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Rivalry under price and quantity uncertainty
Review of Financial Economics 14 (2005) 209 – 224 www.elsevier.com/locate/econbase
Rivalry under price and quantity uncertainty Dean Paxson, Helena PintoT Manchester Business School, Booth Street West, Manchester M15 6PB, United Kingdom Received 3 December 2003; received in revised form 29 September 2004; accepted 25 April 2005 Available online 23 June 2005
Abstract We present a real option model for a duopoly setting where there are two stochastic factors and where the roles of the players are defined both exogenously and endogenously. The two stochastic factors are the number of units (market volume) and the profit per unit, which may have significantly different drifts and volatilities, and different correlations, depending on market structure and (dis)economies of scale. The paper shows that the degree of correlation between unit profits and market volume might result in different value functions and triggers, especially for followers and simultaneous investors in non-pre-emptive games. Monopoly-like volume is a critical determinant of the leader’s trigger in both pre-emptive and non-pre-emptive games. First-mover advantages are significant in the definition of the leader’s optimal entry moment, if the players are fighting for the leader’s position (pre-emptive game). D 2005 Elsevier Inc. All rights reserved. JEL classification: D43; G30; L13 Keywords: Real options; Pre-emption; Stochastic processes; Competitive games
1. Introduction The number of units sold and the price per unit are usually treated in the real options literature as an aggregate variable (revenues). In this paper these two variables are treated separately. Our intuition to justify the decomposition of the revenues into two variables is that the number of units sold and the price per unit can be affected by different factors. As an example, when demand is inelastic a change in price T Corresponding author. Tel.: +44 1612750527. E-mail address: [email protected] (H. Pinto). 1058-3300/$ - see front matter D 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.rfe.2005.04.002
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(and consequently in revenue assuming all other factors are constant) will change the quantity sold by a small proportionate change then when demand is elastic. Correlation between volume and unit price may vary according to market structure, and (dis)economies of scale. Conditions of both excess demand and economies of scale may lead to positive correlation. In this paper we present a real option model for a duopoly market where the profit per unit and the number of units sold are both stochastic variables (marginal cost is assumed to be constant). The first continuous-time real option duopoly model was developed by Smets (1993). Grenadier (1996) applies the model of Smets (1993) to the real estate market. The effect of incomplete information is analysed by Lambrecht and Perraudin (2003); R&D competition in Weeds (2002); network advantages are included in Mason and Weeds (2000), Hoppe (2000) considers second mover advantages; Thijssen, Van Damme, Huisman, and Kort (2001) compare first- and second-mover advantages. Smit and Trigeorgis (2001) use binomial trees to value a sequence of investment decisions of a pioneer firm and the interaction of those investment decisions with competitive Cournot or Bertrand settings. Ex-ante asymmetry of the players is allowed in Huisman (2001) and in Pawlina and Kort (2002a). Mixed strategies equilibria are analysed in Huisman and Kort (1999). Pawlina and Kort (2002b) develop a model where the roles of leader and follower are exogenously defined and where both timing and the quality of products involve flexibility. The previous authors considered only one stochastic factor, typically the price or the profit flow. Shackleton, Tsekrekos and Wojakowski (2004) present a game-theoretic approach to real options in a duopoly framework where the profits of each firm are stochastic. They consider a market that can only accommodate one active firm and where the idle firm has the option to claim the market. We extend the model of Smets (1993) to two stochastic variables. In the next section, we present the traditional deterministic model of price and quantity in a competitive environment. In Section 3, we assume that both quantity and profit per unit follow geometric Brownian motions, and allow the roles of the leader (the first mover) and the follower (the second mover) to be determined either endogenously (pre-emptive game) or exogenously (non-preemptive game). We present closed-form solutions for the value functions of the leader and of the follower, and for the entry trigger of the follower. In Section 4, we show and interpret the sensitivity results to changes in the volatility of the number of units sold, the correlation between profits per unit and number of units sold, the monopoly quantity (and hence profits) of the leader (while alone in the market), and the first-mover advantage. Section 5 concludes and considers avenues for future research.
2. The deterministic case Consider a deterministic model, in which price and quantity are both specified, the actions are taken only on quantity, and there is no stochastic variation in either. We will assume a market where two firms are competing according to a Cournot (1838) setting for a homogeneous good.1 The demand curve is given by: Q ¼ d hP
1
For rigorous descriptions of the Cournot equilibrium, see Fudenberg and Tirole (1991) and Dutta (1999).
ð1Þ
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where d and h are larger than zero, Q = Q 1 + Q 2 is the quantity produced respectively by firms 1 and 2, and P is the (common) price. If a = d / h and b = 1 / h, the price P can be expressed as: P ¼ a bQ:
ð2Þ
Let c denote the marginal cost which is assumed constant and equal for both firms. We also assume that firms are competing on quantity, i.e. each firm has to determine which quantity, conditional on the quantity of the competitor, it should produce to maximise its profit. The best-response (reaction) function for firm i( f), conditional on any quantity produced by firm f(i), R i,f ( Q f,i ), is given by: ( a c bQf ;i ac if Qf ;i V b 2b Ri;f Qf ;i ¼ ð3Þ ac 0 if Qf ;i N b The intersection of the two firms’ reaction functions is a Nash equilibrium, with equilibrium quantity and price for each firm given by: Q4i ¼
ac ; 3b
P4i ¼
1 2 a þ c: 3 3
ð4Þ
The above Cournot model assumes that the firms decide on their quantities simultaneously. If the firms choose their quantities sequentially, the results are different because firm f(i) will decide on its quantity knowing the quantity of firm i( f) (Stackelberg, 1934). The Nash equilibrium is given by: Q41 ¼
ac ; 2b
Q42 ¼
ac 1 3 and P4i ¼ a þ c 4b 4 4
ð5Þ
This model illustrates in a non-stochastic setting the principle of optimal sequential competitive decision-making and shows how different competitive structures in a duopoly can produce different equilibrium levels of output. We now generalise the setting to the case where both the profit and the quantity are stochastic.
3. The case with stochastic profit per unit and number of units Traditionally in the real options literature the profit and the number of units are not disaggregated.2 In reality, the path followed by the profit is not necessarily the same as that followed by the number of units sold, because prices and quantities can be affected by different factors. Due to their different characteristics they can even follow different stochastic processes. Notice that profit per unit is a continuous variable but the number of units is technically a discrete one, though this distinction is less important for high volume products such as raw materials, food or fuel.3 In this paper we are concerned with high volume products, thus the number of units sold will be considered a continuous variable. 2 Many real options models consider that the profits follow a certain stochastic process. Exceptions are rare, for example, in Howell et al. (2001) (page 109) the number of units sold is random (but only one stochastic variable is considered). 3 If a firm sells very large quantities of a certain product, any unit increase in the quantity sold is proportionally very small and thus quantity can be considered continuous. Moreover, if we consider a business like fuel retailing, the litres of fuel sold is indeed a continuous variable.
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We consider below that both the profit per unit and the number of units follow different but possibly correlated geometric Brownian motion processes. Let P t represent the profit per unit sold and Q t the quantity sold in a market by a follower. We assume that each variable follows a geometric Brownian motion of the form:4 dPt ¼ lPt dt þ rPt dz1
ð6Þ
dQt ¼ xQt dt þ aQt dz2
ð7Þ
where l and x are the expected multiplicative trends of P t and Q t , r and a are the volatilities, and dz 1 and dz 2 the increments of a Wiener process. The two variables may be correlated with correlation coefficient q.5 Consider two firms with the same investment cost, K, that are contemplating entering a new market. The firm that enters first (defined as the leader) will acquire a first-mover advantage. The leader is assumed to have a higher share of the market. Non-pre-emptive and pre-emptive games are analysed below. In the non-pre-emptive game, it is assumed that the roles of the leader and the follower are pre-assigned, resulting in a sequential adoption if the players enter separately, or in a collusive adoption if the players enter at the same time. In the preemptive game the roles of leader and follower are defined endogenously, with the final outcome resulting in a sequential or a simultaneous entry of the players depending on the level of profits. The two state variables, P and Q, follow a Markov process and it is assumed that the strategies of both players are also Markovian. Thus integrating all payoff relevant factors in the game, those Markov strategies will yield a Nash equilibrium in every proper sub-game. If one player applies Markovian strategies, the rival has as best reaction a Markovian strategy as well. For this reason, a Markovian equilibrium will remain an equilibrium when history-dependent strategies are also permitted, even if other equilibria exist (Fudenberg and Tirole, 1991). 3.1. The pre-emptive game Consider a portfolio that consists of a long position in the option to enter second in a given market, P 0F(V, M), and a short position consisting of D1 and D2 units of P and Q, respectively. We assume that firms are risk-neutral.6 Applying Ito’s lemma, the following partial differential equation for an idle follower is obtained (where r denotes the risk-free rate): 1 B2 P0F 2 2 1 B2 P0F 2 2 B2 P0F BP0F BP0F r P þ a Q þ PQqra þ lP þ xQ rP0F ¼ 0 2 BP2 2 BQ2 BPBQ BP BQ
ð8Þ
Eq. (8) explains the movements in the value function of an idle follower and is subject to the usual boundary conditions. The first boundary condition is the value-matching that gives the value of P 0F(V, M) 4
Our intuition to use geometric Brownian motion to describe the variables path is the same as in Dixit and Pindyck (1994). They argue that the value of an investment depends on predicted variables, e.g. price or number of units sold. These variables depend on other variables, like technology conditions. Therefore variations in the value of the investment can be based on these primary variables. 5 The time subscripts of P t and Q t will be suppressed from now on. 6 For details on equivalent risk-neutral valuation, see Dixit and Pindyck (1994). The assumption of risk neutrality may be relaxed by adjusting the drifts of P and Q to account for a proper risk premium.
