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European Journal of Operational Research 190 (2008) 834–854 www.elsevier.com/locate/ejor
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Induced effects and technological innovation with strategic environmental policy Atsuyuki Ohyama
a,*
, Motoh Tsujimura
b
a
b
Graduate School of Economics, Kyoto University, Yoshida-honmachi, Sakyo, Kyoto 606-8501, Japan Department of Economics, Ryukoku University, Tsukamoto, Fukakusa, Fushimi, Kyoto 612-8577, Japan Received 11 November 2005; accepted 9 August 2007 Available online 23 August 2007
Abstract This paper investigates an environmental policy designed to reduce the emission of pollutants under uncertainty, where the agents’ problem is formulated as an optimal stopping problem. We first analyze the single-agent’s case according to Pindyck [Pindyck, R.S., 2002. Optimal timing problems in environmental economics. Journal of Economic Dynamics and Control 26, 1677–1697]. We then extend the model to the case in which there are two competing agents. Therefore, we consider the external economic effects that are peculiar to an agent’s environmental policy decision. Finally, we consider the effect of technological innovation. The results of the analysis suggest that if there are two competing agents, they implement environmental policy simultaneously. Furthermore, the threshold for implementing environmental policy is higher when there are two agents, and how long these two agents take to implement environmental policy depends on the magnitude of the external economic effect. Furthermore, when we consider the effect of technological innovation, we show that the incentive to be the leader occurs if an additional condition is satisfied. 2007 Elsevier B.V. All rights reserved. JEL classification: Q53; C72; G19 Keywords: Induced effects; Technological innovation; Environmental policy; Real options; Optimal timing
1. Introduction The Kyoto protocol on global warming came into effect on 16 February 2005. According to this agreement, each country’s emissions target must be achieved during the period 2008–2012. For example, Japan will be obliged to reduce its 1990 level of greenhouse gases emissions by 6% (as for Canada, Hungary and Poland), while Switzerland, most Central and East European states and the European Union must reduce their emissions by 8%. However, there is some concern because the US and a number of other key countries have yet to ratify the protocol. On their part, these countries maintain that there is no clear benefit from implementing the *
Corresponding author. Tel./fax: +81 75 753 3511. E-mail addresses: [email protected] (A. Ohyama), [email protected] (M. Tsujimura).
0377-2217/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.08.006
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Kyoto protocol. Since there are countries which ratified the protocol and countries which have not ratified it, we find the relation between these countries. This paper then examines the problem of implementing strategic environmental policy. In this paper, we investigate environmental policies designed to reduce the emission of a pollutant under uncertainty. It is assumed that agents face a problem by attempting to minimize both the expected total damage from a pollutant and the cost of implementing the environmental policy. To solve the agent problem, we extend the works of Dixit and Pindyck (1994) and Pindyck (2002) concerning the optimal timing for an environmental policy implementation under irreversibility and uncertainty. We also account for the influence of the strategic competition framework on environmental policy timing: that is, we combine real options and a strategic competition. Following the analysis in Kijima and Shibata (2004) of strategic entry decisions in an oligopoly market when the underlying state variable follows a geometric Brownian motion, we formulate the agents’ problem as an optimal stopping problem. See, also Grenadier (1996), Kulatilaka and Perotti (1998) and Lambrecht and Perraudin (2002) for discussion of strategic competition. We assume that the agent receives damage from the pollutant. Following Dixit and Pindyck (1994) and Pindyck (2002), the damage function is then defined as a function of the pollutant stock. In this paper, we consider the external economic effects of an agent’s policy implementation. We examine the technological innovation that may appear some time after the environmental policy is implemented. It is natural that the environmental policy stimulates technological innovations, for example, innovations in automobile engines and electric power plants. See Faffe et al. (2003) for an excellent survey of environmental policy and technological innovations. We assume that agents only commence research after policy implementation, and once any of the agents have discovered the technological innovation, the remaining agents cannot claim the benefits of discovery. This means that a race to discover technological innovation, such as a winner-takesall patent system, occurs. In this paper, we formulate the technological innovations following Weeds (2002). See also, for example, Dasguputa and Stiglitz (1980), Dixit (1988) and Huisman and Kort (2004). We assume that technological innovation occurs randomly and that innovation discovery by each agent is independent. Therefore, we consider the combined effects of real options, strategic competition and technological innovation. When an agent implements environmental policy, we assume that the efficiency of the policy depends on the technological innovation. In this paper, we assume that if one agent implements the environmental policy, this induces environmental improvements for all other agents. That is, the remaining agents’ environment also improves through implementation of this environmental policy. We call this effect the induced effect as in Barrieu and Chesney (2003). We consider three cases: a complete induced effect, a partial induced effect, and no induced effect. The complete induced effect is generally applied to analyze the problem of global warming. To prevent an increase in the earth’s surface temperature, the atmospheric concentration of greenhouse gases (GHGs) (such as carbon dioxide (CO2), methane (CH4), chlorofluorocarbons (CFCs) and nitrous oxide (N2O)) must be reduced. Thus, global emissions of GHGs should be controlled. This means that if a country implements a policy of reducing GHGs emissions, all countries will benefit equally from its reduction. The partial induced effect is often applied to analyze the problem of acid rain. To reduce acid rain, it is necessary to reduce the emission of as its primary causes in the form of SO2 and NOx. While these gases sometimes diffuse over hundreds of kilometers, acid rain is a generally regional problem, unlike global warming. Finally, the no induced effect is usually applied to analyze the problem of soil contamination. Soil contamination results from spilled or buried hazardous substances. Thus, soil contamination problems are localized to places where hazardous substances are spilled or buried. See Hanley et al. (1997) for more details of global warming and acid rain. The results of our analysis are as follows. If there are two competing agents, both agents have an incentive to let the other agent implement the policy first. Therefore, they both will implement the policy at a later date than a single agent would have done. How long these two agents take to implement the environmental policy depends on the magnitude of the induced effect. This implies that a kind of external effect occurs. Also, this has crucial implications for international policy and regulation. The reason is that if each country considers the timing of implementing an environmental policy under a strategic competition framework, the environmental problems are never resolved optimally from a global perspective. This result may give an explanation why countries are not eager to implement the environmental policy in order to meet the Kyoto protocol. However, if we consider the optimal timing for an environmental policy taking into account technological innovation,
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this result changes. Even if the induced effect is complete, and an external authority never adopts the intervention to promote an environmental policy, the effect of the innovation may actually create a spirit of competition in the decision of the environmental policy if an additional condition is satisfied. Therefore, the incentive to be the leader also occurs (first mover advantages arise), depending on the degree of technological innovation and the induced effect. In other words, the result implies that there exist scenarios in which a preemption equilibrium prevails. The paper itself is organized as follows. In the next section, we describe the agent’s problem and analyze the single-agent problem following Dixit and Pindyck (1994) and Pindyck (2002). In Section 3, we examine the case of two agents where there is an induced effect. In Section 4, we consider technological innovation in the presence of two agents. Section 5 concludes. 2. Optimal timing for an environmental policy in the single-agent case We consider the problem faced by agent i, i = 1, 2, whose objective is to control the accumulation of a pollutant. For example, policymakers may aim to stabilize global climate change or the acid rain effect. We assume that agents have to implement the policy, which is to reduce the flow of the pollutant. We follow Dixit and Pindyck (1994) (Chapter 12, Section 3) and Pindyck (2002). Suppose that the agent i emits the pollutant by pit , and that its stock Y it is governed by dY it ¼ ðpit dY it Þdt;
Y i0 ¼ y;
ð1Þ
where d is the rate of natural decay of the stock of pollutant. In this context, we assume that the initial stock of the pollutant does not depend on agents’ attributes. Let Bi ðX t ; Y it Þ : Rþþ Rþ ! R denote the agent i’s damage (or negative benefit) function associated with the stock of the pollutant Y it . It is assumed that Bi ðX t ; Y it Þ ¼ X t Y it ;
ð2Þ
where Xt is a parameter that stochastically shifts over time to reflect damage due to the pollutant and is assumed to be governed by dX t ¼ lX t dt þ rX t dW t ;
X 0 ¼ x;
ð3Þ
where l 2 R and r 2 R n f0g are constants. (Wt)tP0 is a standard Brownian motion process defined on a filtered probability space ðX; F; P; ðFt ÞtP0 Þ satisfying the usual conditions.1 Ft is generated by Wt in R; i.e. Ft ¼ rðW s ; s 6 tÞ. Note that Xt is independent of the properties of agents. For simplicity, assume that pit remains constant at its initial level pi0 until agent i implements the policy. When agent i implements the policy, pi0 reduces to pi1 with 0 6 pi1 < pi0 . Thus, Eq. (1) becomes ( ðpi0 dY it Þdt; 0 6 t < si ; i dY t ¼ ð4Þ ðpi1 dY it Þdt; si 6 t < 1; where si 2 T is the decision time of agent i and T is the class of all decision times relative to ðFt ÞtP0 . Furthermore, we assume that Z 1 rt i E e jBðX t ; Y t Þjdt < 1; ðAS:1Þ 0
where r 2 Rþþ is a discount rate. Let Ki be the constant cost of implementing the environmental policy for agent i. Therefore, the agent’s problem is to choose si 2 T to minimize the expected total discounted cost associated with the environmental policy: Z 1 i rt i i rsi i V ðx; yÞ ¼ inf E e B ðX t ; Y t Þdt þ e K ; ð5Þ i s 2T
0
i
where V is the value function of agent i’s problem.
1
See, for example, Karatzas and Shreve (2001).
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Let us define the operator Li as follows: 8 < 1 r2 x2 o22 þ lx o þ ðpi0 dyÞ o r; x 2 H i ; 2 ox oy ox Li ¼ : 1 r2 x2 o22 þ lx o þ ðpi dyÞ o r; x 2 6 H i; 1 2 ox oy ox
ð6Þ
where Hi is the continuation region defined by H i :¼ fx 2 Rþþ ; V i ðx; yÞ < Gi ðx; yÞg; i
ð7Þ
i
i
where G (x, y) is given by rewriting J (x, y; s ) as follows: "Z i Z 1 Z s rt i i rsi i E e B ðX t ; Y t Þdt þ e K ¼ E ert Bi ðX t ; Y it Þdt þ 0
0
"Z
si
ert Bi ðX t ; Y it Þdt þ e
¼E
#
1 rt
e B si
rsi
Z
¼E
si
e
rt
B
i
ðX t ; Y it Þdt
þe
ðX t ; Y it Þdt
1
e
rðtsi Þ
si
0
"Z
i
rsi
G
i
ðX si ; Y isi Þ
þe
rsi
K
i
Bi ðX t ; Y it Þdt þ K i
#
# :
ð8Þ
0
Thus, the decision time s is defined by si ¼ infft > 0; x 62 H i g:
ð9Þ
We analyze the case in which a single-agent implements an environmental improvement policy. See Pindyck (2000) for details. The single-agent’s problem is to choose sS 2 T to minimize the expected total discounted cost: Z 1 S V S ðx; yÞ ¼ inf E ert BS ðX t ; Y St Þdt þ ers K S : ð10Þ sS 2T
0
Since the agent has not implemented the environmental policy for x 2 HS, we obtain 1 LS V S ðx; yÞ þ BS ðx; yÞ ¼ r2 x2 V Sxx þ lxV Sx þ ðpS0 dyÞV Sy rV S þ xy ¼ 0: 2
ð11Þ
On the other hand, since the agent has implemented the policy for x 62 HS, we have 1 LS V S ðx; yÞ þ BS ðx; yÞ ¼ r2 x2 V Sxx þ lxV Sx þ ðpS1 dyÞV Sy rV S þ xy ¼ 0: 2
ð12Þ
Eqs. (11) and (12) must be solved by using the following appropriate boundary conditions: V ðx; yÞjx¼0 ¼ 0;
ð13Þ
S
S
S
S
S x ðx; y; x
S
S x ðx; y; x
S
S
V ðx; y; x 2 H Þjx¼xS ¼ V ðx; y; x 62 H Þjx¼xS þ K ; V
2 H Þjx¼xS ¼ V
62 H Þjx¼xS ;
ð14Þ ð15Þ
where xS is the critical value of the shift parameter x. When x reaches xS, the single-agent implements the policy. Thus, the decision time is given by sS ¼ infft > 0; x P xS g:
ð16Þ
The conditions given by Eqs. (14) and (15) are the well-known value-matching and smooth-pasting conditions, respectively. Then, the value function of the single agent is given by 8 S < A1 xb1 þ xp0 þ xy ; x < xS ; q1 q2 q2 S V ðx; yÞ ¼ ð17Þ : xpS1 þ xy ; x P xS ; q1 q2 q2
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where A1 is a constant to be determined, q1 = r l and q2 = r + d l. b1 is the positive solution of the following characteristic equation: 1 2 r bðb 1Þ þ lb r ¼ 0: 2
ð18Þ
Thus, we have 1 l b1 ¼ 2 þ 2 r
"
l 1 r2 2
2
2r þ 2 r
#12 > 1:
ð19Þ
The first line of Eq. (17) can be discussed as follows. The first term on the right-hand side represents the value of the option to implement the environmental policy in the future. The sum of the second and third terms on the right-hand side is the present value of the damage. Conversely, the second line of Eq. (17) refers to the following. The sum of the two terms on the right-hand side represents the present value of the damage. Note that the emission flow of the pollutant decreases from pS0 to pS1 due to the policy implementation. From Eqs. (14), (15) and (17), we obtain b1 q1 q2 K S S x ¼ ; ð20Þ b1 1 pS0 pS1 b 1 b b 1 1 pS0 pS1 1 : ð21Þ A1 ¼ 1 S b1 q1 q2 K 3. Optimal timing of an environmental policy under a strategic competition As mentioned in Section 1, there are countries which ratified the protocol and countries which have not ratified the Kyoto protocol. In this section, for simplicity, we study the case in which there are only two competing agents. Suppose that both agents have experienced damage from pollutants and wish to determine when they should implement environmental policy to reduce pollutant emissions. The agent F is the last to implement the policy. This agent is referred to as the follower. The agent L implements the environmental policy before or at the same time as agent F. This agent is referred to as the leader. We assume that if one agent (say L) implements the environmental policy, it affects the environmental improvement of the other agent (say F). That is, the other agent experiences environmental improvement because of the implementation of the environmental policy. This effect is referred to as the induced effect (IE), as in Barrieu and Chesney (2003). Let piN i N j be the emission flow of agent i, where for k 2 {i, j}, we denote 0; if the agent k has not implemented the policy; ð22Þ Nk ¼ 1; if the agent k has implemented the policy: We assume that the total amount of emission reduction does not depend on the magnitude of the induced effect. We also assume that the two agents have the same emission structure as follows: pS0 ¼ pL00 ¼ pF00 ;
pS1 ¼ pL11 ¼ pF11 ;
ðpS0 pS1 Þ ¼ ðpL00 pL10 Þ þ ðpF00 pF01 Þ ¼ ðpL10 pL11 Þ þ ðpF01 pF11 Þ; pS pS1 6 pL10 6 0 ; 2
ðAS:2Þ
pS0 6 pF01 6 pS0 : 2
Furthermore, we assume that the implementation cost in the single-agent case is equal to the sum of the implementation cost of the leader and follower, i.e. 2K S ¼ K L þ K F :
ðAS:3Þ
We now examine the induced effect shown in Fig. 1. Suppose that the agents, L, F, emit GHGs such as CO2 or methane and an increase in GHGs is assumed to raise the earth’s average temperature. To prevent this, the atmospheric concentration of the GHGs must be reduced. This means that if the agent L implements a policy
A. Ohyama, M. Tsujimura / European Journal of Operational Research 190 (2008) 834–854
π 0S
L π 00
F π 00
π 0S
L π 00
F π 00
π 0S
L π 00
839
F π 00
IE F π 01
IE
π 10L
F π 01
π 10L π 1S
π 11L
π 11F
Complete induced effect
π 1S
π 11F
π 11L
Partial induced effect
π 1S
π 10L
π 11F
No induced effect
Fig. 1. Induced effect (IE).
