Analysis of Renewable Energy Policy: Feed-in Tariffs with Minimum Price Guarantees

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1 Analysis of Renewable Energy Policy: Feed-in ariffs with Minimum Price Guarantees Luciana Barbosa a, Paulo Ferrão a, Artur Rodrigues b, and Alberto Sardinha c a MI Portugal Program and Instituto Suerior écnico, Universidade de Lisboa b NIPE and School of Economics and Management, University of Minho c INESC-ID and Instituto Suerior écnico, Universidade de Lisboa June 7, 2016 Abstract Policy intervention can imact the decision making rocess of deloying a renewable energy roject. he feed-in tariff (FI) rogram is a oular olicy for incentivizing new renewable energy rojects, because it establishes a long-term contract with renewable energy roducers. his aer resents a model to analyze a FI contract with a minimum rice guarantee in two distinct scenarios. First, we analyze FI contracts in an oligoolistic market structure. he derivation uses an asymmetric Stackelberg model with a real otions valuation model. Second, we analyze a rice-taker scenario with a real otions valuation model. We also analyze the FI contracts with a eretual and finite duration in both scenarios. With the model, we can find several interesting roerties. For examle, we can first identify the otimal time to deloy a renewable energy roject. Second, we can analyze how the value and duration of a minimum rice guarantee affects the investment threshold and the value of the roject. he results show that a eretual guarantee can induce investment for rices below the minimum rice guarantee, when the roject comensates the investment cost. Another interesting result is that a FI contract with a higher rice and duration induces an earlier investment. From a managerial 1

2 ersective, our model rovides a owerful tool to analyze investment in renewable energy rojects where we take into account managerial flexibilities and a FI olicy with a minimum rice guarantee. JEL Classification: L94, Q42, C72 Keywords: Asymmetric Stackelberg Equilibrium, Peretual guarantee, Finite guarantee, Feed-in tariff 1 Introduction he generation of energy from renewable resources, such as wind and sunlight, is an imortant otion that can mitigate many environmental roblems. In addition, the utilization of deletable resources, such as gas or coal, for generating energy is a roblem from a sustainability ersective, since sulies used now are not available for future generations. herefore, olicymakers must incentivize energy generation through renewable resource, and consequently decrease the cost of scarcity for society and rice of resources, and increase the social welfare. here are many olicies to incentivize renewable energy rojects (Grubb 2004), which in turn might reduce environmental ollution, global warming and ublic health issues. For instance, the Renewable Portfolio Standards (Wiser, Namovicz, Gielecki & Smith 2007) in the US and the RES Directive (Klessmann, Lamers, Ragwitz & Resch 2010) in the EU are olicies that are laying an increasingly imortant role in the decision-making rocess of generation comanies. Policy intervention can have a significant imact on the decision of deloying a renewable energy roject. Couture & Gagnon (2010) state that the feed-in tariff (FI) rogram is considered one of the most imortant olicies for stimulating new renewable energy rojects. FI rogram is a longterm contract with renewable energy roducers (e.g., homeowners, business and organizations such as schools and community grous) to enhance energy generation. In addition, Couture & Gagnon (2010) resent many different remuneration schemes utilized by olicymakers that have evolved over time. his article resents a model to analyze a FI olicy with a minimum rice guarantee in two scenarios, namely an oligooly and a rice-taker scenario. he model includes managerial flexibilities and identifies the otimal timing for the investment in a renewable energy roject. he model also takes into account the effect of the duration of the guarantee: hence, we analyze a eretual and finite guarantee. In articular, the oligoolistic market structure uses an asymmetric Stackelberg model with a real otions valuation model. he rice-taker scenario 2

