stackelberg leadership with demand uncertainty

Copyright # 2005 John Wiley & Sons, Ltd.

MANAGERIAL AND DECISION ECONOMICS

Manage. Decis. Econ. 26: 345–350 (2005)

Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/mde.1226

Stackelberg Leadership withDemand Uncertainty

Zhiyong Liu*

Kellogg School of Management, Northwestern University, USA

We consider a simple Stackelberg model with demand uncertainty only for the first mover in

order to compare the advantages of leadership and flexibility, and use an example to provide

some discussion about the endogenous order of moves in the presence of demand uncertainty.We find that only when the realized demand is in an intermediate zone does the first mover

preserve its advantage; when the realized demand is far from its expected value, the second

mover obtains higher profit than the leading firm, as the leadership advantage is dominated bythe benefit of flexibility when demand fluctuation is significant. Even with this risk of losing

flexibility under significant demand variation, for some parameter values in our model the first

firm still has incentive to choose Stackelberg rather than Cournot competition. Copyright

# 2005 John Wiley & Sons, Ltd.

INTRODUCTION

The Stackelberg model is one of the most widelyused models in industrial organization for analyz-ing firms’ behavior in a competitive environment.It studies the strategic situation where firmssequentially choose their output levels in a market.In microeconomics textbooks, the standard Stack-elberg model is characterized by complete infor-mation and exogenous order of moves, i.e. allfirms know the market demand and each other’scost of production; furthermore, the order of theirquantity decisions is given exogenously. Theleading firm chooses the output level first, takinginto account the follower’s optimal response to itsquantity choice. The follower then sets its outputlevel based on the leader’s choice. In equilibrium,the leading firm obtains higher profit than thefollower; the difference in equilibrium profits is thevalue of the Stackelberg leadership, or the first-mover advantage.

The question we ask is: Do first movers reallyhave a strategic advantage in practice? The beliefof first-mover advantage was widely held amongentreprenuers and venture capitalists, but is nowquestioned by numerous practitioners. Manybelieve that because Amazon was the first tointroduce an online bookstore, it is able tomaintain its number-one status in the market.We do see some examples of successful pioneeringfirms. Dell was the first to introduce the direct-salebusiness model into the PC market, and itachieved great success, growing from Mr. Dell’ssmall-dorm business into a giant in the PC market.However, we can find many counterexamples.During the dotcom booming era, Pets.com,Webvan, Garden.com and eToys were all firstmovers in their respective market segments, butthey all ended up burning through their investmentcapital before attracting enough customers tosustain a business (Stalter, 2002). Why do thesepioneering firms get very different results? Theprobability of success of pioneering in a marketclearly depends on many factors, including tech-nology, marketing strategy and market demand.In this note, we wish to focus on the strategic

*Correspondence to: MEDS, Kellogg School of Management,Northwestern University, 2001 Sheridan Rd., Evanston, IL60208,USA. E-mail: [email protected]

consequence of asymmetric demand informationowned by first and second movers. Usually, thefollowers in markets get more market informationthan first movers before sinking their investments.We argue that, in some industries that we considerto have fairly stable and predictable marketdemand (and thus the asymmetry of marketinformation between the first mover and secondmover does not matter too much), the pioneeringfirm tends to be the biggest player. An examplewould be the copying-machine industry, whereXerox has been always at the top since itintroduced the world’s first plain-paper copier in1959. But in some markets whose demand is ratherunpredictable, such as the markets of first-genera-tion PCs and medical devices or drugs, it seemsthat second-mover advantage dominates. This isbecause in a market with a high degree ofuncertainty, the followers can wait and see thecustomers’ response to the new product introducedby the first movers, as well as move along the‘learning curve’ of innovation. The first moverbears the cost of creating the market, while thesecond mover enjoys the customer recognition ofthe product and the information of customers’response to the technology and pricing of the firstmover’s product. For example, IBM waited afterApple introduced PCs into the market, and it nowdominates Apple in the market. It was recentlyreported that Boston Scientific claims that it seizes70% market share of drug-coated stents just oneyear after Johnson & Johnson first introduced thisproduct (Abelson, 2004).