to reduce the emission of the GHGs, the other agent F receives benefit from the reduction in emissions, just as if the agent F had implemented the policy. That is, the induced effect is complete. Thus, we have pL10 ¼ pF01 ¼ pS0 =2 from the emission structure (AS.2). Next, suppose that the agents emit either SO2 or NOx as a primary cause of acid rain. Prevailing winds blow these gases across state and national borders; sometimes for hundreds of kilometers. However, winds do not carry the gases globally. Thus, acid rain is a regional problem, unlike global warming. Thus, the reduced effect is partial and we have pL10 < pF01 from the emission structure (AS.2). Next, we suppose that the agents emit a hazardous substance. Then, there is no induced effect, and we have pL10 ¼ pS1 and pF01 ¼ pS0 (see Fig. 1). Thus, the induced effect, IE, is defined by IE ¼ pF00 pF01 ¼ pL10 pL11 :
ð23Þ
Since all parameter values and actions are common knowledge, the game is one of complete information. 3.1. The follower’s problem In this subsection, we consider the follower’s problem. As in the standard strategic real-options framework, we assume that the leader has already implemented the environmental policy. See, for example, Kijima and Shibata (2004) and Huisman and Kort (1999). The follower’s problem is then similar to the single-agent problem. Note that the follower receives the induced effect from the leader’s policy implementation. Thus, the dynamics of the pollutant stock, Eq. (4), becomes ( d Yb Ft ¼ ðpF01 d Yb Ft Þdt; 0 6 t < sF ; F dY t ¼ ð24Þ d Ye F ¼ ðpF d Ye F Þdt; sF 6 t < 1; t
11
t
F
where s is the followers policy decision time that is defined by sF ¼ infft > 0; X t P xF g:
ð25Þ F
In this context, we assume that if the shift parameter Xt reaches the threshold x , the follower implements the environmental policy. Then, the follower’s problem is to choose sF 2 T to minimize the expected total discounted costs: "Z F # Z 1 s F F rt F F rs F rt F F E e B ðX t ; Yb t Þdt þ e K þ e B ðX t ; Ye t Þdt : ð26Þ V ðx; yÞ ¼ inf F s 2T
0
sF
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As in the single-agent case, Section 2, we obtain the follower’s value function as 8 F e 1 xb1 þ xp01 þ xy ; x < xF ; 1 and A e and xF by using the value-matching and smooth-pasting conditions as follows: obtain A b1 1 F b p01 pF11 1 e 1 ¼ b1 1 ; ð28Þ A b1 q1 q2 KF b1 q1 q2 K F : ð29Þ xF ¼ b1 1 pF01 pF11 Using the strong Markov property and the tower property of conditional expectations, we have "Z F # Z 1 s F F rt F rs F rt F V ðx; yÞ ¼ E e X t Yb t dt þ e K þ e X t Ye t dt Z ¼E
0 1
sF
Z h Fi ert X t Yb Ft dt þ E ers K F þ E
0 F xp01
1
sF
xy x b1 xF ðpF11 pF01 Þ ¼ þ þ F þ KF : x q1 q2 q1 q2 q2
ert X t Ye Ft X t Yb Ft dt ð30Þ
The detailed derivation of the last term in Eq. (30) is in Appendix. Then, the follower’s value function is i 8 xpF F F F h < q q01 þ qxy þ xxF b1 K F x ðpq01qp11 Þ ; x < xF ; 1 2 2 1 2 ð31Þ V F ðx; yÞ ¼ : xpF11 xy F F þ þ K ; x P x : q1 q2 q2 For x < xF, the sum of the first and the second terms on the right-hand side is the present value of the damage. In this context, since the leader has implemented the environmental policy and the induced effect occurs, the follower’s emission flow of the pollutant decreases from pF00 to pF01 . The remaining terms represent the value of the option to implement the environmental policy in the future. On the other hand, For x P xF, the sum of the first and second terms is also the present value of the damage. Note that the emission flow of the pollutant decreases from pF01 to pF11 due to the implementation of the policy by the follower. The last term represents the policy implementation cost. 3.2. The leader’s problem We now examine the leader’s problem. In this subsection, the leader makes a policy decision under the assumption that the follower will act optimally in the future. Once the leader implements the environmental policy, the leader has no further decisions to make. Moreover, the leader obtains the benefit from the future policy implementation by the follower due to the induced effect. In this case, the dynamics of the stock of the pollutant is given by 8 L L bL b 0 6 t < sL ; > < d Y t ¼ ðp00 d Y t Þdt; dY Lt ¼ d Ye Lt ¼ ðpL10 dt d Ye Lt Þdt; sL 6 t < sF ; ð32Þ > : L L L F dY t ¼ ðp11 dt dY t Þdt; s 6 t < 1; where sL is defined by sL ¼ infft > 0; X t P xL g:
ð33Þ
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841
In this case, we assume that if the shift parameter Xt reaches the threshold xL, the leader implements the environmental policy. Hence, the leader’s problem is to choose sL 2 T so as to minimize the expected total discounted costs: "Z L # Z sF Z 1 s L L rt L L rs L rt L L rt L L V ðx; yÞ ¼ inf E e B ðX t ; Yb Þdt þ e K þ e B ðX t ; Ye Þdt þ e B ðX t ; Y Þdt : ð34Þ t
sL 2T
t
sL
0
sF
t
After the follower implements the policy, the leader’s (and the follower’s) value is given by the present value of the damage. That is, we have that V L ðx; yÞjx¼xF ¼ V F ðx; yÞjx¼xF :
ð35Þ
Note that since there is no optimality on the part of the leader, there is no corresponding smooth-pasting condition (see Weeds, 2002 for more detail). Using the strong Markov property and the tower property of conditional expectations as in Eq. (30), we have "Z L # Z sF Z 1 s L V L ðx; yÞ ¼ E ert X t Yb L dt þ ers K L þ ert X t Ye L dt þ ert X t Y L dt t
0
sL
t
sF
t
Z 1 Z 1 h Li ¼E ert X t Yb Lt dt þ E ers K L þ E ert X t Ye Lt X t Yb Lt dt 0 sL Z 1 ert X t Y Lt X t Ye Lt dt þE sF
F L xpL xy x b1 xL ðpL10 pL00 Þ x b1 x ðp11 pL10 Þ ¼ 00 þ þ L þ KL þ F : x q1 q 2 x q1 q2 q1 q2 q2 It can be shown that the value function for the leader is given by 8 L x b1 h L xL ðpL00 pL10 Þi x b1 hxF ðpL10 pL11 Þi xp00 xy > þ þ K q1 q2 xF ; x < xL ; > xL q1 q2 q2 q1 q2 > > < L b hxF ðpL pL Þi xp 10 11 V L ðx; yÞ ¼ q q10 þ qxy þ K L xxF 1 ; xL 6 x 6 xF ; q1 q2 1 2 2 > > > > : xpL11 þ xy þ K L ; x P xF : q2 q1 q2
ð36Þ
ð37Þ
For x < xL in Eq. (37), the sum of the first and second terms on the right-hand side is the present value of the damage. The third term represents the value of the option to implement the environmental policy in the future. The fourth term is an option-like term that represents the effect of the follower’s future implementation on the leader’s value function. Next, for xL 6 x 6 xF in Eq. (37), the first term represents the present value of the damage from the pollutant emission flow. Note that the emission flow decreases from pL00 to pL10 due to the policy implementation of the leader. Lastly, for x P xF in Eq. (37), the first term represents the present value of the damage from the pollutant emission flow. Note that the emission flow decreases from pL10 to pL11 due to the policy implementation of the follower. 3.3. Results of the two-agents case In this subsection, we examine both the leader’s and follower’s optimal decision times. We find that the leader has no incentive to implement the policy before the follower because of the induced effect. In other words, we show that first mover advantages do not arise in this model. Suppose that the leader implements the environmental policy and reduces pollutant emissions. Although the follower has not implemented the environmental policy, the follower’s environment improves due to the induced effect. Note especially that the degree of environmental improvement depends on the magnitude of the induced effect. To illustrate these
842
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findings, we investigate the shapes of the value functions of the leader and the follower. We first examine the follower’s value function for x < xF. From (30) and the follower’s value function for x < xF is xpF01 xy x b1 F xF ðpF01 pF11 Þ F V ðx; yÞ ¼ þ þ F K : ð38Þ x q1 q2 q1 q2 q2 The first and the second derivatives with respect to x are as follows: x b1 1 1 pF y xF ðpF01 pF11 Þ F K ; V Fx ðx; yÞ ¼ 01 þ þ b1 F x xF q1 q2 q1 q2 q2 x b1 2 1 xF ðpF01 pF11 Þ F V Fxx ðx; yÞ ¼ b1 ðb1 1Þ F K : 2 x q1 q2 ðxF Þ
ð39Þ ð40Þ
The term xF ðpF01 pF11 Þ=q1 q2 is the present value of the benefit of policy implementation. KF denotes the cost of policy implementation. It follows that xF ðpF01 pF11 Þ=q1 q2 P K F . However, since x < xF, V Fx > 0. On the other hand, V Fxx 6 0. Next, we examine the leader’s value function for xL 6 x 6 xF. From Eq. (37), the leader’s value function for L x 6 x 6 xF is x b1 xF ðpL pL Þ xpL10 xy L L 10 11 V ðx; yÞ ¼ þ þK F : ð41Þ x q1 q2 q1 q2 q2 The first and the second derivatives with respect to x are as follows: x b1 1 ðpL pL Þ pL y 10 11 ; V Lx ðx; yÞ ¼ 10 þ b1 F x q1 q2 q1 q2 q2 x b1 2 1 ðpL pL Þ 10 11 V Lxx ðx; yÞ ¼ b1 ðb1 1Þ F : x xF q1 q 2
ð42Þ ð43Þ
For xL 6 x 6 xF, V Lx > 0. On the other hand, V Lxx < 0. Thus, the leader’s value function is strictly concave for xL 6 x 6 xF. Then, we have the following result. Lemma 3.1. Suppose that Assumptions (AS.1)–(AS.3) hold. See Sections 2 and 3. Moreover, assume that KL ¼ KF :
ðAS:4Þ
Then, we obtain V L ðx; yÞ > V F ðx; yÞ;
xL 6 x < xF :
ð44Þ
Lemma 3.1 implies that the leader never implements policy before so does the follower. It means that a kind of attrition game arises in this game. Then, we obtain the following result. Proposition 3.1. Suppose that Assumptions (AS.1)–(AS.4) hold. Then, we have xL ¼ xF :
ð45Þ
Therefore, from Eqs. (42) and (43), the leader and the follower simultaneously implement the environmental policy. Furthermore, given Eqs. (20) and (29), we obtain the following result: Proposition 3.2. Suppose that (AS.1)–(AS.4) hold. Then we obtain xS < xL ¼ xF :
ð46Þ S
F
L
Moreover, the greater the induced effect, the greater the difference between x and x = x .
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Proof. Let h be defined by h :¼ xS xF. From Eqs. (20) and (29), it follows that b1 KS KF : qq h¼ b1 1 1 2 pS0 pS1 pF01 pF11 It follows from (AS.3) and (AS.4) that b1 KF KF b1 1 1 F h¼ ¼ < 0: qq qqK IE þ ðpF01 pF11 Þ pF01 pF11 b1 1 1 2 pS0 pS1 pF01 pF11 b1 1 1 2
843
ð47Þ
ð48Þ
The second equality in Eq. (48) holds because of Eq. (23): pS0 pS1 ¼ IE þ ðpF01 pF11 Þ. Let IEC be the complete induced effect and let IEP be the partial induced effect. Further, let IEN denote the no induced effect. Then, we have pS0 pS1 ¼ IEC > IEP > IEN ¼ 0: 2
ð49Þ
Thus, as the induced effect increases, the difference between xS and xF = xL increases. The proof is completed. h Proposition 3.2 implies that the environmental policy is implemented faster in the single-agent case than that in the two competing-agents case. This has crucial implications for international policy and regulation. It means that if each country considers the timing of implementing an environmental policy under a strategic competition framework, the environmental problems are never resolved optimally from a global perspective. 3.4. Numerical example and comparative statics In this section, we present a comparative static results of r, d, l, r and K. Table 1 presents the base case parameter values. We first calculate b1, AS1 and xS by using the base case parameter values. Then, we obtain that b1 = 1.41, S A1 ¼ 15883:6, and xS = 0.052. Note that, in this paper, we discuss the damage (negative benefits) instead of profit flows. Fig. 2 illustrates the value function for the single agent. This suggests that the environmental policy should be implemented when the shift parameter x reaches xS = 0.05. Similarly, Fig. 3 illustrates the both agents’ value functions: VL and VF. It is obvious from Fig. 3 that the leader’s value function never crosses the follower’s value function. Thus, strategic agents have no incentive to be the leader. Therefore, the two strategic agents simultaneously implement the environmental policy when the shift parameter reaches xF = 0.069. Note that the critical value for the follower is larger than in the single-agent case because there is the induced effect. This implies that inefficiency occurs in the competing-agent’s problem. We next present the results of a comparative-static analysis for r, d, l, r, K and IE by varying parameters by ±30% in Table 2. Table 2 implies as follows. Since the cost K is paid when the policy is implemented, an Table 1 Parameter values Parameter
Value
r d l r Ki p00 ¼ ps0 pL10
The The The The The The The
discount rate natural rate expected percentage rate of growth of x volatility parameters of x cost of implementing the policy for agent i initial pollutant’s flow pollutant’s flow of the leader
0.04 0.02 0.01 0.3 100 20 12.5
pF01 p11 ¼ ps1 y
The pollutant’s flow of the follower The pollutant’s flow after adoption of all policy The stock of the pollutant
17.5 10 100
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600 500
VS
400 300 200 100 0
0
0.01
0.02
0.03
0.04
0.05
x Fig. 2. The value function for the single agent.
