Investment Timing under Uncertainty
in a Competitive Industry : a Social
Welfare Perspective
Dion Hoek
A thesis presented for the degree of
Master of Science
Faculty of Economics and Business
Rijksuniversiteit Groningen
Master’s Thesis Econometrics, Operations Research and Actuarial Studies. Supervisor: dr. G.T.J. Zwart
Investment Timing under Uncertainty
in a Competitive Industry : a Social
Welfare Perspective
Dion Hoek
March 4, 2020
Abstract
1
Introduction
Many corporate decisions can be regarded as call option. Consider for example an oil company that has just bought an oil field. The company now has the option to start drilling for oil at any time. Before the company can start drilling for oil, however, it must first invest in machinery and human resources. The costs resulting from these investments can be viewed as the strike price. The value of the option is mainly derived from the oil price, whose future movements are uncertain. The decision to invest can be regarded as exercising the option. The oil company faces a trade-off in its investment timing. On the one hand, it can decide to invest and thereby claim the benefits of the oil field immediately. On the other hand, it can decide to postpone the investment to await a more favourable oil price.
Because of their similarity to financial options, such corporate decisions are called real options. Following the seminal papers on the pricing of financial options by Black and Scholes (1973) and Merton (1973), a broad body of literature on real op-tion theory has emerged. A key result in this literature is that irreversible investment is delayed under uncertainty.
In a competitive industry, each firm should also take the strategic behaviour of its competitors into account when determining its investment strategy. In the context of the oil company, such strategic interactions can arise from a competitor that owns a nearby oil field. Suppose that the firms do not know exactly the size of the oil deposit on their fields. Both firms now have an incentive to wait for the other firm to start drilling first, since this will reveal more information about the oil deposit on their own field. As an alternative situation, suppose that multiple oil companies have the option to start drilling for oil on the same oil field. Once one firm starts drilling for oil, however, the other firms lose their option. In this scenario, the firms have an incentive to advance their investment in order to pre-empt their competition.
their optimal trigger. We first derive the Nash equilibrium conditions for a general cost distribution and then provide a closed-form solution when the investment cost follows a truncated Pareto distribution. In that case, the optimal investment trigger is a constant proportion of the investment cost.
In practice, a firm may have to do research to determine its investment cost. We assume that firms only know their investment cost distribution before they perform such research. To decide whether to perform such research, a firm will then weigh the resulting cost against the expected value of the investment opportunity. A satisfactory implication of the introduction of this cost is that the number of firms that has the option to invest is no longer exogeneously given. We will derive the number of firms that researches its investment cost. Furthermore, we will show that the ex-ante expected optimal investment trigger is decreasing in the number of firms that researches its investment cost for two reasons. Firstly, when more firms learn their investment cost, the lowest investment cost among these firms is expected to decrease. This then leads to a lower investment trigger. Secondly, firms decrease their trigger as a reaction on increased competition, because they fear being pre-empted.
The preceding analyses lead up to the analysis of the investment opportunity from a different perspective. No longer will we only look at the investment from the perspective of the firms, but also from the perspective of social welfare. We first show that a monopolist always invests too late from a social welfare perspective. The reason is that a monopolist carries the full investment cost, whereas it captures only part of the social welfare as its profit. Therefore, it misses out on less income when it decides to delay investment. When firms face a threat of pre-emption, they will decrease their investment trigger. If the threat is big enough, firms actually invest earlier than is socially optimal.
Finally, we consider the actions of a regulator that aims to maximize social welfare. To that end, the regulator can scale the firms’ investment cost and determine a price for which firms can buy the option to invest. We will derive the optimal option price and investment cost scale and analyse their behaviour for different parameter values. We show that the optimal investment scale is increasing in the degree of uncertainty of the investment opportunity. The reason is that the strategic behaviour of the firms outweighs the option value of waiting. From the perspective of social welfare, firms could do better by delaying the investment.
2
Literature Review
A firm that has an option to invest in an irreversible project under uncertain market conditions faces a trade-off in its investment timing. On the one hand, the firm foregoes some profits of the project when it decides to postpone investment. On the other hand, investing early causes the firm to lose its opportunity to wait for new information about the market conditions that affect the profitability of the project. The classical NPV rule, that states that a firm should invest in a project whenever its value exceeds its investment cost, ignores the option value of waiting to invest. In their highly influential paper, McDonald and Siegel (1986) show that a firm that has an option to invest in an irreversible project, whose value follows geometric Brownian motion, invests whenever the value of the project exceeds some trigger level that is higher than the trigger level that follows from the NPV rule. Indeed, for reasonable parameter values, the trigger level in McDonald and Siegel (1986) is twice as high as the investment cost. Dixit (1989) extends McDonald and Siegel (1986) by allowing a firm to exit operations at any time when the revenue of the project drops significantly below its operating cost. Pindyck (1986) develops a model in which a firm can choose the capacity of an irreversible project, which can continuously be increased but can never be decreased. Later contributions in investment timing under uncertainty include the choice between mutually exclusive projects (Dixit, 1993), the opportunity of stepwise investment (Kort, Murto and Pawlina, 2010) and the joint determination of investment timing and debt structure (Shibata and Nishihara, 2015).
The previous studies all consider a monopoly or, similarly, a naive firm in a competitive market that does not take the action of other firms into consideration. Yet, the competitive environment incentivizes a strategic firm either to lower its monopoly trigger in an attempt to preempt its competitors or to raise its monopoly trigger to wait for its competitors to invest first.1 The first author to include such
strategic considerations in investment timing under uncertainty was Smets (1991), who considers the optimal timing for a firm to switch from exporting to foreign direct investment in a growing foreign market. In line with research on monopolies, he finds that a single firm delays investment under uncertainty. In a duopoly with significant first-mover advantages, however, he finds that this effect is reversed. 1An example of a pre-emption game, in which firms have an incentive to be the first to invest, is
Indeed, Nielsen (2002) shows that when the market exhibits negative externalities, i.e. when the entry of a competitor reduces a firm’s profitability, the firm that invests first times its investment earlier than a monopolist would do. Moreover, he shows that this result still holds in a market with positive externalities, although firms will then choose to invest simultaneously instead of sequentially. Huisman and Kort (1999) relax Nielsen’s assumption that firms are ex-ante not active on the output market. They find that when first-mover advantages are low and uncertainty is high, the investment threshold of both firms is actually higher than that of a monopolist.
The previous studies assume that firms are symmetric, in particular in their investment cost. In reality, there is reason to believe that firms have different in-vestment costs. Such differences may arise from different access to capital markets; a more liquid firm may finance its investment under more favourable terms than an illiquid firm. Moreover, firms with a higher degree of organizational flexibility may find it easier to adopt new innovations, thus incurring less costs (Pawlina and Kort, 2006). Huisman (2000) studies the model in Nielsen (2002) under asymmetrical investment costs and finds that Nielsen’s results still hold, but in a weaker sense. If the difference in the investment cost between both firms is higher, the decrease of the investment threshold of the first firm in comparison to the investment threshold of a monopolist is lower. Pawlina and Kort (2006) combine Huisman and Kort (2000) and Huisman (2000) by considering a setting in which firms are ex-ante active on the output market and have unequal investment costs. As in the single firm case, other contributions to investment timing under uncertainty in a competitive industry include the joint determination of market entry, company foreclosure and capital structure (Lambrecht, 2001) and the joint determination of investment timing and capacity level (Huisman and Kort, 2015).
Perraudin (2003) as a starting point for the analysis. Firstly, the game will be extended to a game with more than two players. Secondly, we will add a preparation cost that each firm must incur before it is able to invest. Lastly, we will analyze the outcome of the game from a social welfare perspective.
3
The investment timing problem under uncertainty
We study the investment decision of multiple strategic firms that compete for invest-ment in a single project, whose future value is uncertain. A natural starting point is the investment decision of a monopolist. The model that we use is due to McDonald and Siegel (1986). After we establish the behaviour of the monopolist, we will extend the framework to accommodate for the strategic interactions of multiple firms that com-pete for the same project. The extension follows Lambrecht and Perraudin (2003), but extends it to an industry with more than two firms. Finally, we will include a fixed cost that firms incur before they have the opportunity to invest. As such, we are able to determine the number of firms that buy the option. Throughout this section, we assume that all investors are risk neutral and can borrow and lend freely at the same risk-free interest rate 𝑟, which is constant over time. Proofs of propositions in this sec-tion mimic the proofs in McDonald and Siegel (1986) and Lambrecht and Perraudin (2003) and will appear in the appendix.