3 uses a real otions valuation model. In both scenarios, the renewable energy roducer receives either the energy market rice or a minimum rice guarantee. We use the model to analyze how the value and duration of a minimum rice guarantee affects the investment threshold and the value of the roject. he results show that a eretual guarantee can induce investment for rices below the minimum rice, when the roject comensates the investment cost. his aer is organized as follows. Section 2 resents the related work and our main contributions to literature. Section 3 analyzes a minimum rice guarantee FI within an oligooly. Section 4 resents a FI analysis with many renewable energy roducers that are Price taker. Finally, Section 5 resents the concluding remarks. 2 Related Work In literature, many aers have used the Cournot or Stackelberg game to model energy markets in order to redict interesting economic results. In fact, Cournot oligoolies are very oular because the majority of electricity markets nowadays are neither a monooly nor a erfect cometition; womey & Neuhoff (2010) state that modeling the market as an oligooly is the most aroriate assumtion. Moreover, these analyses can lead to olicies that may resolve some of the economic and sustainability issues. For instance, Wolfram (1999) analyzes the market ower of generation comanies in the British electricity market, and shows that rices are lower than estimates due to many reasons, such as entry deterrence and actions from the regulator. Chuang, Wu & Varaiya (2001) formulate a Cournot oligooly market for generation exansion lanning, and resent numerical results to analyze industry exansion, generation investment and trends. Murhy & Smeers (2005) resent an oen-loo and closed-loo Cournot model, in which investment and ower disatch decisions occur simultaneously in the former model and in two stages in the latter model; in addition, this work comares both models with a erfect cometition. Nanduri, Das & Rocha (2009) utilize a similar single stage Cournot model as Murhy & Smeers (2005) taking into consideration the network transmission constraints. Chang, Hu & Han (2013) use a Stackelberg game to analyze a constant rice FI contract in aiwan. womey & Neuhoff (2010) examine the case of intermittent generation return under erfect cometition, monooly and oligooly; the results show that, when different technologies are used, the market articiants benefit differently from the increased rice. Moreover, intermittent technologies benefit less from the market ower effect 3

4 than conventional technologies. However, desite the numerous of aers with oligoolies for energy markets, not much has been said (if any) about the imact of the duration of a guarantee, namely a minimum rice guarantee, and the effect of managerial flexibilities. Moreover, although our model focus on the minimum rice guarantee, it can be easily extend to a market deendent Feed in tariff where the firm receives a bonus over the market rice. Our model relates and contributes to the literature in several ways. Our first contribution is to game theory, where we extend the Stackelberg model in order to value managerial flexibilities and identify the otimal timing for the investment in a renewable energy roject. he model also takes into account the effect of minimum rice guarantee FI olicies, using the Shackleton & Wojakowski (2007) framework. In articular, we show how the value and duration of a FI change the investment threshold. Our second contribution is to real otion community, where we extend Shackleton & Wojakowski (2007) and develo two models: the first one with a eretual guarantee and the second with a finite guarantee. he model shows that eretual guarantee can only induce investment for rices below the FI if it at least comensates the investment sunk cost. Our third contribution is to energy research community, where we resent a owerful model taking into account a FI olicy and managerial flexibilities within an oligooly market. We show how a FI may affect the investment decision in renewable energy rojects. 3 Oligoolistic Market Structure and FI Policy he majority of electricity markets nowadays are neither a monooly nor a erfect cometition. In fact, womey & Neuhoff (2010) state that modeling the market as an oligooly is the most aroriate assumtion. We thus analyze a minimum rice guarantee FI within an oligoolistic market. he model has a conventional energy roducer that acts as the leader and the follower is a renewable energy roducer. Hence, we analyze the roducers within a Stackelberg game. We then include managerial flexibilities with a real otion model. We start assuming that the roducers face the linear inverse demand function of the following form: P (Q, t) = ax(t) bq (t) (1) where a > 0 reresents the market size, b > 0 shows how rice changes affect 4

