Güth, W.

Müller, W.

Potters, J.J.M.

English

Research Papers in Economics

Endogenous preemption on both sides of a market
Werner Gu ¨th
a, Wieland Mu ¨ller
b, Jan Potters
b,*
a Max Planck Institute Jena, Germany
b Tilburg University, The Netherlands
Received 14 August 2005; received in revised form 15 March 2006; accepted 18 April 2006
Available online 7 September 2006
Abstract
We study a market in which both buyers and sellers can decide to preempt and set their quantities before market
clearing. Will this lead to preemption on both sides of the market, only one side of the market, or to no preemption
at all? We find that preemption tends to be asymmetric in the sense that it is restricted to only one side of the
market (buyers or sellers).
D 2006 Elsevier B.V. All rights reserved.
Keywords: Preemption; Endogenous timing
JEL classification: C72; D43; L11
1. Introduction
Starting with Saloner (1987) and Hamilton and Slutsky (1990), there has been a growing literature
that analyzes endogenous timing in oligopolistic markets. Generally, these models allow for endogenous
timing on the supply side of the market only (e.g., Anderson and Engers, 1992; van Damme and
Hurkens, 1999; Matsumura, 1999). In this paper we analyze a simple model that allows for endogenous
timing on both sides of the market. Both buyers and sellers can decide whether or not to preempt. The
main question is what the pattern of preemption will be. Will there be preemption by both sellers and
0165-1765/$ - see front matter D 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.econlet.2006.04.004
* Corresponding author. CentER and Department of Economics, P.O. Box 90153, 5000LE Tilburg, The Netherlands. Tel.: +31
13 466 8204; fax: +31 13 466 3042.
E-mail address: j.j.m.potters@uvt.nl (J. Potters).
Economics Letters 93 (2006) 126–131
www.elsevier.com/locate/econbasebuyers? Will only one side of the market preempt? Will all traders on one side of a market preempt, or
only a subset of traders? Or will there perhaps be no preemption at all?
We consider a homogeneous market with complete information so that there can be only one price.
Quantities demanded and supplied are the decision variables which can be determined earlier or later. In
order to prevent rationing we assume a competitive fringe which ensures market clearing. Each trader
outside the competitive fringe is a flexible trader and can either precommit to a certain quantity (move
early)orrefrainfromdoingso(movelate).Inthelattercase,thetraderjoinsthecompetitivefringeandacts
as a price taker. The assumption that only preempting traders act strategically is not innocuous. It implies
that traders who move early consider how their quantity affects the price, while the other traders do not.
We solve the market equilibrium for given numbers of preempting buyers and sellers and analyze then
the stable configurations of (numbers of) preempting traders. We find that an equilibrium in which both
buyers and sellers preempt exists only if there is at most one flexible trader on each side of the market. In
all other cases, the only equilibrium outcome is for all flexible traders on one side of the market to
precommit and for all traders on the other side of the market to abstain.
2. The market model
Let S, resp. B, denote the set of sellers, resp. buyers on a homogenous market. The number of sellers
(buyers) is denoted by S(B) where S, Bz2. Each seller’s payoff function is given by
p ¼ p  
y
2c

y with cN0
where p denotes the market price and y(z0) the individual sales amount of a given seller. Each buyer’s
payoff function is
u ¼
a
b
 
x
2b
  p

x with a;bN0
where x(z0) is a buyer’s individual demand. These payoff functions imply individual supply functions
y ¼ y  p ðÞ ¼ cp ð1Þ
and individual demand functions
x ¼ x  p ðÞ ¼ a   bp: ð2Þ
To render the analysis tractable we set a=b=c=1.
2.1. The preemption game
Can a non-empty subgroup of traders on each market side gain by precommitting to what they will
trade? We consider a two-stage commitment game with observable delay and two production periods
(Hamilton and Slutsky, 1990). It is assumed that all traders but one on each market side have flexibility
in the timing of production. Thus there are B 1 flexible buyers who can choose to state their demand
early (in period 1) or late (in period 2). Likewise, there are S 1 flexible sellers who can choose to
produce early (in period 1) or late (in period 2). The inflexible traders on each market side represent the
W. Gu ¨th et al. / Economics Letters 93 (2006) 126–131 127competitive fringe which guarantees market clearing. Without loss of generality, we assume that it is
seller S (buyer B) who is inflexible.
The preemption game with observable delay has the following stages:
Stage 0: Flexible sellers and buyers choose the period (period 1 or 2) in which they set their quantities.
Stage 1: Flexible sellers and buyers, who chose period 1, decide about their quantity; others wait.
Stage 2: Flexible traders, who chose period 2, as well as the inflexible traders act as price takers and set
their quantities competitively. The market clears and payoffs are realized.
2.1.1. Solution of stage 2
Assume that sVS 1 sellers and bVB 1 buyers are precommited. Let Y(X) be the sum of quantities
that the s committed sellers (b committed buyers) have chosen in period 1. In period 2 non-committed
players (as well as the two bfringeQ traders) choose their quantities competitively such that the market
clears. Thus, using (1) and (2) it must hold that
Y þ S   s ðÞ p ¼ X þ B   b ðÞ 1   p ðÞ
or
p ¼
B   b þ X   Y
S   s þ B   b
:
2.1.2. Solution of stage 1
Anticipating the results of the second stage, a committing seller’s and buyer’s payoff are
p y ðÞ ¼ p  
y
2

