Real option duopolies with quasi-hyperbolic discounting

Real Option Duopolies with Quasi-hyperbolic Discounting

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Real Option Duopolies with Quasi-hyperbolic Discounting Pengfei Luo, Yuan Tian, Zhaojun Yang PII: DOI: Reference:

S0165-1889(19)30224-6 https://doi.org/10.1016/j.jedc.2019.103829 DYNCON 103829

To appear in:

Journal of Economic Dynamics & Control

Received date: Revised date: Accepted date:

5 May 2019 5 October 2019 17 December 2019

Please cite this article as: Pengfei Luo, Yuan Tian, Zhaojun Yang, Real Option Duopolies with Quasi-hyperbolic Discounting, Journal of Economic Dynamics & Control (2019), doi: https://doi.org/10.1016/j.jedc.2019.103829

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Real Option Duopolies with Quasi-hyperbolic Discounting✩ Pengfei Luoa , Yuan Tianb , Zhaojun Yangc,∗ a

School of Finance and Statistics, Hunan University, Changsha, China. b Faculty of Economics, Ryukoku University, Kyoto, Japan. c Department of Finance, Southern University of Science and Technology, Shenzhen, China.

Abstract This paper utilizes a real options and game-theoretic approach to consider the strategic real investment in a duopoly market under uncertainty with timeinconsistent preferences resulting from quasi-hyperbolic discounting. We show that the time-consistent agent becomes the leader when s/he interacts with a time-inconsistent rival. If the rival’s time inconsistency is very significant, the leader will optimally behave as if competition did not exist. If the two rivals have about the same time preferences, the leader will accelerate investment and her/his investment threshold will accord with the rival’s preemptive one, which is higher than that determined in the classical time-consistent model. The inefficiency of investment from preemptive com✩

The authors would like to thank two anonymous referees for their helpful comments and thank the participants at the 2018 FMA Annual Meeting, where an earlier version of the paper, entitled ”option games with time-inconsistent preferences”, was a semifinalist for Best Paper Award. The first and third authors acknowledge support by National Natural Science Foundation of China (Grant No. 71371068). The second author acknowledges support by JSPS KAKENHI (Grant Nos. 16K17151 and 25245046) and the Institute of Economic Research, Kyoto University. ∗ Corresponding author. Tel: +86 755 8801 8603. Email addresses: [email protected] (Pengfei Luo), [email protected] (Yuan Tian), [email protected] (Zhaojun Yang) Preprint submitted to Journal of Economic Dynamics and Control

December 26, 2019

petition is mitigated and even eliminated if the heterogeneity among agents is sufficiently high. Our model provides a behavioral explanation for the empirical fact that preemption occurs in some markets but is not present in others. Keywords: Option game, Duopoly competition, Time-inconsistent preferences, Preemptive investment. JEL: D91, G11, G32 1. Introduction The real options theory has a long research line since Myers (1977) and more often than not, the literature does not take competition into account. Actually, most firms operate in a competitive environment and their investment opportunities are not exclusive. Therefore, it is more interesting to examine how strategic interactions in a competitive environment affect the timing and pricing of the option to invest. Smets (1991) firstly develops a real options model to examine irreversible market entry for a duopoly facing stochastic demand. Grenadier (1996) provides a general and tractable approach for deriving the strategic exercise of options in real estate markets. Weeds (2002) analyzes irreversible investment in competing research projects with uncertain returns under a winner-takesall patent system. Huisman and Kort (2003) utilize a real option game approach to determine when and which technology should be adopted in a duopoly framework, which is further extended by Huisman and Kort (2004) to take into account technological progress. Mason and Weeds (2010) examine the impact of preemption on the relationship between uncertainty and

2

investment. Nishihara and Shibata (2010) consider the investment and financing policies in a duopoly model, and study the interactions between preemptive competition and financing constraint. All of these papers have strengthened the importance of extending real options analysis by including strategic interactions among agents. Lambrecht and Perraudin (2003) incorporate incomplete information into preemption investment where the agents’ costs are private information. Hsu and Lambrecht (2007) incorporate asymmetric information into a model by assuming that the challenger has complete information about the incumbent whereas the incumbent does not know well the challenger. Kong and Kwok (2007) further investigate the strategic investment decisions and option values under asymmetry on both the sunk cost of investment and revenue flows.1 Shibata and Yamazaki (2010) examine the impacts of an asymmetric access charge regulation on competitive investment strategies in a liberalized telecommunication market. To the best of our knowledge, all the papers in the literature on preemptive investment assume an exponential discounting, i.e., a constant rate of time preference. However, agents may have time-inconsistent preferences, which can arise in practice as shown in several empirical studies, see, e.g., Thaler (1981) and Loewenstein and Prelec (1992). Recently, time-inconsistent behavior of agents has received increasing attention. Grenadier and Wang (2007) consider a monopolistic investment problem under uncertainty and time-inconsistent preferences. Lien and Yu (2014) develop a discrete model to study the interplay between firm invest1

For more references, see, e.g., Azevedo and Paxson (2014), who conduct a complete

survey on strategic investment under uncertainty.