5 the demand function, x(t) denotes the industry s demand shock at time t observed by all firms, and Q (t) is the total outut roduced. We assume that the industry s demand shock follows the geometric Brownian motion rocess. dx t = µx t dt + σx t dw t (2) where µ < r is a deterministic drift, σ > 0 is the volatility, and dw t is the standard Brownian motion rocess. he volatility σ can be estimated (e.g., through OLS) from the returns of the historical data. We denote the conventional energy roducer with a subscrit c and the renewable energy roducer with a subscrit r. he rofit function of the conventional energy roducer is: Π c = P (Q )q c c c (3) where q c is the quantity roduced. he cost function has the following form c c = q 2 c /2k c, where k c is the conventional energy roducer s caacity. he rofit function of the renewable energy roducer is: Π r = Max(F, P (Q ))q r c r (4) where, F is a minimum rice guarantee due to a FI contract. he cost function also has the following form c r = q 2 r/2k r where k r is the renewable energy roducer s caacity. he roduction cost of conventional energy is different from the renewable energy. For instance, the roduction cost of wind ower is almost zero (womey & Neuhoff 2010). his rationale leads to a different cost function for each roducer, as described above. In the Stackelberg game, the market equilibrium is: q c = ak c (bk r + 1) 2b(bk r k c + k r + k c ) + 1 x (5) If the renewable energy roducer receives the market rice, the rofit is: Π r = a 2 k r (b(k r (bk c + 2) + k c ) + 1) 2 2(2bk r + 1)(2b(bk r k c + k r + k c ) + 1) 2 x2 = αx 2 (6) When the market rice is below the minimum rice guarantee, the energy roduction yields a different rofit due to the FI contract: Π r = F 2 k r 2 (7) 5

6 Note that the rofit function in Equation 6 is roortional to x 2, hence, it is straightforward to rove by Itô s Lemma that the rofit follows the following stochastic rocess: dπ = (2µ + σ 2 )Π t dt + 2σΠ t dw t (8) Hence, the aroriate discounted rate to find the value of the roject when the market rice,, is greater than the minimum rice guarantee, F, is: R = r 2µ σ 2 (9) he renewable energy roducer maximizes its rofit by choosing an aroriate quantity that deends on x. However, the roducer with a FI contract receives a fixed amount when the market rice is below the minimum rice guarantee. Let us denote x F as the oint where the rofit received by the equilibrium market rice (Equation 6) is equals to the rofit received by FI contract (Equation 7). Hence, x F = F kr 2α (10) 3.1 Peretual FI within an Oligooly In this section, we assume that the duration of the FI contract is eretual. Hence, the renewable energy roducer has a eretual set of otions of selling energy for F instead of selling energy for the market rice. For any value of x at any time t, the otion is exercised when the rofit generated by F is greater than the rofit generated by. his characteristic is consistent with Euroean ut otions and not an American otions, because each otion can only be exercised at its secified instant. In contrast, an American otion can be exercised at any time before maturity. Let V (x) be the value of the roject. herefore the differential equation for the value of the roject is: where V (x) rv = α x x + 0.5σ2 x 2 2 V (x) + Π(x) (11) x 2 Π(x) = F 2 k r 2r αx 2 R x < x F x > x F (12) 6

7 where And the general solution to this ODE is given by: V (x) = Kx β 1 + Bx β 2 (13) Following Dixit & Pindyck (1994), the value of the roject is: Kx β 1 + F 2 k r 2r V (x) = Bx β 2 + αx2 R x < x F x > x F (14) β 2 F 2 k r (β 2 2)(αx 2 F ) K = 2r R (15) (β 1 β 2 )x β 1 F β 1 F 2 k r (β 1 2)(αx 2 F ) B = 2r R (16) (β 1 β 2 )x β 2 F Let us assume now that the firm has a eretual otion to invest, for a given sunk cost I. he value of that otion is given by: F (x) = Ax β 1 (17) For x > x F, the value matching and smooth asting conditions are: Ax β 1 = Bx β 2 + α R x 2 I (18) β 1 Ax β 1 1 = Bβ 2 x β α R x (19) where x is the threshold for investment. hese two equations reduce to the following non-linear equation: (1 β 2 )Bx β 2 α R x 2 I = 0 (20) We can calculate the trigger for investment, x, by numerically solving Equation 20. It is straightforward to show that investment will never occur for x < x F. Value matching and smooth asting conditions are not met for the the first branch of Equation 14. A eretual FI will defer investment until a time when the state variable x is higher than x F, meaning that it is never otimal to invest if the initial rice for selling energy is below the minimum rice guarantee. his aarently counter-intuitive result is due to the eretual guarantee, which is not foregone if investment does not occur. he next section studies the case of a finite-lived guarantee. 7

