Green Capacity Investment Under Subsidy Withdrawal Risk

This article studies the effect of different subsidy characteristics, like its size and withdrawal probability, on optimal investment timing and size. We find that increasing the risk of subsidy withdrawal, as well as increasing subsidy size, increases the firm's incentive to invest earlier but decreases the optimal investment size. We show that the optimal investment size is larger when no subsidy is implemented, although the firm will invest later then. A subsidy increases total welfare when subsidy withdrawal risk is very low or absent. If a policy maker aims at maximizing welfare, we show that the larger the subsidy withdrawal risk, the smaller the optimal subsidy is. When the policy maker aims to increase capacity, a subsidy is ineffective. However, a lump-sum tax can increase the firm's optimal capacity, at the cost of a delay in investment.


Introduction
In an attempt to limit climate change, many countries have set ambitious targets to reduce greenhouse gas emissions during the past two decades. Increasing the share of renewable energy (RE) production to the overall energy mix is recognized as critical in reaching those targets [European Commission, 2017]. As of 2017, 179 countries had renewable energy targets, where, in particular, 90 countries had targets to generate more than 50% of their electricity from renewables no later than 2050 [REN21, 2018b]. The European Union (EU) Commission, for example, has set a recent new target according to the "2030 framework for climate and energy policies", which is to achieve 32% of total energy consumption for the entire EU in 2030 to be delivered by renewable energy sources. Another example is China that has just reached their wind capacity target of 221 gigawatts (GW) in 2019 [World Wind Energy Association, 2019], and aims to increase total renewable power capacity to 680 GW by 2020 [REN21, 2018b].
Many countries have introduced support schemes aimed at accelerating investments in renewable energy over the past two decades, in order to reach these ambitious targets. Governments therewith, want to ensure competitiveness of renewable energy production and encourage investment. As of 2017, 128 countries had power regulatory incentives and mandates [REN21, 2018b]. China, for example, implemented the world's largest emissions trading scheme in 2017 [REN21, 2018b]. However, many support schemes have been retracted or revised suddenly and unexpectedly over the last years. Those policy changes have a severe impact on the profitability of renewable energy projects and investment behavior 1 .
We consider a monopolist that has the option to invest in a renewable energy project. It has to decide both the time to invest, as well as the size of the capacity it wants to install. Once capacity is installed, the monopolist produces a homogeneous good. The demand for the good is linear and subject to a stochastic shock. The cost of installing capacity of a certain size depends on the size of the capacity as well as the availability of support. Support is provided in the form of a lump-sum investment subsidy, which represents a general class of subsidies on investment costs including investment tax credits and capital subsidies. Investment tax credits, for example, constitute the most widespread policy instrument for RE globally 2 , often implemented with the aim to increase the affordability and profitability of RE production [REN21, 2018a, page 70]. We study the effect of policy uncertainty in the form of retraction of a currently provided subsidy.
We first derive the optimal investment decisions of a profit-maximizing firm facing subsidy retraction risk. We then take the viewpoint of a policy maker that aims to set an optimal subsidy size to reach certain targets, taking into account the resulting investment decisions of the firm. We evaluate the outcome of the subsidy in terms of its effect on social welfare. This standard approach in the public economics literature is, however, not necessarily standard in public decision-making, where it is of main importance to set the right goals and targets [Stern, 2018]. We, therefore, also look at the effect of subsidy and policy uncertainty from the perspective of a policy maker that aims to achieve a certain capacity target.
We find that increasing the subsidy size speeds up investment but this goes at the expense of a decreased optimal investment size. Increasing subsidy retraction risk for a given subsidy size has the same effect. Surprisingly, the firm's optimal investment size when there is no subsidy provided is larger than the optimal investment size when subsidy is provided but there is risk of future retraction. Analyzing the effect of policy risk on social welfare, we find that a subsidy can increase total welfare when subsidy retraction risk is small. When subsidy retraction risk is large, it is socially optimal not to implement a subsidy. We also derive the impact of subsidy retraction risk on social welfare and the ability to reach certain policy targets. When a proposed capacity target is smaller than the firm's optimal investment size, a subsidy can be used to speed up investment. If the policy maker aims at a capacity larger than the firm's optimal investment size, a lump-sum tax can be used to delay investments to a time when output prices are higher, which justifies higher capacity investment.
Our paper contributes to different strands of literature. First, we contribute to the literature on incentive regulation of a firm within an uncertain dynamic framework (see, e.g. Brennan and Schwartz [1982], Dobbs [2004], Guthrie [2005, 2012], Guthrie [2006] and Willems and Zwart [2018]). Within this literature regulatory uncertainty is considered by Hassett and Metcalf [1993], Teisberg [1993], and Dixit and Pindyck [1994, Chapter 9]. Motivated by recent frequent occurrences of changes in regulatory policies in the green energy industry, we contribute to this literature by focusing on the effect of policy risk in the form of potential subsidy withdrawal. In addition we determine the optimal subsidy size looking at different aims, welfare maximization and capacity targets, and the role of policy risk.