3

ment and cash flow hedging decisions under time-inconsistent preferences. Tian (2016) extends Grenadier and Wang (2007) to provide an analytically tractable framework for a monopolistic entrepreneurial firm’s capital structure and investment decisions under time-inconsistent preferences. However, all the papers above focus on decision-making problems of a single agent and emphasize that it must be interesting to extend the model to a duopoly case and explore how time-inconsistent agents would interact with time-consistent agents. Along the research line, we aim to fill this gap. Our contributions. The objective of our paper is to integrate the two strands of literature: preemptive investment and time-inconsistent preferences. We provide an analytically tractable game-theoretical real options framework taking into consideration time-inconsistent preferences. We fix the leader’s value function and follower’s option value and define preemption in general. We find that the key determinant of the Nash equilibrium is the relative time-inconsistent degree between two competitors. Therefore, we further investigate two cases in particular: a homogeneous (symmetric) case, i.e. the situation where both players have the same time-inconsistent preferences, and a heterogeneous (asymmetric) one, i.e. the situation where one player is time-inconsistent and the other is time-consistent. The main contribution of our paper is to demonstrate how time inconsistency affects preemptive investment. First, in the homogeneous case, we find that time inconsistency delays preemptive investment. As a result, the inefficiency caused by preemptive competition is mitigated. Our analysis complements the monopolistic investment literature under uncertainty and time-inconsistent preferences. Second, in the heterogeneous case, we 4

show that the time-consistent agent becomes the leader when facing a timeinconsistent rival. In particular, if the degree of the rival’s time-inconsistent preferences is weak, the time-consistent leader must accelerate investment to take the preemptive investment threshold chosen by the time-inconsistent rival. However, if the degree of the rival’s time inconsistency is strong, the time-consistent leader does not need to take an inefficient preemptive investment as if the competition disappeared, i.e., the optimal investment in a non-strategic setting is realized. As a result, we demonstrate that the inefficiency of investment caused by preemptive competition is even eliminated if the difference between the two rivals’ time-inconsistency degrees is sufficiently high. Third, it is well known in the existing literature on competition that the leader’s value function as well as the entry threshold depends on the follower’s entry threshold. Going one step further, we clarify that in the heterogeneous case, the time-consistent agent’s entry threshold also depends on the time-inconsistent rival’s preemptive threshold, the optimal threshold in a non-strategic setting and their ordering as well. As far as we know, this finding is brand new to the literature on preemptive investment and well answers the questions raised by the literature on time-inconsistent preferences that how time inconsistency works in a duopoly case and how time-inconsistent agents would interact with time-consistent agents. Forth, we find and verify that the preemptive investment threshold under the time-inconsistent case is higher than that derived from the classical time-consistent model. Last, our theoretical results provide a behavioral explanation for why preemption occurs in some markets but is not present in others. For example, this phenomenon is observed in a real estate market, which is reported by Grenadier

5

(1996). Our paper is most closely related with Grenadier and Wang (2007) and Chapter 9 of Dixit and Pindyck (1994). We extend the former into a duopoly model addressed by the latter among many others. There are however several major differences shown below. First, the former’s equilibria depend on the intra-personal competition between the current self and future self due to the time inconsistency, while the latter’s equilibria arise from the duopoly competition without considering time-inconsistent preferences. In this paper, we combine both to provide novel insightful findings, which are never discussed before in the literature to our knowledge. Second, we assume that the two players (agents) have heterogeneous beliefs, which are a novel and realistic feature from behavioral finance. Dixit and Pindyck (1994) and Grenadier and Wang (2007) only consider that the agents have homogeneous beliefs. In the heterogeneous case, we find that the leader’s investment policy depends on the time-inconsistent difference between the two players. Our conclusions produce an explanation for an empirical fact. Third, we simultaneously consider the time-inconsistent effect of Grenadier and Wang (2007) and the first-mover advantage effect of Dixit and Pindyck (1994). In the duopoly market, from the view of a rational marketer, who is time-consistent, the time-inconsistent effect can mitigate and even eliminate the investment distortion arising from the first-mover advantage effect, implying that the time-inconsistent effect improves the effectiveness of the market in the duopoly competition. By contrast, Grenadier and Wang (2007) state

6

that the time-inconsistent effect leads to the ineffectiveness in a monopoly market. Last, we examine the duopoly competition while Grenadier and Wang (2007) consider a monopoly market. Our model instead of the latter can capture preemptive action and its effects on the pricing and timing of the option to invest. The structure of the paper is as follows. Section 2 describes the setup of the model. Section 3 briefly reviews the time-consistent benchmark. As the main part of this paper, Section 4 considers the strategic investment under time-inconsistent preferences. In particular, we consider two cases: One is homogeneous and the other is heterogeneous. Section 5 examines the implications of the model by providing numerical examples. Section 6 concludes. 2. Model setup We consider two entrepreneurial firms that are both given an option to enter a market by paying a fixed cost I. The instantaneous after-tax cash flow from the whole market is assumed to be X, which is given by the following geometric Brownian motion: dX(t) = µX(t)dt + σX(t)dZ(t), t ≥ 0, X(0) = x, where Z denotes the standard Brownian motion defined on a probability space (Ω, F, P), µ is the risk-adjusted growth rate, and σ > 0 is the volatility. We call the firm firstly entering the market the leader and the other the follower. We assume that the leader receives π10 X until the follower enters the 7

market and both the leader and the follower receive π11 X after the follower has also entered the market. In our setting, we focus on the interesting case of strategic substitution (π11 < π10 ), meaning that investment is less profitable due to the increase in supply and thus there exists a first-mover advantage.2 We note that our model is time-homogeneous and therefore, without loss of generality, we assume that the current time is zero throughout the text. Moreover, all the investment thresholds discussed in the paper are naturally constant and independent of time and the existing cash flow level. For this reason, we denote by xf the investment threshold chosen by the follower, by xp the investment threshold chosen by the leader if there exists preemption, and by xl the investment threshold chosen by the leader in the non-strategic setting (i.e. the leader role is exogenously given). If the cash flow level X(t) of the market hits from below some investment threshold, the corresponding investment takes place after paying a fixed investment cost I. The corresponding investment entry times are denoted by τp := inf{t ≥ 0; X(t) ≥ xp } for the leader if there exists preemption, by τl := inf{t ≥ 0; X(t) ≥ xl } for the leader in the non-strategic setting, and by τf := inf{t ≥ τp ; X(t) ≥ xf } for the follower, respectively. As experimental and empirical evidences show, decisions being made at different points in time are usually inconsistent with each other. We therefore follow Harris and Laibson (2013) and assume that the temporal discounting interval is divided into two subintervals: the present interval and the future one. Cash flows in the present interval are discounted exponentially with 2

See Graham (2011) and Azevedo and Paxson (2014).