800
PV of Damage VF L V
700 600
V
500 400 300 200 100 0 0
0.01
0.02
0.03
x
0.04
0.05
0.06
0.07
Fig. 3. The value functions of the leader and follower.
Table 2 The results of comparative-static analysis xs
xF
xL
xs
xF
xL
r + 30% r r 30%
0.075 0.052 0.032
0.100 0.069 0.043
0.100 0.069 0.043
r + 30% r r 30%
0.069 0.052 0.038
0.092 0.069 0.051
0.092 0.069 0.051
d + 30% d d 30%
0.058 0.052 0.045
0.077 0.069 0.061
0.077 0.069 0.061
K + 30% K K 30%
0.067 0.052 0.036
0.09 0.069 0.048
0.09 0.069 0.048
l + 30% l l 30%
0.048 0.052 0.056
0.064 0.069 0.075
0.064 0.069 0.075
IE + 30% IE IE 30%
– – –
0.077 0.069 0.063
0.077 0.069 0.063
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xF 1
2
3
4
5
IE
0.09
0.08
0.07
0.06
Fig. 4. Dependence of critical threshold, xF, on IE.
increase in r implies a greater reduction in the present value of that cost. Thus, an increase in the discount rate r increases the value of the option to implement the policy and the thresholds: xS, xF and xL. A higher value of d implies that the environmental damage from emission is more reversible, such that the sunk benefit of implementing the policy now, rather than waiting, is lower. Also, the more uncertainty over the future cost of the pollutant, the greater the incentive to wait, rather than implement the policy now. Hence, an increase in r raises the thresholds. Furthermore, an increase in l implies that each agent can observe the fact that the damage from emissions increases in the future. Thus, an increase in l decreases the thresholds. Finally, we examine the sensitivity of the critical value xF to the magnitude of the induced effects. See Table 2 and Fig. 4. Note that IE = 5 represents the complete induced effect, 0 < IE < 5 represents the partial induced effect and IE = 0 represents no induced effect from the base case parameter values in Table 1. Table 2 and Fig. 4 show that the bigger the induced effects, the bigger the threshold xF. Since the induced effect provides the external effect to agents, an increase in the induced effects mean that the external effects increase. Thus, it is natural that the bigger the induced effects, the higher the threshold xF. 4. Optimal timing for an environmental policy with technological innovation In this section, and in addition to the induced effect we consider the effect of technological innovation following Weeds (2002). When agent i implements the environmental policy with cost Ki, we assume that the agent can take advantage of technological innovation. We assume that technological innovation occurs randomly and that the time in which it occurs follows an exponential distribution with a constant hazard rate ki > 0. See, for example, Dasguputa and Stiglitz (1980), Dixit (1988), and Huisman and Kort (2004). We further assume that the discovery by each agent is independent. We also assume that agents only start researching from the time of policy implementation, and once any of the agents has made the discovery, the others cannot claim further benefits from the discovery. In other words, the agent who has made the discovery can only obtain benefits from technological innovation. All parameter values and actions are assumed to be common knowledge, and hence, the game is one of complete information, as in Section 3. Since the damage from the pollutant depends on the shift parameter x, the value of technological innovation depends on the shift parameter x. We assume that agent i receives a positive benefit from technological innovation by aXt, where a > 0. Thus, the damage function (2) becomes ( X t Y it ; t < sTIi ; i i BTI ðX t ; Y t Þ ¼ ð50Þ X t Y it aX t ; t P sTIi ; where sTIi is the agent i’s policy implementing time.
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4.1. The single-agent’s problem with technological innovation In this section, we examine the single-agent’s problem with technological innovation. The single-agent’s problem is to choose the policy implementing time sTIS to minimize the expected total discounted costs: Z 1 S S rt S rsTIS S VTI ðx; yÞ ¼ inf ¼ E e BTI ðX t ; Y t Þdt þ e K : ð51Þ sTIS 2T
0
Suppose that the single-agent implements the environmental policy when the shift parameter xt reaches the threshold xTIS. Then, as in Section 2, we obtain the value function as follows: 8 S < ASTI xb1 þ xp0 þ xy ; x < xTIS ; q2 q q 1 2 ð52Þ VTIS ðx; yÞ ¼ : xpS1 þ xy akx þ K S ; x P xTIS ; q1 q2 q2 q3 where q3 = r + k l. The parameters ASTI and xTIS are determined by solving the following value-matching and smooth-pasting conditions: b 1 b b 1 1 akq1 q2 þ ðp0 p1 Þq3 1 ASTI ¼ 1 S ; ð53Þ b1 q1 q2 q3 K b1 q1 q2 q3 K S xTIS ¼ : ð54Þ b1 1 akq1 q2 þ ðpS0 pS1 Þq3 The difference in the value function of the single agent from that in Section 2 (between VTIS and VS) represents the present value of the effect of technological innovation. See Eqs. (17) and (52). Furthermore, the threshold xTIS is also to be added the term of technological innovation. See Eqs. (20) and (54). 4.2. The follower’s problem In this section, we investigate the follower’s problem with technological innovation. Suppose that the leader has already implemented the environmental policy. The dynamics of the pollutant stock, Eq. (24), becomes ( d Yb Ft ¼ ðpF01 d Yb Ft Þdt; 0 6 t < sTIF ; F dY t ¼ ð55Þ d Ye F ¼ ðpF d Ye F Þdt; sTIF 6 t < 1; 11
t
t
where sTIF is the follower’s policy implementing time. Then, the follower’s problem is to choose sTIF to minimize the expected total discounted costs: "Z TIF # Z 1 s F F F F ert BTI ðX t ; Yb Ft Þdt þ ers K F þ ert BTI ðX t ; Ye Ft Þdt : ð56Þ VTI ðx; yÞ ¼ inf E sTIF 2T
As in Section 3, we have "Z TIF Z s TIF F VTI ðx; yÞ ¼ E ert X t Yb Ft dt þ ers K F þ Z
sTIF
0
0
1
e sTIF
rt
X t Ye Ft dt
Z
#
1
e
rt 2kt
e
kaX t dt
sTIF
Z 1 Z 1 h Fi F rs F rt F F rt 2kt b e b ¼E e X t Y t dt þ E e e X t ð Y t Y t Þdt E e e kaX t dt K þE 0 sTIF sTIF x b1 akxTIF xpF xy x b1 xTIF ðpF11 pF01 Þ ¼ 01 þ þ TIF þ K F TIF ; ð57Þ x q1 q2 x q1 q2 q2 q4 1
rt
where q4 = r + 2k l and b1 is the positive root of the following characteristic equation: 1 2 r b ðb 1Þ þ lb ðr þ kÞ ¼ 0: 2
ð58Þ
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847
Thus, b1 is b1
1 l ¼ 2þ 2 r
"
l 1 r2 2
2
2ðr þ kÞ þ r2
#12 ð59Þ
> 1:
xTIF is the threshold of follower in which technological innovation is considered. Hence, the follower’s value function is given by
F
VTI ðx; yÞ ¼
8 h i h i xTIF ðpF01 pF11 Þ xpF x b1 x b1 akxTIF > < xTIF KF xTIF þ q1 q012 þ qxy2 ; q4 q1 q2
x < xTIF ;
> : xpF11 þ xy akx þ K F ; q4 q1 q2 q2
x P xTIF :
ð60Þ
The difference between the value functions VTIF and VF comes from the effect of technological innovation in VTIF. Thus, the value function includes the present value of the effect of technological innovation: akxTIF/q4. The threshold xTIF is found by solving the value-matching and smooth-pasting condition for VTIF at x = xTIF: b1 K F xTIF ¼ ðpF pF Þðb 1Þ : akðb 1Þ 1 01 11 þ q14 q1 q2
ð61Þ
The difference between thresholds xTIF and xF comes from the effect of technological innovation. Note that the follower’s value function VTIF has two option-like terms in Eq. (60). The second term means that the follower has an opportunity to obtain benefit by discovering technological innovation. In other words, if the leader discovers the technological innovation before the follower implements the environmental policy, the follower never gets the benefits from discovering the technological innovation. The second term is the same as in Section 3. 4.3. The leader’s problem We now examine the leader’s problem. As in Section 3, we assume that the leader makes a policy decision with the assumption that the follower will act optimally in the future. Once the leader implements the environmental policy, the leader has no further decisions to make. Moreover, the leader receives two opposite effects from the future policy implementation by the follower: positive effects from the induced effects and negative effects from the rival (follower) entry on the competing game for discovery of technological innovation. In this case, Eq. (32) becomes 8 > d Yb L ¼ ðpL00 d Yb Lt Þdt; > > < t dY Lt ¼ d Ye Lt ¼ ðpL10 d Ye Lt Þdt; > > > : L dY t ¼ ðpL11 dY Lt Þdt;
0 6 t < sTIL ; ð62Þ
sTIL 6 t < sTIF ; sTIF 6 t < 1;
where sTIL is the leader’s policy implementing time. The leader’s problem is then to choose sTIL to minimize the expected total discounted costs: "Z
sTIL
VTIL ðx; yÞ ¼ inf E sTIL 2T
0
b L ðX t ; Yb L Þdt þ e ert B t
rsTIL
K Lþ
Z
sTIF sTIL
b L ðX t ; Ye L Þdt þ ert B t
Z
1 sTIF
# b L ðX t ; Y L Þdt : ert B t ð63Þ
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As in Section 3.2, we obtain the following: "Z TIL Z s L rt L rsTIL L b VTI ðx; yÞ ¼ E e X t Y dt þ e K þ t
0
sTIF
e X t ð Ye Lt ekt kaÞdtþ rt
sTIL
Z
1
e sTIF
rt
X t ðY Lt
e
2kt
kaÞdt
Z 1 Z 1 Z 1 h TIL i ¼E ert X t Yb Lt dt þ E ers K L þ E ert X t ð Ye Lt Yb Lt Þdt E ert ekt kaX t dt TIL sTIL 0Z 1 Z 1 s þE ert X t ðY Lt Ye Lt Þdt E ert kaX t ðe2kt ekt Þdt TIF TIF s s xpL00 xy x b1 L x b1 xTIL ðpL10 pL00 Þ x b1 akxTIL ¼ þ þ TIL K þ TIL TIL x x q1 q2 x q1 q2 q2 q3 x b1 xF ðpL pL Þ x b1 akxTIF akxTIF 11 10 þ TIF þ TIF : ð64Þ x q1 q2 x q3 q4 Both agents (leader and follower) scramble for technological innovation, after follower implements the policy. Therefore, note that the value of technological innovation changes from k ekt · aXt into 2k e2kt 12 aX t (see Weeds, 2002 for details). It can be shown that the value function for the leader is 8 L x b1 h L xTIL ðpL00 pL10 Þi x b1 hakxTIL i xp00 xy > > þ þ K xTIL > xTIL q1 q2 q3 > q1 q2 q2 > h i h > x b1 ak2 xTIF x b1 xTIF ðpL10 pL11 Þi > > > þ xTIF xTIF ; x < xTIL ; > q1 q2 q3 q4 > > < L h i x b1 ak2 xTIF ð65Þ VTIL ðx; yÞ ¼ qxpq10 þ qxy akx þ K L þ xTIF q q3 q4 1 2 2 3 > > h i > x b1 xTIF ðpL pL Þ > > 10 11 > xTIF ; xTIL 6 x < xTIF ; > q1 q2 > > > > > : xpL11 þ xy akx þ K L ; x P xTIF : q1 q2
q2
q4
The difference between the value functions VTIL and VL also comes from the effect of technological innovation in VTIL. Thus, the value function has the present values of the effects of technological innovation, akxTIL/q4 and ak2xTIF/(q3q4). Before the follower has implemented the environmental policy, the leader’s value function with respect to the technological innovation is given by the second line of Eq. (65). It is worth noting that, after the follower has implemented the environmental policy, the denominator of the third term of Eq. (65) is replaced by q4. This means that a rival in the research for the technological innovation has made an appearance. 4.4. Results of technological innovation In this section, we assume an additional condition such that an agent has an incentive to become the leader. To this end, we examine the shapes of the value functions of the leader and the follower. First, we examine the follower’s value function. From Eq. (60), the follower’s value function for x < xTIF is xpF01 xy x b1 F xTIF ðpF01 pF11 Þ x b1 akxTIF F VTI ðx; yÞ ¼ þ þ TIF K TIF : ð66Þ x q1 q2 x q1 q2 q2 q4 The first and second derivatives with respect to x are as follows: x b1 1 1 x b1 1 1 akxTIF pF y xTIF ðpF01 pF11 Þ F K VTIFx ðx; yÞ ¼ 01 þ þ b1 TIF b ; ð67Þ 1 x xTIF q1 q2 xTIF xTIF q1 q2 q2 q4 x b1 1 1 x b1 1 1 akxTIF xTIF ðpF01 pF11 Þ F K ðb 1Þ VTIFxx ðx; yÞ ¼ b1 ðb1 1Þ TIF b : 1 1 2 2 x q1 q2 xTIF q4 ðxTIF Þ ðxTIF Þ ð68Þ
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Let qF1 be given by qF1 :¼ ½K F xTIF ðpF01 pF11 Þ=q1 q2 . Since b1 > 1 and pF01 > pF11 , the sign of Eq. (68) depends on the sign of qF1 . The term xTIF ðpF01 pF11 Þ=q1 q2 is the present value of the benefit from policy implementation. The parameter KF is the cost of policy implementation. It follows that xTIF ðpF01 pF11 Þ=q1 q2 P K F . Then, the sign of qF1 is negative. It followers from b1 > 1 that the second term of Eq. (68) is negative. For x < xTIF we obtain VTIFxx < 0. On the other hand, the sign of Eq. (67) depends on the value of a and on the magnitude of the induced effect. We next examine the leader’s value function. From Eq. (65), the leader’s value function for xTIL 6 x 6 xTIF is x b1 xTIF ðpL pL Þ x b1 ak2 xTIF xpL xy akx 10 11 þ K L TIF VTIL ðx; mÞ ¼ 10 þ þ TIF : ð69Þ q3 x q1 q2 x q1 q2 q2 q3 q4 The first and second derivatives with respect to x are as follows: x b1 1 ðpL pL Þ x b1 1 ak2 pL y ak 11 10 VTILx ðx; mÞ ¼ 10 þ þ b1 TIF þ b1 TIF ; x q1 q2 x q1 q2 q2 q3 q3 q4 x b1 2 1 ðpL pL Þ x b1 2 1 ak2 L 10 11 VTIxx ðx; mÞ ¼ b1 ðb1 1Þ TIF þ b1 ðb1 1Þ TIF : x ðxTIF Þ q1 q 2 x ðxTIF Þ q3 q4
ð70Þ ð71Þ
Since b1 > 1, b1 > 1, pL00 > pL10 , and pL10 > pL11 , the signs of Eqs. (70) and (71) depend on the value of a and on the magnitude of the induced effect. Proposition 4.1. Suppose that (AS.1) and (AS.2) hold. If a satisfies the inequality L b1 1 q3 q4 p10 pL11 a> ; b1 1 q1 q2 k2
ð72Þ
then we have VTIL < VTIF for x < xTIF: the curves for VTIL and VTIF cross each other before x reaches xTIF. Then the strategic agents have an incentive to be a leader. Namely, first mover advantages arise. Proof. We choose the value of the parameter a so that VTILx ðxTIF ; yÞ > GTIx ðxTIF ; yÞ, where GTIðx; yÞ ¼ xp11 þ qxy2 akx þ K. Then, we obtain that q1 q2 q4 L p pL11 qq VTILx ðxTIF ; yÞ GTIx ðxTIF ; yÞ ¼ ðb1 1Þ 10 þ ðb1 1Þ 3 2 4 > 0: ð73Þ q1 q2 k It yields inequality (72). This completes the proof.