3.1
A single-firm equilibrium
Consider an investment opportunity for a single firm that yields an uncertain flow of income 𝑃𝑡, which evolves over time 𝑡. The flow of income follows a geometric
Brow-nian motion
𝑑𝑃𝑡 =𝜇𝑃𝑃𝑡𝑑𝑡 + 𝜎𝑃𝑃𝑡𝑑𝑊𝑡,
where the drift rate 𝜇𝑃and the diffusion rate 𝜎𝑃 > 0 are known and constant over time
and 𝑊𝑡 is a Wiener process. This process for 𝑃𝑡is suitable when the flow of income is
expected to grow at an expected rate 𝜇𝑃, but unforeseen shocks in the market
condi-tions affect the flow income in any direction. The firm can invest a lump-sum 𝐼 in the project at any time 𝑡 to earn the flow of income from time 𝑡 onwards.
The firm faces the problem to determine its optimal investment trigger ˜𝑃, i.e. the flow of income 𝑃 that the project generates, for which the firm exercises its option to invest. Such an investment trigger only exists when 𝜇𝑃 < 𝑟. When 𝜇𝑃 ≥ 𝑟, the
outweigh the loss of potential income: the firm will invest immediately. To avoid such a situation, we assume from now on that 𝜇𝑃 < 𝑟.
We will now derive the value 𝑉(𝑃| ˜𝑃) of the investment opportunity when the current income is 𝑃, given that the firm has investment trigger ˜𝑃. Suppose first that 𝑃 ≥ ˜𝑃. Then the firm will invest immediately. Recall that once the firm invests, it will earn the flow of income in perpetuity. Since
E(𝑃𝑡) = 𝑒−𝜇𝑃𝑡𝑃0, we obtain 𝑉(𝑃| ˜𝑃) = ∫ ∞ 0 𝑒−𝑟𝑡E(𝑃𝑡)𝑑𝑡 − 𝐼 = ∫ ∞ 0 𝑒(𝜇𝑃−𝑟)𝑡𝑃𝑑𝑡 − 𝐼 = 𝑃 𝑟 − 𝜇𝑃 −𝐼 for 𝑃 ≥ ˜𝑃. (1)
Now suppose that 𝑃 < ˜𝑃. The firm will invest at the moment when 𝑃 exceeds ˜𝑃 for the first time. The time 𝑇 at which this happens is uncertain. At time 𝑇, the value of the investment opportunity is that in Equation (1). Since the firm is risk-neutral, we can find the firm’s present value by discounting at the risk-free rate and taking the expectation with respect to 𝑇, i.e.
𝑉(𝑃| ˜𝑃) =E𝑇 𝑃 𝑟 − 𝜇𝑃 −𝐼 𝑒−𝑟𝑇 for 𝑃 < ˜𝑃.
Following McDonald and Siegel (1986), we find the following expression for the firm’s value when the current level of income 𝑃 is lower than the investment trigger ˜𝑃 of the firm.
Proposition 1. The value of a monopolist prior to investing under the assumptions in
this section is 𝑉(𝑃| ˜𝑃) = 𝑃˜ 𝑟 − 𝜇𝑃 −𝐼 𝑃 ˜ 𝑃 𝛽 ,
where𝛽 is the positive root of the quadratic equation 𝛽(𝛽 − 1)𝜎
Since 𝜇𝑃 < 𝑟, we have 𝛽 > 12 and it follows that 𝛽/(𝛽 − 1) > 1. Comparing the
optimal investment trigger with the trigger ˜𝑃𝑁 𝑃𝑉 ≡ (𝑟 −𝜇𝑃)𝐼 that follows from the NPV
rule, i.e. the level of income for which the present value of the project equals the in-vestment cost, we arrive at the well-known result by Mcdonald and Siegel (1986) that uncertainty delays irreversible investment. This is caused by the asymmetry between investing and waiting to invest. Investment is irreversible; once a firm has invested, it cannot undo the investment and recover its cost. Waiting to invest, however, is indeed reversible; at any point in time, a firm can decide to reverse its decision of waiting to invest. The flexibility that a firm has when it has not invested yet, which is referred to as the option value of waiting, induces a firm to postpone investment until the net present value of the project is considerably higher than the investment cost. In fact, by differentiating 𝛽 with respect to 𝜎𝑃, it can be shown that the premium a firm
re-quires on top of its investment cost before it is willing to invest, is increasing in the degree of uncertainty 𝜎𝑃.
3.2
A multi-firm equilibrium
Now consider the same investment opportunity as in the previous section, but in a competitive industry. The firms in this industry contend with each other for one in-vestment opportunity. Once one firm has invested, the other firms lose their opportu-nity to invest. The firms do not know each other’s investment cost. Such incomplete information can for example result from firms not being fully informed about the cap-ital structure or organizational flexibility of their competitors. The only information that firms have about the investment cost of their competitors is that it is an inde-pendent draw from some known distribution on a common support. We will follow Lambrecht and Perraudin (2003) in finding a Nash equilibrium over the domain of possible investment costs, and extend their analysis to a setting with more than two firms.
To model the strategic interaction between firms in a competitive industry, let us suppose that each firm 𝑖 (where 1 ≤ 𝑖 ≤ 𝑛 + 1) faces the threat of pre-emption by 𝑛 competitors. Firm 𝑖 does not know the investment triggers ¯𝑃𝑗(where 𝑗 ≠ 𝑖) of its
com-petitors, but instead conjectures that the trigger of firm 𝑗 is an independent draw from a known distribution 𝐹𝑗(𝑃). The rational conjecture follows from the investment cost
distribution of the other firms, but for ease of exposition, we first consider the conjec-tures 𝐹𝑗(𝑃) as given. We assume that the distributions 𝐹𝑗have continuously
differen-2This follows from
tiable density 𝑓𝑗with positive support on a common interval [ ¯𝑃𝐿, ¯𝑃𝑈]. Firm 𝑖 updates
its conjecture according to Bayes’ rule. Whenever 𝑃 hits a new high and firm 𝑗 still does not invest, firm 𝑖 learns that the lower bound of the support of firm 𝑗’s trigger is higher than it first believed. To make this more explicit, we define ˆ𝑃𝑡 ≡ max{𝑃𝜏|0 ≤ 𝜏 ≤ 𝑡} as
the highest level of income up till time 𝑡. Since the strategy of all firms consists of in-vesting once 𝑃 first crosses their investment trigger, ˆ𝑃𝑡is sufficient for everything that
firm 𝑖 has learned about the triggers of its competitors. Then, firm 𝑖’s conjecture of the investment trigger of firm 𝑗, conditional on all information that firm 𝑖 has obtained up till time 𝑡, is
𝐹𝑗(𝑃| ˆ𝑃𝑡) = 𝐹𝑗(𝑃) − 𝐹𝑗( ˆ𝑃𝑡) 1 − 𝐹𝑗( ˆ𝑃𝑡)
.
Using this conjecture, we can modify the firm value and optimal investment trigger in Proposition 1 to account for the threat of pre-emption by 𝑛 competitors.
Proposition 2. Under the assumptions above, the value of firm 𝑖 in a competitive
in-dustry, where it faces the threat of pre-emption by 𝑛 firms, prior to investment is
𝑍𝑖(𝑃, ˆ𝑃| ¯𝑃𝑖) = 𝑃¯ 𝑖 𝑟 − 𝜇𝑃 −𝐼 𝑃 ¯ 𝑃𝑖 𝛽
Ö
𝑗≠𝑖 1 − 𝐹𝑗( ¯𝑃𝑖) 1 − 𝐹𝑗( ˆ𝑃) .Its optimal investment trigger ¯𝑃𝑖is the solution of
¯ 𝑃𝑖 = Í 𝑗≠𝑖ℎ𝑗( ¯𝑃𝑖) +𝛽 Í 𝑗≠𝑖 ℎ𝑗( ¯𝑃𝑖) +𝛽 − 1 𝐼𝑖(𝑟 − 𝜇𝑃),
where ℎ𝑗(𝑃) = 𝑃 𝑓𝑗(𝑃)/(1 − 𝐹𝑗(𝑃)) denotes the hazard rate of firm 𝑗.