8

a constant discount rate ρ, whereas those in the future interval are first discounted exponentially with ρ as usual and then further discounted by an additional factor δ ∈ [0, 1]. For convergence, we assume ρ > µ, see Dixit and Pindyck (1994) and many others. Let Dn (t, s) denote self n’s intertemporal discount function: self n’s value at time t of one unit of wealth received at the future time s. That is   e−ρ(s−t) , if s ∈ [t , t ), n n+1 Dn (t, s) =  δe−ρ(s−t) , if s ∈ [t , ∞), n+1

for s > t and t ∈ [tn , tn+1 ). For each self n, time is divided into two periods, where [tn , tn+1 ) is the present period and [tn+1 , ∞) is the future one. The duration of the present period tn+1 − tn is assumed to be exponentially distributed with parameter λ. Therefore, the expectation of the present period is E[tn+1 − tn ] = 1/λ. The additional discount factor δ reflects the degree of the present bias of the preferences. The parameter δ in conjunction with the intensity λ determines the degree of time inconsistency. The smaller the δ, or the larger the λ, the stronger the degree of the time inconsistency. In particular, if δ = 1 or λ = 0, the discount function Dn (t, s) is reduced to the exponential discounting, i.e., the time-consistent benchmark. In the following, we give a short overview of the time-consistent benchmark model. Then we turn to the strategic investment games under timeinconsistent preferences. 3. Preliminaries: Time-consistent benchmark In this benchmark case, we adapt the complete information model from Chapter 9 of Dixit and Pindyck (1994) to consider the homogeneous situation 9

with two agents. We work backwards and begin by assuming one agent has already invested. We first consider the timing and pricing of the follower’s option to invest. In this case, the follower must solve a standard real options problem with a cash flow π11 X. Therefore, the value of the follower’s option to invest before his/her investment (x < x∗f ) is "Z ∞

F ∗ (x) = E =



τf∗

e−ρu π11 X(u)du − e

π11 x∗f −I ρ−µ



x x∗f

!β

−ρτf∗

I

#

,

(1)

where the superscript ∗ stands for the time-consistent benchmark, E denotes the expectation operator conditional on information available at time 0, the initial cash flow level X(0) = x, and s 2 1 1 µ µ 2ρ β= − 2+ − 2 + 2 > 1. 2 σ 2 σ σ

(2)

The investment entry threshold for the follower, x∗f , is obtained by maximizing Eq. (1) and given by3 x∗f =

β ρ−µ I. β − 1 π11

(3)

Second, we turn to the option value for the leader before investment (x < x∗l ), which is similarly given by "Z ∗ ∗

L (x) = E =

3



τf

e

−ρu

τl∗

π10 x∗l −I ρ−µ

π10 X(u)du − e



x x∗l

β

−ρτl∗

I+

Z

∞

−ρu

e

τf∗

(π10 − π11 )x∗f − ρ−µ

x x∗f

#

π11 X(u)du

!β

.

(4)

Generally speaking, we should use a smooth-pasting condition to derive such result.

10

By the first-order optimality condition, the non-strategic entry threshold for the leader (i.e., the leadership role is exogenously preassigned or the leader has an exclusive right for investing first), is given by x∗l =

β ρ−µ I. β − 1 π10

(5)

We get x∗l < x∗f from (3) and (5) since 0 < π11 < π10 as assumed previously. However, matters are different if the role of the leader or the follower is not preassigned and must be endogenously determined by the competitive equilibrium of the game between the two players. From (4), if the leader invests immediately when the cash flow level X(0) = x < x∗f , the payoff received by the leader is given by (π10 − π11 )x∗f π10 x ¯ L(x) = −I − ρ−µ ρ−µ

x x∗f

!β

.

(6)

Obviously, only if this value is larger than the value F ∗ (x) of the follower’s investment option, there is an incentive for the follower to become a leader by making a preemptive investment. For this reason, we define the preemptive ¯ ≥ F ∗ (x)}. Following Graham investment threshold by x∗p := inf{x > 0 : L(x) (2011) and Huisman (2001), we know that x∗p < x∗l < x∗f , which means that

both agents have incentive to become a leader, who has to invest earlier at x∗p than the non-strategic optimal entry threshold x∗l due to the threat of preemption. 4. Time-inconsistent preferences In this section, we turn to the strategic investment with time-inconsistent preferences. As argued before, laboratory and field studies of time preferences have provided strong evidence that decision-makers generally have 11

time-inconsistent (present-biased) preferences, see, e.g., Thaler (1981) and Loewenstein and Prelec (1992). For instance, if you are given a chance to select one of two rewards which are both far away in the future, you would prefer a later-larger reward to a sooner-smaller one, but if both rewards are brought close enough to the decision-making time, you might choose the sooner-smaller reward instead. This phenomenon explains that there is time inconsistency when you make a decision. In short, the order of two selections strongly depends on when the decision is made, i.e. the decision-making is dynamically time-inconsistent. Formally, we assume that both firms (agents) are time-inconsistent with parameters δ i and λi (i ∈ {1, 2}).4 Let T i denote the arrival time of the future

self, which is exponentially distributed with mean 1/λi . Noting that the current time is zero as assumed before, we thus have the following discount function used by firm i ∈ {1, 2}:   e−ρs , if s ∈ [0, T i ), i D (0, s) =  δ i e−ρs if s ∈ [T i , ∞),

where [0, T i ) and [T i , ∞) represent the present and future interval respectively. The investment entry time selected by firm i is denoted by τfi := inf{t ≥ 4

Two types of agents with time-inconsistent preferences, naive and sophisticated, are

addressed in the literature. However, as mentioned in Grenadier and Wang (2007), it is hard to define a competitive equilibrium for naive agents since there is no standard assumption regarding what naive agents forecast for others’ current and future selves. Therefore, in this paper, we focus on the situation where all the time-inconsistent agents are sophisticated.