h
This proposition implies that there exist scenarios in which a preemption equilibrium prevails. 4.5. Numerical examples and comparative statics with technological innovation In this section, we calculate the thresholds and the value functions for the single agent, the leader, and the follower when technological innovation is considered. We then show the results of a comparative-static analysis. As in Section 3, we assume in this section that the induced effect is partial. First, we show the value function for the single agent. Suppose that a = 3000, such that Proposition 4.1 is satisfied. Then, we obtain that xTIS = 0.04. As in Section 3.4, we show the value function of the single agent in Fig. 5. Fig. 5 means that the single agent should implement the environmental policy when the shift parameter x reaches xTIS = 0.04 in the case of a = 3000. Note that, compared with Section 2, the value of the option to implement the environmental policy increases. This is because the value of technological innovation is taken into account. Next, we show the value functions for the competing two agents: the leader and follower. In this case, both the critical values of the leader and the follower depend on the value of a. Since ax represents technological innovation, the parameter a describes the magnitude of technological innovation. Suppose that k = 0.05. The larger the parameter a, the lower the critical values of both the leader and follower. Moreover, we need to pay attention to the value function of the leader because it is ambiguous whether two competing-agents implement
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400
PV of Damage VTI S
350 300
VTI S
250 200 150 100 50 0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
x Fig. 5. The values function for the single agent with technological innovation.
the environmental policy simultaneously or not. Consequently, we must classify each case and evaluate each value of a. To determine if the strategic agents have an incentive to be the leader, we need to classify the following three cases for each value of a. (1) When a = 0, as is shown in Section 3, agents have no incentive to be the leader. (2) When 0 < a < 3312, an increase in parameter a makes each critical value smaller and the strategic agents have no incentive to be the leader. Therefore, the two strategic agents adopt the policy on the same critical value. (3) If 3312 < a, as in the case shown in Fig. 6 agents have an incentive to be the leader. Fig. 6 shows the thresholds of the leader and the follower for a = 20,000 such that we set up the extreme parameter a in order to make the result clear. In the region of a > 3312, the leader’s value function is below the follower’s value function before the follower’s value function reaches the threshold. Therefore, the critical value of the leader is different from that of the follower. As Fig. 6 shows, in the region of 0.01 < x < 0.02, the leader’s value function is lower than the follower’s value function. The first crossing point is important because an incentive to be the leader appears at this point
200 180
VTI F PV of Damage VTI L
160 140
VTI
120 100 80 60 40 20 0 0
0.005
0.01
0.015
x
0.02
0.025
Fig. 6. The value functions of the leader and follower for a = 20,000.
0.03
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851
(x = 0.01). That is, Fig. 6 means that the leader should implement the environmental policy when the shift parameter x reaches xTIL = 0.01. We now present a comparative-static analysis for the thresholds: xTIS, xTIF, and xTIL, by varying the parameters: r, d, l, r and a. The base case parameter values present in Table 1. Suppose that a = 7000 such that we again choose the appropriate parameter a for this analysis. The results of the comparative-static analysis are presented in Table 3. Note that the parameter a = 7000 implies that an agent have an incentive to be the leader. We assume that the parameter a = 5000 such that we obtain the tangible results of comparativestatic analysis when k is varied. The result in Table 3 illustrates that the influence of the hazard rate k on the leader’s threshold is relatively larger than the influence on the follower’s threshold. This means that an increase of k increases the fear of starting a race to discover technological innovation, such as a winnertakes-all patent system. Therefore, the leader’s threshold becomes lower. Fig. 7 shows the critical value xTIF as a function of the induced effects (IE) for a = 0, 2000, 5000 and 7000. This shows that the larger the induced effects, the greater the threshold xTIF. Since we regard the induced effect in this analysis as the external effect, an increase in the induced effects means that the external effects also increase. Therefore, we can consider it natural that the larger the induced effects, the greater the threshold xTIF. Fig. 8 shows how the critical values xTIS, xTIF and xTIL decrease with respect to a. We find the following from Fig. 8. We find that an incentive to implement the environmental policy rather than to wait becomes
Table 3 The results of comparative-static analysis with TI xTIS
xTIF
xTIL
xTIS
xTIF
xTIL
r + 30% r r 30%
0.038 0.031 0.024
0.042 0.034 0.026
0.030 0.025 0.020
r + 30% r r 30%
0.041 0.031 0.023
0.044 0.034 0.026
0.032 0.025 0.019
d + 30% d d 30%
0.033 0.031 0.029
0.037 0.034 0.032
0.026 0.025 0.024
K + 30% K K 30%
0.041 0.031 0.022
0.045 0.034 0.024
0.032 0.025 0.021
l + 30% l l 30%
0.030 0.031 0.032
0.033 0.034 0.036
0.024 0.025 0.026
k + 30% k k 30%
0.034 0.035 0.037
0.039 0.040 0.044
0.027 0.033 0.041
0.1 0.09 0.08
alpha=0 alpha=2000
x TIF
0.07 0.06
alpha=5000
0.05 0.04 0.03 alpha=7000
0.02
0
1
2
3
4
5
IE Fig. 7. Dependence of critical threshold, xTIF with technological innovation, on IE.
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0.07 TIF
x x TIL x TIS
0.06
xTIS,xTIF,xTIL
0.05
Threshold of F
0.04
Threshold of S
0.03
0.02 Threshold of L
0.01
0
2000
4000
6000
8000
10000
alpha Fig. 8. Dependence of critical thresholds xTIS, xTIL and xTIF on a.
greater. Note that an incentive to be the leader appears when a = 3300. Furthermore, it is also worth noting that the threshold of the leader is lower than that of the single-agent’s threshold of about a = 4100. This means that strategic agents should exercise earlier than the single agent. The reason is that an increase in a implies that the fear of starting a race to discover technological innovation, such as a winnertakes-all patent system, increases. That is, the problem of obtaining benefit from technological innovation is more important than the environmental problem of avoiding loss through the induced effect.
5. Conclusion This paper investigates an environmental policy designed to reduce the emission of pollutants under uncertainty, where the agent problem is formulated as an optimal stopping problem. First, we analyze the case following Dixit and Pindyck (1994) and Pindyck (2002) where there is only one agent. Second, the results of our original analysis are as follows. If there are two competing agents, they implement the environmental policy simultaneously. Moreover, the threshold for implementing the environmental policy is higher than that for the single-agent case. How long these two agents take to implement the environmental policy depends on the magnitude of the induced effect. We also consider the optimal timing of an environmental policy taking into account technological innovation. Even if the induced effect is complete, and an external authority never adopts intervention to promote an environmental policy, the effect of innovation may actually create a spirit of competition in the decision of the environmental policy if the additional condition (Proposition 4.1) is satisfied. Therefore, the incentive to be the leader also depends on the degree of technological innovation and the induced effect (in the model of Section 4). To conclude this paper, we discuss extensions of our model. First, we leave to examine equilibrium of our model. Next, in Section 4, we considered technological innovation such as a patent. However, we can also consider that if one agent (the leader) who has already implemented the environmental policy discovers a technological innovation, then the other agent (the follower), which has not implemented the policy yet, benefits from the technological innovation. That is, the leader does not benefit from the discovery of technological innovation. Lastly, the leader (the follower) is usually able to be assumed to be in the developed country (the developing country). We leave these topics for future research.
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853
Acknowledgements The authors would like to thank Masaaki Kijima and Takashi Shibata for helpful comments. The authors would also like to thank Robert Graham Dyson and two anonymous referees for valuable comments and suggestions. This research was partially supported by Daiwa Securities Group Inc. and the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Grant-in-Aid for 21st Century COE Program ‘‘Interfaces for Advanced Economic Analysis’’. The second-named author was partially supported by the Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Scientific Research (B) (2), 16310118. Appendix. Derivation of the term in Eq. (30) At first, we calculate the YF(t) from Eq. (24): Z t pF01 edðstÞ ds; Yb Ft ¼ edt Y 0 þ 0 Z t Ye Ft ¼ edt Y 0 þ pF11 edðstÞ ds:
ð74Þ
0
R1 Then, we show the term of Eq. (30): E½ sF ert ðX t Ye Ft X t Yb Ft Þdt. The other terms are derived in the same way: Z t Z 1 Z 1 rt F F rt F F dðstÞ dt F dt F e b e X t Y t X t Y t dt ¼ E e Xt ðp01 p11 Þe ds þ e Y ðs Þ e Y ðs Þ dt F0 E sF
sF
sF
ðpF01 pF11 ÞedðstÞ ds dt FsF jF0 sF sF Z t Z 1 rt F lðtsF Þ F F dðstÞ ¼E e x e ðp01 p11 Þe ds dt F0 Z ¼E E
1
ert X t
Z
t
sF
sF
Z ¼ xF ðpF01 pF11 ÞE
1
e
rt lðtsF Þ
sF
¼
Z
t
e
e sF
dðstÞ
ds dt F0
h F i xF ðpF01 pF11 Þ E ers jF0 : ðr lÞðr þ d lÞ ð75Þ
rsF
Thirdly, we focus on the term E½e have F
log xx l 12 r2 sF W sF ¼ : r
jF0 . We follow Øksendal (1998). Since X sF ¼ xF ¼ xe
ðl12r2 ÞsF þrW sF
, we
ð76Þ
1 2
Suppose that Dt ¼ efW t 2f t . Since Dt is martingale, we have h i 1 2 F E efW sF 2f s ¼ 1: From Eqs. (76) and (77), we obtain 8 93 2
F rf < log XxF l 12 r2 sF 1 = l 12 r2 x 1 2 0 2 F 4 5 f sF fs ¼ E exp f E exp f þ ¼ 1: : ; 2 2 r r x We set f > 0 so that r = (1/2)f2 + ((l 1/2r2)/r)f holds. It yields that 1 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 r l l 12 r2 2 þ f¼ þ 2r: r r
ð77Þ
ð78Þ
ð79Þ
854
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It follows from Eqs. (78) and (79) that we have x rf x b1 E½ersF ¼ F ¼ F : x x Therefore, we obtain Z 1 ert X t Ye Ft X t Yb Ft dt ¼ E sF
xF ðpF01 pF11 Þ x b : ðr lÞðr þ d lÞ xF
ð80Þ
ð81Þ
References Barrieu, P., Chesney, M., 2003. Optimal timing for an environmental policy in a strategic framework. Environmental Modeling and Assessment 8, 149–163. Dasguputa, P., Stiglitz, J., 1980. Uncertainty, industrial structure, and the speed of R&D. Bell Journal of Economics 11, 1–28. Dixit, A.K., 1988. A general model of R&D competition and policy. RAND Journal of Economics 19, 317–326. Dixit, A.K., Pindyck, R.S., 1994. Investment under Uncertainty. Princeton University Press, New Jersey. Faffe, A.B., Newell, R.G., Stavins, R.N., 2003. Technological change and the environment. In: Maler, K.-G., Vincent, J.R. (Eds.), Handbook of Environmental Economics, vol. 1. Amsterdam, North-Holland, pp. 461–516. Grenadier, S.R., 1996. The strategic exercise of options: Development cascades and overbuilding in real estate markets. Journal of Finance 51, 1653–1679. Hanley, N., Shogren, J.F., White, B., 1997. Environmental Economics in Theory and Practice. Macmillan Press, London. Huisman, K.J.M., Kort, P.M., 1999. Effects of strategic interactions on the option value of waiting, Discussion Paper, Department of Econometrics and CentER, Tilburg University. Huisman, K.J.M., Kort, P.M., 2004. Strategic technology adoption taking into account future technological improvements; A real options approach. European Journal of Operational Research 159, 705–728. Karatzas, I., Shreve, S.E., 2001. Brownian Motion and Stochastic Calculus, second ed. Springer-Verlag, New York. Kijima, M., Shibata, T., 2004. Real options in an oligopoly market, Discussion Paper, 033, 21COE Interfaces for Advanced Economic Analysis, Kyoto University. Kulatilaka, M.I., Perotti, E.C., 1998. Strategic growth options. Management Science 44, 1021–1031. Lambrecht, B., Perraudin, W., 2002. Real options and preemption under incomplete information. Journal of Economic Dynamics and Control 27, 619–643. Øksendal, B., 1998. Stochastic Differential Equations: An Introduction with Applications, fifth ed. Springer-Verlag, Heidelberg. Pindyck, R.S., 2000. Irreversibilities and the timing of environmental policy. Resource and Energy Economics 22, 233–259. Pindyck, R.S., 2002. Optimal timing problems in environmental economics. Journal of Economic Dynamics and Control 26, 1677–1697. Weeds, H., 2002. Strategic delay in a real options model of R&D competition. Review of Economic Studies 69, 729–747.