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at which the follower should invest. The second boundary condition is the smooth-pasting that assures that the derivatives of the two functions (before and after the follower enters the market) have to be equal at the investment point. Similarity methods can be used to obtain a closed-form solution for Eq. (8). Let X = PQ denote the total profit for the follower, implying that P 0F(X) = P 0F( P, Q). After the appropriate substitutions, Eq. (8) can be written as:7 1 2 d2 P0F ð X Þ 2 dP0F ð X Þ 2 X ½qra þ l þ x rP0F ð X Þ ¼ 0 r þ a þ 2qra þ X 2 dX 2 dX
ð9Þ
Eq. (9) is an Euler’s type ordinary differential equation with the following characteristic quadratic function: 1 2 r þ a2 þ 2qra bðb 1Þ þ ðqra þ l þ xÞb r ¼ 0 2 Eq. (10) has two roots, a positive and a negative one, given by: 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1@ 1 2 1 b1;2 ¼ 2 qra þ l þ x z F 2rz2 qar þ l þ x z2 A z 2 2
ð10Þ
ð11Þ
where z 2 = r 2 + a 2 + 2qra. The solution of Eq. (9) is: P0F ð X Þ ¼ AX b1 þ BX b2 :
ð12Þ
We know that as X increases, the value function of the follower has to increase and that Eq. (12) has to be finite, thus B equals zero. Eq. (12) is subject to the value-matching condition: P0F ðXF Þ ¼
XF K rlx
ð13Þ
where X F is the follower’s trigger value, i.e. the value of X at which the follower should enter the market, and is also subject to the smooth-pasting condition: dP0F ðXF Þ 1 ¼ dXF rlx
ð14Þ
Eqs. (12), (13) and (14) imply that: XF ¼
7
K ðr x lÞb1 b1 1
For details on the computation of (9) see Appendix A.
ð15Þ
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Thus the value function of the follower, P(F), is given by: 8 b1 X > K > < X bXF PðFÞ ¼ b1 1 XF > X > : K X zXF rlx
ð16Þ
Eq. (16) describes the value function of the follower before and after the trigger is hit. Before the trigger X F is hit, the follower has not yet entered the market and its value function is a monopolist American option to invest second in a new market. At the trigger, the follower invests and after that its value function is the net present value. The trigger of the leader is defined considering the strategic interactions between the leader and the follower. Both firms will want to enter in order to obtain the leader’s advantage. Notice that if both firms entered, the game would be one of the simultaneous and not sequential exercises. In order to avoid simultaneous entry (which would be sub-optimal at this point), we will assume that the leader will be determined by a toss of a coin. Both firms have a 50% chance of being a leader or a follower; if they both try to enter a coin will be tossed, and the winner will be the leader entering the market. In order to maximise its value, the follower will wait until the follower’s profit trigger X F is hit.8 After entering the market, the leader has no further actions to take. It will enjoy monopoly profits while it remains alone, and it will share the market with the follower after the latter enters. Let m ¯ be an absolute value larger than one that when multiplied by Q results in the number of units sold by the leader while it is alone in the market (i.e. Qm ¯ is the number of units sold by the leader at a certain moment in time). Let m be an absolute value larger than one, but smaller than m ¯ , that when multiplied by Q results in the number of units sold by the leader after the follower enters the market (i.e. Qm is the number of units sold by the leader after the follower enters the market). Q(m 1) represents the first-mover advantage, in terms of number of units, after the follower enters the market.9 The value function of the leader while alone in the market is given by: " # # "Z TF XF m rs rTF ¯ ds þ E e K ð17Þ e Xm E r lx 0 The first term in Eq. (17) represents the profit that the leader receives while being alone in the market. The second term is the value of the profits of the leader at the moment that the follower enters the market. The value function of the leader, P(L), is given by: 8 b 1 ¯ X Kb1 Xm > > < ¯Þþ K X bXF ðm m X r l x b 1 F 1 PðLÞ ¼ ð18Þ Xm > > : X zXF rlx 8
This barbitraryQ determination of who enters first is also used in Grenadier (1996) and in Weeds (2002). Since the model must define the roles, with identical firms the determination of the leader’s role has to be defined by a random procedure like the outcome of the toss of a coin. 9 Suppose, for example, that while alone in the market, the leader sells 15 units of a certain product and that after the follower enters the leader sells 10 units and the follower sells 5 units. In this case Q = 5, m ¯ = 3, and m = 2. We assumed, in this example, that the number of units sold in the market is not affected by the follower’s entry. The model allows for changes in the number of units sold in the market, before and after the follower enters.
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Value functions
15 10 Follower
5
Leader 0 0.0
10.0
20.0
30.0
40.0
50.0
-5
Number of units (Q ) Fig. 1. Sensitivity of leader and follower value functions to number of units. The parameters are: r = 10%; l = 1%; x = 1%; r = 5%, a = 20%; q = 0; P = 0.01; m = 1.05; m ¯ = 2.05; K = 5 and Q varies from 0 to 50.
Fig. 1 illustrates the sensitivity of the value functions of the follower and of the leader to changes in the number of units sold (eventually) by the follower, Q. The follower and leader value functions have the classical shape. The value function of the leader is almost always larger than that of the follower, except when the number of units is very low. At this point the follower has not yet entered the market, and if the leader were already in the market it would have been better off being a follower. When the follower enters the market the two functions get closer. Additionally, if there were no competitive advantage, the two functions would be exactly the same from X F onwards. Dixit and Pindyck (1994) describe this as a smooth-like-pasting condition of the present values. The value function of the leader is more complicated than the one of the followers. It is concave until the moment the follower enters, and at that moment its slope becomes discontinuous. This happens because although the number of units is increasing, they are also approaching the trigger of the follower, i.e. the negative effect of the entry of the follower increases. It can also be seen in Fig. 1 that there is a point where the two functions meet. Before this point a firm would be better off being a follower, and after that a leader. This point represents the equalization principle of Fudenberg and Tirole (1985). If until this point a firm is better off being a follower and after it a leader, this point should be the trigger of the leader, X L. Although a closed-form solution for X L cannot be obtained because the resulting function is highly non-linear, the trigger can be obtained numerically by solving the following non-linear equation (where X L is the unknown):10 b 1 b1 ¯ XL Kb1 XL m Kb1 XL ¯Þþ ¼0 ð19Þ K ðm m rlx XF b1 1 b1 1 XF 3.2. The non-pre-emptive game In a non-pre-emptive game, the role of the leader and of the follower is assigned exogenously. With the roles pre-assigned, the leader adoption point can now be defined at the value-maximising point for a
10
The leader’s value function has a single root strictly below X F, implying that the trigger point exists and that it is unique. Below that point the firm would be better off being a follower and after that point becoming leader.
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firm who knows that it has the right to enter first (initially earning monopoly profits) or later as it wishes, but that a follower will then enter at the (later) time which maximises the value of the follower. We first assume that the leader and the follower will enter the market at different points in time, resulting in a sequential equilibrium (a collusive equilibrium will be considered later in this section). The follower’s value function is not affected by the non-pre-emptive game. That is, the follower will still enter the market, considering that the leader is already there, in the process of maximising its value function, at the point in time where it is worthwhile to exercise its American call option to enter second. The same arguments can be used for deriving the follower’s value function when the roles are pre-assigned. Thus the optimal time for the follower to enter the market, conditional on previous entrance by the leader, is given by Eq. (15), and the follower’s value function is given by Eq. (16). The value function of the leader can now be described by three components: before it enters the market, the leader holds an American option to enter first; after it enters the market but before the follower enters, the leader receives bmonopoly-likeQ profits. After the follower enters the market, the leader shares the market with the follower, holding a first-mover advantage. The second and the third components are exactly the same as derived earlier; the first component is discussed next. In a non pre-emptive game the leader can delay its market entry. Its value function, before it enters, L P 0 (X), can be explained by Eq. (9), with the following solution: P0L ð X Þ ¼ GX b1 þ HX b2
ð20Þ
where the roots are exactly as defined before. Since b 2 is negative, we know from previous exposition that H = 0. The constant G and the trigger point of the leader are defined by the value-matching condition: b1 ¯ XL Kb1 XL m L ¯Þþ K ð21Þ ðm m P0 ðXL Þ ¼ rlx XF b1 1 and by the smooth-pasting condition: ¯ dP0L ðXL Þ XL b1 1 Kb21 m ¯Þþ ðm m dX L XF ðb1 1ÞXF rlx
ð22Þ
Combining Eqs. (20), (21) and (22), we obtain the trigger of the leader: XL ¼
K ðr l xÞ b1 ¯ m b1 1
ð23Þ
The value function of the leader is described by: 8 b 1 b1 > X Kb X K > 1 > ¯Þþ ðm m > > > X XL b1 1 b 1 > < F b 1 1 ¯ X Kb1 Xm PðLÞ ¼ ¯Þþ K ðm m > > XF rlx b1 1 > > > Xm > > K : rlx
X bXL ½XL ; XF X NXF
ð24Þ
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Value function
14 12 10 8 6 4 2 0 0
0.05
0.1
0.15
0.2
XL
0.25
0.3
0.35
0.4
XF
0.45
0.5
Profits per unit times number of units (X ) Fig. 2. Sensitivity of P(L) to X in a non-pre-emptive game where the number of units and the profits per unit follow separate GBM. The parameters are: r = 10%; l = 1%; x = 1%; r = 5%, a = 20%; q = 0; P = 0.01; K = 5; m = 1.05; m ¯ = 2.05 and X varies from 0 to 0.5 ( Q varies from 0 to 50). For these parameters the trigger of the leader is 0.21 and the trigger of the follower 0.44.