12

0; X(t) ≥ xif }, where xif represents the entry threshold. Following Grenadier

and Wang (2007) and Tian (2016), the value Π∗ (x) of the after-tax cash flow after investment under time-consistent preferences is given by Z ∞  x ∗ −ρu Π (x) = E e X(u)du = . ρ−µ 0 In contrast, as far as the time-inconsistent firm i is concerned, the value Πi (x) is given by Πi (x) = E

"Z

Ti

e−ρu X(u)du +

Z

∞

δ i e−ρu X(u)du = θi

Ti

0

#

x , ρ−µ

where θi =

ρ + δ i λi − µ ∈ (0, 1], i ∈ {1, 2}. ρ + λi − µ

(7)

Again, we use a backward induction method to solve the optimization problems. First, we derive the option value F i for the follower before investment (x < xif ). Under time-inconsistent preferences, the payoff received by the follower upon investment is determined by whether the future self arrives in advance: if it arrives, i.e. τfi > T i , the payoff is π11 Π∗ (xif ) − I; if not, i.e. τfi ≤ T i , the payoff is π11 Πi (xif ) − I. Thus, if x < xif , the value of the

investment option of the follower i is n o i i F i (x) = E I{τfi ≤T i } e−ρτf [π11 Πi (xif ) − I] + I{τfi >T i } e−ρτf δ i [π11 Π∗ (xif ) − I] .

(8)

Similar to Luo et al. (2019), we conclude that the value function F i satisfies the following ordinary differential equation (ODE): 1 2 2 i00 σ x F (x) + µxF i0 (x) − ρF i (x) + λi [δ i F ∗ (x) − F i (x)] = 0, 2 13

(9)

where the symbol ’0’ represents a derivative of the function with respect to x. The last term on the left side of ODE (9) shows the change of the option value resulting from the possible arrive of the future self. Once it occurs, the current self’s continuation value changes from F i (x) to F ∗ (x), which is further discounted by a factor of δ i . The general solution of the ODE (9) is given by: π11 xif −I ρ−µ

F i (x) = δ i where

1 µ β±i = − 2 ± 2 σ

!

s

x xif

!β

1 µ − 2 2 σ

i

i

+ A1 xβ+ + B1 xβ− ,

2

+

2(ρ + λi ) , σ2

(10)

and A1 and B1 are constants to be determined. The following value-matching and smooth-pasting conditions are imposed to fix the entry threshold xif , constants A1 and B1 : i

i

F (x)|x=0 = 0, F (x)|x=xif

π11 xif π11 =θ − I, F i0 (x)|x=xif = θi . ρ−µ ρ−µ i

Second, we consider the leader’s value function Li (x) if s/he preempts the follower immediately. Naturally, the function is given by Li (x)    R T i −ρt R τfj i −ρt j i −ρτf ∗ j = E I{τ j >T i } 0 e π10 X(t)dt + T i δ e π10 X(t)dt + δ e π11 Π (xf )   f  j Rτ j +E I{τ j ≤T i } 0 f e−ρt π10 X(t)dt + e−ρτf π11 Πi (xjf ) − I, i 6= j ∈ {1, 2}. f

In the same way with the derivation of (9), it satisfies the following ODE 1 2 2 i00 ¯ ∗ (x) − Li (x)] + π10 x = 0, σ x L (x) + µxLi0 (x) − ρLi (x) + λi [δ i L 2 14

(11)

¯ ∗ (x) is given by (6) while the threshold x∗ is replaced by xj here. where L f f The general solution of ODE (11) is j

(π10 − π11 )xf π10 x Li (x) = θi − I − δi ρ−µ ρ−µ

x xjf

!β

i

i

+ A2 xβ+ + B2 xβ− ,

where A2 and B2 are constants determined by the following value-matching conditions: i

L (x)|x=0 = −I,

i

L (x)|x=xj

f

π11 xjf − I. =θ ρ−µ i

After standard calculations, we obtain the following value functions for the leader and follower, and the preemptive investment threshold optimally chosen by the two agents. Proposition 4.1. Suppose there are two agents with time preferences characterized by the parameter pair (δ i , λi ), i ∈ {1, 2}. Then the follower’s entry option value is F i (x) =

θ

π xi i 11 f ρ−µ

−I

!

x xif

!β+i

+ δi

π11 xif ρ−µ

−I

! 

x xif

!β

−

x xif

!β+i 

,

(12)

where xif is the follower’s optimal entry threshold given by xif =

δ i β + (1 − δ i )β+i ρ−µ I. i δ i (β − 1) + (θi − δ i )(β+ − 1) π11

If the leader invests immediately, the leader’s value function is !β+i j (π − π )x π x x 10 11 10 f Li (x) = θi − I − θi ρ−µ ρ−µ xjf  !β !β+i  j (π − π )x x x 10 11 f  . −δ i − ρ−µ xjf xjf 15

(13)