European Journal of Operational Research 190 (2008) 834–854 www.elsevier.com/locate/ejor
Interfaces with Other Disciplines
Induced effects and technological innovation with strategic environmental policy Atsuyuki Ohyama
a,*
, Motoh Tsujimura
b
a
b
Graduate School of Economics, Kyoto University, Yoshida-honmachi, Sakyo, Kyoto 606-8501, Japan Department of Economics, Ryukoku University, Tsukamoto, Fukakusa, Fushimi, Kyoto 612-8577, Japan Received 11 November 2005; accepted 9 August 2007 Available online 23 August 2007
Abstract This paper investigates an environmental policy designed to reduce the emission of pollutants under uncertainty, where the agents’ problem is formulated as an optimal stopping problem. We first analyze the single-agent’s case according to Pindyck [Pindyck, R.S., 2002. Optimal timing problems in environmental economics. Journal of Economic Dynamics and Control 26, 1677–1697]. We then extend the model to the case in which there are two competing agents. Therefore, we consider the external economic effects that are peculiar to an agent’s environmental policy decision. Finally, we consider the effect of technological innovation. The results of the analysis suggest that if there are two competing agents, they implement environmental policy simultaneously. Furthermore, the threshold for implementing environmental policy is higher when there are two agents, and how long these two agents take to implement environmental policy depends on the magnitude of the external economic effect. Furthermore, when we consider the effect of technological innovation, we show that the incentive to be the leader occurs if an additional condition is satisfied. 2007 Elsevier B.V. All rights reserved. JEL classification: Q53; C72; G19 Keywords: Induced effects; Technological innovation; Environmental policy; Real options; Optimal timing
1. Introduction The Kyoto protocol on global warming came into effect on 16 February 2005. According to this agreement, each country’s emissions target must be achieved during the period 2008–2012. For example, Japan will be obliged to reduce its 1990 level of greenhouse gases emissions by 6% (as for Canada, Hungary and Poland), while Switzerland, most Central and East European states and the European Union must reduce their emissions by 8%. However, there is some concern because the US and a number of other key countries have yet to ratify the protocol. On their part, these countries maintain that there is no clear benefit from implementing the *
Corresponding author. Tel./fax: +81 75 753 3511. E-mail addresses: [email protected] (A. Ohyama), [email protected] (M. Tsujimura).
0377-2217/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.08.006
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835
Kyoto protocol. Since there are countries which ratified the protocol and countries which have not ratified it, we find the relation between these countries. This paper then examines the problem of implementing strategic environmental policy. In this paper, we investigate environmental policies designed to reduce the emission of a pollutant under uncertainty. It is assumed that agents face a problem by attempting to minimize both the expected total damage from a pollutant and the cost of implementing the environmental policy. To solve the agent problem, we extend the works of Dixit and Pindyck (1994) and Pindyck (2002) concerning the optimal timing for an environmental policy implementation under irreversibility and uncertainty. We also account for the influence of the strategic competition framework on environmental policy timing: that is, we combine real options and a strategic competition. Following the analysis in Kijima and Shibata (2004) of strategic entry decisions in an oligopoly market when the underlying state variable follows a geometric Brownian motion, we formulate the agents’ problem as an optimal stopping problem. See, also Grenadier (1996), Kulatilaka and Perotti (1998) and Lambrecht and Perraudin (2002) for discussion of strategic competition. We assume that the agent receives damage from the pollutant. Following Dixit and Pindyck (1994) and Pindyck (2002), the damage function is then defined as a function of the pollutant stock. In this paper, we consider the external economic effects of an agent’s policy implementation. We examine the technological innovation that may appear some time after the environmental policy is implemented. It is natural that the environmental policy stimulates technological innovations, for example, innovations in automobile engines and electric power plants. See Faffe et al. (2003) for an excellent survey of environmental policy and technological innovations. We assume that agents only commence research after policy implementation, and once any of the agents have discovered the technological innovation, the remaining agents cannot claim the benefits of discovery. This means that a race to discover technological innovation, such as a winner-takesall patent system, occurs. In this paper, we formulate the technological innovations following Weeds (2002). See also, for example, Dasguputa and Stiglitz (1980), Dixit (1988) and Huisman and Kort (2004). We assume that technological innovation occurs randomly and that innovation discovery by each agent is independent. Therefore, we consider the combined effects of real options, strategic competition and technological innovation. When an agent implements environmental policy, we assume that the efficiency of the policy depends on the technological innovation. In this paper, we assume that if one agent implements the environmental policy, this induces environmental improvements for all other agents. That is, the remaining agents’ environment also improves through implementation of this environmental policy. We call this effect the induced effect as in Barrieu and Chesney (2003). We consider three cases: a complete induced effect, a partial induced effect, and no induced effect. The complete induced effect is generally applied to analyze the problem of global warming. To prevent an increase in the earth’s surface temperature, the atmospheric concentration of greenhouse gases (GHGs) (such as carbon dioxide (CO2), methane (CH4), chlorofluorocarbons (CFCs) and nitrous oxide (N2O)) must be reduced. Thus, global emissions of GHGs should be controlled. This means that if a country implements a policy of reducing GHGs emissions, all countries will benefit equally from its reduction. The partial induced effect is often applied to analyze the problem of acid rain. To reduce acid rain, it is necessary to reduce the emission of as its primary causes in the form of SO2 and NOx. While these gases sometimes diffuse over hundreds of kilometers, acid rain is a generally regional problem, unlike global warming. Finally, the no induced effect is usually applied to analyze the problem of soil contamination. Soil contamination results from spilled or buried hazardous substances. Thus, soil contamination problems are localized to places where hazardous substances are spilled or buried. See Hanley et al. (1997) for more details of global warming and acid rain. The results of our analysis are as follows. If there are two competing agents, both agents have an incentive to let the other agent implement the policy first. Therefore, they both will implement the policy at a later date than a single agent would have done. How long these two agents take to implement the environmental policy depends on the magnitude of the induced effect. This implies that a kind of external effect occurs. Also, this has crucial implications for international policy and regulation. The reason is that if each country considers the timing of implementing an environmental policy under a strategic competition framework, the environmental problems are never resolved optimally from a global perspective. This result may give an explanation why countries are not eager to implement the environmental policy in order to meet the Kyoto protocol. However, if we consider the optimal timing for an environmental policy taking into account technological innovation,
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this result changes. Even if the induced effect is complete, and an external authority never adopts the intervention to promote an environmental policy, the effect of the innovation may actually create a spirit of competition in the decision of the environmental policy if an additional condition is satisfied. Therefore, the incentive to be the leader also occurs (first mover advantages arise), depending on the degree of technological innovation and the induced effect. In other words, the result implies that there exist scenarios in which a preemption equilibrium prevails. The paper itself is organized as follows. In the next section, we describe the agent’s problem and analyze the single-agent problem following Dixit and Pindyck (1994) and Pindyck (2002). In Section 3, we examine the case of two agents where there is an induced effect. In Section 4, we consider technological innovation in the presence of two agents. Section 5 concludes. 2. Optimal timing for an environmental policy in the single-agent case We consider the problem faced by agent i, i = 1, 2, whose objective is to control the accumulation of a pollutant. For example, policymakers may aim to stabilize global climate change or the acid rain effect. We assume that agents have to implement the policy, which is to reduce the flow of the pollutant. We follow Dixit and Pindyck (1994) (Chapter 12, Section 3) and Pindyck (2002). Suppose that the agent i emits the pollutant by pit , and that its stock Y it is governed by dY it ¼ ðpit dY it Þdt;
Y i0 ¼ y;
ð1Þ
where d is the rate of natural decay of the stock of pollutant. In this context, we assume that the initial stock of the pollutant does not depend on agents’ attributes. Let Bi ðX t ; Y it Þ : Rþþ Rþ ! R denote the agent i’s damage (or negative benefit) function associated with the stock of the pollutant Y it . It is assumed that Bi ðX t ; Y it Þ ¼ X t Y it ;
ð2Þ
where Xt is a parameter that stochastically shifts over time to reflect damage due to the pollutant and is assumed to be governed by dX t ¼ lX t dt þ rX t dW t ;
X 0 ¼ x;
ð3Þ
where l 2 R and r 2 R n f0g are constants. (Wt)tP0 is a standard Brownian motion process defined on a filtered probability space ðX; F; P; ðFt ÞtP0 Þ satisfying the usual conditions.1 Ft is generated by Wt in R; i.e. Ft ¼ rðW s ; s 6 tÞ. Note that Xt is independent of the properties of agents. For simplicity, assume that pit remains constant at its initial level pi0 until agent i implements the policy. When agent i implements the policy, pi0 reduces to pi1 with 0 6 pi1 < pi0 . Thus, Eq. (1) becomes ( ðpi0 dY it Þdt; 0 6 t < si ; i dY t ¼ ð4Þ ðpi1 dY it Þdt; si 6 t < 1; where si 2 T is the decision time of agent i and T is the class of all decision times relative to ðFt ÞtP0 . Furthermore, we assume that Z 1 rt i E e jBðX t ; Y t Þjdt < 1; ðAS:1Þ 0
where r 2 Rþþ is a discount rate. Let Ki be the constant cost of implementing the environmental policy for agent i. Therefore, the agent’s problem is to choose si 2 T to minimize the expected total discounted cost associated with the environmental policy: Z 1 i rt i i rsi i V ðx; yÞ ¼ inf E e B ðX t ; Y t Þdt þ e K ; ð5Þ i s 2T
0
i
where V is the value function of agent i’s problem.
1
See, for example, Karatzas and Shreve (2001).
A. Ohyama, M. Tsujimura / European Journal of Operational Research 190 (2008) 834–854
837
Let us define the operator Li as follows: 8 < 1 r2 x2 o22 þ lx o þ ðpi0 dyÞ o r; x 2 H i ; 2 ox oy ox Li ¼ : 1 r2 x2 o22 þ lx o þ ðpi dyÞ o r; x 2 6 H i; 1 2 ox oy ox
ð6Þ
where Hi is the continuation region defined by H i :¼ fx 2 Rþþ ; V i ðx; yÞ < Gi ðx; yÞg; i
ð7Þ
i
i
where G (x, y) is given by rewriting J (x, y; s ) as follows: "Z i Z 1 Z s rt i i rsi i E e B ðX t ; Y t Þdt þ e K ¼ E ert Bi ðX t ; Y it Þdt þ 0
0
"Z
si
ert Bi ðX t ; Y it Þdt þ e
¼E
#
1 rt
e B si
rsi
Z
¼E
si
e
rt
B
i
ðX t ; Y it Þdt
þe
ðX t ; Y it Þdt
1
e
rðtsi Þ
si
0
"Z
i
rsi
G
i
ðX si ; Y isi Þ
þe
rsi
K
i
Bi ðX t ; Y it Þdt þ K i
#
# :
ð8Þ
0
Thus, the decision time s is defined by si ¼ infft > 0; x 62 H i g:
ð9Þ
We analyze the case in which a single-agent implements an environmental improvement policy. See Pindyck (2000) for details. The single-agent’s problem is to choose sS 2 T to minimize the expected total discounted cost: Z 1 S V S ðx; yÞ ¼ inf E ert BS ðX t ; Y St Þdt þ ers K S : ð10Þ sS 2T
0
Since the agent has not implemented the environmental policy for x 2 HS, we obtain 1 LS V S ðx; yÞ þ BS ðx; yÞ ¼ r2 x2 V Sxx þ lxV Sx þ ðpS0 dyÞV Sy rV S þ xy ¼ 0: 2
ð11Þ
On the other hand, since the agent has implemented the policy for x 62 HS, we have 1 LS V S ðx; yÞ þ BS ðx; yÞ ¼ r2 x2 V Sxx þ lxV Sx þ ðpS1 dyÞV Sy rV S þ xy ¼ 0: 2
ð12Þ
Eqs. (11) and (12) must be solved by using the following appropriate boundary conditions: V ðx; yÞjx¼0 ¼ 0;
ð13Þ
S
S
S
S
S x ðx; y; x
S
S x ðx; y; x
S
S
V ðx; y; x 2 H Þjx¼xS ¼ V ðx; y; x 62 H Þjx¼xS þ K ; V
2 H Þjx¼xS ¼ V
62 H Þjx¼xS ;
ð14Þ ð15Þ
where xS is the critical value of the shift parameter x. When x reaches xS, the single-agent implements the policy. Thus, the decision time is given by sS ¼ infft > 0; x P xS g:
ð16Þ
The conditions given by Eqs. (14) and (15) are the well-known value-matching and smooth-pasting conditions, respectively. Then, the value function of the single agent is given by 8 S < A1 xb1 þ xp0 þ xy ; x < xS ; q1 q2 q2 S V ðx; yÞ ¼ ð17Þ : xpS1 þ xy ; x P xS ; q1 q2 q2
838
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where A1 is a constant to be determined, q1 = r l and q2 = r + d l. b1 is the positive solution of the following characteristic equation: 1 2 r bðb 1Þ þ lb r ¼ 0: 2
ð18Þ
Thus, we have 1 l b1 ¼ 2 þ 2 r
"
l 1 r2 2
2
2r þ 2 r
#12 > 1:
ð19Þ
The first line of Eq. (17) can be discussed as follows. The first term on the right-hand side represents the value of the option to implement the environmental policy in the future. The sum of the second and third terms on the right-hand side is the present value of the damage. Conversely, the second line of Eq. (17) refers to the following. The sum of the two terms on the right-hand side represents the present value of the damage. Note that the emission flow of the pollutant decreases from pS0 to pS1 due to the policy implementation. From Eqs. (14), (15) and (17), we obtain b1 q1 q2 K S S x ¼ ; ð20Þ b1 1 pS0 pS1 b 1 b b 1 1 pS0 pS1 1 : ð21Þ A1 ¼ 1 S b1 q1 q2 K 3. Optimal timing of an environmental policy under a strategic competition As mentioned in Section 1, there are countries which ratified the protocol and countries which have not ratified the Kyoto protocol. In this section, for simplicity, we study the case in which there are only two competing agents. Suppose that both agents have experienced damage from pollutants and wish to determine when they should implement environmental policy to reduce pollutant emissions. The agent F is the last to implement the policy. This agent is referred to as the follower. The agent L implements the environmental policy before or at the same time as agent F. This agent is referred to as the leader. We assume that if one agent (say L) implements the environmental policy, it affects the environmental improvement of the other agent (say F). That is, the other agent experiences environmental improvement because of the implementation of the environmental policy. This effect is referred to as the induced effect (IE), as in Barrieu and Chesney (2003). Let piN i N j be the emission flow of agent i, where for k 2 {i, j}, we denote 0; if the agent k has not implemented the policy; ð22Þ Nk ¼ 1; if the agent k has implemented the policy: We assume that the total amount of emission reduction does not depend on the magnitude of the induced effect. We also assume that the two agents have the same emission structure as follows: pS0 ¼ pL00 ¼ pF00 ;
pS1 ¼ pL11 ¼ pF11 ;
ðpS0 pS1 Þ ¼ ðpL00 pL10 Þ þ ðpF00 pF01 Þ ¼ ðpL10 pL11 Þ þ ðpF01 pF11 Þ; pS pS1 6 pL10 6 0 ; 2
ðAS:2Þ
pS0 6 pF01 6 pS0 : 2
Furthermore, we assume that the implementation cost in the single-agent case is equal to the sum of the implementation cost of the leader and follower, i.e. 2K S ¼ K L þ K F :
ðAS:3Þ
We now examine the induced effect shown in Fig. 1. Suppose that the agents, L, F, emit GHGs such as CO2 or methane and an increase in GHGs is assumed to raise the earth’s average temperature. To prevent this, the atmospheric concentration of the GHGs must be reduced. This means that if the agent L implements a policy
A. Ohyama, M. Tsujimura / European Journal of Operational Research 190 (2008) 834–854
π 0S
L π 00
F π 00
π 0S
L π 00
F π 00
π 0S
L π 00
839
F π 00
IE F π 01
IE
π 10L
F π 01
π 10L π 1S
π 11L
π 11F
Complete induced effect
π 1S
π 11F
π 11L
Partial induced effect
π 1S
π 10L
π 11F
No induced effect
Fig. 1. Induced effect (IE).