Our paper also contributes to an increasing strand of literature that studies the effect of subsidies on green investment (e.g., Pizer [2002], Eichner and Runkel [2014], Abrell et al. [2019], Bigerna et al. [2019]). Pennings [2000] and Danielova and Sarkar [2011] focus on the combination of subsidy and tax rate reduction. None of these papers analyze how policy risk intervenes with the effect of policy measures. Some recent literature related to renewable energy accounts for policy uncertainty related to random provision, revision or retraction of a subsidy (see, e.g., Boomsma et al. [2012], Boomsma and Linnerud [2015], Adkins and Paxson [2016], Eryilmaz and Homans [2016], Ritzenhofler and Spinler [2016] and Chronopoulos et al. [2016]). We contribute to this literature by not solely focusing on the firm's investment behavior, but also studying the effect of policy risk on the goals of the social planner and welfare.
The remainder of this paper is organized as follows. Section 2 presents the model and characterizes the optimal investment decisions both from a profit-maximizing firm and social welfare point of view. In Section 3, we study the optimal investment decision of a firm in more detail by providing comparative statics and numerical experiments. Section 4 focuses on the effect of both the subsidy size and the likelihood of withdrawal of a subsidy, on total welfare as well as reaching certain environmental targets. Section 5 concludes.

Model
We consider a risk-neutral, profit-maximizing firm that holds the option to invest in a project with an uncertain future revenue stream. The firm has to determine the optimal timing and size of the investment. After the firm has invested, all production is sold at the given output price at time t, P (t), given by where K is the firm's installed capacity, and η > 0 is a constant. The firm produces up to capacity, and all output is instantly sold on the market. This assumption is in the literature often referred to as market clearing assumption (for papers making the same assumption see, for example, Goyal and Netessine [2007], Huisman and Kort [2015] or Huberts et al. [2019]). Furthermore, the output price P (t) depends on an exogenous shock X(t), which is assumed to follow a geometric Brownian motion process given by where µ is the drift rate, σ the uncertainty parameter and dW (t) the increment of a Wiener process. The inverse demand function (2.1) is a special case of the one used by Dixit and Pindyck [1994, Chapter 9], which assumes P = XD(K) with an unspecified demand function D(K), and is frequently used in the literature (see, e.g., Pindyck [1988], He and Pindyck [1992], and Huisman and Kort [2015]). The cost of one unit of investment is set equal to δ. Hence, installing a production capacity of size K yields an investment cost of δK when no subsidy is in effect. Subsidy provides a one time discount at rate θ on the investment cost, so that the investment costs are then equal to (1 − θ)δK.
Initially, the lump-sum subsidy 3 is assumed to be available, but the subsidy will be withdrawn permanently at an uncertain time in the future. We model the subsidy withdrawal process by an exponential jump with parameter λ. This implies that the probability that the subsidy will be retracted in the next time interval dt is equal to λdt.
The optimization problem for the profit-maximizing firm is then given by an optimal stopping problem in which it aims to find the optimal time τ to invest in a capacity of optimal size K: if subsidy retraction has occurred at time t or earlier, 1 otherwise. (2.4) When investing, the firm pays a lump-sum investment cost and obtains the revenue stream P (t)K from time τ on. r is the risk-free rate, where we assume r > µ. In case r ≤ µ, the problem is trivial as it would always be optimal to wait with investment. Obviously, it is optimal for the firm to invest when the output price P is large enough, where (2.1) learns that P (t) is proportional to X. It follows that the investment rule is of a threshold type. In particular, there exists a threshold value of X at which the firm is indifferent between investing and waiting with investment 4 . It is intuitively clear that when the price is below a certain threshold level, denoted by X 1 , the firm will not invest, independently of whether the subsidy is available or not. Furthermore, when the price is high enough, i.e. above a threshold X 0 > X 1 , the firm will always invest, independent of the availability of the subsidy. For X in the interval [X 1 , X 0 ], the firm will only invest when the subsidy is active, and it will not do so when the subsidy has been withdrawn. Therefore, X 1 (X 0 ) is the value of the geometric Brownian motion at which the firm is indifferent between investing and not investing, while the policy is (not) in effect. Figure 1 summarizes the above.