8 3.2 Finite FI within an Oligooly In this section, we derive the value of a renewable energy roject that has a FI contract with a finite duration. he comlete derivation is resented in Aendix A, where we extend the model from Shackleton & Wojakowski (2007) to include an equilibrium analysis and FI olicy. Equation 21 resents the value of the roject until (when the firm benefits from the FI guarantee) and Equation 23 resents the value of the roject with a finite FI contract. Kx β 1 N(d β1 ) + F 2 k r ( 1 e r (1 N(d 0 )) ) 2r Bx β 2 N(d β2 ) αx2 R e R N(d 2 ) V G (x) = x < x F Kx β 1 (1 N(d β1 )) F 2 k r 2r e r (1 N(d 0 )) +Bx β 2 (1 N(d β2 )) + αx2 ( 1 e R N(d 2 ) ) x > x F R where N(.) is the cumulative normal integral and ln x ( + (µ + σ 2 β 1 )) x F 2 d β = σ (21) (22) V (x) = V G (x) + αx2 R e R (23) When there is no guarantee ( = 0) the value of the roject reduces to αx, which is the resent value of the rofits in Equation 6. When the R guarantee is eretual ( = ), the value reduces to Equation he value of the otion to invest is: F (x) = Ax β 1 (24) he value matching and smooth asting conditions are: 1 Proofs are resented in Aendix B. Ax β 1 = V (x ) I (25) β 1 Ax β 1 1 = V (x ) (26) 8

9 hese two equations reduce to the following nonlinear equations, that must be solved numerically to find the investment threshold x (Aendix C): ( ) (β 1 β 2 )Bx β 2 αx 2 N(d β2 ) (β 1 2) R e R N(d 2 ) ( ) F 2 k r +β 1 2r (1 e r (1 N(d 0 ))) I x < x F ( ) (β 1 β 2 )Bx β 2 αx 2 (1 N(d β2 )) + (β 1 2) R (1 e R N(d 2 )) ( ) F 2 k r β 1 2r e r (1 N(d 0 )) + I x > x F (27) he value of the otion to invest is given by: ( x ) β1 (V (x ) I) x < x x F (x) = (28) V (x) I x > x 3.3 Numerical analysis of the FI Policy within an Oligooly In this section we resent a comarative statics analysis of the main drivers of a renewable energy roducer s otion to invest and its threshold. We use the base-case arameters resented in able 1. able 1: Base-case arameters used to calculate the threshold a 10 b 0.1 kl 1 kf 0.2 r 0.05 F 1 / KWh 5 years µ σ 0.2 I 10 9

10 Figure 1 illustrates the value of the threshold x for a finite guarantee (ink curve) and for a eretual guarantee (blue curve) as a function of the minimum rice guarantee F. he gray line is the value of x F as a function of F. Recall that x F is the value of x when the roducer receives the market rice instead of the guarantee x Figure 1: Investment thresholds as a function of the minimum rice guarantee For the eretual case, the threshold reduces and converges to x F from above as we increase F. At the oint where the two lines converge, the eretual guarantee roduces a ositive net resent value (NPV) for every x and a null NPV for x = 0. Hence, a eretual FI is sufficient to induce investment for every x. However, it is not economically sound to increase the FI guarantee above the oint where the two lines converge, as it will roduce always a risk-free rofit. On the contrary, a finite FI guarantee is able to induce investment below x F without roducing a risk-free rofit. For the base-case arameter, a minimum rice guarantee around 4.8 is the maximum needed to induce investment for every x. As exected, the investment threshold is higher for the finite case. Figure 2 shows that a higher duration of the guarantee induces an earlier investment. he threshold also converges to the eretual case as increases. In Figure 3, we can see that a higher volatility defers investment for both cases. 4 Price aker Scenario and FI Policy In this section, the scenario has a large number of renewable energy roducers with small scale rojects. A renewable energy roducer is not influential enough to affect the market rice. In other words, the renewable energy 10