The preemptive investment threshold is determined by xip := inf{x > 0 : Li (x) ≥ F i (x)}. Several observations can be derived from the proposition. First, if δ i = 1 or λi = 0, i ∈ {1, 2}, then the time-inconsistent model boils down to the previous time-consistent benchmark and we recover the conclusions in the last section. Second, this proposition states that the follower’s investment option value (12) under time-inconsistent preferences consists of two parts (cf. (1)): One is the value of the investment option exercised by the current self, the present value of future cash flow reduced by the factor θi due to the time inconsistency. Since the arrival time of the future self remains uncertain, the i

investment probability (x/xif )β+ in the current period should include the influence of the parameter λi (see (10) for the option factor β+i corresponding to the discount rate ρ + λi ). The other part is the value of the investment option exercised by the future self, the present value of future uncertain cash flow additionally discounted by the factor δ i . On account of that the arrival time of the future self has already been realized, the parameter λi has no direct influence on the investment probability (x/xif )β in the future period (see (2) for the option factor β corresponding to the discount rate ρ). The investment option we consider here can be exercised only once and therefore, i

i

the amount of (x/xif )β+ , i.e. (x/xif )β+ (xif /xif )β , is subtracted to exclude the probability that the cash flow process again hits the investment threshold xif in the future period after initially hitting the threshold in the current period. The leader’s value function (13) can be similarly interpreted. Last but most interestingly, Proposition 4.1 implies that agent 1 is the 16

leader if and only if x1p < x2p , which is completely determined by who is more time-inconsistent between the two agents specified by the parameter pair (δ i , λi ), i ∈ {1, 2}. Therefore, to make a sharp contrast, in the following text, we focus on preemption investment in two cases: a homogeneous one where all the agents have the same time preference parameter pair (δ i , λi ), and a heterogeneous one where one agent is time-consistent and the other is not. 4.1. The homogeneous case In this case, both agents (firms) have the same time-inconsistent preferences and therefore we take δ i = δ, λi = λ, i ∈ {1, 2}. Naturally, we immediately conclude from Proposition 4.1 the following conclusions: Corollary 4.2. If the two agents have the same time-inconsistent preferences specified by (δ, λ), the follower’s entry option value is  !β s !β     s s π x π x x x 11 11 f f −I −I  s +δ − F s (x) = θ ρ−µ xsf ρ−µ xf

x xsf

!β s 

,

(14)

where β s is just β+i defined by (10) except that λi is replaced with λ, and θ is just θi defined by (7) except that λi is replaced with λ and δ i is replaced with δ. The follower’s optimal entry threshold is given by xsf =

δβ + (1 − δ)β s ρ−µ I. δ(β − 1) + (θ − δ)(β s − 1) π11

(15)

If the leader invests immediately, the leader’s value function is  !β s !β s s (π − π )x (π − π )x π10 x x x 10 11 f 10 11 f s  L (x) = θ −I−θ −δ − ρ−µ ρ−µ xsf ρ−µ xsf

x xsf

(16)

17

!β s 

.

The preemptive investment threshold is determined by xsp := inf{x > 0 : Ls (x) ≥ F s (x)}. Therefore, one of the two agents (leader) invests once the

cash flow level hits xsp from below while the other (follower) invests once the level hits xsf . Proof. The last assertion of the corollary can be easily proved according to Section 3 of Graham (2011). The remaining conclusions are directly derived from Proposition 4.1. Roughly speaking, the smaller the intensity parameter λ, the more beneficial to preempt. Moreover, under our model assumption (π11 < π10 ), if the current cash flow level is higher than the optimal entry threshold of the follower, all agents should invest immediately. Therefore the preemptive investment threshold must be less than the follower’s entry threshold. More formally, we have Proposition 4.3. If all agents have the same time-inconsistent preferences specified by (δ, λ) and the intensity parameter satisfies λ <

1 ρ−µ , δ β−1

there exists

a unique preemptive threshold xsp ∈ (0, xsf ). Proof. The proof is relegated to the Appendix. Remark 1. We guess and our numerical experiments show that Proposition 4.3 holds for all agents specified by 0 < δ ≤ 1 and λ > 0 instead of only ones specified by λ <

1 ρ−µ . δ β−1

However, we cannot prove this in general after many

attempts and so we leave it open for the moment. To provide us with intuitions, Figure 1 shows the leader’s value function if s/he invests immediately and the value function of the follower’s option to 18

30

Fs Ls

25

Value

20

15

10

5

0

−5

xsf

xsp 0

0.5

1

1.5

x

2

2.5

3

3.5

Figure 1: The leader’s value function and the follower’s option value with regard to the cash flow x in a homogeneous case. The parameter values are ρ = 0.08, µ = 0.06, σ = 0.2, δ = 0.3, λ = 0.33, I = 5, π10 = 1, and π11 = 0.5.

enter the market with regard to the cash flow x. The two functions intersect at two points: the smaller one is xsp and the bigger one is xsf indicating that there exists a unique preemptive threshold xsp ∈ (0, xsf ). In addition, the figure explains that once the cash flow is higher than the threshold xsf , the two functions coincide as we expected. 4.2. The heterogeneous case As pointed out previously, the relative degree of the time-inconsistency between the two agents determined by (δ i , λi ) is the determinant of who is the leader. Intuitively, the higher the degree of the time-inconsistency, the less the project value, the less the investment option value, and therefore the agent would be more reluctant to invest, i.e. s/he is more unlikely to be a leader. Naturally, in the heterogeneous case, without loss of generality, we 19

assume agent 1 has time-inconsistent preferences defined by 0 < δ 1 = δ < 1 with λ1 = λ > 0 and by contrast, agent 2 is time-consistent, i.e. δ 2 = 1 or λ2 = 0. We assume that xap and xa∗ p are agent 1’s and agent 2’s preemptive investment threshold respectively, which are determined according to Proposition 4.1 with a few minor changes for an obvious reason. For example, we have xap := inf{x > 0 : La (x) ≥ F s (x)}, where F s (x) is given by (14) and La (x) ≡ θ