We wish to investigate how uncertainty willaffect the first-mover advantage. In this note, weconsider a simple Stackelberg model with demanduncertainty only for the first mover in order tocompare the advantages of leadership and flex-ibility, and use an example to provide somediscussion about the endogenous order of moves.In our model, only the Stackelberg leading firm(first mover) faces demand uncertainty. By thetime the second mover chooses its output level,that uncertainty is resolved. Therefore, the leadingfirm possesses first-mover advantage, but thesecond mover enjoys an informational advantagebecause it can adjust the production level afterobserving the realized demand (flexibility). Whenthe leading firm makes the quantity choice, it doesnot know the exact demand, but only its distribu-tion. We assess the effect of this single-sideddemand uncertainty on the first-mover advantage

and find that when the realized demand is veryhigh or very low (close to the two ends of thepossible range of demand), i.e. when the ex antedemand estimation is very inaccurate, the first-mover advantage vanishes and can even become adisadvantage. Only when the realized demand is inan intermediate zone (not too far from its mean)does the first mover sustain the leadershipadvantage. The intuition is very simple, the leadingfirm determines its quantity according to theexpected demand, while the second mover choosesits quantity according to the realized demand.Consequently when the realized demand is not toofar from its expected value, the flexibility is notvery valuable, because the difference between theex ante and the ex post (assuming that the leadingfirm also chooses its quantity after observing thedemand) optimal decisions by the first mover isnot big; in this case, the first mover’s leadershipadvantage dominates the second mover’s informa-tional advantage. However, when the realizeddemand is far from its mean, the value of flexibilityis high, since the difference between the Stackel-berg leader’s ex ante choice and ex post choice willbe significant (the first mover would have made agreater decision error from the ex post perspec-tive); thus in this situation, the second moverobtains higher profit in equilibrium.

Moreover, we use an example to discuss theendogenous order of moves. For some parametervalues of our model, the first firm will endogen-ously choose to be a Stackelberg leader rather thana Cournot competitor even when it has the optionto make the output decision after the demanduncertainty is resolved.

We present our model and analysis in the nextsection, and offer a review of the related literatureand our concluding remarks in the last section.

THE MODEL AND EQUILIBRIUM

ANALYSIS

Set up

The inverse demand function is p ¼ a�Q, whereQ is the total industry output; the demandparameter a is uniformly distributed in the interval½a0; a1�, with a1 > a050. The distribution of a iscommon knowledge. The constant marginal pro-duction cost is normalized to 0, which is the samefor both firms.1

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Copyright # 2005 John Wiley & Sons, Ltd. Manage. Decis. Econ. 26: 345–350 (2005)

The model is a two-firm, two-period game, ofwhich the timing and structure are as follows: Inperiod 1 when a is not known, firm 1 sets its outputlevel, q1, with the knowledge that firm 2 will enterthe market in the next period if entry is profitable.In period 2 when a is realized, firm 2 observes q1and a, chooses its own quantity, q2, (it can beequal to 0, which means not entering), and themarket clears. In other words, firm 1 is the firstmover (Stackelberg leader), but the second mover(follower), firm 2, has an informational advantagesince it can choose the optimal quantity afterobserving the realized demand. As in the standardStackelberg model, we assume that in period 2 firm1’s pre-chosen quantity is inflexible (this dependson the credibility of firm 1’s quantity commitmentin period 1, and we assume that this commitmentis exogenously given credible,2 i.e. firm 1 cannotadjust its quantity choice after the uncertainty isresolved in period 2). This credible-quantity-commitment assumption is valid with respect toindustries in which once an investment in capacityis made, the cost of adjusting capacity or opera-tional levels in the short run is prohibitive (e.g. thealumina-refining industry; see Ghemawat andNalebuff, 1985, p. 186); thus running at fullcapacity is most efficient.

Therefore, in our model only the leading firmfaces demand uncertainty. We use this variation ofthe information structure (which is reasonablesince the resolution of random demand takes time,and the follower naturally obtains more abundantand accurate information than the pioneeringfirm) to investigate how the single-sided demanduncertainty affects the Stackelberg equilibriumoutcome. Furthermore, we will show using anexample, that when firm 1 has the option to waitand produce after the resolution of demanduncertainty, it may choose not to do so inequilibrium because the benefit of pre-emptingthe follower outweighs the disadvantage of inferiordemand information.