Never invest
Only invest if subsidy available Always invest Assuming the initial value of the geometric Brownian motion process, x, meets the requirement 5 x < X 1 , then the firm invests either when the geometric Brownian motion hits the threshold X 1 for the first time while the subsidy has not been retracted, or when 3 In this paper, we also use subsidy to refer to the lump-sum subsidy. 4 See, for example, Dixit and Pindyck [1994] or Huisman and Kort [2015]. 5 If x ≥ X 1 , it is optimal for the firm to invest immediately, and the problem is trivial. the process hits the threshold X 0 for the first time after the subsidy has been retracted. Therefore, the expected time to investment follows from the investment thresholds and is equal to (2.5) in which P[τ > τ 1 ] is the probability that the subsidy withdrawal occurs after threshold X 1 is hit, and E[τ 1 ] (E[τ 0 ]) is the expected first hitting time of threshold X 1 (X 0 ). 6 To determine the values of the thresholds X 1 and X 0 , the value function of the firm needs to be determined. We denote the value of the firm given it has invested by V 0 in case the subsidy has been retracted, and V 1 in case the subsidy is in effect 7 : (2.7) Using the value functions (2.6) and (2.7) the optimal investment size for a given value of X can be straightforwardly derived. The result is presented in Proposition 1. Proposition 1. Let K 1 (X) (K 0 (X)) denote the optimal investment size while the policy is (not) in effect. When the firm decides to invest at X, the optimal investment size are equal to The proofs of all propositions can be found in Appendix A.
Using similar steps as in Dixit and Pindyck [1994] and Huisman and Kort [2015], the value of the investment option with and without the subsidy can be derived. These are stated in Proposition 2.
Proposition 2. Let F 1 (X, K) (F 0 (X, K)) denote the value of the option to invest at X while the policy is (not) in effect. When the firm decides to invest at X, it invests in capacity K. The value of the option to invest at X after the subsidy has been retracted is equal to where A 0 is a (positive) constant and β 01 is the positive solution to 1 2 σ 2 β 2 + (µ − 1 2 σ 2 )β − r = 0, β 01 > 1.
The value of the option to invest at X while the subsidy is available is equal to where A 1 is a (positive) constant and β 11 is the positive solution to 1 2 σ 2 β 2 + (µ − 1 2 σ 2 )β − (r + λ) = 0, β 11 > β 01 > 1. 6 Explicit derivation of the expected time to investment is shown in Appendix B.2 7 We write X instead of X(t) for convenience.
When the subsidy is (not) available, it is optimal to invest when X > X 1 (X > X 0 ), yielding equation (2.7) (equation (2.6)) as the value of the investment option. When it is optimal to wait, the value of the investment option consists of two parts: the value of holding the option to invest while the subsidy is available and the option to invest after the subsidy has been retracted. When the subsidy is retracted, the former value is lost as the subsidy will not be re-enacted again in the future.
After the subsidy has been abolished, policy uncertainty will not influence the investment decision anymore. The problem to be solved in such a situation is already analyzed in Huisman and Kort [2015]. Proposition 3 presents the optimal investment decision in this case.
Proposition 3. When the subsidy is abolished, the optimal investment threshold satisfies: whereas the corresponding investment size 8 is given by: Proposition 4 presents the firm's optimal investment decision when the subsidy is still available.
Proposition 4. If the investment subsidy has not been retracted yet, the optimal investment threshold X 1 is implicitly given by: in which K 1 is the optimal capacity under subsidy when investing at X = X 1 , i.e.: In the special case in which there is no subsidy retraction risk, equation (2.14) can be solved explicitly. Corollary 1 presents the optimal investment decisions under a lump-sum subsidy without retraction risk.
Corollary 1. In case of a subsidy with no subsidy retraction risk, the optimal investment timing and size are given by:

16)
and Comparing the investment decision under a subsidy and the one without subsidy, we observe that the optimal investment sizes are the same (K 1 = K 0 ), but the timing threshold with subsidy is actually smaller than the one without subsidy (X 1 = (1 − θ)X 0 < X 0 ). The reason behind this is that lower investment costs allow for investment at lower output prices, i.e. earlier. The decrease in investment costs has two effects on the optimal size. First, there is a direct effect. The lower the investment costs, the more the firm likes to invest for a given level of X. Second, there is an indirect effect via the timing. As investment is done sooner, i.e. at a lower output price, the firm can only justify a smaller investment size. The two effects cancel out when the firm invests at the optimal time. Now, we consider the problem from the perspective of a social planner with the objective to maximize social welfare. The social planner maximizes the total surplus (TS), which consists of the sum of the consumer (CS) and producer surplus (PS) minus the subsidy costs of θδK.
The total surplus when investing at X with capacity K is equal to 9 : Note that the total surplus does not directly depend on the subsidy. This is the result of the fact that the subsidy is solely a welfare-transfer with a zero-sum contribution to total surplus. We can determine the socially optimal timing and capacity using similar steps as before. The following Proposition 5 states the first-best social optimum: Proposition 5. The social optimal capacity for a given level of X is equal to The total surplus (TS) is then given by: in which A S is a (positive) constant, and X S is the threshold value at which the social planner is indifferent between investing and not investing. The optimal timing maximizing the total surplus, X S , is equal to The social optimal capacity, K S , is given by We find that the investment timing of the social planner and the firm are identical when there is no subsidy (i.e. X S = X 0 ). Regarding the size of investment, we conclude that it is socially optimal to invest twice as much as the profit-maximizing firm (i.e. K S = 2K 0 ). The reason is that the social planner also takes the consumer surplus into account, where consumers value a larger quantity, because this lowers the price they need to pay to acquire energy. Thus, to obtain the first-best solution, the social planner should introduce the subsidy with the aim to achieve a larger investment size of the firm without changing its timing decision. The next section investigates whether this is possible.