11 x Peretual Finite Figure 2: Investment thresholds as a function of x Peretual Finite Figure 3: Investment thresholds as a function of σ roducers are Price taker. he energy rice follows a geometric Brownian rocess, which is the usual assumtion in real otions models. d t = µ t dt + σ t dw t (29) 4.1 Peretual FI within the Price aker Scenario We first resent a model that assumes a FI contract with a eretual duration. In this scenario, the value of the roject is: K β 1 + F < F r V () = B β 2 + (30) r µ > F 11

12 he two regions above meet when = F. Recall that V () must be continuously differentiable across F. Equating the values and derivatives gives: K = B = F β 2 F (β 2 1) r r µ F β 1 (β1 β 2 ) F β 1 F (β 1 1) r r µ F β 2 (β1 β 2 ) (31) (32) Following the assumtion that a firm has a eretual otion to invest, for a sunk cost of I, will lead to a value of the otion that is given by: F (x) = Ax β 1 (33) he value matching and smooth asting conditions for > F are: A β 1 = B β 2 + r µ I (34) Aβ 1 β 1 1 = Bβ 2 β r µ he two equations reduce to the following non-linear equation: (35) (β 1 β 2 )B β 2 + (β 1 1) r µ Iβ 1 = 0 (36) he trigger for investment, is the numerical solution to Equation 36. It is straightforward to show that the results are the same as in the Oligooly scenario. Hence, a eretual FI defers investment until a time when the state variable is higher than F. his means that it is never otimal to invest if the rice at which the firm will start selling energy is bellow the minimum rice guarantee. he next section studies the case of a FI with a finite guarantee, assuming roducers are Price taker. 4.2 Finite FI within the Price aker Scenario Similar to the revious section, we use Shackleton & Wojakowski (2007) to derive the value of the roject for a FI contract with a finite duration. he comlete derivation is resented in Aendix D. 12

13 Equation 37 resents the value of the roject until, when the firm benefits from the FI guarantee. Equation 39 resents the value of the roject with a finite FI contract. K β 1 N(d β1 ) + F r (1 e r (1 N(d 0 ))) B β 2 N(d β2 ) r µ e (r µ) N(d 1 ) V G () = K β 1 (1 N(d β1 )) F r e r (1 N(d 0 )) +B β 2 (1 N(d β2 )) + r µ (1 e (r µ) N(d 1 )) where N(.) is the cumulative normal integral and ln ( ( F + µ + σ 2 β 1 )) 2 d β = σ < F > F (37) (38) V () = V G () + r µ e (r µ) (39) When there is no guarantee (i.e., = 0) the value of the roject reduces to the resent value. In the case of a eretual guarantee (i.e., = ), r µ the value reduces to Equation he value of the otion to invest is: F () = A β 1 (40) he value matching and smooth asting conditions are: A β 1 = V ( ) I (41) β 1 A β 1 1 = V ( ) (42) hese two equations reduce to the following nonlinear equations, that must be solved numerically to find the investment threshold (Aendix 2 Proofs are resented in Aendix E. 13

14 F): ( ) (β 1 β 2 )B β 2 N(d β2 ) (β 1 1) r µ e (r µ) N(d 1 ) ( ) F +β 1 r (1 e r (1 N(d 0 ))) I ( ) (β 1 β 2 )B β 2 (1 N(d β2 )) + (β 1 1) r µ (1 e (r µ) N(d 1 )) ( ) F β 1 r e r (1 N(d 0 )) + I he value of the otion to invest is given by: ( ) β1 (V ( ) I) < F () = V () I > < F > F (43) (44) 4.3 Numerical analysis within the Price aker Scenario Similar to the revious section, we resent omarative statics to analyze the otion to invest and the threshold that triggers a renewable energy investment. We use the base-case arameters resented in able 2. able 2: Base-case arameters used to calculate the threshold r 0.05 F 1 / KWh 5 years µ σ 0.2 I 30 Figure 4 resents the value of the threshold for a finite guarantee (ink curve) and for a eretual guarantee (blue curve) as a function of the minimum rice guarantee F. he gray line is the value of the minimum rice guarantee F. 14