As a result of the threat of pre-emption, the value of firm 𝑖 decreases for any trigger ¯
𝑃𝑖. In addition, since the hazard rates ℎ𝑗 are non-negative, the optimal investment trigger of firm 𝑖 is always lower than its trigger would be if it was a monopolist. Since firm 𝑖 will never invest when its net present value is negative, we find that its optimal investment trigger satisfies ˜𝑃𝑁 𝑃𝑉 ≤ ¯𝑃𝑖 ≤ ˜𝑃𝑖.
The previous analysis puts us in a position to incorporate incomplete informa-tion into the investment timing problem. To that end, assume that the investment cost of each firm 𝑖 is an independent draw from a known distribution 𝐺𝐼𝑖(𝐼), which has a continuously differentiable density 𝑔𝐼𝑖(𝐼). The distributions 𝐺𝐼𝑖need not be the same, as long as they share the same support [𝐼𝐿, 𝐼𝑈]. Firms can observe their own
investment cost, but not that of their competitors.
invest-ment cost 𝐼𝑖to its optimal trigger ¯𝑃𝑖(𝐼). Lambrecht and Perraudin (2003) show that the
mappings ¯𝑃𝑖(𝐼) are increasing in 𝐼 and that their values coincide at both ends of the
𝐺𝐼𝑖support, i.e. ¯𝑃𝑖(𝐼𝐿) = ¯𝑃𝑗(𝐼𝐿)and ¯𝑃𝑖(𝐼𝑈) = ¯𝑃𝑗(𝐼𝑈)for all 1 ≤ 𝑖, 𝑗 ≤ 𝑛 +1. If, in addition, 𝐼 𝑔𝐼𝑖(𝐼)/(1 − 𝐺𝐼𝑖(𝐼)) is increasing in 𝐼, the mappings ¯𝑃𝑖(𝐼) are continuous. This, together with the property that the mappings ¯𝑃𝑖(𝐼) are strictly increasing in 𝐼, implies that the
mappings ¯𝑃𝑖(𝐼) are invertible. Now, firm 𝑖’s rational conjecture of firm 𝑗’s investment
trigger is
𝐹𝑗(𝑃) = 𝐺𝐼𝑗(𝐼𝑗(𝑃)),
where 𝐼𝑗(𝑃) is the inverse of ¯𝑃𝑗(𝐼). In words, this means that the probability that firm 𝑗’s
trigger is smaller than 𝑃 equals the probability that firm 𝑗’s investment cost is smaller than the investment cost for which 𝑃 would be its investment trigger. By combining this rational conjecture with the results in Proposition 2, we derive a system of differ-ential equations that the mappings ¯𝑃𝑖(𝐼) should satisfy in equilibrium.
Proposition 3. In equilibrium, the mappings 𝐼𝑖(𝑃), that give the investment cost of firm
𝑖 for which 𝑃 is its optimal investment trigger, satisfy 𝐼0 𝑖(𝑃) = 1 − 𝐺𝐼𝑖(𝐼𝑖(𝑃)) 𝑛𝑃 𝑔𝐼𝑖(𝐼𝑖(𝑃)) © « −(𝑛 − 1)𝑃 𝑃 − 𝐼𝑖(𝑃)(𝑟 − 𝜇𝑃) +
Õ
𝑖≠𝑗 𝑃 𝑃 − 𝐼𝑗(𝑃)(𝑟 − 𝜇𝑃) −𝛽 ª ® ¬ subject to 𝐼𝑖((𝑟 − 𝜇𝑃)𝐼𝑢) = 𝐼𝑢.When all firms are symmetric, in the sense that their investment costs are a draw from the same distribution, the equilibrium condition reduces to a single differential equa-tion.
Corollary 3.1. When all firms share the same cost distribution, a symmetric
equilib-rium exists in which all firms have the same mapping 𝐼(𝑃) that gives their investment cost for which 𝑃 is their investment trigger, that satisfies
𝐼0(𝑃) = 1 − 𝐺𝐼(𝐼(𝑃)) 𝑛𝑃 𝑔𝐼(𝐼(𝑃)) 𝑃 𝑃 − 𝐼(𝑃)(𝑟 − 𝜇𝑃) −𝛽 subject to 𝐼((𝑟 −𝜇𝑃)𝐼𝑢) = 𝐼𝑢.
For most cost distributions 𝐺(𝐼), this differential equation has no closed-form solu-tion and one should instead resort to numerical methods. An excepsolu-tion to this is the truncated Pareto distribution, whose distribution function is
0.0 0.1 0.2 0.3 0.4 0.0 2.5 5.0 7.5 10.0 Investment Cost (𝐼) Inv estment Tr igger ( ¯𝑃) Monopoly Trigger NPV Trigger 𝛼 = 1 𝛼 = 3 𝛼 = 5
(a) The investment trigger ¯𝑃(𝐼) for different values of 𝛼, with 𝑛 = 1. 0.0 0.1 0.2 0.3 0.4 0.0 2.5 5.0 7.5 10.0 Investment Cost (𝐼) Inv estment Tr igger ( ¯𝑃) Monopoly Trigger NPV Trigger 𝑛 = 1 𝑛 = 3 𝑛 = 5
(b) The investment trigger ¯𝑃(𝐼) for different values of 𝑛, with 𝛼 = 2.
Figure 1: The investment trigger ¯𝑃(𝐼) as a function of the investment cost 𝐼 for (a) different values of cost distribution parameter 𝛼 and (b) different values of the number of competitors
𝑛. The other parameters are: 𝑟 =0.03, 𝜇𝑃 =0and 𝜎𝑃 =0.1.
with scale parameters 0 < 𝐼𝐿 < 𝐼𝑈 < ∞ and shape parameter 𝛼 ≠ 0. By letting 𝐼𝑈 →
∞ and 𝛼 > 0, an explicit solution for the equilibrium mappings ¯𝑃(𝐼) exists, which takes on a particularly simple form. Indeed, the optimal investment trigger then is a constant proportion of the investment cost.
Proposition 4. Assume that for all firms the investment cost follows a truncated Pareto
distribution with𝛼 > 0 and 𝐼𝑢 → ∞. Then, in equilibrium, the investment trigger ¯𝑃(𝐼)
corresponding to investment cost 𝐼 is
¯
𝑃(𝐼) = 𝑛𝛼 + 𝛽
𝑛𝛼 + 𝛽 − 1𝐼(𝑟 − 𝜇𝑃). (2)
Thus, the optimal trigger is decreasing in 𝑛 and 𝛼. When a firm has a higher number of competitors 𝑛, it faces a bigger threat of being pre-empted. This reduces the firm’s incentive to delay investment. In the limiting case that the number of competitors tends to infinity, i.e. 𝑛 → ∞, the firm’s optimal trigger approaches its NPV trigger
˜
𝑃𝑁 𝑃𝑉(𝐼) = 𝐼(𝑟 − 𝜇𝑃). Similarly, since a higher 𝛼 shifts probability to lower investment costs, an increase in 𝛼 increases a firm’s threat of being pre-empted. In the limiting case that 𝛼 → 0, a firm faces a low threat of pre-emption and its optimal trigger therefore approaches its monopoly trigger ˜𝑃(𝐼). On the contrary, in the limiting case that 𝛼 → ∞, the firm’s optimal trigger approaches its NPV trigger ˜𝑃𝑁 𝑃𝑉(𝐼) = 𝐼(𝑟 − 𝜇𝑃).
The characteristics of the investment trigger ¯𝑃(𝐼) are illustrated in Figure 1.
non-linearity of the system of differential equations in Proposition 3 makes it hard to find a closed-form expression for the equilibrium mappings ¯𝑃𝑖(𝐼). If, however,
the investment cost of firm 𝑖 follows a truncated Pareto distribution with parameter 𝛼𝑖 > 0 and 𝐼𝑈 → ∞, the mappings ¯𝑃𝑖(𝐼) do indeed have a closed-form expression.
Motivated by the solution in Proposition 4, where the parameters 𝛼𝑖 were assumed
to be the same for all firms 𝑖, we obtain the following equilibrium.
Proposition 5. Assume that the investment cost of firm 𝑖 follows a truncated Pareto
distribution with 𝛼𝑖 > 0 and 𝐼𝑢 → ∞. Then, in equilibrium, the investment triggers
¯
𝑃𝑖(𝐼) corresponding to investment cost 𝐼 are ¯ 𝑃(𝐼) = Í 𝑗≠𝑖𝛼𝑗+𝛽 Í 𝑗≠𝑖𝛼𝑗+𝛽 − 1𝐼(𝑟 − 𝜇𝑃 ).