y ¼
Y   X   B þ b
s   S þ b   B
 
y
2

y
ux ðÞ ¼ 1  
x
2
  p

x ¼ 1  
x
2
 
Y   X   B þ b
s   S þ b   B

x
From B
By p ¼ 0 and B
Bx u ¼ 0 as well as from the obvious symmetry of the equilibrium one gets:
1
yc s;b ðÞ ¼
BBþ S   b   s þ 1 ðÞ   b
B þ S þ 1 ðÞ B þ S   b   s þ 1 ðÞ
if sz1
xc s;b ðÞ ¼
SB þ S   b   s þ 1 ðÞ   s
B þ S þ 1 ðÞ B þ S   b   s þ 1 ðÞ
if bz1
and a market price of
p ¼
BB þ S   b   s þ 1 ðÞ   b
B þ S þ 1 ðÞ B þ S   b   s ðÞ
:
The individual sales quantity of a non-committed seller is equal to the price p,o r
ync s;b ðÞ ¼
BBþ S   b   s þ 1 ðÞ   b
B þ S þ 1 ðÞ B þ S   b   s ðÞ
if sVS   2:
1 Note that these payoff functions are strictly concave in y and x, respectively, such that the first-order conditions are sufficient.
W. Gu ¨th et al. / Economics Letters 93 (2006) 126–131 128The individual quantity of a non-committed buyer is
xnc s;b ðÞ ¼
SB þ S   b   s þ 1 ðÞ   s
B þ S þ 1 ðÞ B þ S   b   s ðÞ
if bVB   2:
The total quantity sold and bought is
syc þ S   s ðÞ ync ¼ bxc þ B   b ðÞ xnc
¼
SSþ B   s   b þ 1 ðÞ   s ðÞ BSþ B   s   b þ 1 ðÞ   b ðÞ
S þ B þ 1 ðÞ S þ B   s   b ðÞ S þ B   s   b þ 1 ðÞ
:
A committed seller earns
pc s;b ðÞ ¼
1
2
B þ S   b   s þ 2 ðÞ BBþ S   b   s þ 1 ðÞ   b ðÞ
2
B þ S þ 1 ðÞ
2 B þ S   b   s þ 1 ðÞ
2 B þ S   s   b ðÞ
if sz1
and a non-committed seller
pnc s;b ðÞ ¼
1
2
BBþ S   b   s þ 1 ðÞ   b ðÞ
2
B þ S   s   b ðÞ
2 B þ S þ 1 ðÞ
2 if sVS   2:
A committed buyer earns
uc s;b ðÞ ¼
1
2
SB þ S   b   s þ 1 ðÞ   s ðÞ
2 B þ S   b   s þ 2 ðÞ
B þ S þ 1 ðÞ
2 B þ S   b   s þ 1 ðÞ
2 B þ S   b   s ðÞ
if bz1
and a non-committed buyer
unc s;b ðÞ ¼
1
2
SB þ S   b   s þ 1 ðÞ   s ðÞ
2
B þ S   b   s ðÞ
2 B þ S þ 1 ðÞ
2 if bVB   2:
Note that y
c(s,b)by
nc(s,b) and x
c(s,b)bx
nc(s,b). Due to the assumption of a homogeneous market, this
implies p
nc(s,b)Np
c(s,b) and u
nc(s,b)Nu
c(s,b). Hence, taking s and b as given, both, sellers and buyers,
would prefer to be non-committed. However, when deciding whether or not to precommit, a trader
cannot take s and b as given. If a seller (buyer) decides not to commit s(b) will be reduced by 1. This
simple fact determines the equilibrium values for s and b.
2.2. Precommitment in stage 0
With the help of the results above we can derive the equilibrium numbers b* and s* (with
0Vb*VB 1 and 0Vs*VS 1) of committing buyers and sellers. For an inner equilibrium, that is for
1VsVS 2 and 1VbVB 2 the following four conditions have to be satisfied:
Committed seller: pc s;b ðÞ zpnc s   1;b ðÞ or
1
2
BB  2b ðÞ B þ S   b   s þ 2 ðÞ þ 2b2
B þ S þ 1 ðÞ
2 B þ S   b   s þ 1 ðÞ
2 B þ S   b   s ðÞ
z0
Non committed seller: pnc s;b ðÞ zpc s þ 1;b ðÞ or
 