15 6 4 Peretual Finite F Figure 4: Investment thresholds as a function of the FI Similar to the oligooly scenario, the threshold of the eretual case starts above F and then converges to F. Notice that the eretual guarantee roduces a ositive NPV when the two lines converge. Hence, increasing the minimum rice guarantee above that oint is not economically sound, because it will roduce always a risk-free rofit. A finite FI guarantee is able to induce investment below F without roducing a risk-free rofit. Considering the base-case arameter of able 2, a minimum rice guarantee around 6.8 is the maximum needed to induce investment for every. As exected, the investment threshold is higher for the finite case. A higher duration of the guarantee induces an earlier investment, and the threshold converges to the eretual case (Figure 5). A higher volatility defers investment for both cases (Figure 6) Peretual Finite Figure 5: Investment thresholds as a function of 15

16 6 5 Peretual Finite Figure 6: Investment thresholds as a function of σ 5 Concluding Remarks his work analyzes a FI contract with a minimum rice guarantee in two different scenarios, namely an oligooly and rice-taker scenario. In the oligooly, we use a Stackelberg game and include managerial flexibilities with a real otions model. While the rice-taker scenario is based on a real otions model. In both scenarios, the renewable energy roducer receives either the energy market rice or a minimum rice guarantee. he key results of this aer is twofold. First, both scenarios show that a eretual guarantee can induce investment for rices below the minimum rice guarantee, when the roject comensates the investment cost. Second, FI contracts with a higher rice and duration induces an earlier investment. 16

17 Aendix A Value of the roject for a finite FI his is an extension of Shackleton & Wojakowski (2007) to a renewable energy roject, where we include a FI and an equilibrium analysis. he value of the roject with eretual guarantee is: Kx β 1 + F 2 k r x < x F 2r V (x, ) = (A.1) Bx β 2 + αx2 x > x F R Now, the value of the roject is extended with finite guarantee. For more details see Aendix A of Shackleton & Wojakowski (2007). V (x, ) = V (x, ) V (x, ) = V (x, ) e r E Q 0 [V (x, )] (A.2) V (x, ) = V (x, ) = [ Kx β 1 + F ] [ ] 2 k r 1 x<xf + Bx β 2 αx r r 2µ σ 2 x>xf (A.3) [ Kx β 1 + F ] [ ] 2 k r 1 x <x 2r F + Bx β 2 + αx 2 1 r 2µ σ 2 x >x F (A.4) d β = e r E Q 0 e r E Q 0 ln x ( + (µ + σ 2 β 1 )) x F 2 σ q(β) = 1 2 σ2 β(β 1) + βµ r (A.5) (A.6) ] [x β 1 x <xf = e q(β) x β N( d β ) (A.7) ] [x β 1 x >xf = e q(β) x β N(d β ) (A.8) 17

18 d β1 = e r E Q 0 ln x x F + β = β 1 q(β 1 ) = 0 ( (µ + σ 2 β )) σ [ Kx β 1 ] = Kx β 1 N( d β1 ) (A.9) e r E Q 0 d 0 = [ F 2 k r 2r β = 0 q(0) = r ln x x F ] (µ σ2 2 σ ) = e r F 2 k r 2r N( d 0) (A.10) d β2 = ln x x F e r E Q 0 β = β 2 q(β 2 ) = 0 ( (µ + σ 2 β 2 1 )) 2 σ [ Bx β 2 ] = Bx β 2 N(d β2 ) (A.11) 18