3.3
The number of firms that buys the option
Up till now, we have taken the number of firms that have the opportunity to invest as given. Before a firm has the opportunity to invest, however, it may have to carry out research to determine its investment cost or undertake some additional prepa-rations. We assume that the preparation cost 𝐼𝑓 that results from these activities are
known and equal for all firms. Moreover, we assume that firms can incur this cost only at one point in time, which we will call time 𝑡 = 0. Before each firm 𝑖 decides to incur the preparation cost, it knows that its own investment cost 𝐼𝑖and that of its
competi-tors 𝐼𝑗 are independent draws from a truncated Pareto distribution with 𝛼 > 0 and
𝐼𝑈 → ∞. To avoid a situation in which multiple firms exercise the option to invest immediately after incurring the preparation cost, we will assume that the flow of in-come 𝑃0at time 𝑡 = 0 is smaller than the lowest possible investment trigger ¯𝑃𝐿. Table
1 summarizes the timing of the model.
In deciding whether or not to incur the preparation cost, the firms compare this cost with the expected value of the investment opportunity. Whenever the ex-pected value of the investment opportunity exceeds the preparation cost, a firm will incur this cost to have the option to invest at a later point in time. We now obtain the following result.
Proposition 6. Under the assumptions in this section, the value of firm 𝑖 before it learns
Table 1: Timeline of the model. 𝑇𝑚𝑖𝑛denotes the first time that the flow of income 𝑃𝑡exceeds
the investment trigger of the firm with the lowest trigger.
𝑡 = 0 • Firms decide whether they want to incur preparation cost 𝐼𝑓 to
be able to invest at a later point in time. Immediately thereafter, all firms that incurred the preparation cost learn their
investment cost 𝐼𝑖.
0< 𝑡 < 𝑇𝑚𝑖𝑛 • The firms wait for the flow of income 𝑃𝑡to exceed their
investment trigger ¯𝑃(𝐼𝑖).
𝑡 = 𝑇𝑚𝑖𝑛 • The flow of income exceeds the investment trigger of the firm
with the lowest trigger. That firm will now invest and earn the flow of income 𝑃𝑡 in perpetuity. The other firms lose their
opportunity to invest. is E𝐼𝑖(𝑍𝑖(𝑃| ¯𝑃𝑖, 𝑛)) = 𝑛𝛼 + 𝛽 𝑛𝛼 + 𝛽 − 1 − 1 𝑛𝛼 + 𝛽 − 1 𝑛𝛼 + 𝛽 𝛽 𝛼 (𝑛 + 1)𝛼 + 𝛽 − 1 × 𝑃 𝑟 − 𝜇𝑃 𝛽 𝐼𝐿1−𝛽. (3)
Since all firms are risk-neutral, they incur the preparation cost 𝐼𝑓 whenever the
ex-pected value in Proposition 6 exceeds this cost. The number of firms that will prepare for the option is thus 𝑛 + 1, where 𝑛 satisfies
E𝐼𝑖(𝑍𝑖(𝑃| ¯𝑃𝑖, 𝑛)) ≥ 𝐼𝑓 > E𝐼𝑖(𝑍𝑖(𝑃| ¯𝑃𝑖, 𝑛 + 1)). (4)
3.4
The ex-ante expected investment trigger
In Section 3.2, we established that all firms in a competitive industry have a lower investment trigger than had they faced no competition. The rationale behind this decrease was that firms lose their opportunity to invest if they are pre-empted by another firm. To prevent that from happening, firms lower their investment trigger. The more competition a firm faces, the lower its investment trigger will be. Before firms learn their investment cost, however, there is another effect that advances in-vestment. To understand why, recall that the optimal investment trigger ¯𝑃𝑖(𝐼) is
To quantify the factors that decrease the investment trigger, we first consider the ex-ante expected investment trigger of a monopolist, i.e. before it learns its in-vestment cost. To ensure that the expectation exists, we will assume that 𝛼 > 1. The ex-ante expected investment trigger of a monopolist is then given by
E𝐼 ˜ 𝑃(𝐼) = 𝛽 𝛽 − 1(𝑟 − 𝜇𝑃)E𝐼(𝐼) = 𝛼 𝛼 − 1 𝛽 𝛽 − 1(𝑟 − 𝜇𝑃)𝐼𝐿, (5)
where we used that a truncated Pareto distribution with 𝛼 > 1 and 𝐼𝑈 → ∞has
ex-pectation 𝛼𝐼𝐿/(𝛼 − 1).
Now consider the triggers of 𝑛 + 1 non-strategic firms, that invest as if they were a monopolist. To find the expectation of the lowest trigger, define 𝐼𝑚𝑖𝑛 ≡ min(𝐼𝑖 : 1 ≤ 𝑖 ≤ 𝑛 + 1) as the lowest investment cost out of 𝑛 + 1 draws from the cost distribution. For the distribution 𝐺𝐼𝑚𝑖𝑛 of the lowest investment trigger, we find
𝐺𝐼𝑚𝑖𝑛(𝐼) = 1 − (1 − 𝐺𝐼(𝐼))𝑛+1= 1 − 𝐼
−(𝑛+1)𝛼
𝐼−(𝑛+1)𝛼 𝐿
.
Thus, 𝐼𝑚𝑖𝑛also follows a truncated Pareto distribution, but with shape parameter (𝑛 +
1)𝛼. The ex-ante expectation of the lowest investment trigger of 𝑛 + 1 non-strategic firms is now given by
E𝐼𝑚𝑖𝑛 ˜ 𝑃(𝐼𝑚𝑖𝑛) = 𝛽 𝛽 − 1(𝑟 − 𝜇𝑃)E𝐼𝑚𝑖𝑛(𝐼𝑚𝑖𝑛) = (𝑛 + 1)𝛼 (𝑛 + 1)𝛼 − 1 𝛽 𝛽 − 1(𝑟 − 𝜇𝑃)𝐼𝐿. (6) Comparing this expected trigger with the expected trigger of a monopolist in Equa-tion (5), it follows that introducing 𝑛 extra non-strategic firms causes the expected trigger to drop by a factor 𝑛/((𝑛 + 1)𝛼 − 1).
The strategic interaction between the firms further decreases the investment trigger. By evaluating the trigger in Equation (2) at 𝐼𝑚𝑖𝑛and taking the expectation, we
find that the expectation of the lowest trigger of 𝑛 + 1 strategic firms is given by E𝐼𝑚𝑖𝑛 𝑃(𝐼¯ 𝑚𝑖𝑛) = (𝑛 + 1)𝛼 (𝑛 + 1)𝛼 − 1 𝑛𝛼 + 𝛽 𝑛𝛼 + 𝛽 − 1(𝑟 − 𝜇𝑃)𝐼𝐿. (7)
Comparing this expected trigger with the expectation of the lowest trigger of 𝑛 + 1 non-strategic firms in Equation (6), it follows that the strategic behaviour of the firms causes the expected trigger to drop by an additional factor 𝑛𝛼/(𝛽(𝑛𝛼 + 𝛽 − 1)).
Table 2 shows the value of the ex-ante expected lowest investment trigger in a competitive industry, relative to the expected monopoly triggerE𝐼( ˜𝑃(𝐼)). Table 2a
Table 2: The effect of the number of competitors 𝑛 on the ex-ante expected lowest
investment triggerE𝐼𝑚𝑖𝑛( ¯𝑃(𝐼𝑚𝑖𝑛)). The effect is presented as the proportion of the expected
monopoly triggerE𝐼( ˜𝑃(𝐼)), and is split into the effect due to having more draws from the cost
distribution and the effect of the firms’ strategic behaviour. The effect is considered for (a) a low cost distribution parameter 𝛼 and a low degree of uncertainty 𝜎𝑃and (b) a high cost
distribution parameter 𝛼 and a high degree of uncertainty 𝜎𝑃. The other parameters are:
𝑟 =0.03and 𝜇𝑃 =0.
(a) 𝛼 = 1.5 and 𝜎𝑃= 0.05.
Number of competitors Effect of more draws Effect of strategic behaviour Total effect
𝑛 = 1 0.50 0.95 0.48
𝑛 = 3 0.40 0.91 0.36
𝑛 = 5 0.38 0.88 0.33
𝑛 = 10 0.35 0.86 0.30
(b) 𝛼 = 5 and 𝜎𝑃= 0.2.