Economic analysis
This section analyzes the effect of the size of the subsidy and of retraction risk on the firm's optimal investment decision. The following proposition states how the optimal investment decision is affected by subsidy retraction risk.
Proposition 6 states that a higher subsidy retraction risk decreases both the optimal investment threshold and the optimal investment size. A firm speeds up investment under a higher subsidy retraction risk in order to make use of the subsidy now, as it is less likely it will be available in the future. Investing at a lower threshold implies that the firm invests when the output price is lower, which leads to a smaller optimal investment size. There is no direct effect of subsidy retraction risk on optimal investment size, but only an indirect effect via the timing, as can be easily concluded from expression (2.9). The intuition behind this is that the investment subsidy only affects the investment payoff at the moment of the investment, so that the optimal investment size does not depend on whether the subsidy will be withdrawn very soon after investing or remains for a long period of time.
The result that a higher probability of retraction of a subsidy speeds up investment is in accordance with findings of Hassett and Metcalf [1999] and Dixit and Pindyck [1994]. Chronopoulos et al. [2016] however finds that subsidy retraction risk increases the investment threshold for high levels of subsidy retraction risk. This is because Chronopoulos et al. [2016] study a subsidy in the form of a price premium, and this keeps on having an effect after the investment has been undertaken. This is opposite to the subsidy considered by us, because, as we just stated, our lump-sum subsidy just affects the investment payoff at the moment of the investment.
We find that investment size decreases with subsidy retraction risk, which is opposed to Hassett and Metcalf [1999]. This is due to the fact that Hassett and Metcalf [1999] do not account for the interaction between investment timing and size. Chronopoulos et al. [2016] only derive the conclusion that investment size decreases with subsidy retraction risk for low levels of subsidy retraction risk.
Proposition 7 presents the influence of the size of the subsidy on the optimal investment decision.
Proposition 7. The effects of the subsidy size θ on the optimal investment threshold and the investment size are given by: if and only if condition (3.2) holds.
Proposition 7 shows that a larger size of the subsidy speeds up investment and decreases the investment size. Increasing the subsidy size has two different effects on the optimal investment decision. First, providing a larger subsidy gives some incentive to invest more for a given output price. Second, as the lower costs make the investment profitable at lower output prices, it gives also some incentive to invest earlier, and as result of the dependency between timing and size, invest in a smaller capacity. We find that the second effect always dominates the first, leading to the result in Proposition 7.
In order to gain more insights on the effect of subsidy retraction risk and subsidy size, we provide a numerical example. We consider the following parameter values: µ = 0, σ = 0.1, r = 0.05, δ = 0.1, η = 0.05 and θ = 0.1. Figure 2 presents the investment timing thresholds X 0 and X 1 , and the investment sizes K 0 and K 1 as functions of the subsidy retraction risk λ. Figure 2 is in accordance with the results presented in Proposition 6. Investment timing X 1 and size K 1 decrease with subsidy retraction risk λ. Furthermore, as X 0 and K 0 are the investment threshold and capacity size after retraction of the lump-sum subsidy, these do not depend on λ.
More importantly, Figure 2 shows that the optimal investment size when there is no subsidy available (K 0 ) is in fact larger than the optimal investment size when the subsidy is available (K 1 ) but exposed to retraction risk (i.e. λ > 0). This means that when there is a risk of subsidy retraction, the firm's optimal investment size at the corresponding investment threshold is larger without subsidy than it is with subsidy, but it is equal if there is no subsidy retraction risk. There are three underlying opposing effects of receiving subsidy that influence the firm's optimal investment decision and lead to the aforementioned observation. The first two effects, the direct effect of subsidy on investment size (increasing the optimal size) and the indirect effect of subsidy on investment size via timing (decreasing the optimal size), cancel each other out, as discussed when presenting Corollary 1. The third effect is that retraction risk speeds up investment, as the firm prefers to obtain the subsidy over not obtaining subsidy. This causes the optimal investment size under subsidy to be smaller than without subsidy.
Based on Figure 2, we generate some important policy advice regarding green investment projects. Investors in green investment projects usually have long-term goals and high investment costs. Given that a subsidy has been implemented and the policy maker wants the firm to invest as much as possible, the optimal situation for the policy maker would be that there is no subsidy retraction risk (i.e. λ = 0). Without subsidy retraction risk, the firm invests in the same size as it would without subsidy, but undertakes the investment sooner. Note that subsidy retraction risk λ represents the firm's belief about how reliable the policy maker is. Therewith, λ is not a decision variable of the social planner. At best, a policy maker can increase subsidy retraction risk λ via signals. Decreasing λ is a reputation issue and cannot be influenced on a short-term.