to reduce the emission of the GHGs, the other agent F receives benefit from the reduction in emissions, just as if the agent F had implemented the policy. That is, the induced effect is complete. Thus, we have pL10 ¼ pF01 ¼ pS0 =2 from the emission structure (AS.2). Next, suppose that the agents emit either SO2 or NOx as a primary cause of acid rain. Prevailing winds blow these gases across state and national borders; sometimes for hundreds of kilometers. However, winds do not carry the gases globally. Thus, acid rain is a regional problem, unlike global warming. Thus, the reduced effect is partial and we have pL10 < pF01 from the emission structure (AS.2). Next, we suppose that the agents emit a hazardous substance. Then, there is no induced effect, and we have pL10 ¼ pS1 and pF01 ¼ pS0 (see Fig. 1). Thus, the induced effect, IE, is defined by IE ¼ pF00 pF01 ¼ pL10 pL11 :
ð23Þ
Since all parameter values and actions are common knowledge, the game is one of complete information. 3.1. The follower’s problem In this subsection, we consider the follower’s problem. As in the standard strategic real-options framework, we assume that the leader has already implemented the environmental policy. See, for example, Kijima and Shibata (2004) and Huisman and Kort (1999). The follower’s problem is then similar to the single-agent problem. Note that the follower receives the induced effect from the leader’s policy implementation. Thus, the dynamics of the pollutant stock, Eq. (4), becomes ( d Yb Ft ¼ ðpF01 d Yb Ft Þdt; 0 6 t < sF ; F dY t ¼ ð24Þ d Ye F ¼ ðpF d Ye F Þdt; sF 6 t < 1; t
11
t
F
where s is the followers policy decision time that is defined by sF ¼ infft > 0; X t P xF g:
ð25Þ F
In this context, we assume that if the shift parameter Xt reaches the threshold x , the follower implements the environmental policy. Then, the follower’s problem is to choose sF 2 T to minimize the expected total discounted costs: "Z F # Z 1 s F F rt F F rs F rt F F E e B ðX t ; Yb t Þdt þ e K þ e B ðX t ; Ye t Þdt : ð26Þ V ðx; yÞ ¼ inf F s 2T
0
sF
840
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As in the single-agent case, Section 2, we obtain the follower’s value function as 8 F e 1 xb1 þ xp01 þ xy ; x < xF ; 1 and A e and xF by using the value-matching and smooth-pasting conditions as follows: obtain A b1 1 F b p01 pF11 1 e 1 ¼ b1 1 ; ð28Þ A b1 q1 q2 KF b1 q1 q2 K F : ð29Þ xF ¼ b1 1 pF01 pF11 Using the strong Markov property and the tower property of conditional expectations, we have "Z F # Z 1 s F F rt F rs F rt F V ðx; yÞ ¼ E e X t Yb t dt þ e K þ e X t Ye t dt Z ¼E
0 1
sF
Z h Fi ert X t Yb Ft dt þ E ers K F þ E
0 F xp01
1
sF
xy x b1 xF ðpF11 pF01 Þ ¼ þ þ F þ KF : x q1 q2 q1 q2 q2
ert X t Ye Ft X t Yb Ft dt ð30Þ
The detailed derivation of the last term in Eq. (30) is in Appendix. Then, the follower’s value function is i 8 xpF F F F h < q q01 þ qxy þ xxF b1 K F x ðpq01qp11 Þ ; x < xF ; 1 2 2 1 2 ð31Þ V F ðx; yÞ ¼ : xpF11 xy F F þ þ K ; x P x : q1 q2 q2 For x < xF, the sum of the first and the second terms on the right-hand side is the present value of the damage. In this context, since the leader has implemented the environmental policy and the induced effect occurs, the follower’s emission flow of the pollutant decreases from pF00 to pF01 . The remaining terms represent the value of the option to implement the environmental policy in the future. On the other hand, For x P xF, the sum of the first and second terms is also the present value of the damage. Note that the emission flow of the pollutant decreases from pF01 to pF11 due to the implementation of the policy by the follower. The last term represents the policy implementation cost. 3.2. The leader’s problem We now examine the leader’s problem. In this subsection, the leader makes a policy decision under the assumption that the follower will act optimally in the future. Once the leader implements the environmental policy, the leader has no further decisions to make. Moreover, the leader obtains the benefit from the future policy implementation by the follower due to the induced effect. In this case, the dynamics of the stock of the pollutant is given by 8 L L bL b 0 6 t < sL ; > < d Y t ¼ ðp00 d Y t Þdt; dY Lt ¼ d Ye Lt ¼ ðpL10 dt d Ye Lt Þdt; sL 6 t < sF ; ð32Þ > : L L L F dY t ¼ ðp11 dt dY t Þdt; s 6 t < 1; where sL is defined by sL ¼ infft > 0; X t P xL g:
ð33Þ
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841
In this case, we assume that if the shift parameter Xt reaches the threshold xL, the leader implements the environmental policy. Hence, the leader’s problem is to choose sL 2 T so as to minimize the expected total discounted costs: "Z L # Z sF Z 1 s L L rt L L rs L rt L L rt L L V ðx; yÞ ¼ inf E e B ðX t ; Yb Þdt þ e K þ e B ðX t ; Ye Þdt þ e B ðX t ; Y Þdt : ð34Þ t
sL 2T
t
sL
0
sF
t
After the follower implements the policy, the leader’s (and the follower’s) value is given by the present value of the damage. That is, we have that V L ðx; yÞjx¼xF ¼ V F ðx; yÞjx¼xF :
ð35Þ
Note that since there is no optimality on the part of the leader, there is no corresponding smooth-pasting condition (see Weeds, 2002 for more detail). Using the strong Markov property and the tower property of conditional expectations as in Eq. (30), we have "Z L # Z sF Z 1 s L V L ðx; yÞ ¼ E ert X t Yb L dt þ ers K L þ ert X t Ye L dt þ ert X t Y L dt t
0
sL
t
sF
t
Z 1 Z 1 h Li ¼E ert X t Yb Lt dt þ E ers K L þ E ert X t Ye Lt X t Yb Lt dt 0 sL Z 1 ert X t Y Lt X t Ye Lt dt þE sF
F L xpL xy x b1 xL ðpL10 pL00 Þ x b1 x ðp11 pL10 Þ ¼ 00 þ þ L þ KL þ F : x q1 q 2 x q1 q2 q1 q2 q2 It can be shown that the value function for the leader is given by 8 L x b1 h L xL ðpL00 pL10 Þi x b1 hxF ðpL10 pL11 Þi xp00 xy > þ þ K q1 q2 xF ; x < xL ; > xL q1 q2 q2 q1 q2 > > < L b hxF ðpL pL Þi xp 10 11 V L ðx; yÞ ¼ q q10 þ qxy þ K L xxF 1 ; xL 6 x 6 xF ; q1 q2 1 2 2 > > > > : xpL11 þ xy þ K L ; x P xF : q2 q1 q2
ð36Þ
ð37Þ
For x < xL in Eq. (37), the sum of the first and second terms on the right-hand side is the present value of the damage. The third term represents the value of the option to implement the environmental policy in the future. The fourth term is an option-like term that represents the effect of the follower’s future implementation on the leader’s value function. Next, for xL 6 x 6 xF in Eq. (37), the first term represents the present value of the damage from the pollutant emission flow. Note that the emission flow decreases from pL00 to pL10 due to the policy implementation of the leader. Lastly, for x P xF in Eq. (37), the first term represents the present value of the damage from the pollutant emission flow. Note that the emission flow decreases from pL10 to pL11 due to the policy implementation of the follower. 3.3. Results of the two-agents case In this subsection, we examine both the leader’s and follower’s optimal decision times. We find that the leader has no incentive to implement the policy before the follower because of the induced effect. In other words, we show that first mover advantages do not arise in this model. Suppose that the leader implements the environmental policy and reduces pollutant emissions. Although the follower has not implemented the environmental policy, the follower’s environment improves due to the induced effect. Note especially that the degree of environmental improvement depends on the magnitude of the induced effect. To illustrate these
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findings, we investigate the shapes of the value functions of the leader and the follower. We first examine the follower’s value function for x < xF. From (30) and the follower’s value function for x < xF is xpF01 xy x b1 F xF ðpF01 pF11 Þ F V ðx; yÞ ¼ þ þ F K : ð38Þ x q1 q2 q1 q2 q2 The first and the second derivatives with respect to x are as follows: x b1 1 1 pF y xF ðpF01 pF11 Þ F K ; V Fx ðx; yÞ ¼ 01 þ þ b1 F x xF q1 q2 q1 q2 q2 x b1 2 1 xF ðpF01 pF11 Þ F V Fxx ðx; yÞ ¼ b1 ðb1 1Þ F K : 2 x q1 q2 ðxF Þ
ð39Þ ð40Þ
The term xF ðpF01 pF11 Þ=q1 q2 is the present value of the benefit of policy implementation. KF denotes the cost of policy implementation. It follows that xF ðpF01 pF11 Þ=q1 q2 P K F . However, since x < xF, V Fx > 0. On the other hand, V Fxx 6 0. Next, we examine the leader’s value function for xL 6 x 6 xF. From Eq. (37), the leader’s value function for L x 6 x 6 xF is x b1 xF ðpL pL Þ xpL10 xy L L 10 11 V ðx; yÞ ¼ þ þK F : ð41Þ x q1 q2 q1 q2 q2 The first and the second derivatives with respect to x are as follows: x b1 1 ðpL pL Þ pL y 10 11 ; V Lx ðx; yÞ ¼ 10 þ b1 F x q1 q2 q1 q2 q2 x b1 2 1 ðpL pL Þ 10 11 V Lxx ðx; yÞ ¼ b1 ðb1 1Þ F : x xF q1 q 2
ð42Þ ð43Þ
For xL 6 x 6 xF, V Lx > 0. On the other hand, V Lxx < 0. Thus, the leader’s value function is strictly concave for xL 6 x 6 xF. Then, we have the following result. Lemma 3.1. Suppose that Assumptions (AS.1)–(AS.3) hold. See Sections 2 and 3. Moreover, assume that KL ¼ KF :
ðAS:4Þ
Then, we obtain V L ðx; yÞ > V F ðx; yÞ;
xL 6 x < xF :
ð44Þ
Lemma 3.1 implies that the leader never implements policy before so does the follower. It means that a kind of attrition game arises in this game. Then, we obtain the following result. Proposition 3.1. Suppose that Assumptions (AS.1)–(AS.4) hold. Then, we have xL ¼ xF :
ð45Þ
Therefore, from Eqs. (42) and (43), the leader and the follower simultaneously implement the environmental policy. Furthermore, given Eqs. (20) and (29), we obtain the following result: Proposition 3.2. Suppose that (AS.1)–(AS.4) hold. Then we obtain xS < xL ¼ xF :
ð46Þ S
F
L
Moreover, the greater the induced effect, the greater the difference between x and x = x .