The first line of Eq. (24) gives the value of an American option to enter first in a new market, the second line gives the bmonopoly-likeQ profits less the effect of the follower’s option to invest, and the third line gives the value of the market when shared with the follower. Fig. 2 shows the sensitivity of the value function of the leader to increases in total profits (X).11 Before the leader enters, the function behaves like an option, and is almost a straight line when plotted against total profits (X). After the leader enters, the function is concave, reflecting not only the increases in value due to increases in profits but also decreases due to an imminent entrance of the follower. After the follower enters, the leader shares the market. We next consider the alternative setting where the leader and the follower collude on investment timing. It is assumed that if the firms agree to invest simultaneously, the total number of units sold by both the leader and the follower will be Q (the previous follower’s unit sales) multiplied by a factor m S that is larger than one but smaller than m, and that when m S is multiplied by Q results in the number of units sold by a colluder investor. This framework allows for a setting where it is more beneficial for the follower to collude than to enter sequentially.12 Following the same procedure described above for the follower, the trigger, X S, and the value function of each firm when the investors collude, P(S), are given by: XS ¼
K ðr l xÞ b1 mS ðb1 1Þ 8 > > <
b 1 K X b X 1 S 1 PðSÞ ¼ > X mS > : K rlx
X bXS
ð25Þ
ð26Þ
X zXS
Eq. (26) is very similar to the follower’s value function; the main differences arise from the advantage of collusive investment, Q(m S 1).
11 12
See Appendix B, for Deltas. The factor m S can result, for example, from network advantages.
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In general it would seem that collusion, i.e. an implicit agreement to invest jointly at X S N X L, would be especially attractive to both firms in the pre-emptive game. Here we only consider collusion as a special case of the non-pre-emptive game. The state of collusion, even though the endogenously determined leader has a right to initiate it, potentially creates a form of prisoner’s dilemma under preemptive games: both firms have a high expected value if they collude in entering after X S N X L, but either firm has an incentive to break the agreement by investing after X L but before X reaches X S.
4. Sensitivity analysis and a comparison of the pre-emptive and non-pre-emptive equilibria As described previously, in a pre-emptive game the roles of the leader and the follower are endogenous to the model (each firm having equal probability of becoming a leader or a follower). In a pre-emptive game the resulting outcome can be sequential (if the players enter separately in time) or simultaneous (if one player enters and the other immediately follows). If the initial value of the total profits X is smaller than X L both firms will wait, until X L is hit. When X L is hit one of the firms will enter the market, so if X a [X L, X F] a sequential equilibrium will arise. One firm will enter at X L and in order to maximise its profits the other firm will wait and enter at X F. If the initial value of the state variable X is greater than X F, and none of the firms have already entered the market, then a simultaneous equilibrium will arise. In a non-pre-emptive game, either a collusive or a sequential equilibrium may arise. A sequential equilibrium in non-pre-emptive games may arise in markets where one of the firms that is considering entering is so strong that the other firm would prefer to wait and see the reaction of the market. A collusive equilibrium may arise if both firms agree on the advantage of entering together. In Fig. 3 the sensitivities of the triggers to volatility of number of units sold, a, in pre-emptive and non-pre-emptive games are presented. The trigger values of the follower in both games, and of the leader in the non-pre-emptive game, and of the collusive firms, all tend to behave in a similar way. The trigger of the collusive investor is smaller than that of the follower due to m S, since while investing at the same time as the other firm, the follower will sell more units than if it was investing alone, and will enter the market sooner. The trigger of the
Trigger values
2.0 1.5
Follower's trigger with and without pre-emption
1.0
Leader's trigger with preemption
0.5
Leader's Trigger without pre-emption
0.0 0.0
0.2
0.4
0.6
0.8
Simultaneous trigger without pre-emption
Volatility of number of units ( α ) Fig. 3. Triggers sensitivity to Q volatility when the number of units and the profits follow separate GBM. The parameters are: r = 20%; l = 1%; x = 1%; r = 5%; q = 0; m S = 1.1; K = 5; Q = 30; m = 1.5; m ¯ = 2.5; P = 0.05 and a varies from 0.0001 to 0.781.
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1.4
Trigger values
1.2
Follower's trigger with and without pre-emption
1 0.8
Leader's trigger with pre-emption
0.6 0.4
Leader's Trigger without pre-emption
0.2 0 -1
-0.5
0
0.5
1
Simultaneous trigger without pre-emption
Correlation coeficient ρ Fig. 4. Triggers sensitivity to correlation coefficient. The parameters are: r = 10%; l = 1%; x = 1%; a = 20%; r = 5%; Q = 30; P = 0.05; m = 1.5; m S = 1.1; m ¯ = 2.5; K = 5 and q varies from 1 to 1.
leader in a non-pre-emptive game is even smaller, because the advantage of being first and alone exceeds the advantage (in game terms) of entering at exactly the same time. As expected in a pre-emptive game the trigger of the leader is less sensitive to volatility because the leader’s trigger is defined by the current value of the follower’s option to enter later (which is small), and not by the value of the leader’s option to enter later (which does not exist in pre-emption). In other words, in a pre-emption game the leader acts under the fear of pre-emption, not at the moment when it would be optimal for a monopolist to exercise the option to invest first, but at the moment where both firms are indifferent to being a leader or a follower. The sensitivity of the triggers to the correlation between the profits per unit P, and quantity sold Q, is presented in Fig. 4. As the correlation coefficient increases, the aggregate volatility involving the number of units and the profits per unit also increases, increasing the triggers. Comparing Fig. 4 with Fig. 3 we can see that the triggers react similarly to changes in the volatility. The triggers and value functions may vary considerably from the case of considering only the aggregate profit parameter, especially if correlations are low, and the drifts and volatilities of P and Q are different.13 In Fig. 5 we present the sensitivity of the leader’s triggers to the number of units sold while it is alone in the market, m ¯ Q. Both the leader’s trigger in a pre-emptive game and the leader’s trigger in a non-pre-emptive game decrease as the number of units sold while the leader is alone in the market, m ¯ Q, increase, i.e. the leader will enter earlier if the monopoly-like number of units sold increase. Notice that as the number of units sold, while the leader is alone in the market, m ¯ Q, increase, the difference between the leader’s trigger in the pre-emptive game and the leader’s trigger in the non-pre-emptive game narrows. As m ¯ increases, both the value of being an active non-pre-emptive leader (but alone in the market), which is given by the second line of Eq. (24), and the value of the option to invest, which is given by the first line of Eq. (24), increase. Nevertheless, the increase in the value of an active (but alone in the market) leader, which is the same for both pre-emptive and non-pre-emptive games, is higher than the increase in the value of the 13
Although we present in Fig. 4, for illustration purposes, changes in the correlation coefficient from 1 to 1, in many circumstances the correlation coefficient between the number of units sold and the profits per unit is expected to be positive. Because of economies of scale, as the number of units sold increases, the profit per unit might also increase. Alternatively, in a highly competitive environment, lower per unit profits might be associated with high unit sales.
