, and is atomless below this point, where the firm produces more than a half,
but less than the entire industry demand. (For details, see [6].)
Ideally, if the Gertner’s model and our model from §2 are consistent, the expected profit πi is the
same when calculated both by plugging (9) into the profit formula (1), and by taking the expectation of
πi with respect to the firm-specific demand, for any combination of prices and quantities. However, as
we argue below, this may not be the case in general.
Firstly, it is obvious that the definition of di in (9) is supposed to give the expected value of firmspecific demand; this is the case for pi > pj , as can easily be verified by comparison with (4), but not
for pi = pj , as can be easily seen from Proposition 2. This is not as serious a problem in Gertner’s
treatment, as the equilibrium price distributions are mostly atomless, and therefore P{Pi = Pj } is zero.
(With increasing marginal cost, the price distribution has a mass point, but the corresponding quantities
are such that there is no spillover demand.) It might, however, change the results significantly in case we
switch from continuous prices and quantities to discrete ones, as in [8] and [2].
Secondly, even if the formula for the case pi = pj is replaced with EDi , the resulting profit will in
general be different from the expectation of the profit in the model where Di is treated as random. To
see why, note that in the profit equation sales are calculated as the minimum of production and demand.
Broadly speaking, it matters whether the expectation is taken before or after the minimum operator.
More precisely, it is not difficult to show that
E min{qi , Di } < min{qi , EDi } ⇔ P{Di < qi } > 0 and P{Di > qi } > 0,
(10)
which implies that using (9) in the profit formula will underestimate the expected profit whenever the
probabilities on the right in (10) are non-zero. It is thus natural to ask whether these probabilities are
in fact non-zero in the region of interest, i.e. near the equilibrium combinatoin of strategies. Gertner’s
equilibrium results for increasing marginal cost suggest the answer is positive: the NE prices and quantities
[in any realization of the NE mixed strategy (Pi , Qi )] always satisfy δ(pi )/2 ≤ qi < δ(pi ) in his model.
This automatically guarantees that if firm i is the high-priced one, P{Di < qi } > 0 (since Di can be as
low as δ(pi ) − qj < δ(pi )/2 ≤ qi ). Moreover, pi = δ(pi )/2 at the upper support of the price distribution,
which makes it likely that P{Di > qi } may indeed be non-zero.6
5
Conclusions
In this paper, we proposed a stochastic rationing scheme in PQ oligopolies based on a notion of randomly
ordered consumers with varying acceptance thresholds, which is often used as a justification for the wellknown deterministic proportional rationing scheme. First, we derive the probabilistic properties of our
rationing scheme for the duopoly case. Second, we show that expected sales and profits determined from
our stochastic model are not fully consistent with those obtained from the deterministic proportionate
rationing scheme, suggesting that the justification for deterministic proportional rationing is not quite
correct. Whether or not this substantially changes equilibrium results remains an open question that
should be addressed in future research.
Acknowledgements
I would like to thank Harold Houba, who took charge of my fellowship at VU University Amsterdam, for
his hospitality and inspiring ideas; it was one of the many fruitful discussions we had in the corridors of
VU that drew my attention to rationing schemes in PQ oligopolies. This research was supported by the
Grant Agency of the Czech Republic, project no. GA402/12/P241.
References
[1] Binmore K.: Playing for Real: A Text on Game Theory. Oxford University Press, New York, 2007.
[2] Cracau D., Franz B.: Mixed strategies in PQ-duopolies, in: Proceedings of the 19th International
Congress on Modelling and Simulation, Perth, 2011, 1414–1420.
[3] Davidson C., Deneckere, R.: Long-run competition in capacity, short-run competition in price, and
the Cournot model, Rand Journal of Economics, 17(3) (1986), 404–415.
[4] Dixit A. K.: A Model of duopoly suggesting a theory of entry barriers, Bell Journal of Economics,
Vol. 10(1) (1979), 20–32.
[5] Friedman J. W.: On the strategic importance of prices versus quantities, Rand Journal of Economics,
19(4) (1988), 607–622.
[6] Gertner R.: Essays in Theoretical Industrial Organization. Ph.D. thesis, Massachusetts Institute of
Technology, Dept. of Economics, 1986. URL: http://dspace.mit.edu/handle/1721.1/14892
[7] Kreps D., Scheinkman J.: Quantity precommitment and Bertrand competition yield Cournot outcomes, Bell Journal of Economics, 14(2) (1983), 326–338.
[8] McCulloch, H.: PQ-Nash duopoly: A computational characterization, submitted, 2011. URL:
http://www.econ.ohio-state.edu/jhm/papers/PQNash.pdf
6 Note however that analytic solution to Gertner’s problem is not known, and numeric experiments that would verify
this are not easily carried out; e.g., it is neccessary to turn the problem into a discrete-price one to run the simulation, and
even for a relatively coarse price scale, the computational burden is huge, see [8].