Analyzing the role of large policy risk, we set λ = 1. This refers to the subsidy being expected to be retracted after one year. The investment timing thresholds X 0 and X 1 , and the investment sizes K 0 and K 1 are shown as functions of subsidy size θ in Figure  3. In accordance with Proposition 7, both timing and size decrease when increasing subsidy size. The results shown in Figure 3 have important implications for policy recommendations when a government aims to speed up investment by threatening to remove the subsidy soon. Whether the firm will invest immediately under large subsidy withdrawal risk, depends on the size of the subsidy and the current output price level. When the government has implemented a large subsidy (i.e. θ close to one), threatening to take away the subsidy soon results in firms investing immediately to still receive the large investment cost subsidy. However, it could happen that then, if the current output price is low, firms will invest in a small capacity.
However, when the subsidy size is relatively small, the approach to make the firm invest immediately by threatening to remove the subsidy soon is not always effective. For example, consider a subsidy size of θ = 0.15. Figure 3 shows that the optimal timing threshold while the subsidy is available, X 1 , is equal to 0.5474. Increasing the subsidy withdrawal risk even further than λ = 1 makes the threshold eventually converge to a value of approximately 0.5153. Therefore, when the current value of the demand intercept is smaller than 0.5153, trying to make the firm investment immediately by threatening to remove the subsidy is ineffective as it is never optimal to invest immediately, independent of the subsidy withdrawal risk.

Welfare and capacity target
We now study how a policy maker can influence and steer the decisions of the firm towards a socially optimal (first-best) decision. In the following we consider two different types of objectives for the social planner. We first assume that the policy maker's aim is to increase social welfare, and analyze how the policy maker can set its subsidy size to steer the firm's decisions towards the social optimum. Alternatively, especially relevant considering green energy, we consider that a social planner strives to achieve a certain capacity target as soon as possible.
To analyze the effect of subsidy retraction risk and subsidy size on the total surplus (TS), we study the relative difference between welfare generated by the first-best solution and welfare under the investment decision made by the firm. This relative difference is called the relative welfare loss (RWL), and depends on the policy parameters λ and θ. In case there is no subsidy in effect, we can show that the RWL is always equal to: See Appendix B.1 for the derivation details. This implies that a subsidy only has value in terms of increasing total surplus if it can decrease RWL below 25%. We find that the first-best outcome can in fact not be obtained with a lump-sum subsidy. To achieve the first-best outcome, according to Proposition 5, we know that the subsidy should double the size the firm would optimally chose without affecting the investment timing. Since in case subsidy is provided, the firm's optimal investment size is less than or equal to the optimal investment size without subsidy, steering the firm towards the first best outcome by providing a subsidy is not possible.
We present further results illustrated by a numerical example. To do so, we choose the following parameter values: µ = 0.02, σ = 0.1, r = 0.05, η = 0.05 and δ = 10. Figure  4 plots the total surplus and relative welfare loss as a function of subsidy retraction risk λ. For any given subsidy level, we find that the higher the likelihood of subsidy retraction (i.e. larger λ), the lower the total surplus. The more likely the subsidy retraction, the higher the incentive to invest sooner to still obtain the subsidy. As the profit-maximizing firm under subsidy retraction risk invests earlier than the socially optimal timing, it also invests less, therewith departing even further from the socially optimal capacity level. Therefore, subsidy retraction risk harms total surplus. We, in fact, find that no subsidy retraction risk is optimal in terms of total surplus. Already very small increases in subsidy retraction risk drastically decrease total surplus. Next, we turn our analysis to the social optimal subsidy size θ. Figure 5 plots the total surplus as a function of subsidy size θ. We obtain that providing subsidy can increase total welfare as illustrated in both the left and middle panel of Figure 5. The left panel of Figure 5 shows that in case of no subsidy retraction risk the total surplus is highest when θ = 0.20, i.e. the lump-sum subsidy is equal to 20% of the firm's total investment costs. At θ = 0.20, the total surplus is equal to 1.2860, while the first-best outcome leads to a total surplus of 1.6461. This results in a RWL of 21.9% opposed to the 25% when the subsidy is not provided. By implementing the subsidy, the relative welfare loss decreases by approximately 12%. The increase in welfare is the result of the fact that, under no withdrawal risk, the firm invests earlier and in the same size. This increases both the discounted consumer surplus and the discounted producer surplus, and these increases outweigh the costs of providing the subsidy. This result holds when there is no policy risk. We now study how policy risk affects this result. The middle panel of Figure 5 shows the total surplus if there is a low subsidy retraction risk. If we introduce only a small probability of subsidy withdrawal by setting λ = 0.002, corresponding to an expected subsidy retraction after 500 years, the optimal subsidy size is slightly smaller and equal to θ * = 0.163 compared to when there is no risk of subsidy retraction (θ * = 0.20). Introducing a probability of a subsidy retraction, results in that the investment is done sooner and, therefore, with a smaller capacity. Decreasing the subsidy size makes the firm postpone investment. When it invests, it, therefore, invests in a larger size. Thus, decreasing the subsidy size counters the effect of the increased probability of subsidy retraction. Comparing the middle panel with the left panel in Figure 5, we observe that for any given subsidy size the total surplus decreases when there is subsidy retraction risk.