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Trigger values
0.2 0.15 Leader's trigger with preemption
0.1
Leader's trigger without preemption 0.05 0 60
260
460
660
Monopoly Like Quantity (mQ) Fig. 5. Leader’s triggers sensitivity to monopoly-like quantity (m ¯ Q). The parameters are: r = 10%; l = 1%; x = 1%; a = 20%; r = 5%; Q = 30; P = 0.01; m = 1.05; q = 0; K = 5 and m ¯ Q varies from 60 to 720 (m ¯ varies from 2 to 24).
option to invest. Thus, the non-pre-emptive leader will be less eager to wait and its trigger will approach the investment trigger of the pre-emptive leader. Conversely, Eq. (15) shows that the follower’ trigger (which is not displayed in the figure) is not affected by the number of units sold by the leader while alone 14 in the market m ¯ Q. The sensitivity of the leader’s trigger, in a pre-emptive game, to the first-mover advantage Q(m 1) is presented in Fig. 6. As the first-mover advantage, Q(m 1), increases, the leader enters the market sooner. As the parameter m increases (and consequently the first-mover advantage also increases) the negative effect of a potential follower’s entry (latter in the game) decreases. Thus, the pre-emptive leader will enter sooner in a market with a higher first-mover advantage, because the follower’s entry will be less damaging in terms of number of units sold. The follower’s trigger (which is the same in pre-emptive and non-preemptive games), and the non-pre-emptive leader’s trigger (which are not plotted in the figure) are not affected by the leader’s first-mover advantage Q(m 1). The follower defines its optimal entry moment, knowing that the leader is already in the market, and that the latter will have a first-mover advantage. The number of units sold by the active follower, Q, is not affected by the number of units sold by the active leader (after the follower enters the market), Qm, and thus the follower is indifferent to the leader’s first-mover advantage. More interesting, and less intuitive, is the indifference of the non-pre-emptive leader’s trigger (which is defined by Eq. (23)) to the leader’s first-mover advantage. The option value of the leader and its value function (before the follower enters the market) increase with m, in exactly the same proportion. The value of the leader’s value function, before it enters the market is given by the first line of Eq. (24), and the value of the leader’s value function after it enters the market, but before the follower enters, is given by the second line of Eq. (24). Notice that the derivative of both expressions with respect to m is X XF
b1
Kb1 b1 1 .
Thus, as the first-mover advantage increases, the option value increases and the value of the
leader, after it enters the market (but before the follower does), increases in exactly the same proportion.
14
Notice that the follower’s optimal entry policy, defined by the trigger X F, depends on the number of units sold, but it does not depend on the parameter m ¯ . The parameter m ¯ is not an indication for the follower of the number of units that it can (potentially) sell in the future (if it becomes active).
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Trigger's value
0.13 0.12 0.11 0.1 0.09 5.0
15.0
25.0
35.0
45.0
55.0
First Mover Advantage (Q(m-1)) Fig. 6. Leader’s trigger sensitivity to first-mover advantage in a pre-emptive game. The parameters are: r = 10%; l = 1%; x = 1%; a = 20%; r = 5%; Q = 100; P = 0.01; m ¯ = 2.05; q = 0; K = 5 and Q(m 1) varies from 5 to 60 (m varies from 1.05 to 1.6).
Consequently, the leader defines its optimal entry rule, as if it was a monopolist, knowing nevertheless that this monopoly is of (stochastic) finite duration.
5. Conclusions We present a real options rivalry model in which the profits per unit and the number of units sold are both stochastic variables. Closed-form solutions were obtained, under the assumptions of possibly correlated geometric Brownian motion, for the value functions of the leader and the follower for three different games. Analytical solutions for the trigger of the leader (except for pre-emption) and for the follower were also obtained. The sensitivity of value functions and trigger levels to changes in critical parameters such as number of units sold, volatilities, correlation, monopoly volume and first-mover advantage was examined. The degree of correlation between unit profits and market volume might result in different value functions and triggers, especially for followers and simultaneous investors without pre-emption. Monopoly-like volume (relative to Q) is a critical determinant of the leader’s trigger in both pre-emptive and non-pre-emptive games. First-mover advantages are significant in the definition of the leader’s optimal entry moment, if the players are fighting for the leader’s position (pre-emptive game). The model herein considers only two stochastic factors with constant drift, volatilities and correlation. The market volume multiple is based on the follower’s volume after entry, allowing for a constant monopoly-like leader’s volume multiple before the follower’s entry, and a constant first-mover advantage multiple afterwards. Relaxing these assumptions for pre-emptive, non-pre-emptive and collusion games, and comparing the implications with the results in this paper, is a challenge for future research.
Acknowledgements We appreciate the comments of an anonymous referee, Sydney Howell, Alan Jones, Mark Shackleton, Lenos Trigeorgis, Helen Weeds and the participants of the 2003 Real Options conference on parts of this
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paper. Helena Pinto gratefully acknowledges financial support from Fundac¸a˜o para a Cieˆncia e Tecnologia.
Appendix A. Derivation of PDE (9) Let Eq. (8) represent the value function of an idle follower: 1 B2 P0F 2 2 1 B2 P0F 2 2 B2 P0F BP0F BP0F r P þ a Q þ PQqra þ lP þ xQ rP0F ¼ 0: 2 BP2 2 BQ2 BPBQ BP BQ
ð8Þ
In order to obtain a closed-form solution for this equation, similarity methods can be used. Let X = PQ implying that: P0F ð X Þ ¼ P0F ð P; QÞ BP0F ðP; QÞ BP0F ð X Þ ¼ P BQ BX BP0F ðP; QÞ BP0F ð X Þ ¼ Q BP BX B2 P0F ð P; QÞ B2 P0F ð X Þ 2 ¼ P BQ2 BX 2 B2 P0F ðV ; M Þ B2 P0F ð X Þ 2 ¼ Q BP2 BX 2 B2 P0F ðV ; M Þ B2 P0F ð X Þ BP0F ð X Þ ¼ PQ þ BPBQ BX 2 BX Substituting back into Eq. (8), we obtain Eq. (9): 1 2 F X P0 Wð X Þ r2 þ a2 þ 2qra þ X P0F Vð X Þ½qra þ l þ x rP0F ð X Þ ¼ 0: 2
ð9Þ
Appendix B. Slopes of the value functions The first derivative of the value function of the follower, in the pre-emption game, where the number of units and the profit per unit are the state variables, is given by: 8 b1 1 X > > < N0 X bXF dPðFÞ ¼ r l x XF > 1 dX > : N0 X zXF rlx
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Delta behaves as expected, i.e. as total profit increases the follower’s value function increases. The sensitive of the leader’s value function to the state variable is: 8 > ¯ Þ X b1 ¯ b1 ð m m m > < Nb0 X bXF þ dPðLÞ r l x XF ¼ rlx m > dX > N0 X zXF : rlx After the follower enters the market, the first derivative of the value function of the leader to X is similar to the first derivative of the follower, increasing linearly with the state variable. Before the follower enters the market, the function can be either negative or positive depending on the magnitude of b 1 ¯ Þð XXF Þ 1 .15 As X approaches the trigger, the ratio X/X F increases implying that the term ¯ þ b1 ð m m m that can have a negative effect (m m ¯ ) is also increasing. In the game without pre-emption the delta of the leader will be:16 8 ¯ Þb1 X b1 1 1b1 ¯ X b1 ðm m m > 1b > XL 1 þ XF N0 X bXL > > > r l x r l x > < dPðLÞ ¯ Þ X b1 ¯ b1 ð m m m ¼ Nb0 ½XL ; XF þ > dX > r l x XF rlx > > m > > : N0 X NXF rlx The value function of the leader after it enters the market is exactly the same as in the pre-emption case. Before the leader enters the market the function is different. While the leader is not in the market, it has the option to wait to invest and therefore we expect an increasing function with total profit. The first derivative of the value function of the leader before it enters the market is indeed positive. Notice that 1b 1b what is different between the two terms is: m ¯ )b 1X F 1. Since X F = X Lm ¯ and with the ¯ X L 1 and (m m 1b 1 1b 1 1b 1 in the proportion m , the difference between the triggers taken to a negative power; X L N X F ¯ b1 two terms will be: m ¯ )b 1. The first term will always be larger than the second, and since the ¯ N (m m second is the one representing the negative effect, the delta of the leader before it enters the market will always be positive. If the players agree to enter simultaneously the delta is given by: 8 b1 > 1 X > < N0 X bXS dPðLÞ r l x X S ¼ mS > dX > N0 X zXS : rlx As expected, delta increases with total profits.
15
In Fig. 1 we can see that the value function of the leader is initially increasing with Q, then as it approaches the follower’s trigger the rate of increase declines, and then it is a linear function of Q after the follower enters the market. 16 Notice that the value function of the follower is the same as in the pre-emption model.