A. Ohyama, M. Tsujimura / European Journal of Operational Research 190 (2008) 834–854
Proof. Let h be defined by h :¼ xS xF. From Eqs. (20) and (29), it follows that b1 KS KF : qq h¼ b1 1 1 2 pS0 pS1 pF01 pF11 It follows from (AS.3) and (AS.4) that b1 KF KF b1 1 1 F h¼ ¼ < 0: qq qqK IE þ ðpF01 pF11 Þ pF01 pF11 b1 1 1 2 pS0 pS1 pF01 pF11 b1 1 1 2
843
ð47Þ
ð48Þ
The second equality in Eq. (48) holds because of Eq. (23): pS0 pS1 ¼ IE þ ðpF01 pF11 Þ. Let IEC be the complete induced effect and let IEP be the partial induced effect. Further, let IEN denote the no induced effect. Then, we have pS0 pS1 ¼ IEC > IEP > IEN ¼ 0: 2
ð49Þ
Thus, as the induced effect increases, the difference between xS and xF = xL increases. The proof is completed. h Proposition 3.2 implies that the environmental policy is implemented faster in the single-agent case than that in the two competing-agents case. This has crucial implications for international policy and regulation. It means that if each country considers the timing of implementing an environmental policy under a strategic competition framework, the environmental problems are never resolved optimally from a global perspective. 3.4. Numerical example and comparative statics In this section, we present a comparative static results of r, d, l, r and K. Table 1 presents the base case parameter values. We first calculate b1, AS1 and xS by using the base case parameter values. Then, we obtain that b1 = 1.41, S A1 ¼ 15883:6, and xS = 0.052. Note that, in this paper, we discuss the damage (negative benefits) instead of profit flows. Fig. 2 illustrates the value function for the single agent. This suggests that the environmental policy should be implemented when the shift parameter x reaches xS = 0.05. Similarly, Fig. 3 illustrates the both agents’ value functions: VL and VF. It is obvious from Fig. 3 that the leader’s value function never crosses the follower’s value function. Thus, strategic agents have no incentive to be the leader. Therefore, the two strategic agents simultaneously implement the environmental policy when the shift parameter reaches xF = 0.069. Note that the critical value for the follower is larger than in the single-agent case because there is the induced effect. This implies that inefficiency occurs in the competing-agent’s problem. We next present the results of a comparative-static analysis for r, d, l, r, K and IE by varying parameters by ±30% in Table 2. Table 2 implies as follows. Since the cost K is paid when the policy is implemented, an Table 1 Parameter values Parameter
Value
r d l r Ki p00 ¼ ps0 pL10
The The The The The The The
discount rate natural rate expected percentage rate of growth of x volatility parameters of x cost of implementing the policy for agent i initial pollutant’s flow pollutant’s flow of the leader
0.04 0.02 0.01 0.3 100 20 12.5
pF01 p11 ¼ ps1 y
The pollutant’s flow of the follower The pollutant’s flow after adoption of all policy The stock of the pollutant
17.5 10 100
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600 500
VS
400 300 200 100 0
0
0.01
0.02
0.03
0.04
0.05
x Fig. 2. The value function for the single agent.
800
PV of Damage VF L V
700 600
V
500 400 300 200 100 0 0
0.01
0.02
0.03
x
0.04
0.05
0.06
0.07
Fig. 3. The value functions of the leader and follower.
Table 2 The results of comparative-static analysis xs
xF
xL
xs
xF
xL
r + 30% r r 30%
0.075 0.052 0.032
0.100 0.069 0.043
0.100 0.069 0.043
r + 30% r r 30%
0.069 0.052 0.038
0.092 0.069 0.051
0.092 0.069 0.051
d + 30% d d 30%
0.058 0.052 0.045
0.077 0.069 0.061
0.077 0.069 0.061
K + 30% K K 30%
0.067 0.052 0.036
0.09 0.069 0.048
0.09 0.069 0.048
l + 30% l l 30%
0.048 0.052 0.056
0.064 0.069 0.075
0.064 0.069 0.075
IE + 30% IE IE 30%
– – –
0.077 0.069 0.063
0.077 0.069 0.063
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xF 1
2
3
4
5
IE
0.09
0.08
0.07
0.06
Fig. 4. Dependence of critical threshold, xF, on IE.
increase in r implies a greater reduction in the present value of that cost. Thus, an increase in the discount rate r increases the value of the option to implement the policy and the thresholds: xS, xF and xL. A higher value of d implies that the environmental damage from emission is more reversible, such that the sunk benefit of implementing the policy now, rather than waiting, is lower. Also, the more uncertainty over the future cost of the pollutant, the greater the incentive to wait, rather than implement the policy now. Hence, an increase in r raises the thresholds. Furthermore, an increase in l implies that each agent can observe the fact that the damage from emissions increases in the future. Thus, an increase in l decreases the thresholds. Finally, we examine the sensitivity of the critical value xF to the magnitude of the induced effects. See Table 2 and Fig. 4. Note that IE = 5 represents the complete induced effect, 0 < IE < 5 represents the partial induced effect and IE = 0 represents no induced effect from the base case parameter values in Table 1. Table 2 and Fig. 4 show that the bigger the induced effects, the bigger the threshold xF. Since the induced effect provides the external effect to agents, an increase in the induced effects mean that the external effects increase. Thus, it is natural that the bigger the induced effects, the higher the threshold xF. 4. Optimal timing for an environmental policy with technological innovation In this section, and in addition to the induced effect we consider the effect of technological innovation following Weeds (2002). When agent i implements the environmental policy with cost Ki, we assume that the agent can take advantage of technological innovation. We assume that technological innovation occurs randomly and that the time in which it occurs follows an exponential distribution with a constant hazard rate ki > 0. See, for example, Dasguputa and Stiglitz (1980), Dixit (1988), and Huisman and Kort (2004). We further assume that the discovery by each agent is independent. We also assume that agents only start researching from the time of policy implementation, and once any of the agents has made the discovery, the others cannot claim further benefits from the discovery. In other words, the agent who has made the discovery can only obtain benefits from technological innovation. All parameter values and actions are assumed to be common knowledge, and hence, the game is one of complete information, as in Section 3. Since the damage from the pollutant depends on the shift parameter x, the value of technological innovation depends on the shift parameter x. We assume that agent i receives a positive benefit from technological innovation by aXt, where a > 0. Thus, the damage function (2) becomes ( X t Y it ; t < sTIi ; i i BTI ðX t ; Y t Þ ¼ ð50Þ X t Y it aX t ; t P sTIi ; where sTIi is the agent i’s policy implementing time.
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4.1. The single-agent’s problem with technological innovation In this section, we examine the single-agent’s problem with technological innovation. The single-agent’s problem is to choose the policy implementing time sTIS to minimize the expected total discounted costs: Z 1 S S rt S rsTIS S VTI ðx; yÞ ¼ inf ¼ E e BTI ðX t ; Y t Þdt þ e K : ð51Þ sTIS 2T
0
Suppose that the single-agent implements the environmental policy when the shift parameter xt reaches the threshold xTIS. Then, as in Section 2, we obtain the value function as follows: 8 S < ASTI xb1 þ xp0 þ xy ; x < xTIS ; q2 q q 1 2 ð52Þ VTIS ðx; yÞ ¼ : xpS1 þ xy akx þ K S ; x P xTIS ; q1 q2 q2 q3 where q3 = r + k l. The parameters ASTI and xTIS are determined by solving the following value-matching and smooth-pasting conditions: b 1 b b 1 1 akq1 q2 þ ðp0 p1 Þq3 1 ASTI ¼ 1 S ; ð53Þ b1 q1 q2 q3 K b1 q1 q2 q3 K S xTIS ¼ : ð54Þ b1 1 akq1 q2 þ ðpS0 pS1 Þq3 The difference in the value function of the single agent from that in Section 2 (between VTIS and VS) represents the present value of the effect of technological innovation. See Eqs. (17) and (52). Furthermore, the threshold xTIS is also to be added the term of technological innovation. See Eqs. (20) and (54). 4.2. The follower’s problem In this section, we investigate the follower’s problem with technological innovation. Suppose that the leader has already implemented the environmental policy. The dynamics of the pollutant stock, Eq. (24), becomes ( d Yb Ft ¼ ðpF01 d Yb Ft Þdt; 0 6 t < sTIF ; F dY t ¼ ð55Þ d Ye F ¼ ðpF d Ye F Þdt; sTIF 6 t < 1; 11
t
t
where sTIF is the follower’s policy implementing time. Then, the follower’s problem is to choose sTIF to minimize the expected total discounted costs: "Z TIF # Z 1 s F F F F ert BTI ðX t ; Yb Ft Þdt þ ers K F þ ert BTI ðX t ; Ye Ft Þdt : ð56Þ VTI ðx; yÞ ¼ inf E sTIF 2T
As in Section 3, we have "Z TIF Z s TIF F VTI ðx; yÞ ¼ E ert X t Yb Ft dt þ ers K F þ Z
sTIF
0
0
1
e sTIF
rt
X t Ye Ft dt
Z
#
1
e
rt 2kt
e
kaX t dt
sTIF
Z 1 Z 1 h Fi F rs F rt F F rt 2kt b e b ¼E e X t Y t dt þ E e e X t ð Y t Y t Þdt E e e kaX t dt K þE 0 sTIF sTIF x b1 akxTIF xpF xy x b1 xTIF ðpF11 pF01 Þ ¼ 01 þ þ TIF þ K F TIF ; ð57Þ x q1 q2 x q1 q2 q2 q4 1
rt
where q4 = r + 2k l and b1 is the positive root of the following characteristic equation: 1 2 r b ðb 1Þ þ lb ðr þ kÞ ¼ 0: 2
ð58Þ
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847
Thus, b1 is b1
1 l ¼ 2þ 2 r
"
l 1 r2 2
2
2ðr þ kÞ þ r2
#12 ð59Þ
> 1:
xTIF is the threshold of follower in which technological innovation is considered. Hence, the follower’s value function is given by
F
VTI ðx; yÞ ¼
8 h i h i xTIF ðpF01 pF11 Þ xpF x b1 x b1 akxTIF > < xTIF KF xTIF þ q1 q012 þ qxy2 ; q4 q1 q2
x < xTIF ;
> : xpF11 þ xy akx þ K F ; q4 q1 q2 q2
x P xTIF :
ð60Þ
The difference between the value functions VTIF and VF comes from the effect of technological innovation in VTIF. Thus, the value function includes the present value of the effect of technological innovation: akxTIF/q4. The threshold xTIF is found by solving the value-matching and smooth-pasting condition for VTIF at x = xTIF: b1 K F xTIF ¼ ðpF pF Þðb 1Þ : akðb 1Þ 1 01 11 þ q14 q1 q2
ð61Þ
The difference between thresholds xTIF and xF comes from the effect of technological innovation. Note that the follower’s value function VTIF has two option-like terms in Eq. (60). The second term means that the follower has an opportunity to obtain benefit by discovering technological innovation. In other words, if the leader discovers the technological innovation before the follower implements the environmental policy, the follower never gets the benefits from discovering the technological innovation. The second term is the same as in Section 3. 4.3. The leader’s problem We now examine the leader’s problem. As in Section 3, we assume that the leader makes a policy decision with the assumption that the follower will act optimally in the future. Once the leader implements the environmental policy, the leader has no further decisions to make. Moreover, the leader receives two opposite effects from the future policy implementation by the follower: positive effects from the induced effects and negative effects from the rival (follower) entry on the competing game for discovery of technological innovation. In this case, Eq. (32) becomes 8 > d Yb L ¼ ðpL00 d Yb Lt Þdt; > > < t dY Lt ¼ d Ye Lt ¼ ðpL10 d Ye Lt Þdt; > > > : L dY t ¼ ðpL11 dY Lt Þdt;
0 6 t < sTIL ; ð62Þ
sTIL 6 t < sTIF ; sTIF 6 t < 1;
where sTIL is the leader’s policy implementing time. The leader’s problem is then to choose sTIL to minimize the expected total discounted costs: "Z
sTIL
VTIL ðx; yÞ ¼ inf E sTIL 2T
0
b L ðX t ; Yb L Þdt þ e ert B t
rsTIL
K Lþ
Z
sTIF sTIL
b L ðX t ; Ye L Þdt þ ert B t
Z
1 sTIF
# b L ðX t ; Y L Þdt : ert B t ð63Þ
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As in Section 3.2, we obtain the following: "Z TIL Z s L rt L rsTIL L b VTI ðx; yÞ ¼ E e X t Y dt þ e K þ t
0
sTIF
e X t ð Ye Lt ekt kaÞdtþ rt
sTIL
Z
1
e sTIF
rt
X t ðY Lt
e
2kt
kaÞdt
Z 1 Z 1 Z 1 h TIL i ¼E ert X t Yb Lt dt þ E ers K L þ E ert X t ð Ye Lt Yb Lt Þdt E ert ekt kaX t dt TIL sTIL 0Z 1 Z 1 s þE ert X t ðY Lt Ye Lt Þdt E ert kaX t ðe2kt ekt Þdt TIF TIF s s xpL00 xy x b1 L x b1 xTIL ðpL10 pL00 Þ x b1 akxTIL ¼ þ þ TIL K þ TIL TIL x x q1 q2 x q1 q2 q2 q3 x b1 xF ðpL pL Þ x b1 akxTIF akxTIF 11 10 þ TIF þ TIF : ð64Þ x q1 q2 x q3 q4 Both agents (leader and follower) scramble for technological innovation, after follower implements the policy. Therefore, note that the value of technological innovation changes from k ekt · aXt into 2k e2kt 12 aX t (see Weeds, 2002 for details). It can be shown that the value function for the leader is 8 L x b1 h L xTIL ðpL00 pL10 Þi x b1 hakxTIL i xp00 xy > > þ þ K xTIL > xTIL q1 q2 q3 > q1 q2 q2 > h i h > x b1 ak2 xTIF x b1 xTIF ðpL10 pL11 Þi > > > þ xTIF xTIF ; x < xTIL ; > q1 q2 q3 q4 > > < L h i x b1 ak2 xTIF ð65Þ VTIL ðx; yÞ ¼ qxpq10 þ qxy akx þ K L þ xTIF q q3 q4 1 2 2 3 > > h i > x b1 xTIF ðpL pL Þ > > 10 11 > xTIF ; xTIL 6 x < xTIF ; > q1 q2 > > > > > : xpL11 þ xy akx þ K L ; x P xTIF : q1 q2
q2
q4
The difference between the value functions VTIL and VL also comes from the effect of technological innovation in VTIL. Thus, the value function has the present values of the effects of technological innovation, akxTIL/q4 and ak2xTIF/(q3q4). Before the follower has implemented the environmental policy, the leader’s value function with respect to the technological innovation is given by the second line of Eq. (65). It is worth noting that, after the follower has implemented the environmental policy, the denominator of the third term of Eq. (65) is replaced by q4. This means that a rival in the research for the technological innovation has made an appearance. 4.4. Results of technological innovation In this section, we assume an additional condition such that an agent has an incentive to become the leader. To this end, we examine the shapes of the value functions of the leader and the follower. First, we examine the follower’s value function. From Eq. (60), the follower’s value function for x < xTIF is xpF01 xy x b1 F xTIF ðpF01 pF11 Þ x b1 akxTIF F VTI ðx; yÞ ¼ þ þ TIF K TIF : ð66Þ x q1 q2 x q1 q2 q2 q4 The first and second derivatives with respect to x are as follows: x b1 1 1 x b1 1 1 akxTIF pF y xTIF ðpF01 pF11 Þ F K VTIFx ðx; yÞ ¼ 01 þ þ b1 TIF b ; ð67Þ 1 x xTIF q1 q2 xTIF xTIF q1 q2 q2 q4 x b1 1 1 x b1 1 1 akxTIF xTIF ðpF01 pF11 Þ F K ðb 1Þ VTIFxx ðx; yÞ ¼ b1 ðb1 1Þ TIF b : 1 1 2 2 x q1 q2 xTIF q4 ðxTIF Þ ðxTIF Þ ð68Þ
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Let qF1 be given by qF1 :¼ ½K F xTIF ðpF01 pF11 Þ=q1 q2 . Since b1 > 1 and pF01 > pF11 , the sign of Eq. (68) depends on the sign of qF1 . The term xTIF ðpF01 pF11 Þ=q1 q2 is the present value of the benefit from policy implementation. The parameter KF is the cost of policy implementation. It follows that xTIF ðpF01 pF11 Þ=q1 q2 P K F . Then, the sign of qF1 is negative. It followers from b1 > 1 that the second term of Eq. (68) is negative. For x < xTIF we obtain VTIFxx < 0. On the other hand, the sign of Eq. (67) depends on the value of a and on the magnitude of the induced effect. We next examine the leader’s value function. From Eq. (65), the leader’s value function for xTIL 6 x 6 xTIF is x b1 xTIF ðpL pL Þ x b1 ak2 xTIF xpL xy akx 10 11 þ K L TIF VTIL ðx; mÞ ¼ 10 þ þ TIF : ð69Þ q3 x q1 q2 x q1 q2 q2 q3 q4 The first and second derivatives with respect to x are as follows: x b1 1 ðpL pL Þ x b1 1 ak2 pL y ak 11 10 VTILx ðx; mÞ ¼ 10 þ þ b1 TIF þ b1 TIF ; x q1 q2 x q1 q2 q2 q3 q3 q4 x b1 2 1 ðpL pL Þ x b1 2 1 ak2 L 10 11 VTIxx ðx; mÞ ¼ b1 ðb1 1Þ TIF þ b1 ðb1 1Þ TIF : x ðxTIF Þ q1 q 2 x ðxTIF Þ q3 q4
ð70Þ ð71Þ
Since b1 > 1, b1 > 1, pL00 > pL10 , and pL10 > pL11 , the signs of Eqs. (70) and (71) depend on the value of a and on the magnitude of the induced effect. Proposition 4.1. Suppose that (AS.1) and (AS.2) hold. If a satisfies the inequality L b1 1 q3 q4 p10 pL11 a> ; b1 1 q1 q2 k2
ð72Þ
then we have VTIL < VTIF for x < xTIF: the curves for VTIL and VTIF cross each other before x reaches xTIF. Then the strategic agents have an incentive to be a leader. Namely, first mover advantages arise. Proof. We choose the value of the parameter a so that VTILx ðxTIF ; yÞ > GTIx ðxTIF ; yÞ, where GTIðx; yÞ ¼ xp11 þ qxy2 akx þ K. Then, we obtain that q1 q2 q4 L p pL11 qq VTILx ðxTIF ; yÞ GTIx ðxTIF ; yÞ ¼ ðb1 1Þ 10 þ ðb1 1Þ 3 2 4 > 0: ð73Þ q1 q2 k It yields inequality (72). This completes the proof.