Number of competitors Effect of more draws Effect of strategic behaviour Total effect
𝑛 = 1 0.89 0.53 0.47
𝑛 = 3 0.84 0.48 0.40
𝑛 = 5 0.83 0.47 0.39
𝑛 = 10 0.81 0.46 0.37
trigger is caused by the bigger number of draws from the investment cost distribution. The probability mass of the investment cost distribution is more spread out over its support when the shape parameter 𝛼 is low. Therefore, having more draws from the distribution significantly increases the probability of receiving a much lower invest-ment cost. The marginal reduction in the investinvest-ment trigger is decreasing in the num-ber of competitors 𝑛 of each firm. Table 2b shows the triggers in an industry where the cost distribution parameter 𝛼 and the degree of uncertainty 𝜎𝑃are both high. In this
industry, the major part of the decrease in the expected trigger is caused by the strate-gic behaviour of the firms. In an industry without competitors, the firms have a large option value of waiting because of the high degree of uncertainty. The threat of pre-emption in a competitive industry, however, induces firms to significantly decrease their investment trigger. Again, the marginal reduction in the investment trigger is decreasing in the number of competitors 𝑛 of each firm.
4
Procurement of the investment opportunity
on social welfare. Thereafter, we discuss the discrepancy between the actual number of firms versus the socially optimal number of firms that prepares for the option. Fi-nally, we consider the decisions of a regulator that aims to maximize social welfare. The regulator can determine an option price 𝑐 and scale the investment cost of each firm by a factor 𝑘 to shift the outcome of the game towards its own interests. Through-out this chapter, we assume that the investment cost of each firm is an independent draw from a truncated Pareto distribution with parameters 𝛼 > 0 and 𝐼𝑈 → ∞.
4.1
Social welfare and investment timing
As described in Section 3.1, the flow of income 𝑃𝑡 generated by the investment
op-portunity follows geometric Brownian motion. To make explicit what drives the dy-namics of the flow of income, we assume that the firms operate in an industry where demand 𝑄𝑡 at time 𝑡 is iso-elastic, i.e.
𝑄𝑡 = 𝐴𝑡𝑃𝑡−𝛾,
where 𝛾 > 1 is the elasticity of demand and 𝐴𝑡 is a demand shift parameter that
be-haves as geometric Brownian motion 𝑑𝐴𝑡 =𝜇𝐴𝐴𝑡𝑑𝑡 + 𝜎𝐴𝐴𝑡𝑑𝑊𝑡.
Since the firms cannot choose the size of the project, we normalize the demand to 1, so that we obtain 𝑃𝑡 = 𝐴1/𝑡 𝛾. This implies that the flow of income also follows geometric
Brownian motion3 which is in line with our assumption in Section 3.1. The flow of
consumer surplus is then given by 𝐶𝑆𝑡 =
∫ ∞
𝐴𝑡1𝛾
𝐴𝑃−𝑡 𝛾𝑑𝑃𝑡 = 𝑃𝑡 𝛾 − 1.
Furthermore, we assume that firms do not have variable costs after investing, which implies that their profit equals the flow of income 𝑃𝑡. Then, the flow of social welfare
3Indeed, Itô’s lemma informs us that
𝑑𝑃𝑡= 𝜇𝐴+𝜎 2 𝐴 2 1 𝛾− 1 ! 𝑃𝑡𝑑𝑡 + 𝜎𝐴𝑃𝑡𝑑𝑊𝑡.
Thus, 𝑃𝑡follows geometric Brownian motion with drift term 𝜇𝑃 = 𝜇𝐴+ 𝜎
after investment is equal to
𝑆𝑊𝑡 = 𝛾
𝛾 − 1𝑃𝑡.
We can now analyse the effect of the firms’ optimal investment timing on social welfare. Since the mappings ¯𝑃(𝐼) are increasing in 𝐼, the firm with the lowest invest-ment cost 𝐼𝑚𝑖𝑛has the lowest investment trigger ¯𝑃𝑚𝑖𝑛 ≡ ¯𝑃(𝐼𝑚𝑖𝑛)and will be the first to
invest. Similarly as in Section 3.1, once a firm invests, society earns the flow of social welfare 𝑆𝑊𝑡 in perpetuity. Thus, the present value of social welfare is given by
𝑆𝑊 (𝑃| ¯𝑃𝑚𝑖𝑛) =E𝑇 𝛾 ¯𝑃 𝑚𝑖𝑛 (𝛾 − 1)(𝑟 − 𝜇) −𝐼𝑚𝑖𝑛 𝑒−𝑟𝑇 for 𝑃 < ¯𝑃𝑚𝑖𝑛,
where 𝑇 denotes the first time that 𝑃 crosses ¯𝑃𝑚𝑖𝑛. Using our results from Proposition
1, we can rewrite this as 𝑆𝑊 (𝑃| ¯𝑃𝑚𝑖𝑛) = 𝛾 ¯𝑃 𝑚𝑖𝑛 (𝛾 − 1)(𝑟 − 𝜇)−𝐼𝑚𝑖𝑛 𝑃 ¯ 𝑃𝑚𝑖𝑛 𝛽 . (8)
Taking the derivative with respect to ¯𝑃𝑚𝑖𝑛 and solving the resulting first-order
condi-tion for ¯𝑃𝑚𝑖𝑛then gives the optimal trigger from a social-welfare perspective
¯
𝑃𝑚𝑖𝑛 = 𝛾 − 1 𝛾
𝛽
𝛽 − 1(𝑟 − 𝜇)𝐼𝑚𝑖𝑛.
Comparing this trigger with the firm’s optimal trigger in equation (2), we find that investment occurs too early whenever
𝛾 − 1 𝛾 𝛽 𝛽 − 1 > 𝑛𝛼 + 𝛽 𝑛𝛼 + 𝛽 − 1. (9)
and too late otherwise.
4.2
Social welfare and the number of firms that enters into the
op-tion
In determining whether to prepare for the option, a firm weighs its expected value of the investment opportunity against the preparation cost 𝐼𝑓. The number of firms that
prepares for the option satisfies equation (4). Since firms act in their own interests, the number of firms that enters into the option is not necessarily optimal from a social welfare perspective. Given the number of firms that find it profitable to enter into the option, we can calculate the ex-ante expectation of social welfare as follows.
Proposition 7. The expectation of the social welfare before firms learn their investment
cost, given that 𝑛 + 1 firms decide to enter into the option, equals
E𝐼𝑚𝑖𝑛(𝑆𝑊 (𝑃| ¯𝑃𝑚𝑖𝑛, 𝑛)) = 𝛾 𝛾 − 1 𝑛𝛼 + 𝛽 𝑛𝛼 + 𝛽 − 1 − 1 𝑛𝛼 + 𝛽 − 1 𝑛𝛼 + 𝛽 𝛽 (𝑛 + 1)𝛼 (𝑛 + 1)𝛼 + 𝛽 − 1 × 𝑃 𝑟 − 𝜇𝑃 𝛽 𝐼1−𝛽 𝐿 − (𝑛 + 1)𝐼𝑓. (10)
4.3
The decisions of a regulator
In Section 4.1, we learnt that the firm with the lowest investment cost may invest ear-lier or later than is socially optimal, depending on its level of competition. Moreover, in Section 4.2, we discussed that the number of firms that enters into the option is not necessarily socially optimal. We now introduce a regulator that can change the outcome of the game to its own advantage. The regulator can scale the investment cost of all firms upwards or downwards by a factor 𝑘 by imposing a tax or giving a sub-sidy on the investment cost 𝐼. In addition, the regulator can sell the option at a price 𝑐, which is possibly negative. Effectively, the option price 𝑐 changes the preparation cost 𝐼𝑓 each firm that enters into the option incurs. A negative option price
encour-ages firms to enter into the option that otherwise would not do so. The timing of the model is now the same as in Table 1, but immediately before the firms decide whether to enter into the option, the regulator will set the investment cost scale 𝑘 and option price 𝑐.