Equilibrium

We use backward induction to obtain the subgameperfect equilibrium. In period 2, given firm 1’sprecommitted quantity, q1, and the realizeddemand parameter, a, firm 2’s problem is

maxq250ða� q1 � q2Þq2;

FOC : a� q1 � 2q2 ¼ 0) q�2 ða; q1Þ

¼maxa� q1

2; 0

� �: ð1Þ

It follows that, in period 1, firm 1’s problem is

maxq150

Ef½a� q1 � q�2 ða; q1Þ�q1g;

where E(*) is to take expectation with respect to a.Since, in period 2, if a5q1, then q�2 ¼ ða� q1Þ=2,and if a5q1, then q�2 ¼ 0, firm 1’s decisionproblem can be rewritten as follows:

maxq150

Z a1

q1

a� q1 �a� q1

2

h iq1 f ðaÞ da

þZ q1

a0

ða� q1Þq1f ðaÞ da;

where f ð�Þ is the density function of a’s distribu-tion. The first-order condition is

FOC :

Z a1

q1

a

2� q1

h if ðaÞ da

þZ q1

a0

ða� 2q1Þf ðaÞ da ¼ 0: ð2Þ

Since a is uniformly distributed in ½a0; a1�,Equation ð2Þ ) �3q21 � 4ða1 � 2a0Þq1 þ ða21 � 2a20Þ ¼ 0.

Therefore, we have:3

q�1 ¼4a0 � 2a1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10a20 � 16a0a1 þ 7a21

q3

; ð3Þ

and

q�2 ðaÞ ¼ maxa� q�1

2; 0

� �

¼ max3a� 4a0 þ 2a1 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10a20 � 16a0a1 þ 7a21

q6

; 0

0@

1A: ð4Þ

With Equations (3) and (4) we get the followingmain result:

Proposition:

Let D � 4a0 � 2a1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10a20 � 16a0a1 þ 7a21

q, then

the equilibrium of the model is as follows:

(i) if a4 D3, q�1 ¼

D3> 0, q�2 ¼ p�2 ¼ 0;

p�1 ¼ ða� q�1 Þq�140,

(ii) if a > D3, q�1 ¼

D3> 0,

q�2 ðaÞ ¼ p�ðaÞ ¼ ð3a� DÞ=6 > 0,p�1 ¼ p�ðaÞq�1 > 0;p�2 ¼ p�ðaÞq�2 ðaÞ > 0,

(ii-A) if D35a4D, q�15q�2 ; p

�15p�2 and

(ii-B) if a > D, q�15q�2 ; p�15p�2 .

STACKELBERG LEADERSHIP 347


Proof:

It is obvious upon examination of Equations (3)and (4). It is easy to show that D > a050, and thecomparison of the equilibrium quantities andprofits in (ii-A) and (ii-B) is based on the followingexpressions: q�1 � q�2 ðaÞ ¼ ðD� aÞ=2; p�1 � p�2 ¼p�ðaÞðq�1 � q�2 ðaÞÞ. &

The potentially negative price and firm 1’s profitin equilibrium outcome (i) can be interpreted asfollows: Firm 1, an innovator, introduces a newproduct into the market, but the realized marketdemand for that product turns out to be very low,and the consumers’ marginal willingness to pay iseven lower than the marginal production cost;therefore, firm 1 suffers strict losses due to the(perhaps not technologically, but certainly eco-nomically) unsuccessful innovation. For example,the now-bankrupt pioneering dotcoms}Webvanand eToys}in our view both over-estimated thecustomers’ willingness (at least currently and inthe near future) to shop for groceries and toysonline, and they not only failed to reap the benefitsof their advantageous position, but also wentbankrupt.

The Proposition shows that, with single-sideddemand uncertainty, only if the realized demand isin an intermediate zone does firm 1 secure its first-mover advantage. In very high- or very low-demand states, firm 2 has second-mover advantagebecause it can fine-tune its production levelaccording to the realized demand, which is verydifferent from its ex ante estimation. Benefit offlexibility dominates the first-mover advantagewhen the demand variation is significant. There-fore, which advantage (leadership or flexibility)dominates depends on the extent of demandshock. The intuition is very straightforward: TheStackelberg leader made its quantity choiceaccording to the expected demand (ex ante choice),while the follower sets its output level according tothe realized demand (ex post choice). The leadingfirm’s ex ante output choice generally is notoptimal when evaluated ex post. If the leadingfirm could change its output choice after itobserves the realized demand but before firm 2sets its quantity, it would adjust q1 according tothe discrepancy between the realized demand andits mean. Therefore, when the realized demand isnot far from its mean (demand estimation is quiteaccurate), firm 1 does not need to make a bigadjustment to achieve optimality ex post. Its

ex ante choice is close to the ex post optimal level,so it still obtains higher profit in equilibrium.Conversely, when the realized demand is far fromits mean, the gap between firm 1’s ex ante and expost optimal choices becomes large, its ex antechoice is found ex post to have been so poor that ityields less profit in competition than that of firm 2,which exploits the advantage of flexibility.