h
This proposition implies that there exist scenarios in which a preemption equilibrium prevails. 4.5. Numerical examples and comparative statics with technological innovation In this section, we calculate the thresholds and the value functions for the single agent, the leader, and the follower when technological innovation is considered. We then show the results of a comparative-static analysis. As in Section 3, we assume in this section that the induced effect is partial. First, we show the value function for the single agent. Suppose that a = 3000, such that Proposition 4.1 is satisfied. Then, we obtain that xTIS = 0.04. As in Section 3.4, we show the value function of the single agent in Fig. 5. Fig. 5 means that the single agent should implement the environmental policy when the shift parameter x reaches xTIS = 0.04 in the case of a = 3000. Note that, compared with Section 2, the value of the option to implement the environmental policy increases. This is because the value of technological innovation is taken into account. Next, we show the value functions for the competing two agents: the leader and follower. In this case, both the critical values of the leader and the follower depend on the value of a. Since ax represents technological innovation, the parameter a describes the magnitude of technological innovation. Suppose that k = 0.05. The larger the parameter a, the lower the critical values of both the leader and follower. Moreover, we need to pay attention to the value function of the leader because it is ambiguous whether two competing-agents implement
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400
PV of Damage VTI S
350 300
VTI S
250 200 150 100 50 0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
x Fig. 5. The values function for the single agent with technological innovation.
the environmental policy simultaneously or not. Consequently, we must classify each case and evaluate each value of a. To determine if the strategic agents have an incentive to be the leader, we need to classify the following three cases for each value of a. (1) When a = 0, as is shown in Section 3, agents have no incentive to be the leader. (2) When 0 < a < 3312, an increase in parameter a makes each critical value smaller and the strategic agents have no incentive to be the leader. Therefore, the two strategic agents adopt the policy on the same critical value. (3) If 3312 < a, as in the case shown in Fig. 6 agents have an incentive to be the leader. Fig. 6 shows the thresholds of the leader and the follower for a = 20,000 such that we set up the extreme parameter a in order to make the result clear. In the region of a > 3312, the leader’s value function is below the follower’s value function before the follower’s value function reaches the threshold. Therefore, the critical value of the leader is different from that of the follower. As Fig. 6 shows, in the region of 0.01 < x < 0.02, the leader’s value function is lower than the follower’s value function. The first crossing point is important because an incentive to be the leader appears at this point
200 180
VTI F PV of Damage VTI L
160 140
VTI
120 100 80 60 40 20 0 0
0.005
0.01
0.015
x
0.02
0.025
Fig. 6. The value functions of the leader and follower for a = 20,000.
0.03
A. Ohyama, M. Tsujimura / European Journal of Operational Research 190 (2008) 834–854
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(x = 0.01). That is, Fig. 6 means that the leader should implement the environmental policy when the shift parameter x reaches xTIL = 0.01. We now present a comparative-static analysis for the thresholds: xTIS, xTIF, and xTIL, by varying the parameters: r, d, l, r and a. The base case parameter values present in Table 1. Suppose that a = 7000 such that we again choose the appropriate parameter a for this analysis. The results of the comparative-static analysis are presented in Table 3. Note that the parameter a = 7000 implies that an agent have an incentive to be the leader. We assume that the parameter a = 5000 such that we obtain the tangible results of comparativestatic analysis when k is varied. The result in Table 3 illustrates that the influence of the hazard rate k on the leader’s threshold is relatively larger than the influence on the follower’s threshold. This means that an increase of k increases the fear of starting a race to discover technological innovation, such as a winnertakes-all patent system. Therefore, the leader’s threshold becomes lower. Fig. 7 shows the critical value xTIF as a function of the induced effects (IE) for a = 0, 2000, 5000 and 7000. This shows that the larger the induced effects, the greater the threshold xTIF. Since we regard the induced effect in this analysis as the external effect, an increase in the induced effects means that the external effects also increase. Therefore, we can consider it natural that the larger the induced effects, the greater the threshold xTIF. Fig. 8 shows how the critical values xTIS, xTIF and xTIL decrease with respect to a. We find the following from Fig. 8. We find that an incentive to implement the environmental policy rather than to wait becomes
Table 3 The results of comparative-static analysis with TI xTIS
xTIF
xTIL
xTIS
xTIF
xTIL
r + 30% r r 30%
0.038 0.031 0.024
0.042 0.034 0.026
0.030 0.025 0.020
r + 30% r r 30%
0.041 0.031 0.023
0.044 0.034 0.026
0.032 0.025 0.019
d + 30% d d 30%
0.033 0.031 0.029
0.037 0.034 0.032
0.026 0.025 0.024
K + 30% K K 30%
0.041 0.031 0.022
0.045 0.034 0.024
0.032 0.025 0.021
l + 30% l l 30%
0.030 0.031 0.032
0.033 0.034 0.036
0.024 0.025 0.026
k + 30% k k 30%
0.034 0.035 0.037
0.039 0.040 0.044
0.027 0.033 0.041
0.1 0.09 0.08
alpha=0 alpha=2000
x TIF
0.07 0.06
alpha=5000
0.05 0.04 0.03 alpha=7000
0.02
0
1
2
3
4
5
IE Fig. 7. Dependence of critical threshold, xTIF with technological innovation, on IE.
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A. Ohyama, M. Tsujimura / European Journal of Operational Research 190 (2008) 834–854
0.07 TIF
x x TIL x TIS
0.06
xTIS,xTIF,xTIL
0.05
Threshold of F
0.04
Threshold of S
0.03
0.02 Threshold of L
0.01
0
2000
4000
6000
8000
10000
alpha Fig. 8. Dependence of critical thresholds xTIS, xTIL and xTIF on a.
greater. Note that an incentive to be the leader appears when a = 3300. Furthermore, it is also worth noting that the threshold of the leader is lower than that of the single-agent’s threshold of about a = 4100. This means that strategic agents should exercise earlier than the single agent. The reason is that an increase in a implies that the fear of starting a race to discover technological innovation, such as a winnertakes-all patent system, increases. That is, the problem of obtaining benefit from technological innovation is more important than the environmental problem of avoiding loss through the induced effect.
5. Conclusion This paper investigates an environmental policy designed to reduce the emission of pollutants under uncertainty, where the agent problem is formulated as an optimal stopping problem. First, we analyze the case following Dixit and Pindyck (1994) and Pindyck (2002) where there is only one agent. Second, the results of our original analysis are as follows. If there are two competing agents, they implement the environmental policy simultaneously. Moreover, the threshold for implementing the environmental policy is higher than that for the single-agent case. How long these two agents take to implement the environmental policy depends on the magnitude of the induced effect. We also consider the optimal timing of an environmental policy taking into account technological innovation. Even if the induced effect is complete, and an external authority never adopts intervention to promote an environmental policy, the effect of innovation may actually create a spirit of competition in the decision of the environmental policy if the additional condition (Proposition 4.1) is satisfied. Therefore, the incentive to be the leader also depends on the degree of technological innovation and the induced effect (in the model of Section 4). To conclude this paper, we discuss extensions of our model. First, we leave to examine equilibrium of our model. Next, in Section 4, we considered technological innovation such as a patent. However, we can also consider that if one agent (the leader) who has already implemented the environmental policy discovers a technological innovation, then the other agent (the follower), which has not implemented the policy yet, benefits from the technological innovation. That is, the leader does not benefit from the discovery of technological innovation. Lastly, the leader (the follower) is usually able to be assumed to be in the developed country (the developing country). We leave these topics for future research.
A. Ohyama, M. Tsujimura / European Journal of Operational Research 190 (2008) 834–854
853
Acknowledgements The authors would like to thank Masaaki Kijima and Takashi Shibata for helpful comments. The authors would also like to thank Robert Graham Dyson and two anonymous referees for valuable comments and suggestions. This research was partially supported by Daiwa Securities Group Inc. and the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Grant-in-Aid for 21st Century COE Program ‘‘Interfaces for Advanced Economic Analysis’’. The second-named author was partially supported by the Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Scientific Research (B) (2), 16310118. Appendix. Derivation of the term in Eq. (30) At first, we calculate the YF(t) from Eq. (24): Z t pF01 edðstÞ ds; Yb Ft ¼ edt Y 0 þ 0 Z t Ye Ft ¼ edt Y 0 þ pF11 edðstÞ ds:
ð74Þ
0
R1 Then, we show the term of Eq. (30): E½ sF ert ðX t Ye Ft X t Yb Ft Þdt. The other terms are derived in the same way: Z t Z 1 Z 1 rt F F rt F F dðstÞ dt F dt F e b e X t Y t X t Y t dt ¼ E e Xt ðp01 p11 Þe ds þ e Y ðs Þ e Y ðs Þ dt F0 E sF
sF
sF
ðpF01 pF11 ÞedðstÞ ds dt FsF jF0 sF sF Z t Z 1 rt F lðtsF Þ F F dðstÞ ¼E e x e ðp01 p11 Þe ds dt F0 Z ¼E E
1
ert X t
Z
t
sF
sF
Z ¼ xF ðpF01 pF11 ÞE
1
e
rt lðtsF Þ
sF
¼
Z
t
e
e sF
dðstÞ
ds dt F0
h F i xF ðpF01 pF11 Þ E ers jF0 : ðr lÞðr þ d lÞ ð75Þ
rsF
Thirdly, we focus on the term E½e have F
log xx l 12 r2 sF W sF ¼ : r
jF0 . We follow Øksendal (1998). Since X sF ¼ xF ¼ xe
ðl12r2 ÞsF þrW sF
, we
ð76Þ
1 2
Suppose that Dt ¼ efW t 2f t . Since Dt is martingale, we have h i 1 2 F E efW sF 2f s ¼ 1: From Eqs. (76) and (77), we obtain 8 93 2
F rf < log XxF l 12 r2 sF 1 = l 12 r2 x 1 2 0 2 F 4 5 f sF fs ¼ E exp f E exp f þ ¼ 1: : ; 2 2 r r x We set f > 0 so that r = (1/2)f2 + ((l 1/2r2)/r)f holds. It yields that 1 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 r l l 12 r2 2 þ f¼ þ 2r: r r
ð77Þ
ð78Þ
ð79Þ
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A. Ohyama, M. Tsujimura / European Journal of Operational Research 190 (2008) 834–854
It follows from Eqs. (78) and (79) that we have x rf x b1 E½ersF ¼ F ¼ F : x x Therefore, we obtain Z 1 ert X t Ye Ft X t Yb Ft dt ¼ E sF
xF ðpF01 pF11 Þ x b : ðr lÞðr þ d lÞ xF
ð80Þ
ð81Þ
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