Assuming a slightly larger subsidy withdrawal risk by setting λ = 0.007 (which gives an expected subsidy retraction after 143 years), it in fact becomes optimal not to introduce a subsidy at all. This is because the firm has a strong incentive to invest early, but therefore, in a small capacity. The investment is done too early and at a too small scale from a welfare-maximizing point of view. Therefore, when policy risk is large, it is best for social welfare not to offer a subsidy at all.
We now focus on the case where, instead of maximizing social welfare, the social planner has the aim to reach a certain capacity targetK as soon as possible. Figure 6 illustrates the optimal subsidy size required to reach a certain capacity target (left panel) and the resulting investment timing (left panel) as a function of subsidy retraction risk λ. 10 In case the target is lower than the firm's optimal investment without subsidy (i.e. K < K 0 ), the social planner can use the policy instrument to speed up the firm's investment. In this scenario a subsidy can be used to reach the capacity target earlier, as illustrated in Figure 6. The smaller the capacity target, the sooner investment will take place, which is accelerated by offering a larger subsidy. When the subsidy withdrawal risk increases, the subsidy required to reach a certain capacity target decreases. The optimal investment threshold, however, increases as a result of the smaller subsidy size.
As a subsidy can only be used to speed up investment and as a side effect it decreases the firm's optimal investment size, it cannot be applied to reach a capacity target that is larger than the firm's optimal investment size if no subsidy is provided. In this case, the social planner needs to delay the firm's investment in order to wait for better economic circumstances, which allows the firm to increase the investment size. To accomplish such a delay, a tax can be used, as shown in Figure 7. This figure shows the results for a lump-sum tax, i.e. θ is now the percentage of the total investment costs the investor has to pay to the government additional to paying the full investment cost δK. The model that accounts for this tax as as well as the corresponding derivations are presented in Appendix C. The left panel in Figure 7 illustrated how the tax size should be optimally set in order to reach a certain capacity target given retraction probability λ. The right panel in Figure 7 shows the corresponding investment timing. Given the tax rate, a larger subsidy retraction probability λ makes the firm invest earlier and less. So in order to keep the investment size at the same level when there is subsidy retraction risk, the tax rate must be changed in such a way that the firm invests later and thus under better market conditions, which allows the firm invests more. To reach this aim, the tax rate must be higher.

Conclusions
This paper studies the effect of a lump-sum subsidy subject to risk of retraction on optimal investment decisions in terms of timing and capacity size installed. We find that increasing the likelihood of subsidy withdrawal gives the firm an incentive to invest sooner to still obtain the subsidy. As the firm invests sooner, it also invests in a smaller size. Also, increasing the subsidy size speeds up investment and decreases investment size.
Furthermore, we obtain that the firm's optimal investment size under subsidy is smaller than the firm's optimal investment size without subsidy. This results from the fact that subsidy retraction risk drives the firm to invest at a lower output price, which leads to a lower optimal capacity.
We further conclude that the optimal timing of the profit-maximizing firm and the welfare maximizing social planner are equal. However, the firm underinvests from a social perspective. A lump-sum subsidy can increase welfare when there is no or only very little subsidy retraction risk. When subsidy retraction risk increases, the optimal subsidy size decreases, and welfare decreases rapidly as the firm invests in a much too small size from a social optimal point of view. When there is substantial subsidy retraction risk, any lump-sum subsidy will only decrease welfare.
When the policy maker aims to reach a capacity target that is smaller than the firm's optimal investment size, implementing a lump-sum subsidy can speed up the firm's investment size. When the capacity target decreases, the subsidy can be increased to give the firm the incentive to invest earlier. If the policy maker sets a capacity target that is larger than the firm's optimal investment size, a subsidy cannot be employed to achieve this target. In this case, a lump-sum tax can be used. The tax increases the total costs for the investor and therewith, delays investment to economically better times, which increases the firm's optimal investment size. The larger the capacity target, the larger the required tax, and, as a result, the later the firm invests. Furthermore, the required tax increases with the tax withdrawal risk.
Our model can be extended for the case in which the firm is able to receive signals on future government decisions, so that it can update its beliefs about the possibility of a subsidy retraction. In that light Pawlina and Kort [2005] propose a model with consistent authority behavior, which takes into account that the government will only intervene at a certain price level, but that article only considers the timing of the investment and not the size. Dalby et al. [2018] account that firms receive signals and can learn about the timing of subsidy revision but do not account for a firm's investment timing and capacity size decisions.

A Proofs of theorems and propositions
A.1 Proof of Proposition 1 Proof of Proposition 1. This proof shows that the expression for K 1 (X) (expression (2.9)) holds for X > X 1 . The proof that equation (2.8) is correct for X > X 0 follows the same steps.