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References Cournot, A. (1838). Recherches sur les principes mathe´matiques de la the´orie des richesses. In N. Bacon (Ed.), Researches into the mathematical principles of the theory of wealth, English edition. Dixit, A., & Pindyck, R. (1994). Investment under uncertainty. Princeton7 Princeton University Press. Dutta, P. (1999). Strategies and games—theory and practice. Cambridge, Massachusetts7 The MIT Press. Fudenberg, D., & Tirole, J. (1985). Pre-emption and rent equalization in the adoption of a new technology. Review of Economic Studies, 52(3), 383 – 401. Fudenberg, D., & Tirole, J. (1991). Game theory. Cambridge. Massachusetts7 The MIT Press. Grenadier, S. (1996). The strategic exercise of options: Development cascades and overbuilding in real estate markets. Journal of Finance, 51(5), 1653 – 1679. Hoppe, H. (2000). Second-mover advantages in the strategic adoption of new technology under uncertainty. International Journal of Industrial Organization, 18(2), 315 – 338. Howell, S., Stark, A., Newton, D., Paxson, D., Cavus, M., Pereira, J., et al. (2001). Real options—evaluating corporate investment opportunities in a dynamic world. London7 Financial Times—Prentice Hall. Huisman, K. (2001). Technology investment: A game theoretic real options approach. Boston7 Kluwer Academic Publishers. Huisman, K. & Kort, P. (1999). Effects of Strategic Interactions on the Option Value of Waiting. Working paper from WWW.SSRN.COM Lambrecht, B., & Perraudin, W. (2003). Real options under incomplete information. Journal of Economic Dynamics and Control, 77(4), 619 – 643. Mason, R. & Weeds, H., (2000). Networks, Options and Pre-Emption. Working Paper, Cambridge University. Pawlina, G. & Kort, P., (2002a). Real Options in an Asymmetric Duopoly: Who Benefits from your Competitive Disadvantage? Working Paper from WWW.SSRN.COM Pawlina, G. & Kort, P., (2002b). The Strategic Value of Flexible Quality Choice: A Real Options Analysis. Working Paper from WWW.SSRN.COM Shackleton, M. B., Tsekrekos, A. E., & Wojakowski, R. (2004). Strategic entry and market leadership in a two-player real options game. Journal of Banking and Finance, 28(1), 179 – 201. Smets, F. (1993). Essays on Foreign Direct Investment. PhD Thesis, Yale University. Smit, H., & Trigeorgis, L. (2001). Flexibility and commitment in strategic investment. In E. Schwartz, & L. Trigeorgis (Eds.), Real options and investment under uncertainty: Classical readings and recent contributions (pp. 451 – 498). Massachusetts7 The MIT Press. Cambridge. Stackelberg, H. (1934). Marktform und Gleichgewicht. J. Springer. Berlin. Translated by Alan Peacock (1952) as The Theory of the Market Economy. William Hodge. London. Thijssen, J., Van Damme, E., Huisman, K. & Kort, P., (2001). Investment Under Vanishing Uncertainty Due to Information Arriving over Time. Working Paper, Tilburg University. Weeds, H. (2002). Strategic delay in a real options model of R&D competition. Review of Economic Studies, 69(3), 729 – 747.
Rivalry under price and quantity uncertainty Dean Paxson, Helena PintoT Manchester Business School, Booth Street West, Manchester M15 6PB, United Kingdom Received 3 December 2003; received in revised form 29 September 2004; accepted 25 April 2005 Available online 23 June 2005
Abstract We present a real option model for a duopoly setting where there are two stochastic factors and where the roles of the players are defined both exogenously and endogenously. The two stochastic factors are the number of units (market volume) and the profit per unit, which may have significantly different drifts and volatilities, and different correlations, depending on market structure and (dis)economies of scale. The paper shows that the degree of correlation between unit profits and market volume might result in different value functions and triggers, especially for followers and simultaneous investors in non-pre-emptive games. Monopoly-like volume is a critical determinant of the leader’s trigger in both pre-emptive and non-pre-emptive games. First-mover advantages are significant in the definition of the leader’s optimal entry moment, if the players are fighting for the leader’s position (pre-emptive game). D 2005 Elsevier Inc. All rights reserved. JEL classification: D43; G30; L13 Keywords: Real options; Pre-emption; Stochastic processes; Competitive games
1. Introduction The number of units sold and the price per unit are usually treated in the real options literature as an aggregate variable (revenues). In this paper these two variables are treated separately. Our intuition to justify the decomposition of the revenues into two variables is that the number of units sold and the price per unit can be affected by different factors. As an example, when demand is inelastic a change in price T Corresponding author. Tel.: +44 1612750527. E-mail address: [email protected] (H. Pinto). 1058-3300/$ - see front matter D 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.rfe.2005.04.002
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(and consequently in revenue assuming all other factors are constant) will change the quantity sold by a small proportionate change then when demand is elastic. Correlation between volume and unit price may vary according to market structure, and (dis)economies of scale. Conditions of both excess demand and economies of scale may lead to positive correlation. In this paper we present a real option model for a duopoly market where the profit per unit and the number of units sold are both stochastic variables (marginal cost is assumed to be constant). The first continuous-time real option duopoly model was developed by Smets (1993). Grenadier (1996) applies the model of Smets (1993) to the real estate market. The effect of incomplete information is analysed by Lambrecht and Perraudin (2003); R&D competition in Weeds (2002); network advantages are included in Mason and Weeds (2000), Hoppe (2000) considers second mover advantages; Thijssen, Van Damme, Huisman, and Kort (2001) compare first- and second-mover advantages. Smit and Trigeorgis (2001) use binomial trees to value a sequence of investment decisions of a pioneer firm and the interaction of those investment decisions with competitive Cournot or Bertrand settings. Ex-ante asymmetry of the players is allowed in Huisman (2001) and in Pawlina and Kort (2002a). Mixed strategies equilibria are analysed in Huisman and Kort (1999). Pawlina and Kort (2002b) develop a model where the roles of leader and follower are exogenously defined and where both timing and the quality of products involve flexibility. The previous authors considered only one stochastic factor, typically the price or the profit flow. Shackleton, Tsekrekos and Wojakowski (2004) present a game-theoretic approach to real options in a duopoly framework where the profits of each firm are stochastic. They consider a market that can only accommodate one active firm and where the idle firm has the option to claim the market. We extend the model of Smets (1993) to two stochastic variables. In the next section, we present the traditional deterministic model of price and quantity in a competitive environment. In Section 3, we assume that both quantity and profit per unit follow geometric Brownian motions, and allow the roles of the leader (the first mover) and the follower (the second mover) to be determined either endogenously (pre-emptive game) or exogenously (non-preemptive game). We present closed-form solutions for the value functions of the leader and of the follower, and for the entry trigger of the follower. In Section 4, we show and interpret the sensitivity results to changes in the volatility of the number of units sold, the correlation between profits per unit and number of units sold, the monopoly quantity (and hence profits) of the leader (while alone in the market), and the first-mover advantage. Section 5 concludes and considers avenues for future research.
2. The deterministic case Consider a deterministic model, in which price and quantity are both specified, the actions are taken only on quantity, and there is no stochastic variation in either. We will assume a market where two firms are competing according to a Cournot (1838) setting for a homogeneous good.1 The demand curve is given by: Q ¼ d hP
1
For rigorous descriptions of the Cournot equilibrium, see Fudenberg and Tirole (1991) and Dutta (1999).
ð1Þ
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where d and h are larger than zero, Q = Q 1 + Q 2 is the quantity produced respectively by firms 1 and 2, and P is the (common) price. If a = d / h and b = 1 / h, the price P can be expressed as: P ¼ a bQ:
ð2Þ
Let c denote the marginal cost which is assumed constant and equal for both firms. We also assume that firms are competing on quantity, i.e. each firm has to determine which quantity, conditional on the quantity of the competitor, it should produce to maximise its profit. The best-response (reaction) function for firm i( f), conditional on any quantity produced by firm f(i), R i,f ( Q f,i ), is given by: ( a c bQf ;i ac if Qf ;i V b 2b Ri;f Qf ;i ¼ ð3Þ ac 0 if Qf ;i N b The intersection of the two firms’ reaction functions is a Nash equilibrium, with equilibrium quantity and price for each firm given by: Q4i ¼
ac ; 3b
P4i ¼
1 2 a þ c: 3 3
ð4Þ
The above Cournot model assumes that the firms decide on their quantities simultaneously. If the firms choose their quantities sequentially, the results are different because firm f(i) will decide on its quantity knowing the quantity of firm i( f) (Stackelberg, 1934). The Nash equilibrium is given by: Q41 ¼
ac ; 2b
Q42 ¼
ac 1 3 and P4i ¼ a þ c 4b 4 4
ð5Þ
This model illustrates in a non-stochastic setting the principle of optimal sequential competitive decision-making and shows how different competitive structures in a duopoly can produce different equilibrium levels of output. We now generalise the setting to the case where both the profit and the quantity are stochastic.
3. The case with stochastic profit per unit and number of units Traditionally in the real options literature the profit and the number of units are not disaggregated.2 In reality, the path followed by the profit is not necessarily the same as that followed by the number of units sold, because prices and quantities can be affected by different factors. Due to their different characteristics they can even follow different stochastic processes. Notice that profit per unit is a continuous variable but the number of units is technically a discrete one, though this distinction is less important for high volume products such as raw materials, food or fuel.3 In this paper we are concerned with high volume products, thus the number of units sold will be considered a continuous variable. 2 Many real options models consider that the profits follow a certain stochastic process. Exceptions are rare, for example, in Howell et al. (2001) (page 109) the number of units sold is random (but only one stochastic variable is considered). 3 If a firm sells very large quantities of a certain product, any unit increase in the quantity sold is proportionally very small and thus quantity can be considered continuous. Moreover, if we consider a business like fuel retailing, the litres of fuel sold is indeed a continuous variable.