π11 )x∗f

(π10 − π10 x −I−θ ρ−µ ρ−µ

x x∗f

!β s

−δ

π11 )x∗f

(π10 − ρ−µ

 

x x∗f

!β

−

x x∗f

!β s 

(17)

for x < x∗f with x∗f being the follower’s entry threshold given by (3). We conclude Lemma 4.4. Regarding to the four preemptive investment thresholds defined a∗ ∗ a∗ ∗ s a in the previous text: x∗p , xsp , xap , xa∗ p , we have xp < xf and xp ≤ xp ≤ xp ≤ xp .

a In particular, xa∗ p ≤ xp means that the time-consistent agent is the leader in

the heterogeneous case. Proof. The proof is relegated to the Appendix. a From the Lemma 4.4 we conclude that xa∗ p = xp holds if and only if the

two agents are both time-consistent, meaning that the model degenerates into the time-consistent benchmark. Proposition 4.5. In the heterogeneous case, the strategy profile (xsf , min{xap , x∗l }) is the Nash equilibrium, where x∗l and xsf are given by (5) and (15) respectively.

20

,

6

20

Fs

s

F

a

L

4

La

15

Value

Value

2

0

10

5 −2

0

−4 *

xap

xf −6

0

0.2

0.4

x

0.6

0.8

−5

1

(a) Value versus x

0

0.2

0.4

x

0.6

0.8

1

(b) Value versus x

Figure 2: For a time-inconsistent agent, her/his value function if the agent is a leader and her/his option value if the agent is a follower with the cash flow x in a heterogeneous case. The parameter values are ρ = 0.08, µ = 0.06, σ = 0.2, λ = 0.33, I = 5, π10 = 1, π11 = 0.5 and (a) δ = 0.3 or (b) δ = 0.8.

Proof. Thanks to the last lemma, it is sufficient to prove that the optimal entry threshold, denoted by xa∗ l , of the time-consistent agent (leader) equals min{xap , x∗l }. First, we clearly have xa∗ ≤ xap . Second, for a given entry l threshold y of the leader, the value, denoted by J a∗ (x, y), of the leader’s option to enter is given by J a∗ (x, y) =



π10 y −I ρ−µ

  β (π10 − π11 )xsf x − y ρ−µ

x xsf

!β

.

Differentiating J a∗ (x, y) with respect to y leads to   ∂J a∗ (x, y) xβ π10 y = β+1 (1 − β) + βI > 0 (< 0) ⇔ y < x∗l (y > x∗l ). ∂y y ρ−µ a ∗ a∗ a Therefore we get xa∗ l = min{xp , xl } since xl ≤ xp .

21

60

F* La*

50

Value

40

30

20

10

0

−10

xa* p 0

0.5

1

x

1.5

2

Figure 3: For a time-consistent agent, her/his value function if the agent is a leader and her/his option value if the agent is a follower with the cash flow x in a heterogeneous case. The parameter values are ρ = 0.08, µ = 0.06, σ = 0.2, δ = 0.3, λ = 0.33, I = 5, π10 = 1, and π11 = 0.5.

Remark 2. Depending on the time-inconsistent parameter δ, the preemptive threshold xap may surpass x∗f , see Figure 2(a) where δ = 0.3, meaning that ∗ ∗ ∗ a xa∗ l = xl since xl < xf (< xp ) as pointed out before. In the same way, the

preemptive threshold xap may be less than x∗f as well, see Figure 2(b) where δ = 0.8. Moreover, Figure 3 shows that there exists a unique preemptive investment threshold xa∗ p , of which the proof is presented in the Appendix. a The assertion of x∗p ≤ xsp and of xa∗ p ≤ xp predicts that the time-consistent

agent becomes the leader. However, should the leader invest preemptively at xa∗ p ? Interestingly, the answer is negative. Remember that in a non-strategic setting, where the leader has an exclusive right to invest first, the optimal entry threshold is x∗l (≥ xa∗ p ), given by (5). The lower the preemptive threshold xa∗ p , the more significant the inefficiency if we view the non-strategic 22

threshold x∗l as an efficient one. In other words, the time-consistent agent has to sacrifice the option value of waiting in order to become the leader and a enjoy the first-mover advantage. In fact, the ordering xa∗ p ≤ xp not only im-

plies who is the leader but also that it is unnecessary for the time-consistent agent to enter the market at xa∗ p , a potentially much lower threshold than the non-strategic threshold x∗l . The time-consistent agent can delay her/his investment strategically and still be the leader. If the rival’s time-inconsistent preference is strong enough, the preemptive threshold xap would be higher than x∗l , meaning that the time-consistent agent can freely choose the optimal entry threshold x∗l in a non-strategic case. Definitely, if the rival’s time-inconsistent preference is weak, the preemptive threshold xap would be lower than x∗l , meaning that the rational time-consistent agent has to ’preemptively’ invest by taking xap as her/his investment threshold rather than xa∗ p . Proposition 4.5 is insightful and interesting. First, it demonstrates that in a heterogeneous case, the time-consistent agent’s entry threshold depends on the time-inconsistent rival’s preemptive threshold xap , the optimal threshold x∗l in a non-strategic setting, and the ordering of the two thresholds as well. This is in sharp contrast to the classical conclusion that the leader’s value function and entry threshold depend only on the follower’s value function and follower’s entry threshold. Second and more importantly, we know from Proposition 4.5 that if the rival is time-inconsistent, the time-consistent agent does not need to invest as early as the cash flow level xa∗ p is reached, i.e. the agent can delay investment up to the time when the cash flow level min{xap , x∗l } is hit. In other words, the inefficiency caused by the preemptive 23

competition is mitigated (if xap < x∗l ) or perfectly eliminated (if x∗l < xap ). Formally, we have Corollary 4.6. The preemptive investment threshold under time-inconsistent case is higher than the preemptive threshold derived from the time-consistent benchmark model (classical case). Proof. On one hand, according to Lemma 4.4, the preemptive investment threshold xsp under the homogeneous and time-inconsistent case is higher than the preemptive investment threshold x∗p under the classical case, i.e. x∗p ≤ xsp . On the other hand, under the heterogeneous and time-inconsistent case, we have x∗p ≤ x∗l following Graham (2011) and Huisman (2001). Additionally,

we have x∗p ≤ xap thanks to Lemma 4.4. Therefore, we have x∗p ≤ min{xap , x∗l }. Hence, the corollary holds from Proposition 4.5.