1
2
BB  2b ðÞ B þ S   b   s þ 1 ðÞ þ 2b2
B þ S þ 1 ðÞ
2 B þ S   b   s ðÞ
2 B þ S   b   s   1 ðÞ
z0
W. Gu ¨th et al. / Economics Letters 93 (2006) 126–131 129Committed buyer: uc s;b ðÞ zunc s;b   1 ðÞ or
1
2
SS  2s ðÞ B þ S   b   s þ 2 ðÞ þ 2s2
B þ S þ 1 ðÞ
2 B þ S   b   s þ 1 ðÞ
2 B þ S   b   s ðÞ
z0
Non committed buyer: unc s;b ðÞ zuc s;b þ 1 ðÞ or
 
1
2
SS  2s ðÞ B þ S   b   s þ 1 ðÞ þ 2s2
B þ S þ 1 ðÞ
2 B þ S   b   s ðÞ
2 B þ S   b   s   1 ðÞ
z0:
Since all denominators are strictly positive for 0VsVS 1 and 0VbVB 1, these four conditions are
equivalent to
Committed seller: BB  2b ðÞ B þ S   b   s þ 2 ðÞ þ 2b2z0 ð3Þ
Non committed seller:   BB  2b ðÞ B þ S   b   s þ 1 ðÞ   2b2z0 ð4Þ
Committed buyer: SS  2s ðÞ B þ S   b   s þ 2 ðÞ þ 2s2z0 ð5Þ
Non committed buyer:   SS  2s ðÞ B þ S   b   s þ 1 ðÞ   2s2z0 ð6Þ
From these conditions we derive
Proposition. The only equilibrium configurations (s*,b *) of the commitment game are:
(i) If S=B=2 then s*=1 and b*=1, i.e., the two flexible traders (one on each side of the
market) precommit.
(ii) If Sz3o rB z3 then [s*=S 1 and b*=0] or [s*=0 and b*=B 1].
Proof. There are nine possible equilibrium configurations, with s*=0, 1Vs*VS 2o rs*=S 1 and
b*=0, 1Vb*VB 2o rb*=B 1. The proof proceeds by checking these configurations. We illustrate
this by checking three. The others follow along similar lines. First, to check whether there is an inner
solution as defined above, note that adding inequalities (3) and (4) as well (5) and (6) yields the
conditions B(B 2b)z0 and S(S 2s)z0. Thus, necessary conditions for an inner solution are bVB/2
and sVS/2. But for these restrictions on s and b it is straightforward that inequalities (4) and (6) cannot
be satisfied. Thus, there is no inner equilibrium. Second, consider the possibility that no flexible trader
precommits (i.e. s=0 and b=0). In this case conditions (4) and (6) have to be satisfied. They reduce to
 (B+S+1)B
2z0 and  (B+S+1) S
2z0. These conditions are never fulfilled. Hence, there is no
equilibrium in which no trader precommits. Finally, consider the possibility that all flexible traders
precommit (i.e., s=S 1 and b=B 1): In this case conditions (3) and (5) have to be satisfied. They
reduce to  2(B
2 2B 1)z0a n d 2(S
2 2S 1)z0 for the committed sellers and buyers
respectively. These conditions will be satisfied simultaneously if and only if SV2 and BV2. 5
The Proposition states that if Sz3o rBz3 then all flexible traders on one side of the market
precommit while no trader on the other side of the market precommits. Traders who preempt set lower
quantities than those who do not preempt. Preempting sellers raise the price; preempting buyers lower
W. Gu ¨th et al. / Economics Letters 93 (2006) 126–131 130the price. The marginal benefit of an effectuated price change decreases with the quantity traded,
however. If many traders on the other side of the market preempt the equilibrium quantity is low which
discourages attempts to change the price by the other side of the market. Thus, preemption on one
market side causes the other side to abstain (and vice versa).
3. Conclusion
We analyze endogenous preemption on both sides of a market and show that preemption tends to be
restricted to one side of the market. Either the buyers or the sellers preempt, but not both sides of the
markets at the same time. Also it is not an equilibrium for no trader to preempt.
To simplify matters we relied on a symmetric model with quadratic utility and cost functions. More
crucial is our assumption that traders who do not preempt join the competitive fringe. This suggests an
alternative interpretation of our model as one that endogenizes the number of strategic traders in a
market. It could be interesting to analyze how results change when flexible traders, who do not preempt,
act strategically rather than competitively.
References
Anderson, S., Engers, M.P., 1992. Stackelberg vs. Cournot oligopoly equilibrium. International Journal of Industrial
Organization 10, 127–135.
Hamilton, J.H., Slutsky, S.M., 1990. Endogenous timing in duopoly games: Stackelberg or Cournot equilibria. Games and
Economic Behavior 2, 29–46.
Matsumura, T., 1999. Quantity setting oligopoly with endogenous sequencing. International Journal of Industrial Organization
17, 289–296.
Saloner, G., 1987. Cournot duopoly with two production periods. Journal of Economic Theory 42, 183–187.
van Damme, E., Hurkens, S., 1999. Endogenous Stackelberg leadership. Games and Economic Behavior 28, 105–129.
W. Gu ¨th et al. / Economics Letters 93 (2006) 126–131 131

Endogenous preemption on both sides of a market.

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