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We consider below that both the profit per unit and the number of units follow different but possibly correlated geometric Brownian motion processes. Let P t represent the profit per unit sold and Q t the quantity sold in a market by a follower. We assume that each variable follows a geometric Brownian motion of the form:4 dPt ¼ lPt dt þ rPt dz1
ð6Þ
dQt ¼ xQt dt þ aQt dz2
ð7Þ
where l and x are the expected multiplicative trends of P t and Q t , r and a are the volatilities, and dz 1 and dz 2 the increments of a Wiener process. The two variables may be correlated with correlation coefficient q.5 Consider two firms with the same investment cost, K, that are contemplating entering a new market. The firm that enters first (defined as the leader) will acquire a first-mover advantage. The leader is assumed to have a higher share of the market. Non-pre-emptive and pre-emptive games are analysed below. In the non-pre-emptive game, it is assumed that the roles of the leader and the follower are pre-assigned, resulting in a sequential adoption if the players enter separately, or in a collusive adoption if the players enter at the same time. In the preemptive game the roles of leader and follower are defined endogenously, with the final outcome resulting in a sequential or a simultaneous entry of the players depending on the level of profits. The two state variables, P and Q, follow a Markov process and it is assumed that the strategies of both players are also Markovian. Thus integrating all payoff relevant factors in the game, those Markov strategies will yield a Nash equilibrium in every proper sub-game. If one player applies Markovian strategies, the rival has as best reaction a Markovian strategy as well. For this reason, a Markovian equilibrium will remain an equilibrium when history-dependent strategies are also permitted, even if other equilibria exist (Fudenberg and Tirole, 1991). 3.1. The pre-emptive game Consider a portfolio that consists of a long position in the option to enter second in a given market, P 0F(V, M), and a short position consisting of D1 and D2 units of P and Q, respectively. We assume that firms are risk-neutral.6 Applying Ito’s lemma, the following partial differential equation for an idle follower is obtained (where r denotes the risk-free rate): 1 B2 P0F 2 2 1 B2 P0F 2 2 B2 P0F BP0F BP0F r P þ a Q þ PQqra þ lP þ xQ rP0F ¼ 0 2 BP2 2 BQ2 BPBQ BP BQ
ð8Þ
Eq. (8) explains the movements in the value function of an idle follower and is subject to the usual boundary conditions. The first boundary condition is the value-matching that gives the value of P 0F(V, M) 4
Our intuition to use geometric Brownian motion to describe the variables path is the same as in Dixit and Pindyck (1994). They argue that the value of an investment depends on predicted variables, e.g. price or number of units sold. These variables depend on other variables, like technology conditions. Therefore variations in the value of the investment can be based on these primary variables. 5 The time subscripts of P t and Q t will be suppressed from now on. 6 For details on equivalent risk-neutral valuation, see Dixit and Pindyck (1994). The assumption of risk neutrality may be relaxed by adjusting the drifts of P and Q to account for a proper risk premium.
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at which the follower should invest. The second boundary condition is the smooth-pasting that assures that the derivatives of the two functions (before and after the follower enters the market) have to be equal at the investment point. Similarity methods can be used to obtain a closed-form solution for Eq. (8). Let X = PQ denote the total profit for the follower, implying that P 0F(X) = P 0F( P, Q). After the appropriate substitutions, Eq. (8) can be written as:7 1 2 d2 P0F ð X Þ 2 dP0F ð X Þ 2 X ½qra þ l þ x rP0F ð X Þ ¼ 0 r þ a þ 2qra þ X 2 dX 2 dX
ð9Þ
Eq. (9) is an Euler’s type ordinary differential equation with the following characteristic quadratic function: 1 2 r þ a2 þ 2qra bðb 1Þ þ ðqra þ l þ xÞb r ¼ 0 2 Eq. (10) has two roots, a positive and a negative one, given by: 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1@ 1 2 1 b1;2 ¼ 2 qra þ l þ x z F 2rz2 qar þ l þ x z2 A z 2 2
ð10Þ
ð11Þ
where z 2 = r 2 + a 2 + 2qra. The solution of Eq. (9) is: P0F ð X Þ ¼ AX b1 þ BX b2 :
ð12Þ
We know that as X increases, the value function of the follower has to increase and that Eq. (12) has to be finite, thus B equals zero. Eq. (12) is subject to the value-matching condition: P0F ðXF Þ ¼
XF K rlx
ð13Þ
where X F is the follower’s trigger value, i.e. the value of X at which the follower should enter the market, and is also subject to the smooth-pasting condition: dP0F ðXF Þ 1 ¼ dXF rlx
ð14Þ
Eqs. (12), (13) and (14) imply that: XF ¼
7
K ðr x lÞb1 b1 1
For details on the computation of (9) see Appendix A.
ð15Þ
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Thus the value function of the follower, P(F), is given by: 8 b1 X > K > < X bXF PðFÞ ¼ b1 1 XF > X > : K X zXF rlx
ð16Þ
Eq. (16) describes the value function of the follower before and after the trigger is hit. Before the trigger X F is hit, the follower has not yet entered the market and its value function is a monopolist American option to invest second in a new market. At the trigger, the follower invests and after that its value function is the net present value. The trigger of the leader is defined considering the strategic interactions between the leader and the follower. Both firms will want to enter in order to obtain the leader’s advantage. Notice that if both firms entered, the game would be one of the simultaneous and not sequential exercises. In order to avoid simultaneous entry (which would be sub-optimal at this point), we will assume that the leader will be determined by a toss of a coin. Both firms have a 50% chance of being a leader or a follower; if they both try to enter a coin will be tossed, and the winner will be the leader entering the market. In order to maximise its value, the follower will wait until the follower’s profit trigger X F is hit.8 After entering the market, the leader has no further actions to take. It will enjoy monopoly profits while it remains alone, and it will share the market with the follower after the latter enters. Let m ¯ be an absolute value larger than one that when multiplied by Q results in the number of units sold by the leader while it is alone in the market (i.e. Qm ¯ is the number of units sold by the leader at a certain moment in time). Let m be an absolute value larger than one, but smaller than m ¯ , that when multiplied by Q results in the number of units sold by the leader after the follower enters the market (i.e. Qm is the number of units sold by the leader after the follower enters the market). Q(m 1) represents the first-mover advantage, in terms of number of units, after the follower enters the market.9 The value function of the leader while alone in the market is given by: " # # "Z TF XF m rs rTF ¯ ds þ E e K ð17Þ e Xm E r lx 0 The first term in Eq. (17) represents the profit that the leader receives while being alone in the market. The second term is the value of the profits of the leader at the moment that the follower enters the market. The value function of the leader, P(L), is given by: 8 b 1 ¯ X Kb1 Xm > > < ¯Þþ K X bXF ðm m X r l x b 1 F 1 PðLÞ ¼ ð18Þ Xm > > : X zXF rlx 8
This barbitraryQ determination of who enters first is also used in Grenadier (1996) and in Weeds (2002). Since the model must define the roles, with identical firms the determination of the leader’s role has to be defined by a random procedure like the outcome of the toss of a coin. 9 Suppose, for example, that while alone in the market, the leader sells 15 units of a certain product and that after the follower enters the leader sells 10 units and the follower sells 5 units. In this case Q = 5, m ¯ = 3, and m = 2. We assumed, in this example, that the number of units sold in the market is not affected by the follower’s entry. The model allows for changes in the number of units sold in the market, before and after the follower enters.
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Value functions
15 10 Follower
5
Leader 0 0.0
10.0
20.0
30.0
40.0
50.0
-5
Number of units (Q ) Fig. 1. Sensitivity of leader and follower value functions to number of units. The parameters are: r = 10%; l = 1%; x = 1%; r = 5%, a = 20%; q = 0; P = 0.01; m = 1.05; m ¯ = 2.05; K = 5 and Q varies from 0 to 50.
Fig. 1 illustrates the sensitivity of the value functions of the follower and of the leader to changes in the number of units sold (eventually) by the follower, Q. The follower and leader value functions have the classical shape. The value function of the leader is almost always larger than that of the follower, except when the number of units is very low. At this point the follower has not yet entered the market, and if the leader were already in the market it would have been better off being a follower. When the follower enters the market the two functions get closer. Additionally, if there were no competitive advantage, the two functions would be exactly the same from X F onwards. Dixit and Pindyck (1994) describe this as a smooth-like-pasting condition of the present values. The value function of the leader is more complicated than the one of the followers. It is concave until the moment the follower enters, and at that moment its slope becomes discontinuous. This happens because although the number of units is increasing, they are also approaching the trigger of the follower, i.e. the negative effect of the entry of the follower increases. It can also be seen in Fig. 1 that there is a point where the two functions meet. Before this point a firm would be better off being a follower, and after that a leader. This point represents the equalization principle of Fudenberg and Tirole (1985). If until this point a firm is better off being a follower and after it a leader, this point should be the trigger of the leader, X L. Although a closed-form solution for X L cannot be obtained because the resulting function is highly non-linear, the trigger can be obtained numerically by solving the following non-linear equation (where X L is the unknown):10 b 1 b1 ¯ XL Kb1 XL m Kb1 XL ¯Þþ ¼0 ð19Þ K ðm m rlx XF b1 1 b1 1 XF 3.2. The non-pre-emptive game In a non-pre-emptive game, the role of the leader and of the follower is assigned exogenously. With the roles pre-assigned, the leader adoption point can now be defined at the value-maximising point for a
10
The leader’s value function has a single root strictly below X F, implying that the trigger point exists and that it is unique. Below that point the firm would be better off being a follower and after that point becoming leader.