It is straightforward to show that:

(1) If a153a0, then D5a1, D=35a0. Therefore,cases (i) and (ii-B) in the proposition are impos-sible since a 2 ½a0; a1�, and equilibrium is onlypossible in (ii-A); thus, we have the followingcorollary.

Corollary:

If a153a0, then firm 2 enters the market withcertainty, but firm 1 has first-mover advantage(obtains at least the same, if not higher, profits asthose of firm 2) in equilibrium even with single-sided demand uncertainty.

(2) If a153a0, then D4a1, D=35a0. Therefore,all three cases may occur in equilibrium (in parti-cular, firm 2 might choose not to enter the market).

Since varðaÞ ¼ ða1 � a0Þ2=12, we can say that

when the demand uncertainty is moderateða1 2 ða0; 3a0ÞÞ, firm 1 obtains higher profit inequilibrium. As that uncertainty increasesða153a0Þ, firm 1 may lose its first-mover advan-tage, and it may even suffer strict losses inequilibrium (when it produced too much and firm2 does not even enter the market in period 2;consequently firm 1 is the monopolist whose levelof production is too high when it turns out ex postthat the demand for the product is very low}thisis when the quantity commitment becomes risky).

We can also compare the two firms’ expectedequilibrium profits.

Ep1 ¼Z q�

1

a0

ða� q�1 Þq�1 f ðaÞ da

þZ a1

q�1

a� q�12

q�1 f ðaÞ ds;

Ep2 ¼Z a1

q�1

p�ðaÞq�2 ðaÞf ðaÞ da

¼Z a1

q�1

a� q�12

� �2f ðaÞ da:

Z. LIU348


As a is uniformly distributed, f ðaÞ ¼1=ða1 � a0Þ, and Ep1 � Ep2 ¼ ð�2q�31 þ 3q�21 ð4a0�3a1Þ þ 6q�1 ða

21 � a20Þ � a31Þ=12ða1 � a0Þ. For exa-

mple, if a0 ¼ 1, and a1 ¼ 4, then Ep1 � Ep2 ¼ð587

ffiffiffiffiffi58

p� 4388Þ=486 > 0, firm 1’s expected profit

is higher than that of firm 2, making the leadershipposition valuable ex ante. However, firm 1’srealized profit may be lower than that of firm 2as stated in the above proposition (since witha0 ¼ 1, and a1 ¼ 4, a1 > 3a0, the equilibrium mayoccur in any of the three cases in the Proposition).If a0 ¼ 0, and a1 ¼ 1, then Ep1 � Ep2 ¼ð31

ffiffiffi7

p� 83Þ=8150. In this case, the value of

leadership is negative ex ante.

An Example of Endogenous Leadership

If in period 1 firm 1 can decide whether to commitright now or to wait and produce in period 2after the uncertainty is resolved, would it giveup the opportunity of quantity leadership andchoose to wait until period 2 to produce? In otherwords, would firm 1 choose Stackelberg orCournot competition? Firm 2 is sure to enterthe market if the demand is not too low. There-fore, if firm 1 waits until period 2 to produce,and engages in a Cournot competition withfirm 2 after the demand is realized, its equilibriumprofit is *pp1 ¼ a2=9, and its expected profit inperiod 1 is E *pp1 ¼

R a1a0

a2

9f ðaÞ da. Firm 1’s

choice will depend on the difference between theexpected profits resulted from two competitionmodes,

dpða0; a1Þ �Ep1 � E *pp1

¼ q�1

Z a1

q�1

a� q�12

f ðaÞ da

(

þZ q�

1

a0

ða� q�1 Þf ðaÞ da

)�Z a1

a0

a2

9f ðaÞ da

¼ q�1

Z a1

q�1

a

2� q�1

� �f ðaÞ da

(

þZ a1

q�1

q�12f ðaÞ daþ

Z q�1

a0

ða� 2q�1 Þf ðaÞ da

þZ q�

1

a0

q�1 f ðaÞ da

)�Z a1

a0

a2

9f ðaÞ da

¼ q�1

Z a1

q�1

q�12f ðaÞ daþ

Z q�1

a0

q�1 f ðaÞ da

( )

�Z a1

a0

a2

9f ðaÞ da

ðby the FOC; Equationð2ÞÞ

¼ q�1q�12½1þ Fðq�1 Þ� �

Z a1

a0

a2

9f ðaÞ da:

