Optimal Stopping Game with Investment Spillover Effect for. Energy Infrastructure

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1 Optimal Stopping Game with Investment Spillover Effect for Energy Infrastructure Akira aeda Professor, The University of Tokyo Komaba, eguro, Tokyo , Japan Abstract The purpose of this study is to analyze the consequence of the existence of positive externality in the optimal choice of a firm s time of investment in a gaming environment. I consider a situation where firms can increase their subsequent revenue stream by investing at a period of time. The investment opportunity is assumed to be one-time opportunity and once it is made, it cannot be redone, i.e., the investment is irreversible. This indicates that the choice of optimal time of investment is formulated as a so-called optimal stopping problem in mathematics. oreover, I assume that the investment has a spillover effect to other firms. amely, once a firm makes an expensive investment, all other firms can obtain the same result of the investment at no cost from the next time period. This positive externality allows firms to enter the game of identifying their own optimal investment time vis-à-vis their opponent s choice of time. Key words: Optimal stopping, Externality, Subgame perfect ash equilibrium JEL Classification Code: D92, H54, L32 1. Introduction Energy and environmental policy solution sometimes requires substantial investment for infrastructure that includes not only physical but also intellectual bases, i.e., development of intellectual properties, R&D, etc. In developed countries with market economy, such investment is presumed to be primarily carried out by privately-owned firms, but not governmental bodies or centrally-controlled firms. The role of the government is limited to provide a guide and/or impose regulation to the firm s activity. The question is how to shape governmental policy for the guidance and/or regulation. Investment for infrastructure both physical and intellectual is typically associated with positive externalities because the infrastructure can be utilized as public goods. For example, once transmission network is established by some firm and is made available later to all new entrant firms in accordance with some liberalization policy, it definitely benefits these new entrants. It is noted, however, that existence of positive externality does not always mean benefits to the public. If the transmission network is set freely open to all firms, it results in a 1

2 free lunch that dampens motivation for further investment of any firm. On the contrary, if the network is only possessed by the builder, the investment competition by firms may be described by the typical story of the theory of natural monopoly: In the well-known theory, firms are supposed to compete to each other to establish a monopolistic position in the market by investing more than others. In the process of establishing the monopolistic position by one firm, there will be double, triple, or multiple capital investment by those competing firms, resulting in the accumulation of excessive capital. That situation with excessive capital is not socially efficient. In any way, whether or not existence of positive externality benefits the economy is thus dependent upon a prediction of which and how many firms are willing to launch investment activity. The purpose of this study is to analyze the consequence of the existence of positive externality in the optimal choice of a firm s time of investment in a gaming environment. The paper is organized as follows. As a preliminary to the formal analysis, Section 2 formulates an optimal stopping problem for making an investment decision. Using the model of a firm s choice of time, Section 3 develops a game model in which two firms compete for investment. A key ingredient added in this section is positive externality or the spillover effect of the investment. The main results are addressed as two propositions. Section 4 discusses the implications of the propositions. Section 5 provides concluding remarks. 2. Preliminary: Optimal stopping problem for investment decision This section is intended to provide a basis for the next section, by formulating an optimization problem for the choice of investment time. The model helps the firm identify when to conduct the investment. In mathematics, this is a simplified version of the so-called optimal stopping problem. Consider a firm that earns a net revenue of X t dollars in time t (t = to infinity). ote that the time can be considered as year, quarter, or month. The present time is represented by t =. Further, assume that the firm s net revenue grows at a rate of g, i.e., the following equation holds true: ( ) 1 t Xt = X + g, where X is the initial value of the net revenue. Assume that a prevailing interest rate is r, and r > g holds true. The net present value of the firm s cash stream from the present to the future (PV) at time is represented as follows: g 1+ g 1+ g PV = X + X + X + X r 1+ r 1+ r This is equivalent to the following: 2

3 PV = X + ( 1+ g) X r g. (1) Suppose that the firm reserves an investment opportunity. Once the decision to take it is made, it allows the firm to raise a net revenue in each year (time) to ax t from the next year to forever at the cost of I dollars. ote that a > 1 is assumed. Figure 1 depicts the cash flow of the firm when the investment decision is made at time. Figure 1. Cash flow after the investment decision at time ote that the investment cost I is assumed to be a sunk cost, indicating that once the investment activity is completed, the cost cannot be recovered. Therefore, the investment decision is irreversible. Let PV ( ) denote the PV for the firm that made the investment decision at time. It is calculated as follows: ( 1+ g)( a 1) 1+ r 1+ g 1 PV ( ) = + X I. (2) r g 1+ r r g 1+ r The derivation is shown in the appendix. ote that the following relation holds true: PV ( ) PV =. Optimal choice of investment time for the firm is identified by solving the following optimization problem: max PV ( ) The following lemma is obtained. (3) Lemma1. The optimal solution for the problem (3), ( ) = arg max PV (4) exists and it is unique. oreover, 3