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firm who knows that it has the right to enter first (initially earning monopoly profits) or later as it wishes, but that a follower will then enter at the (later) time which maximises the value of the follower. We first assume that the leader and the follower will enter the market at different points in time, resulting in a sequential equilibrium (a collusive equilibrium will be considered later in this section). The follower’s value function is not affected by the non-pre-emptive game. That is, the follower will still enter the market, considering that the leader is already there, in the process of maximising its value function, at the point in time where it is worthwhile to exercise its American call option to enter second. The same arguments can be used for deriving the follower’s value function when the roles are pre-assigned. Thus the optimal time for the follower to enter the market, conditional on previous entrance by the leader, is given by Eq. (15), and the follower’s value function is given by Eq. (16). The value function of the leader can now be described by three components: before it enters the market, the leader holds an American option to enter first; after it enters the market but before the follower enters, the leader receives bmonopoly-likeQ profits. After the follower enters the market, the leader shares the market with the follower, holding a first-mover advantage. The second and the third components are exactly the same as derived earlier; the first component is discussed next. In a non pre-emptive game the leader can delay its market entry. Its value function, before it enters, L P 0 (X), can be explained by Eq. (9), with the following solution: P0L ð X Þ ¼ GX b1 þ HX b2
ð20Þ
where the roots are exactly as defined before. Since b 2 is negative, we know from previous exposition that H = 0. The constant G and the trigger point of the leader are defined by the value-matching condition: b1 ¯ XL Kb1 XL m L ¯Þþ K ð21Þ ðm m P0 ðXL Þ ¼ rlx XF b1 1 and by the smooth-pasting condition: ¯ dP0L ðXL Þ XL b1 1 Kb21 m ¯Þþ ðm m dX L XF ðb1 1ÞXF rlx
ð22Þ
Combining Eqs. (20), (21) and (22), we obtain the trigger of the leader: XL ¼
K ðr l xÞ b1 ¯ m b1 1
ð23Þ
The value function of the leader is described by: 8 b 1 b1 > X Kb X K > 1 > ¯Þþ ðm m > > > X XL b1 1 b 1 > < F b 1 1 ¯ X Kb1 Xm PðLÞ ¼ ¯Þþ K ðm m > > XF rlx b1 1 > > > Xm > > K : rlx
X bXL ½XL ; XF X NXF
ð24Þ
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Value function
14 12 10 8 6 4 2 0 0
0.05
0.1
0.15
0.2
XL
0.25
0.3
0.35
0.4
XF
0.45
0.5
Profits per unit times number of units (X ) Fig. 2. Sensitivity of P(L) to X in a non-pre-emptive game where the number of units and the profits per unit follow separate GBM. The parameters are: r = 10%; l = 1%; x = 1%; r = 5%, a = 20%; q = 0; P = 0.01; K = 5; m = 1.05; m ¯ = 2.05 and X varies from 0 to 0.5 ( Q varies from 0 to 50). For these parameters the trigger of the leader is 0.21 and the trigger of the follower 0.44.
The first line of Eq. (24) gives the value of an American option to enter first in a new market, the second line gives the bmonopoly-likeQ profits less the effect of the follower’s option to invest, and the third line gives the value of the market when shared with the follower. Fig. 2 shows the sensitivity of the value function of the leader to increases in total profits (X).11 Before the leader enters, the function behaves like an option, and is almost a straight line when plotted against total profits (X). After the leader enters, the function is concave, reflecting not only the increases in value due to increases in profits but also decreases due to an imminent entrance of the follower. After the follower enters, the leader shares the market. We next consider the alternative setting where the leader and the follower collude on investment timing. It is assumed that if the firms agree to invest simultaneously, the total number of units sold by both the leader and the follower will be Q (the previous follower’s unit sales) multiplied by a factor m S that is larger than one but smaller than m, and that when m S is multiplied by Q results in the number of units sold by a colluder investor. This framework allows for a setting where it is more beneficial for the follower to collude than to enter sequentially.12 Following the same procedure described above for the follower, the trigger, X S, and the value function of each firm when the investors collude, P(S), are given by: XS ¼
K ðr l xÞ b1 mS ðb1 1Þ 8 > > <
b 1 K X b X 1 S 1 PðSÞ ¼ > X mS > : K rlx
X bXS
ð25Þ
ð26Þ
X zXS
Eq. (26) is very similar to the follower’s value function; the main differences arise from the advantage of collusive investment, Q(m S 1).
11 12
See Appendix B, for Deltas. The factor m S can result, for example, from network advantages.
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In general it would seem that collusion, i.e. an implicit agreement to invest jointly at X S N X L, would be especially attractive to both firms in the pre-emptive game. Here we only consider collusion as a special case of the non-pre-emptive game. The state of collusion, even though the endogenously determined leader has a right to initiate it, potentially creates a form of prisoner’s dilemma under preemptive games: both firms have a high expected value if they collude in entering after X S N X L, but either firm has an incentive to break the agreement by investing after X L but before X reaches X S.
4. Sensitivity analysis and a comparison of the pre-emptive and non-pre-emptive equilibria As described previously, in a pre-emptive game the roles of the leader and the follower are endogenous to the model (each firm having equal probability of becoming a leader or a follower). In a pre-emptive game the resulting outcome can be sequential (if the players enter separately in time) or simultaneous (if one player enters and the other immediately follows). If the initial value of the total profits X is smaller than X L both firms will wait, until X L is hit. When X L is hit one of the firms will enter the market, so if X a [X L, X F] a sequential equilibrium will arise. One firm will enter at X L and in order to maximise its profits the other firm will wait and enter at X F. If the initial value of the state variable X is greater than X F, and none of the firms have already entered the market, then a simultaneous equilibrium will arise. In a non-pre-emptive game, either a collusive or a sequential equilibrium may arise. A sequential equilibrium in non-pre-emptive games may arise in markets where one of the firms that is considering entering is so strong that the other firm would prefer to wait and see the reaction of the market. A collusive equilibrium may arise if both firms agree on the advantage of entering together. In Fig. 3 the sensitivities of the triggers to volatility of number of units sold, a, in pre-emptive and non-pre-emptive games are presented. The trigger values of the follower in both games, and of the leader in the non-pre-emptive game, and of the collusive firms, all tend to behave in a similar way. The trigger of the collusive investor is smaller than that of the follower due to m S, since while investing at the same time as the other firm, the follower will sell more units than if it was investing alone, and will enter the market sooner. The trigger of the
Trigger values
2.0 1.5
Follower's trigger with and without pre-emption
1.0
Leader's trigger with preemption
0.5
Leader's Trigger without pre-emption
0.0 0.0
0.2
0.4
0.6
0.8
Simultaneous trigger without pre-emption
Volatility of number of units ( α ) Fig. 3. Triggers sensitivity to Q volatility when the number of units and the profits follow separate GBM. The parameters are: r = 20%; l = 1%; x = 1%; r = 5%; q = 0; m S = 1.1; K = 5; Q = 30; m = 1.5; m ¯ = 2.5; P = 0.05 and a varies from 0.0001 to 0.781.
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1.4
Trigger values
1.2
Follower's trigger with and without pre-emption
1 0.8
Leader's trigger with pre-emption
0.6 0.4
Leader's Trigger without pre-emption
0.2 0 -1
-0.5
0
0.5
1
Simultaneous trigger without pre-emption
Correlation coeficient ρ Fig. 4. Triggers sensitivity to correlation coefficient. The parameters are: r = 10%; l = 1%; x = 1%; a = 20%; r = 5%; Q = 30; P = 0.05; m = 1.5; m S = 1.1; m ¯ = 2.5; K = 5 and q varies from 1 to 1.
leader in a non-pre-emptive game is even smaller, because the advantage of being first and alone exceeds the advantage (in game terms) of entering at exactly the same time. As expected in a pre-emptive game the trigger of the leader is less sensitive to volatility because the leader’s trigger is defined by the current value of the follower’s option to enter later (which is small), and not by the value of the leader’s option to enter later (which does not exist in pre-emption). In other words, in a pre-emption game the leader acts under the fear of pre-emption, not at the moment when it would be optimal for a monopolist to exercise the option to invest first, but at the moment where both firms are indifferent to being a leader or a follower. The sensitivity of the triggers to the correlation between the profits per unit P, and quantity sold Q, is presented in Fig. 4. As the correlation coefficient increases, the aggregate volatility involving the number of units and the profits per unit also increases, increasing the triggers. Comparing Fig. 4 with Fig. 3 we can see that the triggers react similarly to changes in the volatility. The triggers and value functions may vary considerably from the case of considering only the aggregate profit parameter, especially if correlations are low, and the drifts and volatilities of P and Q are different.13 In Fig. 5 we present the sensitivity of the leader’s triggers to the number of units sold while it is alone in the market, m ¯ Q. Both the leader’s trigger in a pre-emptive game and the leader’s trigger in a non-pre-emptive game decrease as the number of units sold while the leader is alone in the market, m ¯ Q, increase, i.e. the leader will enter earlier if the monopoly-like number of units sold increase. Notice that as the number of units sold, while the leader is alone in the market, m ¯ Q, increase, the difference between the leader’s trigger in the pre-emptive game and the leader’s trigger in the non-pre-emptive game narrows. As m ¯ increases, both the value of being an active non-pre-emptive leader (but alone in the market), which is given by the second line of Eq. (24), and the value of the option to invest, which is given by the first line of Eq. (24), increase. Nevertheless, the increase in the value of an active (but alone in the market) leader, which is the same for both pre-emptive and non-pre-emptive games, is higher than the increase in the value of the 13
Although we present in Fig. 4, for illustration purposes, changes in the correlation coefficient from 1 to 1, in many circumstances the correlation coefficient between the number of units sold and the profits per unit is expected to be positive. Because of economies of scale, as the number of units sold increases, the profit per unit might also increase. Alternatively, in a highly competitive environment, lower per unit profits might be associated with high unit sales.