Since a is uniformly distributed, dpða0; a1Þ ¼ ð27q�31 þ27q�21 ða1 � 2a0Þ � 2ða31 � a30ÞÞ=54ða1 � a0Þ. Generally,

for any ða0; a1Þ such that dpða0; a1Þ50, firm 1 will

choose to commit in period 1 rather than wait until

period 2 to produce (i.e. leadership is endogenously

chosen); for ða0; a1Þ such that dpða0; a1Þ50; firm 1 will

choose to wait and engage in a Cournot competition

with firm 2. For example, if a0 ¼ 1; and a1 ¼ 2, then

dpða0; a1Þ ¼ ð3ffiffiffi6

p� 7Þ=27 > 0. In this case, even if

firm 1’s choice of leadership is endogenous (it can

choose to set q1 in period 1 or 2) and subject to the risk

of demand shock, its expected first-mover advantage

outweighs the cost of quantity-setting stickiness, and

its choice of leadership is therefore rational in period

1. Firm 1 faces the following trade-off: it can commit

quantity in advance before the resolution of random

demand, and thus, assure itself of the leadership

advantage with the risk of losing flexibility under

drastic demand change; or it can set quantity after the

demand is known, and thus, enjoy the benefit of

flexibility. In this particular example, with a0 ¼ 1, and

a1 ¼ 2, firm 1 will choose Stackelberg rather than

Cournot competition to maximize its expected profit.

Thus, given that firm 2 is sure to enter the market in

period 2 if the demand is not too low, this will be a

Stackelberg competition, endogenously determined by

firm 1’s period 1 decision.

CONCLUSION

When the Stackelberg leading firm faces demanduncertainty, but the follower does not, it turns outthat the first mover does not necessarily haveadvantage over the second mover. If the demandshock is very significant, the benefit of flexibility ofthe second mover outweighs the first-moveradvantage, and, furthermore, if the realizeddemand is very low, the first mover may evensuffer losses (negative profits). However, for someparameter values in our model the first firm stillhas the incentive to choose Stackelberg rather thanCournot competition, even if it is aware of the riskof losing flexibility under significant demandvariation; in other words, it chooses to commit

STACKELBERG LEADERSHIP 349


its capacity in period 1 rather than wait for theuncertainty being resolved in period 2.

Gal-Or (1987) studies the case where both theStackelberg leader and follower have privateinformation on the random demand, and thequantity choice of the leading firm reveals itsprivate information to the follower, thus providingthe second mover with an informational advantage(in addition to using its own private information,firm 2 can update its belief regarding the demandusing the signal from the leader’s choice). How-ever, in our model, the leading firm does not haveany private information regarding the randomdemand when making its decision. We can extendour analysis to consider more general cases withdifferential costs across firms (see Pal and Sarkar,2001) and more general distribution functions ofthe demand parameter, a: Also, Hirokawa (2001)analyzes the endogenous Stackelberg leadership(credibility of precommitment of quantity) moregenerally. But in this short note we merely discussthe simplest version to highlight the trade-off bet-ween leadership and flexibility advantages. In thesimple model we only analyzed the case of homo-geneous products with quantity as the strategicvariable, we can extend the model to considerproduct differentiation, quality competition, in-novation race and technology adoption. With suchextensions, as well, the first mover has the advan-tage of preemption, but the second mover hasbetter information. Hoppe and Lehmann-Grube(2001; forthcoming) already made great contribu-tions along that line. In future research, we can ingreater detail explore the strategic consequences ofasymmetry of information between the earlier andlater movers in these contexts.

Acknowledgements

I would like to thank Professor Paul Rubin (the Editor), twoanonymous referees and Professor Morton Kamien for veryhelpful comments. I am especially grateful to a referee for verydetailed suggestions that greatly improved the exposition. Allerrors are mine.

NOTES

1. If the marginal production cost is positive, then a canbe thought of as a minus the marginal cost. Alsonotice that the assumption of a050 and zeromarginal cost implies that a monopolist is profitableex ante.

2. We know that, in the standard Stackelberg modelwith complete information, the credibility of thepre-chosen quantity is valuable, because if thequantity commitment is not credible, the competitionturns from Stackelberg into Cournot mode, and thefirst firm’s equilibrium profit decreases. In this case,flexibility is undesirable for the leading firm.

3. We can verify that the second-order condition issatisfied with q�1 ; another possible solution toEquation (2) does not satisfy the second-ordercondition.

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