As a result of the investment cost scale 𝑘, each firm 𝑖 times its investment as if its investment cost was 𝑘𝐼𝑖 instead of 𝐼𝑖. Then it follows from Equation (2) that the
firms’ optimal investment trigger ¯𝑃(𝐼|𝑘) corresponding to investment cost 𝐼, given 𝑘, equals
¯
𝑃(𝐼 |𝑘) = 𝑛𝛼 + 𝛽
Moreover, it is easily verified that the firm value in Equation (3) changes into E𝐼𝑖(𝑍𝑖(𝑃| ¯𝑃𝑖, 𝑛, 𝑘)) = 𝑛𝛼 + 𝛽 𝑛𝛼 + 𝛽 − 1 − 1 𝑛𝛼 + 𝛽 − 1 𝑛𝛼 + 𝛽 𝛽 𝛼 (𝑛 + 1)𝛼 + 𝛽 − 1 × 𝑃 𝑟 − 𝜇𝑃 𝛽 (𝑘𝐼𝐿)1−𝛽 (11)
and that the ex-ante social welfare in Equation (10) changes into E𝐼𝑚𝑖𝑛(𝑆𝑊 (𝑃| ¯𝑃𝑚𝑖𝑛, 𝑛, 𝑘)) = 𝛾 𝛾 − 1 𝑛𝛼 + 𝛽 𝑛𝛼 + 𝛽 − 1𝑘 − 1 𝑛𝛼 + 𝛽 − 1 𝑛𝛼 + 𝛽 𝛽 ( 𝑛 + 1)𝛼 (𝑛 + 1)𝛼 + 𝛽 − 1 × 𝑃 𝑟 − 𝜇𝑃 𝛽 𝑘−𝛽𝐼1−𝛽 𝐿 − (𝑛 + 1)𝐼𝑓. (12)
Note that the proceeds from the option price 𝑐 and investment cost scale 𝑘 are not included in the social welfare, since these are just transfers of welfare.
The objective of the regulator is to maximize social welfare. When the optimum is not unique, we assume that the regulator chooses the optimum that yields the highest consumer surplus. We now obtain the following result.
Proposition 8. A regulator, whose objective is maximizing the expected social welfare,
sets the investment cost scale 𝑘 according to
𝑘∗ = 𝛽 𝛽 − 1 𝛾 − 1 𝛾 𝑛𝛼 + 𝛽 − 1 𝑛𝛼 + 𝛽 .
Social welfare is maximized when 𝑛∗+ 1firms buy the option, where 𝑛∗satisfies E𝐼𝑚𝑖𝑛(𝑆𝑊 (𝑃| ¯𝑃𝑚𝑖𝑛, 𝑛 ∗, 𝑘∗)) ≥ E𝐼𝑚𝑖𝑛(𝑆𝑊 (𝑃| ¯𝑃𝑚𝑖𝑛, 𝑛 ∗ − 1, 𝑘∗))and E𝐼𝑚𝑖𝑛(𝑆𝑊 (𝑃| ¯𝑃𝑚𝑖𝑛, 𝑛 ∗, 𝑘∗)) > E𝐼𝑚𝑖𝑛(𝑆𝑊 (𝑃| ¯𝑃𝑚𝑖𝑛, 𝑛 ∗ + 1, 𝑘∗)).
The regulator can induce 𝑛∗ + 1firms to buy the option by setting the option price 𝑐 according to
𝑐∗
=E𝐼𝑖(𝑍𝑖(𝑃| ¯𝑃𝑖, 𝑛
∗, 𝑘∗)) −𝐼 𝑓.
Moreover, the option price is smaller than zero for most parameter values. In that way, the regulator induces more firms to enter into the option.
Table 3a shows that the investment cost scale 𝑘∗ is increasing in the degree of
uncertainty 𝜎𝑃. As the degree of uncertainty increases, the option value of waiting
for the regulator and for the firms both increase. However, the competitive industry causes firms to forego this option value in an attempt to pre-empt their competitors. To counteract this, the regulator raises the investment scale as the degree of uncer-tainty increases. Indeed, when the degree of unceruncer-tainty is very high, the regulator sets a positive investment scale, because the firms invest too early from a social wel-fare perspective. Another effect of a higher degree of uncertainty is that more firms are inclined to enter into the option. For our parameters, this rise is the sharpest when 𝜎𝑃increases from 0.05 to 0.10.
Table 3b illustrates that the number of firms that enters into the option is de-creasing in the cost shape parameter 𝛼. As 𝛼 increases, the firms face a higher haz-ard rate of being pre-empted. As such, the value of their investment opportunity de-creases and therefore less firms are willing to enter into the option.
Table 3c shows that the socially optimal number of firms that enters into the op-tion is decreasing in the elasticity of demand 𝛾. When the elasticity of demand is low, a large portion of the social welfare consists of consumer surplus. Then, the regulator wants firms to invest significantly earlier, since social benefits are much larger than the firms’ benefits. Therefore, the regulator sets a low investment cost scale. A side effect of a lower investment cost scale is that more firms are willing to enter into the option, since the value of their investment opportunity increases. As the elasticity of demand increases, a larger fraction of the social welfare comes from the firms’ profit. Then, the optimal investment timing of the regulator becomes more in line with the timing of the firms. Therefore, the regulator sets the investment cost scale closer to 1. In addition, the regulator then sets a positive option price to transfer wealth from the firms to the consumer.
5
Conclusion
In this paper, we have considered the investment timing problem under uncertainty in a competitive setting. We assume that the firms know their own investment cost, but have incomplete information on the investment cost of their competitors. In-stead, they only know the distribution function of their competitors’ investment cost. We first derive the equilibrium conditions for a general distribution function and then provide a closed-form solution when the investment cost follows a truncated Pareto distribution.
Table 3: The optimal investment cost scale 𝑘∗
, number of competitors 𝑛∗
of each firm, and option price 𝑐∗from a social welfare perspective. The optima are reported for (a) different
values of the degree of uncertainty 𝜎𝑃, (b) different values of the cost distribution parameter
𝛼 and (c) different values of the elasticity of demand 𝛾. The other parameter values are: 𝑟 =0.03, 𝜇 =0, 𝑃0 =1, 𝐼𝐿=100and 𝐼𝑓 =0.01.
(a) The optima for different values of the degree of uncertainty, where 𝛼 = 2 and 𝛾 = 2.
Degree of uncertainty Cost scale 𝑘∗
Number of competitors 𝑛∗ Option price 𝑐∗ 𝜎𝑃 = 0.05 0.53 1 -0.028 𝜎𝑃 = 0.10 0.69 5 -0.028 𝜎𝑃 = 0.15 0.85 6 -0.009 𝜎𝑃 = 0.25 1.24 6 0.046
(b) The optima for different values of the shape parameter of the cost distribution, where 𝜎𝑃= 0.1
and 𝛾 = 2.
Cost shape parameter Cost scale 𝑘∗ Number of competitors 𝑛∗ Option price 𝑐∗
𝛼 = 1.5 0.68 5 -0.014
𝛼 = 2 0.69 5 -0.028
𝛼 = 3 0.70 4 -0.023
𝛼 = 5 0.71 3 -0.017
(c) The optima for different values of the elasticity of demand, where 𝜎𝑃= 0.1and 𝛼 = 2.
Elasticity of demand Cost scale 𝑘∗
investment cost and are able to invest at a later period in time. Firms then weigh the preparation cost against their expected value of the investment opportunity to de-cide whether to enter into the option; whenever the latter exceeds the preparation cost, firms will enter. We show that the ex-ante expected investment trigger, i.e. be-fore firms learn their investment cost, is decreasing in the number of firms for two reasons. First, when more firms learn their investment cost, the lowest investment cost among these firms is expected to be lower. This then leads to a lower investment trigger. Second, firms decrease their trigger as a reaction on increased competition, because they fear being pre-empted.
Finally, we consider the outcome of the game from a social welfare perspective. We show that, depending on the level of competition, the firms may invest earlier or later than is socially optimal. Indeed, when firms face a big threat of pre-emption, they invest too early from a social welfare perspective. Thereafter, we consider a reg-ulator that can sell the option and can scale the firms’ investment cost. We derive the optimality conditions for a regulator that aims to maximize social welfare, and present the optima for some typical parameter values. We show that the optimal in-vestment scale is increasing in the degree of uncertainty of the inin-vestment opportu-nity and the elasticity of demand. Furthermore, we show that option price is increas-ing in the elasticity of demand.
In this paper, we assume that the flow of income follows a geometric Brownian motion. Further research can consider a more sophisticated process for the flow of income. For example, one could add jumps from a Poisson process to the geomet-ric Brownian motion to obtain a more realistic evolution of the flow of income. In the context of the oil company in the introduction, such jumps can then represent shocks in the oil price as a result of major news events. Moreover, in the analysis of the so-cial welfare implications in this paper, we assume that demand is iso-elastic. Further research can consider a more general demand function to test the robustness of our conclusions. Finally, we give a regulator limited options to change the outcome of the game. Indeed, the regulator can only set the option price and a fixed investment cost scale at one point in time. Further research can consider a more flexible set of tools for the regulator, such as an investment scale that is allowed to change over time.