4 1 X r if ( )( ) = otherwise, and ( ) < I, PV is monotonously decreasing (increasing) in for all (rsp. < ). (Proof is omitted) 3. A model of optimal stopping game This section extends the model of a firm s irreversible investment decision to that of a two-firm game. We keep working on the same PV structure with and without investment, but this time, let us consider two identical firms. Again, by the investment, the firm s net revenue at each year t is raised to ax t. The investment cost remains I. A key ingredient added in this section is positive externality or spillover effect of the investment. It is assumed that if one of the two firms invests first, the other firm can enjoy the same effect of net revenue increase from the next time period without further investment of their own. Let g ( σ, ) denote the firm s net present value of cash flow when the firm invests at time, while the other firm invests at time σ. Both and σ are integer values and non-negative. The assumption of positive externality indicates that if σ holds true, the firm needs to make the investment at a positive cost, I; however, when the timing is reversed, i.e., σ <, the cost of the investment for the firm is null (i.e., I = ) and the same effect of net revenue raising becomes available to the firm. Using the same mathematical expressions of Equation (2) in the previous section, we have the following equation: PV ( ) if σ g ( σ, ) = 1. PV ( ) + I if σ r It is noted that thanks to the assumption of identical firms, any concept of equilibrium in the game of these two firms is defined in a symmetric manner. A definition of subgame perfect ash equilibrium (SPE) in this game is formally introduced as follows 1 : 1 The use of the term SPE here may appear strange to some readers. This can simply be ash equilibrium. evertheless, the reason that the term is selected here is as follows. For the firm, an investment decision can be made anytime as long as it has not been done yet. Thus, each decision at one time must be made in a forward-looking manner in accordance with possible future cash streams. In this respect, the current investment choice game for the two firms contains possible future games. 4

5 Definition. SPE is (, σ ) such that g(, σ ) g( σ, ) and g(, ) g(, ) σ σ σ. Let denote the optimal time of investment for one firm when the opponent is assumed to never invest, i.e., g(, ) g(, ). This is exactly the same as what is defined in Equation (4) for which Lemma 1 holds true. The following proposition is obtained for the case of non-zero : Proposition 1. Assume that ( )( ) X r < I. SPEs exist and (, ) σ = (, 1) + and ( 1, ) + if ( 1 g)( a 1) g 1 X 1+ r I. SPE does not exist otherwise. (Proof is omitted.) This proposition indicates that under a certain condition for I, that a double investment by these two firms never happens. σ holds true, indicating For the case of =, we obtain the following proposition: Proposition 2. Assume that ( )( ) X (, ) σ = ( ) r ( )( ) 1+ g, if 1 X 1+ r I. SPEs may exist as follows: I, (, ) σ = (,1) and ( ) 1, if ( 1 g)( a 1) g ( 1 g)( a 1) g 1+ 1 X I > 1 X. 1+ r r g 1+ r Otherwise, i.e., if ( 1+ g)( a 1) 1+ g 2 ( 1+ g)( a 1) X I > 1 X, SPE does not exist. r 1+ r (Proof is omitted.) The significant difference between Propositions 1 and 2 is that while the former rejects the possibility of double investment of two firms, the latter allows such a possibility. If ( 1+ g)( a 1) 1+ g 1 1+ r X I holds true, the two firms immediately go for investment regardless of whether their opponent is going to do the same thing or not. 5

6 4. Discussion In Section 3, I considered a situation where firms can increase their subsequent revenue stream by making an invest decision at one time. The investment opportunity is assumed to be a onetime opportunity, and once the decision is made, it cannot be redone and is thus irreversible. oreover, I assumed that the investment has a spillover effect to other firms. In other words, once a firm makes the expensive investment, all other firms can enjoy the same effect of the investment with no cost from the next time period. This positive externality allows firms to enter a game of identifying their own optimal time of investment vis-à-vis their opponent s choice of time. This setting is realistic in that it reflects a firm s investment strategy on public goods. Typical examples include competition for social infrastructure such as electric power networks, telecommunication networks, roads between/among cities, regional environmental pollution abatement activities, and so on. The two propositions addressed the main results. The indication of Proposition 1 is remarkable when we compare it to the theory of natural monopoly. As was mentioned in the introduction, the theory of natural monopoly leads to the conclusion that competition among firms investing for monopolistic position may result in the accumulation of excessive capital, which is not socially efficient. In contrast, Proposition 1 rejects such a possibility, implying that the competition leads to social efficiency. Only one firm will make the investment that is necessary and sufficient for the society to enjoy the outcome. This result is, of course, conditional. The condition is that the investment cost is lower than a certain level at the time of investment as was specified in the proposition. Otherwise, equilibrium would not exist; thus, we cannot tell anything about the consequence of the game. otwithstanding, the result is insightful. Proposition 2 is also remarkable. However, the significance of the proposition is two-fold. One is that it provides the same implication as Proposition 1; it denies the possibility of a double investment under a certain condition for the investment cost. In this regard, this proposition supports the above-mentioned main result. To the contrary, Proposition 2 also indicates that there is a possibility of double investment when the investment cost is very low and below a certain level. The result makes sense in that if the cost for the investment is sufficiently low, the benefit from taking the action as soon as possible overwhelms any other strategies. 5. Conclusion This study analyzed the consequence of the existence of positive externality in the optimal choice of a firm s time of investment in a gaming environment. A situation where firms can increase their subsequent revenue stream by making an investment decision at one time is 6

7 considered. The investment opportunity is assumed to be one-time opportunity and the decision is irreversible. Furthermore, it has a spillover effect to other firms. Propositions obtained identify conditions for the existence of SPE. Among several features of these SPEs, the most significant is that it is necessary to have a very low investment cost for two firms to immediately make the investment simultaneously. Otherwise, the investment is made by only one firm if an SPE arises, and it is a socially efficient outcome. This result supports policy formulations that promote competition rather than intervening between firms in the economy with externality. 7