A.2 Proof of Proposition 2
Proof of Proposition 2. Firstly, looking at the value of the investment option without the subsidy, we can follow Huisman and Kort [2015] as there is no subsidy uncertainty in this case. When X > X 0 , it is optimal to invest, and we have: When X < X 0 , it is optimal to wait with investing. It can be shown that the following holds for V 0 (X), the value of the investment at level X when the policy has been withdrawn (see e.g. Dixit and Pindyck [1994]): Solving this ordinary differential equation yields V 0 (X) = A 0 X β 01 + B 0 X β 02 . In this expression, A 0 and B 0 are constants that remain to be determined. β 01 (β 02 ) is the positive (negative) solution to 1 2 σ 2 β 2 + (µ − 1 2 σ 2 )β − r = 0. Since V 0 (0) = 0 and β 02 < 0, it follows that B 0 = 0, hence: Combining expressions (A.3) and (A.5) yields the expression (2.10) for V 0 . Secondly, we derive expression (2.11) for V 1 . When X > X 1 , it is optimal to invest and the value of the option to invest when the subsidy is in effect is equal to For X < X 1 , it holds that it is best to wait. The investment option while the policy is active satisfies the following ordinary differential equation: The main difference with equation (A.4) is the addition of the term λ(V 0 (X) − V 1 (X)), which has been added as the value of the option to invest can drop from V 1 to V 0 if the subsidy is retracted while we wait. Since X < X 1 means X < X 0 , we have V 0 (X) = A 0 X β 01 for X < X 1 . Solving the homogeneous part of the above ordinary differential equation yields solution V H 1 (X) = A 1 X β 11 + B 1 X β 12 . β 11 (β 12 ) is the positive (negative) solution to 1 2 σ 2 β 2 +(µ− 1 2 σ 2 )β −(r +λ) = 0. To find a particular solution to the ordinary differential equation in (A.7), one can try V P 1 (X) = C 1 X β 01 , as the in-homogeneous part is A 0 X β 01 . From this it follows that C 1 = A 0 . Combining the homogeneous and particular solution gives V 1 (X) = A 1 X β 11 + B 1 X β 12 + A 0 X β 01 . However, as V 1 (0) = 0 and β 12 < 0, it follows that B 1 = 0. This results in the following expression for V 1 (X): where A 1 and A 0 are constants that needs be determined. As before, β 01 is the positive solution to 1 2 σ 2 β 2 + (µ − 1 2 σ 2 )β − r = 0. Combining expressions (A.6) and (A.8) yields expression (2.11) for V 1 .

A.3 Proof of Proposition 3
Proof of Proposition 3. The constant A 0 and thresholds X 0 satisfy the value matching and smooth pasting condition for V 0 . The value matching equation for V 0 is (A.9), which guarantees that the value for V 0 (X 0 , K 0 ) is uniquely defined.
Apart from value matching condition, there is also a smooth pasting condition for V 0 . Equation (A.10) guarantees that dV 0 dX has a unique value at X = X 0 .
Multiplying (A.9) by β 01 and subtracting X 0 times (A.10) from it yields: Plugging the expression for the optimal capacity K 0 (see expression (2.8)) into (A.12) and rewriting this equation results in: Substituting the expression (A.13) for X 0 into (2.8) yields an expression for the optimal capacity when the subsidy is not available.

A.4 Proof of Proposition 4
Proof of Proposition 4. The constant A 1 and threshold X 1 satisfy the value matching and smooth pasting conditions for V 1 . The value matching equation is (A.15), which guarantees that the value for V 1 (X 1 , K 1 ) is uniquely defined.
Apart from value matching condition, there is also smooth pasting condition (A.16), which guarantees that dV 1 dX has a unique value at X = X 1 .
Subtracting X 1 β 11 times equation (A.16) from (A.15) yields: Rearranging terms in (A.17) leads to: In the above, an expression for A 0 can be derived by rewriting equation (A.10) and subsequently substituting the derived expressions for X 0 and K 0 : A.5 Proof of Proposition 5 Proof of Proposition 5. To derive the optimal capacity from a social welfare point of view, we take the first order condition of TS with respect to K, similar to deriving the optimal capacity for the profit-maximizing firm, see the proof in Appendix A.1. We take the same steps as the proof in Appendix A.2 when determining the expression for V 0 to derive the value of the option to invest for the social planner.
The threshold for the social planner X S satisfies the value matching and smooth pasting conditions. The value matching equation is: .20) and the smooth pasting condition is: The interpretation of the value matching and smooth pasting conditions are the same as the value matching and smooth pasting conditions for the profit-maximizer, which are discussed in Section 2.