5. Model implications In this section, we provide comparative static analysis and highlight the impact of time inconsistency on preemptive investment. The baseline parameter values are set as: ρ = 0.08 according to Nishihara and Shibata (2010); µ = 0.06 and σ = 0.2 from Shibata and Yamazaki (2010); π10 = 1 and π11 = 0.5 from Shibata and Yamazaki (2010) after normalized; δ = 0.3 and λ = 0.33 thanks to Grenadier and Wang (2007). The investment cost (sunk cost) is set as I = 5. Figure 4 plots the preemptive investment thresholds versus time inconsistency parameters δ and λ. It says that the smaller the time-inconsistent discount factor δ or/and the larger the intensity λ, i.e., the stronger the 24

2

1.6

xa* p

1.4

xap

1.8 1.6

Investment threshold

Investment threshold

2

xsp

1.8

x*p x*l

1.2 1 0.8 0.6

1.2 1 0.8 0.6 0.4

0.4 0.2

1.4

0.2 0

0.2

0.4

δ

0.6

0.8

0

1

(a) investment trigger versus δ

0

0.2

0.4

λ

0.6

0.8

1

(b) investment trigger versus λ

Figure 4: The investment thresholds with time inconsistency parameters (a) δ and (b) λ, where x∗p and xsp are the preemptive investment thresholds in the benchmark and homogeneous case respectively, and xap and xa∗ p are the time-inconsistent agent’s and the time-consistent agent’s preemptive threshold in the heterogeneous case respectively with x∗l being the optimal investment threshold in a non-strategic setting. The parameter values are ρ = 0.08, µ = 0.06, σ = 0.2, δ = 0.3, λ = 0.33, π10 = 1, π11 = 0.5 and I = 5.

25

time-inconsistency, the later the preemptive investment. We believe such conclusion holds true in general but we cannot strictly prove this at present. However, if there is no competition, one can strictly prove that the stronger the agent’s time-inconsistency, the later the investment, i.e. ∂xsf ∂xsf < 0 and > 0, ∂δ ∂λ where xsf is given by (15). For more details, please see Grenadier and Wang (2007) and Tian (2016). In particular, Figure 4 indicates that both the preemptive investment threshold (xsp ) derived from the homogeneous case and that (min{xap ; x∗l }) derived from the heterogeneous case are higher than the preemptive threshold x∗p derived from the time-consistent benchmark model. This numerical conclusion is verified by Corollary 4.6, i.e. it holds for all parameter sets. The economic intuition is as follows. On one hand, the competition in a duopoly market induces preemptive investment due to the first-mover advantage. On the other hand, time inconsistency has two opposite effects on investment: One is to accelerate investment. This is because time inconsistency brings intra-personal competition between the current self and future self. The agent has an incentive to investment in a hurry to avoid suboptimal decisions (from the current self’s viewpoint) made in the future. The other effect is to delay investment due to the decrease in the value upon investment. This is obvious, since there is an extra discount factor δ in the future interval. However, the second effect dominates the first and hence investment under time-inconsistent preferences is delayed at last. In this paper, we consider both intra-personal competition between the current self and future self due to time inconsistency and interpersonal competition due to the 26

first-mover advantage. We find that time inconsistency delays the preemptive investment and thus plays a role of mitigating investment inefficiency due to preemption. Figure 4 also shows that the preemptive investment threshold (xap ) of the time-inconsistent agent is higher than the preemptive threshold (xa∗ p ) of the time-consistent agent and the latter becomes the leader, whose entry threshold is delayed to min{xap , x∗l }. This numerical conclusion holds true actually in general as confirmed by Proposition 4.5. Moreover, Figure 4 displays that there exists a unique critical level δ¯ = ¯ = 0.1, such that if δ < δ¯ or λ > λ, ¯ we have xa > x∗ , i.e. the 0.6 or λ p l entry threshold of the time-consistent agent is x∗l . On the other hand, if the rival’s time-inconsistent preference is weak enough, we have x∗l > xap , meaning that the time-consistent leader strategically selects the investment threshold exactly equal to the rival’s preemptive threshold xap . In short, if the rival’s time-inconsistent degree is low, the leader accelerates investment and her/his investment threshold accords with the rival’s preemptive threshold; in the opposite situation, the leader chooses an optimal investment threshold as if s/he had an exclusive right to invest first. These conclusions must hold generally in the option game for any two agents defined by parameter pair (δ, λ), since the parameter pair is the determinant of the Nash equilibrium. In practice, it is difficult to estimate time-inconsistent parameters δ and λ. In view of this, our results are quite helpful in that we do not need to know the exact parameter values but the relative degree of the time-inconsistent preferences between the two players.