19 d 2 = [ αx 2 e r E Q 0 β = 2 q(2) = σ 2 + 2µ r ln x + (µ + 3σ2 x F 2 R ) σ ] = αx2 R N(d 2)e R (A.12) Recall that V (x, ) = V (x, ) e r E Q 0 [V (x, )] For x < x F Kx β 1 + F 2 k r [ 2r Kx β 1 N( d β1 ) + F ] 2 k r 2r e r N( d 0 ) + Bx β 2 N(d β2 ) + αx2 R e R N(d 2 ) N( d β ) = 1 N(d β ). Hence, for x < x F (A.13) Kx β 1 N(d β1 ) + F 2 k r 2r ( 1 e r (1 N(d 0 )) ) Bx β 2 N(d β2 ) αx2 R e R N(d 2 ) (A.14) For x > x F Bx β 2 + αx2 [ R Kx β 1 N( d β1 ) + F ] 2 k r 2r e r N( d 0 ) + Bx β 2 N(d β2 ) + αx2 R e R N(d 2 ) (A.15) 19

20 Kx β 1 (1 N(d β1 )) F 2 k r 2r e r (1 N(d 0 )) + Bx β 2 (1 N(d β2 )) + αx2 R Kx β 1 N(d β1 ) + F 2 k r ( 1 e r (1 N(d 0 )) ) 2r Bx β 2 N(d β2 ) αx2 R e R N(d 2 ) V G (x) = ( 1 e R N(d 2 ) ) (A.16) x < x F Kx β 1 (1 N(d β1 )) F 2 k r 2r e r (1 N(d 0 )) +Bx β 2 (1 N(d β2 )) + αx2 ( 1 e R N(d 2 ) ) x > x F R (A.17) Now, we add the art without guarantee V (x) = V G (x) + αx2 R e R (A.18) 20

21 Aendix B Limits of the value of roject for a finite FI Checking the limit for + lim d 0 = + ln x + x F lim + lim d 2 = + ) (µ σ2 2 σ ln x + x F lim + µ σ2 2 < 0 = + µ σ2 2 > 0 ) (µ + 3σ2 2 σ (B.1) = + (B.2) ln x ( + (µ + σ 2 β 1 1 )) lim d x F 2 β 1 = lim + + σ = + (B.3) ln x ( + ((µ + σ 2 (β 2 1 )) lim d x F 2 β 2 = lim + + σ = (B.4) Recall that β 2 < 0 and β 1 > 1. In addition, β 1 and β 2 is the solution of the following quadratic equation: 0.5σ 2 β(β 1)+µβ r. Hence lim + d β2 is always. 0.5σ 2 β(β 1) + µβ r = 0 (B.5) µ = r 0.5σ2 β 2 (β 2 1) β 2 (B.6) µ + σ 2 (β 2 0.5) = r 0.5σ2 β 2 (β 2 1) β 2 + σ 2 (β 2 0.5) (B.7) µ + σ 2 (β 2 0.5) = r + 0.5σ2 β 2 2 β 2 < 0 (B.8) N(+ ) = 1 (B.9) 21

22 N( ) = 0 (B.10) lim + for x < x F [ lim Kx β 1 N(d β1 ) + F 2 k r ( 1 e r (1 N(d 0 )) ) + 2r Bx β 2 N(d β2 ) αx2 R e R N(d 2 ) + αx2 R e R ] = Hence the same result for eretual otion lim + for x > x F = Kx β 1 + F 2 k r 2r (B.11) [ lim Kx β 1 (1 N(d β1 )) F 2 k r + 2r e r (1 N(d 0 )) +Bx β 2 (1 N(d β2 )) + αx2 R ( 1 e R N(d 2 ) ) + αx2 R e R ] = = Bx β 2 + αx2 R (B.12) lim d 0 = 0 { x < x F (B.13) + x > x F lim 0 for x < x F [ lim Kx β 1 N(d β1 ) + F 2 k r ( 1 e r (1 N(d 0 )) ) 0 2r Bx β 2 N(d β2 ) αx2 R e R N(d 2 ) + αx2 R e R ] = αx2 R (B.14) lim 0 for x > x F [ lim Kx β 1 (1 N(d β1 )) F 2 k r 0 2r e r (1 N(d 0 )) +Bx β 2 (1 N(d β2 )) + αx2 R ( 1 e R N(d 2 ) ) + αx2 R e R ] = αx2 R (B.15) 22