The threshold X S can be derived using the same steps as in Appendix A.3 and even yields: A.6 Proof of Proposition 6 Proof of Proposition 6. We start by proving the first statement of this proposition: To derive the effect of subsidy retraction risk λ on timing threshold X 1 , we only have to look at the direct effect of λ on X 1 , as there is no indirect effect via investment size, since ∂K 1 ∂λ = 0. Therefore: Let implicit equation (2.14) be denoted by f . To derive ∂X 1 ∂λ , we apply total differentiation to f : We are going to show that ∂f ∂λ < 0 always holds, and ∂f ∂X < 0 if and only if condition (3.2) holds.
(A.38) Thus, sign( ∂f ∂X ) = sign(g). Substituting the expression for K 1 as a function of X 1 (see (2.9)) into (A.38), it can be shown that the sign of g and the sign of h are the same, where h is defined as follows: It is straightforward that h is a parabola that opens upwards for β 11 ≥ β 01 > 1. The two zeros are at: Since dX 1 dλ ≤ 0 if and only if h(X 1 ) < 0, we can conclude that dX 1 dλ ≤ 0 if and only if X 1 ∈ (X L , X R ) always holds. Since X 1 ≤ X R is the condition (3.2), only a lower bound on X 1 , X min , meeting the requirement X L ≤ X min needs to be shown.
Deriving the conditions for dK 1 dθ < 0 if condition (3.2) holds, can be done by starting with total differentiation, as before: Note that all terms have already been used previously: We can rewrite expression (A.70) to: When (A.60) holds, it follows that dK 1 dθ < 0.
B Additional derivations B.1 Derivation of constant relative welfare loss under no subsidy Let X S and K S denote the socially optimal timing and capacity, and let X 0 and K 0 be the firm's optimal timing and capacity without any subsidy. Using that X S = X 0 and K S = 2K 0 , and expressions (2.12) and (2.13) for X 0 and K 0 , the relative welfare loss is equal to RWL = TS(X S , K S ) − TS(X 0 , K 0 ) TS(X S , K S ) = X S (2−ηK S )K S 2(r−µ) − δK S − ( X 0 (2−ηK 0 )K 0 2(r−µ) − δK 0 ) X S (2−ηK S )K S 2(r−µ) − δK S = 4X 0 (1−ηK 0 )K 0 2(r−µ) − 2δK 0 − X 0 (1−ηK 0 )K 0 +X 0 K 0 2(r−µ) B.2 Stochastic discount factor and expected time to investment When analyzing the effect of a subsidy on total welfare, we need to take into account that a subsidy speeds up investment, and thus investment is done at a different time under subsidy than without the subsidy. As we compare welfare outcomes under different times, we need to discount both the welfare with and without subsidy properly. This subsection shows that the discount factor for investment without subsidy is equal to We also derive that when investment is influenced by a subsidy subject to subsidy retraction risk, the discount factor is equal to 3) The discount factor for discounting investment without subsidy risk has been derived in Dixit and Pindyck [1994] and has been addressed in Huisman and Kort [2015].
To derive the stochastic discount factor for investment under subsidy subject to subsidy retraction risk, we need to derive the expected time to investment. We define the first hitting times of the thresholds as follows: τ 0 = min{t : X(t) ≥ X 0 }, (B.4) τ 1 = min{t : X(t) ≥ X 1 }.
(B.5) Furthermore, let τ be the time at which the exponential jump process with parameter λ has its first jump. Then the expected time to investment can be written as follows: Expected time to investment = P[τ > τ 1 ] · E[τ 1 ] + (1 − P[τ > τ 1 ]) · E[τ 0 ] (B.6) The first part of the sum takes the scenario in which the first exponential jump occurs after the first hitting time of investment threshold X 1 . In that case, the first hitting time of X 1 is relevant for our solution. The second part of the sum takes the scenario in which the first exponential jump occurs before the first hitting time of investment threshold X 1 . Then, the policy is withdrawn before we invest and we are no longer interested in the first time the GBM process reaches X 1 , but the first hitting time of threshold X 0 is the relevant stochastic variable. In equation (B.6), the analytic expressions for E[τ 0 ] and E[τ 1 ] are known from e.g. Dixit and Pindyck [1994]: . (B.8) P[τ > τ 1 ] is the probability that the exponential jump occurs after the first time the GBM process X hits the threshold X 1 . Thus, we compare two first passage times of two independent random processes. In general, this problem is solved by solving the following integral: ∞ 0 e −λs f τ 1 (s)ds, (B.9) where f τ 1 (s) is the density function of the hitting time of the GBM. Valenti et al. [2007] state that the distribution of time τ 1 for a GBM process X starting at x (see equation (2.2)) to reach threshold X 1 is given by the inverse Gaussian: f (X 1 , x) = X 1 − x 2πσ 2 τ 3 1 · e −(X 1 −x−µτ 1 ) 2 2σ 2 τ 1 . (B.10) To simplify the derivation, we rewrite (B.10) into the standard form of an inverse Gaussian pdf: Expression (B.12) is an inverse Gaussian pdf with parametersλ andμ, whereλ = The probability the exponential jump occurs after threshold X 1 is hit, is equal to the expression (B.17), in which x is the starting value of the GBM.