27

Grenadier (1996) documents empirically that in real estate markets, some of their developments are more gradual than others. The phenomenon can be explained by our model: The investors in a soaring housing market have more approximate parameter pair (δ, λ) among them than those in a gradual development one. That is, in the former case, all investors have roughly the same investment thresholds as our model predicts and in the latter one, investors would take different investment thresholds, inducing that preemption seldom happens. In this sense, our theoretical results are consistent with empirical evidences and provide a behavioral explanation for why preemption occurs in some markets but is not present in others. 6. Conclusion In this paper, we shed light on strategic real investment under uncertainty in a duopoly market with time-inconsistent preferences resulting from quasihyperbolic discounting. We use a game-theoretical real options approach. We show that a time-consistent agent becomes the leader when s/he interacts with a time-inconsistent rival. If the rival’s time inconsistency is sufficiently high, the time-consistent agent would behave in a non-strategic way, just as if there did not exist preemptive competition, i.e. the agent had an exclusive right to invest first. In contrast, if the two rivals have about the same time preferences, let alone that they have the same time preference, the leader will invest at the preemptive investment threshold selected by the player who has a higher degree of time-inconsistency, leading to overinvestment. The preemptive investment threshold under the time-inconsistent case is higher than that determined in the classical time-consistent model. 28

We demonstrate that time inconsistency delays preemptive investment. As a result, due to time inconsistency, the inefficiency caused by preemption competition is mitigated and even eliminated if the difference of time preferences between the two competitors is sufficiently high. We find that there are two opposite effects in the option game: one is the preemption effect due to the first-mover advantage, which gives rise to overinvestment; the other is the time-inconsistent effect, leading to underinvestment. We show that the time-inconsistent effect would dominate the preemption effect if the time preferences between the two players have a major difference. Appendix Proof of Proposition 4.3. Let N (x) = Ls (x) − F s (x) = N1 (x) + N2 (x), where    π10 xsf π10 x  N1 (x) = (1 − δ) −I − −I ρ+λ−µ ρ+λ−µ

and



N2 (x) = δ 

π10 x −I − ρ−µ



π10 xsf −I ρ−µ



x xsf

x xsf

!β s 

!β  



for all x < xsf and N (x) = 0 for all x ≥ xsf . Differentiating function N (·) 00

00

twice with respect to x, we obtain N 00 (x) = N1 (x) + N2 (x), where   s π10 xsf xβ −2 00 s s N1 (x) = −(1 − δ)β (β − 1) s β s −I , (xf ) ρ+λ−µ and xβ−2 N2 (x) = −δβ(β − 1) s β (xf ) 00

29



 π10 xsf −I . ρ−µ

1 ρ−µ , δ β−1

Since π10 > π11 , β > 1, β s > 1 and λ <

00

we have N1 (x) < 0 and

00

N2 (x) < 0, and thus, we have N 00 (x) < 0, implying that function N (·) is strictly concave on the interval [0, xsf ]. In addition, N (0) = −I < 0 and

N (xsf ) = 0, and therefore there exists a unique point, xsp , within the interval (0, xsf ) such that N (xsp ) = 0. Proof of Lemma 4.4. ∗ We first verify the existence and uniqueness of xa∗ p , which is less than xf . We

define function P (x) := La∗ (x) − F ∗ (x), where F ∗ (x) is given by (1) and !β (π10 − π11 )xsf π10 x x a∗ L (x) = −I − . ρ−µ ρ−µ xsf We have (π10 − π11 )xsf π10 P (x) = −I − ρ−µ ρ−µ

x xsf

!β

−



π11 x∗f −I ρ−µ



x x∗f

!β

for all x < x∗f and P (x) = 0 for all x ≥ x∗f . Differentiating function P (·) twice with respect to x, we obtain     π11 x∗f π10 − π11 s 1−β 00 β−2 ∗ −β P (x) = −β(β − 1)x (xf ) + − I (xf ) − . ρ−µ ρ−µ 00

Since π10 > π11 and β > 1, we have P (x) < 0 and thus P (x) is strictly concave on the interval [0, x∗f ]. Since P (0) = −I < 0 and P (x∗f ) > 0, there ∗ a∗ exists a unique point, xa∗ p , within the interval (0, xf ) such that P (xp ) = 0.

To prove the remaining part of the lemma, let L(x; xf ) =

10 x θ πρ−µ

−I −

10 x = θ πρ−µ −I +

(π −π )x θ 10 ρ−µ11 f

 β s x xf

(δ−θ)(π10 −π11 )xf ρ−µ

−  β x xf

30

(π −π )x δ 10 ρ−µ11 f

−δ

  β

(π10 −π11 )xf ρ−µ

x xf

−

 β x xf

.

 β s  x xf

Differentiating function L(x; xf ) with respect to the independent variable xf , we have ∂L(x;xf ) ∂xf

=

(δ−θ)(π10 −π11 )xf (1 ρ−µ

− β s)

 −β s x xf

−δ

(π10 −π11 )xf (1 ρ−µ

− β)

 β x xf

.

Since δ ∈ [0, 1], we have δ − θ = (δ − 1)(ρ − µ)/(ρ + λ − µ) ≤ 0. Noting that

π10 > π11 and β s ≥ β > 1, we have ∂L(x; xf )/∂xf ≥ 0. Grenadier and Wang (2007) have already proved that time inconsistency delays investment, i.e., xsf > x∗f . We can therefore prove that Ls (x; xsf ) ≥ La (x; x∗f ), where function

Ls (·; xsf ) is given by (16) and La (·; x∗f ) is given by (17).

From the definition of xap and xsp , we conclude La (xap ; x∗f ) = F s (xap ; xsf ) (≤ Ls (xap ; xsf )) and Ls (xsp ; xsf ) = F s (xsp ; xsf ), where function F s (·; xsf ) is given by (14). Hence, we finally obtain xap ≥ xsp .

s a∗ a∗ ∗ Similarly, from the definition of xa∗ p and xp , we conclude L (xp ; xf ) =

s ∗ a∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ F ∗ (xa∗ p ; xf ) (≥ L (xp ; xf )) and L (xp ; xf ) = F (xp ; xf ). Comparing them ∗ leads to xa∗ p ≤ xp .

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