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Appendix
Proof of Proposition 1. Whenever 𝑃 > ˜𝑃, the monopolist will invest immediately. Its
corresponding value is 𝑉(𝑃| ˜𝑃) = 𝑃
𝑟 − 𝜇𝑃
−𝐼.
When 𝑃 < ˜𝑃, the monopolist will postpone investment. Its value then satisfies the Bellman equation
𝑉(𝑃| ˜𝑃) =E𝑒−𝑟𝑑𝑡𝑉(𝑃 + 𝑑𝑃| ˜𝑃). (A.13)
Equation (A.13) says that the monopolist’s value prior to investing is equal to the dis-counted expectation of its value after an infinitesimal period of time. Itô’s Lemma informs us that 𝑉(𝑃 + 𝑑𝑃| ˜𝑃) = 𝑉(𝑃| ˜𝑃) + 𝜇𝑃𝑃𝑉0(𝑃| ˜ 𝑃)𝑑𝑡 + 𝜎𝑃𝑃𝑉0(𝑃| ˜𝑃)𝑑𝑊 + 𝜎 2 𝑃 2 𝑃 2𝑉00(𝑃| ˜ 𝑃)𝑑𝑡. (A.14) By taking the expectation of the expression in (A.14), using that 𝑒−𝑟𝑑𝑡
= 1 − 𝑟 𝑑𝑡 as 𝑑𝑡 → 0 and dropping all terms that go to zero faster than 𝑑𝑡, we find that
𝑟𝑉(𝑃| ˜𝑃) = 𝜇𝑃𝑃𝑉0(𝑃| ˜𝑃) + 𝜎 2 𝑃 2 𝑃 2𝑉00(𝑃| ˜𝑃). (A.15) To solve this differential equation, we try a solution of the form 𝑉(𝑃| ˜𝑃) = 𝑃𝛽. Plugging
Thus, the general solution of the differential equation in (A.15) is
𝑉(𝑃| ˜𝑃) = 𝐴𝑃𝛽1+𝐵𝑃𝛽2, (A.16)
where 𝛽1 > 1 and 𝛽2 < 0 are the roots of the quadratic equation 𝑟 = 𝛽(𝛽 − 1) 𝜎2
𝑃
2 +𝛽𝜇𝑃.
The monopolist’s value should be increasing in 𝑃. This is consistent with the general solution in (A.16) only if 𝐵 = 0. To find 𝐴 and ˜𝑃, we use that at the investment trigger
˜
𝑃, the monopolist’s value of postponing the investment and investing immediately should match smoothly, i.e.
𝑉(𝑃| ˜𝑃) = 𝐴𝑃𝛽1 = ˜ 𝑃 𝑟 − 𝜇𝑃 −𝐼 and 𝑉0(𝑃| ˜𝑃) = 𝛽1𝐴𝑃𝛽1−1= 1 𝑟 − 𝜇𝑃 .
By solving for 𝐴 and ˜𝑃, we find that the monopolist’s value prior to investing is 𝑉(𝑃| ˜𝑃) = 𝑃˜ 𝑟 − 𝜇𝑃 −𝐼 𝑃 ˜ 𝑃 𝛽1 , where the optimal investment trigger is
˜
𝑃 ≡ 𝛽1
𝛽1− 1
(𝑟 − 𝜇𝑃)𝐼.
Proof of Proposition 2. If firm 𝑖 has the lowest investment trigger, its value will be that
of the monopolist in Proposition 1. If firm 𝑖 does not have the lowest investment trig-ger, it will be pre-empted by another firm. In that case, its value will be zero. The law of total probability then informs us that
𝑍𝑖(𝑃, ˆ𝑃| ¯𝑃𝑖) = 𝑃¯ 𝑖 𝑟 − 𝜇𝑃 −𝐼 𝑃 ¯ 𝑃𝑖 𝛽
P(firm 𝑖 has the lowest trigger)
= 𝑃¯𝑖 𝑟 − 𝜇𝑃 −𝐼 𝑃 ¯ 𝑃𝑖 𝛽
Ö
𝑗≠𝑖 1 − 𝐹𝑗( ¯𝑃𝑖) 1 − 𝐹𝑗( ˆ𝑃) .Defining ℎ𝑗(𝑃) ≡ 𝑃 𝑓𝑗(𝑃)/(1 − 𝐹𝑗(𝑃)) as the hazard rate of firm 𝑗, it follows that ¯ 𝑃𝑖 = Í 𝑗≠𝑖ℎ𝑗( ¯𝑃𝑖) +𝛽 Í 𝑗≠𝑖 ℎ𝑗( ¯𝑃𝑖) +𝛽 − 1𝐼(𝑟 − 𝜇 𝑃). (A.17) Proof of Proposition 3. Using the chain rule to express the hazard rate of firm 𝑗 as
ℎ𝑗(𝑃) ≡ 𝑃 𝑓𝑗(𝑃) (1 − 𝐹𝑗(𝑃)) = 𝑃 𝑔𝐼𝑗(𝐼𝑗(𝑃)) 1 − 𝐺𝐼𝑗(𝐼𝑗(𝑃)) 𝐼0 𝑗(𝑃),
we can rewrite the first-order condition in (A.17) as
Õ
𝑗≠𝑖 𝑃 𝑔𝐼 𝑗(𝐼𝑗(𝑃)) 1 − 𝐺𝐼𝑗(𝐼𝑗(𝑃)) 𝐼0 𝑗(𝑃) = 𝑃 𝑃 − 𝐼𝑖(𝑃)(𝑟 − 𝜇𝑃) −𝛽 .Repeated substitution then gives us the system of differential equations that must hold in equilibrium: 𝐼0 𝑖(𝑃) = 1 − 𝐺𝐼𝑖(𝐼𝑖(𝑃)) 𝑛𝑃 𝑔𝐼𝑖(𝐼𝑖(𝑃)) © « −(𝑛 − 1)𝑃 𝑃 − 𝐼𝑖(𝑃)(𝑟 − 𝜇𝑃) +
Õ
𝑗≠𝑖 𝑃 𝑃 − 𝐼𝑗(𝑃)(𝑟 − 𝜇𝑃) −𝛽 ª ® ¬ . (A.18)When 𝑃 approaches 𝑃𝑢 and the other firms have not yet invested, firm 𝑖 knows that
the investment cost of the other firms is close to 𝐼𝑢. Moreover, the hazard of being
pre-empted goes to infinity. As a consequence, the option value of waiting is elimi-nated and firm 𝑖 will invest whenever the discounted expected value of the revenue is equal to its investment cost, i.e. whenever 𝑃 = 𝐼𝑖(𝑟 − 𝜇𝑃). This also follows from
equation (A.17) by letting ℎ𝑗(𝑃) → ∞. Hence, the boundary conditions for the system
of differential equations in (A.18) are 𝐼𝑖(𝐼𝑢(𝑟 − 𝜇𝑃)) = 𝐼𝑢.
Proof of Proposition 4. Substituting the density function 𝑔𝐼(𝐼) and the distribution
function 𝐺𝐼(𝐼) into the differential equation in Corollary 3.1 and taking the inverse,
we find that in equilibrium, ¯𝑃(𝐼) satisfies ¯ 𝑃0(𝐼) = 𝑛𝛼 ¯𝑃(𝐼) 𝐼 ¯ 𝑃(𝐼) − 𝐼(𝑟 − 𝜇𝑃) (1 −𝛽) ¯𝑃(𝐼) + 𝛽𝐼(𝑟 − 𝜇𝑃) subject to ¯𝑃(𝐼 𝑢) = (𝑟 −𝜇𝑃)𝐼𝑢
𝑣(𝐼) ≡ ¯𝑃(𝐼)/𝐼 and using that ¯𝑃0(𝐼) = 𝑣0(𝐼) + 𝑣(𝐼)𝐼, we obtain 𝑣0(𝐼) = −𝑣(𝐼) 𝐼 (1 −𝛽 − 𝑛𝛼)𝑣(𝐼) + (𝑛𝛼 + 𝛽)(𝑟 − 𝜇𝑃) (1 −𝛽)𝑣(𝐼) + 𝛽(𝑟 − 𝜇𝑃) .