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Trigger values
0.2 0.15 Leader's trigger with preemption
0.1
Leader's trigger without preemption 0.05 0 60
260
460
660
Monopoly Like Quantity (mQ) Fig. 5. Leader’s triggers sensitivity to monopoly-like quantity (m ¯ Q). The parameters are: r = 10%; l = 1%; x = 1%; a = 20%; r = 5%; Q = 30; P = 0.01; m = 1.05; q = 0; K = 5 and m ¯ Q varies from 60 to 720 (m ¯ varies from 2 to 24).
option to invest. Thus, the non-pre-emptive leader will be less eager to wait and its trigger will approach the investment trigger of the pre-emptive leader. Conversely, Eq. (15) shows that the follower’ trigger (which is not displayed in the figure) is not affected by the number of units sold by the leader while alone 14 in the market m ¯ Q. The sensitivity of the leader’s trigger, in a pre-emptive game, to the first-mover advantage Q(m 1) is presented in Fig. 6. As the first-mover advantage, Q(m 1), increases, the leader enters the market sooner. As the parameter m increases (and consequently the first-mover advantage also increases) the negative effect of a potential follower’s entry (latter in the game) decreases. Thus, the pre-emptive leader will enter sooner in a market with a higher first-mover advantage, because the follower’s entry will be less damaging in terms of number of units sold. The follower’s trigger (which is the same in pre-emptive and non-preemptive games), and the non-pre-emptive leader’s trigger (which are not plotted in the figure) are not affected by the leader’s first-mover advantage Q(m 1). The follower defines its optimal entry moment, knowing that the leader is already in the market, and that the latter will have a first-mover advantage. The number of units sold by the active follower, Q, is not affected by the number of units sold by the active leader (after the follower enters the market), Qm, and thus the follower is indifferent to the leader’s first-mover advantage. More interesting, and less intuitive, is the indifference of the non-pre-emptive leader’s trigger (which is defined by Eq. (23)) to the leader’s first-mover advantage. The option value of the leader and its value function (before the follower enters the market) increase with m, in exactly the same proportion. The value of the leader’s value function, before it enters the market is given by the first line of Eq. (24), and the value of the leader’s value function after it enters the market, but before the follower enters, is given by the second line of Eq. (24). Notice that the derivative of both expressions with respect to m is X XF
b1
Kb1 b1 1 .
Thus, as the first-mover advantage increases, the option value increases and the value of the
leader, after it enters the market (but before the follower does), increases in exactly the same proportion.
14
Notice that the follower’s optimal entry policy, defined by the trigger X F, depends on the number of units sold, but it does not depend on the parameter m ¯ . The parameter m ¯ is not an indication for the follower of the number of units that it can (potentially) sell in the future (if it becomes active).
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Trigger's value
0.13 0.12 0.11 0.1 0.09 5.0
15.0
25.0
35.0
45.0
55.0
First Mover Advantage (Q(m-1)) Fig. 6. Leader’s trigger sensitivity to first-mover advantage in a pre-emptive game. The parameters are: r = 10%; l = 1%; x = 1%; a = 20%; r = 5%; Q = 100; P = 0.01; m ¯ = 2.05; q = 0; K = 5 and Q(m 1) varies from 5 to 60 (m varies from 1.05 to 1.6).
Consequently, the leader defines its optimal entry rule, as if it was a monopolist, knowing nevertheless that this monopoly is of (stochastic) finite duration.
5. Conclusions We present a real options rivalry model in which the profits per unit and the number of units sold are both stochastic variables. Closed-form solutions were obtained, under the assumptions of possibly correlated geometric Brownian motion, for the value functions of the leader and the follower for three different games. Analytical solutions for the trigger of the leader (except for pre-emption) and for the follower were also obtained. The sensitivity of value functions and trigger levels to changes in critical parameters such as number of units sold, volatilities, correlation, monopoly volume and first-mover advantage was examined. The degree of correlation between unit profits and market volume might result in different value functions and triggers, especially for followers and simultaneous investors without pre-emption. Monopoly-like volume (relative to Q) is a critical determinant of the leader’s trigger in both pre-emptive and non-pre-emptive games. First-mover advantages are significant in the definition of the leader’s optimal entry moment, if the players are fighting for the leader’s position (pre-emptive game). The model herein considers only two stochastic factors with constant drift, volatilities and correlation. The market volume multiple is based on the follower’s volume after entry, allowing for a constant monopoly-like leader’s volume multiple before the follower’s entry, and a constant first-mover advantage multiple afterwards. Relaxing these assumptions for pre-emptive, non-pre-emptive and collusion games, and comparing the implications with the results in this paper, is a challenge for future research.
Acknowledgements We appreciate the comments of an anonymous referee, Sydney Howell, Alan Jones, Mark Shackleton, Lenos Trigeorgis, Helen Weeds and the participants of the 2003 Real Options conference on parts of this
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paper. Helena Pinto gratefully acknowledges financial support from Fundac¸a˜o para a Cieˆncia e Tecnologia.
Appendix A. Derivation of PDE (9) Let Eq. (8) represent the value function of an idle follower: 1 B2 P0F 2 2 1 B2 P0F 2 2 B2 P0F BP0F BP0F r P þ a Q þ PQqra þ lP þ xQ rP0F ¼ 0: 2 BP2 2 BQ2 BPBQ BP BQ
ð8Þ
In order to obtain a closed-form solution for this equation, similarity methods can be used. Let X = PQ implying that: P0F ð X Þ ¼ P0F ð P; QÞ BP0F ðP; QÞ BP0F ð X Þ ¼ P BQ BX BP0F ðP; QÞ BP0F ð X Þ ¼ Q BP BX B2 P0F ð P; QÞ B2 P0F ð X Þ 2 ¼ P BQ2 BX 2 B2 P0F ðV ; M Þ B2 P0F ð X Þ 2 ¼ Q BP2 BX 2 B2 P0F ðV ; M Þ B2 P0F ð X Þ BP0F ð X Þ ¼ PQ þ BPBQ BX 2 BX Substituting back into Eq. (8), we obtain Eq. (9): 1 2 F X P0 Wð X Þ r2 þ a2 þ 2qra þ X P0F Vð X Þ½qra þ l þ x rP0F ð X Þ ¼ 0: 2
ð9Þ
Appendix B. Slopes of the value functions The first derivative of the value function of the follower, in the pre-emption game, where the number of units and the profit per unit are the state variables, is given by: 8 b1 1 X > > < N0 X bXF dPðFÞ ¼ r l x XF > 1 dX > : N0 X zXF rlx
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Delta behaves as expected, i.e. as total profit increases the follower’s value function increases. The sensitive of the leader’s value function to the state variable is: 8 > ¯ Þ X b1 ¯ b1 ð m m m > < Nb0 X bXF þ dPðLÞ r l x XF ¼ rlx m > dX > N0 X zXF : rlx After the follower enters the market, the first derivative of the value function of the leader to X is similar to the first derivative of the follower, increasing linearly with the state variable. Before the follower enters the market, the function can be either negative or positive depending on the magnitude of b 1 ¯ Þð XXF Þ 1 .15 As X approaches the trigger, the ratio X/X F increases implying that the term ¯ þ b1 ð m m m that can have a negative effect (m m ¯ ) is also increasing. In the game without pre-emption the delta of the leader will be:16 8 ¯ Þb1 X b1 1 1b1 ¯ X b1 ðm m m > 1b > XL 1 þ XF N0 X bXL > > > r l x r l x > < dPðLÞ ¯ Þ X b1 ¯ b1 ð m m m ¼ Nb0 ½XL ; XF þ > dX > r l x XF rlx > > m > > : N0 X NXF rlx The value function of the leader after it enters the market is exactly the same as in the pre-emption case. Before the leader enters the market the function is different. While the leader is not in the market, it has the option to wait to invest and therefore we expect an increasing function with total profit. The first derivative of the value function of the leader before it enters the market is indeed positive. Notice that 1b 1b what is different between the two terms is: m ¯ )b 1X F 1. Since X F = X Lm ¯ and with the ¯ X L 1 and (m m 1b 1 1b 1 1b 1 in the proportion m , the difference between the triggers taken to a negative power; X L N X F ¯ b1 two terms will be: m ¯ )b 1. The first term will always be larger than the second, and since the ¯ N (m m second is the one representing the negative effect, the delta of the leader before it enters the market will always be positive. If the players agree to enter simultaneously the delta is given by: 8 b1 > 1 X > < N0 X bXS dPðLÞ r l x X S ¼ mS > dX > N0 X zXS : rlx As expected, delta increases with total profits.
15
In Fig. 1 we can see that the value function of the leader is initially increasing with Q, then as it approaches the follower’s trigger the rate of increase declines, and then it is a linear function of Q after the follower enters the market. 16 Notice that the value function of the follower is the same as in the pre-emption model.
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