This is a separable differential equation that, after partial fraction decomposition, can be written as 𝛽 (𝑛𝛼 + 𝛽)𝑣(𝐼) + 𝑛𝛼 𝑛𝛼 + 𝛽 (1 − 𝑛𝛼 − 𝛽)𝑣(𝐼) + (𝑛𝛼 + 𝛽)(𝑟 − 𝜇𝑃) ! 𝑑𝑣(𝐼) = −1𝐼𝑑𝐼 Integrating both sides, taking exponentials and substituting back 𝑣(𝐼) ≡ ¯𝑃(𝐼)/𝐼 yields
𝐶𝐼 = 𝑃(𝐼)¯ 𝐼 −𝛽 𝑛𝛼+𝛽 𝑃(𝐼)¯ 𝐼 + 𝑛𝛼 + 𝛽 1 − 𝑛𝛼 − 𝛽(𝑟 − 𝜇𝑃) −𝑛𝛼 (𝑛𝛼+𝛽)(1−𝑛𝛼−𝛽) , 𝐶 ∈ R. (A.19)
From the boundary condition, we have that ¯𝑃(𝐼) = (𝑟 − 𝜇𝑃)𝐼 as 𝐼 → ∞. Plugging this
into the right-hand side of equation (A.19), we find for 𝐼 → ∞ 𝐶𝐼 = (𝑟 − 𝜇𝑃) −𝛽 𝑛𝛼+𝛽 𝑟 − 𝜇 𝑃 1 − 𝑛𝛼 − 𝛽 (𝑛𝛼+𝛽)(1−𝑛𝛼−𝛽)−𝑛𝛼 , 𝐶 ∈ R.
But as 𝐼 → ∞, this can only hold when 𝐶 → 0. This in turn implies that the square-bracketed term in equation (A.19) is equal to zero. It follows that the investment trig-ger ¯𝑃(𝐼) corresponding to investment cost 𝐼 is
¯
𝑃(𝐼) = 𝑛𝛼 + 𝛽
𝑛𝛼 + 𝛽 − 1𝐼(𝑟 − 𝜇𝑃). (A.20)
Proof of Proposition 5. It is straightforward to verify that the equations in
Proposi-tion 5 satisfy the system of differential equaProposi-tions in ProposiProposi-tion 3. Moreover, these equations satisfy the condition that they are bounded by the NPV-trigger and the
monopoly trigger.
Proof of Proposition 6. From Proposition 2 and using that ˆ𝑃 < 𝑃𝐿, we know that the
value of firm 𝑖, given that it faces pre-emption by 𝑛 symmetric firms, is 𝑍𝑖(𝑃| ¯𝑃𝑖, 𝑛) = 𝑃¯ 𝑖 𝑟 − 𝜇𝑃 −𝐼𝑖 𝑃 ¯ 𝑃𝑖 𝛽 (1 − 𝐹( ¯𝑃𝑖))𝑛. (A.21)
op-tion, since firm 𝑖 does not yet know its investment cost. Substituting in the optimal investment trigger of firm 𝑖 given in equation (A.20), and taking the expectation with respect to 𝐼𝑖of the resulting expression yields
E𝐼𝑖(𝑍𝑖(𝑃| ¯𝑃𝑖, 𝑛)) = 𝑛𝛼 + 𝛽 𝑛𝛼 + 𝛽 − 1 − 1 𝑛𝛼 + 𝛽 − 1 𝑛𝛼 + 𝛽 𝛽 𝑃 𝑟 − 𝜇𝑃 𝛽 𝐼𝐿𝑛𝛼E𝐼𝑖(𝐼1−𝑛𝛼−𝛽𝑖 ). Since E𝐼𝑖(𝐼 1−𝑛𝛼−𝛽 𝑖 ) = ∫ ∞ 𝐼𝐿 𝐼𝑖1−𝑛𝛼−𝛽𝛼𝐼 −𝛼−1 𝑖 𝐼𝐿𝛼 𝑑𝐼𝑖 = 𝛼 (𝑛 + 1)𝛼 + 𝛽 − 1𝐼 1−𝑛𝛼−𝛽 𝐿 , we obtain E𝐼𝑖(𝑍𝑖(𝑃| ¯𝑃𝑖, 𝑛)) = 𝑛𝛼 + 𝛽 𝑛𝛼 + 𝛽 − 1 − 1 𝑛𝛼 + 𝛽 − 1 𝑛𝛼 + 𝛽 𝛽 𝛼 (𝑛 + 1)𝛼 + 𝛽 − 1 𝑃 𝑟 − 𝜇𝑃 𝛽 𝐼1−𝐿 𝛽. (A.22) Proof of Proposition 7. The ex-ante social welfare is equal to the social welfare in
equation (8) resulting from the investment, subtracted by the preparation cost that the 𝑛 + 1 firms incur, i.e.
𝑆𝑊 (𝑃| ¯𝑃𝑚𝑖𝑛, 𝑛) = 𝛾 ¯𝑃 𝑚𝑖𝑛 (𝛾 − 1)(𝑟 − 𝜇) −𝐼𝑚𝑖𝑛 𝑃 ¯ 𝑃𝑚𝑖𝑛 𝛽 − (𝑛 + 1)𝐼𝑓.
Substituting in the optimal investment trigger given in equation (A.20), evaluated at the lowest investment cost 𝐼𝑚𝑖𝑛, and taking the expectation with respect to 𝐼𝑚𝑖𝑛of the
we obtain E𝐼𝑚𝑖𝑛(𝑆𝑊 (𝑃| ¯𝑃𝑚𝑖𝑛, 𝑛)) = 𝛾 𝛾 − 1 𝑛𝛼 + 𝛽 𝑛𝛼 + 𝛽 − 1 − 1 𝑛𝛼 + 𝛽 − 1 𝑛𝛼 + 𝛽 𝛽 (𝑛 + 1)𝛼 (𝑛 + 1)𝛼 + 𝛽 − 1 × 𝑃 𝑟 − 𝜇𝑃 𝛽 𝐼𝐿1−𝛽− (𝑛 + 1)𝐼𝑓.
Proof of Proposition 8. Taking the derivative of the expression for the ante
ex-pected social welfare in equation (12) with respect to 𝑘, setting it equal to zero, and solving for 𝑘 yields
𝑘∗ = 𝛽 𝛽 − 1 𝛾 − 1 𝛾 𝑛𝛼 + 𝛽 − 1 𝑛𝛼 + 𝛽 .
Plugging 𝑘∗into equation (12) then gives
E𝐼𝑚𝑖𝑛(𝑆𝑊 (𝑃| ¯𝑃𝑚𝑖𝑛, 𝑛, 𝑘 ∗ )) = 𝛽 𝛽 − 1 − 1 𝛾(𝛽 − 1) 𝛽(𝛾 − 1) 𝛽 ( 𝑛 + 1)𝛼 (𝑛 + 1)𝛼 + 𝛽 − 1 𝑃 𝑟 − 𝜇𝑃 𝛽 𝐼𝐿1−𝛽−(𝑛+1)𝐼𝑓, which is concave in 𝑛. Therefore, social welfare is maximized when 𝑛∗
+ 1firms buy the option, where 𝑛∗satisfies
E𝐼𝑚𝑖𝑛(𝑆𝑊 (𝑃| ¯𝑃𝑚𝑖𝑛, 𝑛 ∗, 𝑘∗)) ≥ E𝐼𝑚𝑖𝑛(𝑆𝑊 (𝑃| ¯𝑃𝑚𝑖𝑛, 𝑛 ∗ − 1, 𝑘∗))and E𝐼𝑚𝑖𝑛(𝑆𝑊 (𝑃| ¯𝑃𝑚𝑖𝑛, 𝑛 ∗, 𝑘∗)) > E𝐼𝑚𝑖𝑛(𝑆𝑊 (𝑃| ¯𝑃𝑚𝑖𝑛, 𝑛 ∗ + 1, 𝑘∗)).
A firm will buy the option whenever the expected value of the investment opportunity in equation (11) exceeds the option price 𝑐 plus the preparation cost 𝐼𝑓. Since the
regulator prefers the option price 𝑐 that increases consumer surplus when multiple optima exist, the regulator will set 𝑐 according to
𝑐∗
=E𝐼𝑖(𝑍𝑖(𝑃| ¯𝑃𝑖, 𝑛
∗, 𝑘∗)) −𝐼 𝑓.