23 Hence for both the lim 0 is the value of the roject without guarantee, in other words is the NPV. 23

24 Aendix C Investment thresholds Aendix C.1 he value-matching condition is: First interval: x < x F Ax β 1 = Kx β 1 N(d β1 ) + F 2 k r 2r (1 e r (1 N(d 0 ))) Bx β 2 N(d β2 ) αx 2 R e R N(d 2 ) + αx 2 R e R I he smooth-asting condition is: (C.1) β 1 Ax β 1 1 = Kx β 1 N(d β 1 ) + β 1 Kx β1 1 N(d β1 ) + F 2 k r N(d 0) 2r e r β 2 Bx β 2 1 N(d β2 ) Bx β 2 N(d β 2 ) 2αx R e R N(d 2 ) αx 2 N(d 1) R e R + 2αx R e R (C.2) As in Shackleton & Wojakowski (2007) (Aendix B), the artial derivatives of the cumulative distribution function cancel across the betas, reducing the value-matching and smooth-asting conditions to the following nonlinear equation: ( ) (β 1 β 2 )Bx β 2 αx 2 N(d β2 ) (β 1 2) R e R N(d 2 ) + β 1 ( F 2 k r 2r (1 e r (1 N(d 0 ))) I ) (C.3) Equation C.3 must be solved numerically to find otimal exercise threshold, x. Aendix C.2 he value-matching condition is: Second interval: x > x F A 1 β 1 = Kx β 1 (1 N(d β1 )) F 2 k r 2r e r (1 N(d 0 )) + Bx β 2 (1 N(d β2 )) + αx 2 R (1 e R N(d 2 ) + αx 2 R e R I (C.4) 24

25 he smooth-asting condition is: β 1 A 1 β 1 1 = β 1 Kx β 1 1 N(d β1 ) Kx β 1 N(d β 1 ) + e r F 2 k r 2r + β 2 Bx β 2 1 (1 N(d β2 )) Bx β 2 N( d β 2 ) N(d 0 ) + 2αx R (1 e R N(d 2 )) αx 2 N(d 1) R e R + 2αx R e R (C.5) As before, the artial derivatives of the cumulative distribution function cancel across the betas, reducing the value-matching and smooth-asting conditions to the following nonlinear equation: ( ) (β 1 β 2 )Bx β 2 αx 2 (1 N(d β2 )) + (β 1 2) R (1 e R N(d 2 )) β 1 ( F 2 k r 2r e r (1 N(d 0 )) + I ) (C.6) Equation C.6 must be solved numerically to find otimal exercise threshold, x. 25

26 Aendix D Value of the roject for a finite FI: Price taker his is an extension of Shackleton & Wojakowski (2007) to a renewable energy roject, where we include a FI. he value of the roject with eretual guarantee is: K β 1 + F < F r V (, ) = B β 2 + r µ > F (D.1) Now, the value of the roject is extended with finite guarantee. For more details see Aendix A of Shackleton & Wojakowski (2007). V (, ) = V (, ) V (, ) = V (, ) e r E Q 0 [V (, )] (D.2) V (, ) = V (, ) = [ K β 1 + F ] [ 1 <F + B β 2 + ] 1 >F (D.3) r r µ [ K β 1 + F ] [ 1 <F + B β 2 r + ] 1 >F (D.4) r µ d β = e r E Q 0 ln ( ( F + µ + σ 2 β 1 )) 2 σ q(β) = 1 2 σ2 β(β 1) + βµ r e r E Q 0 (D.5) (D.6) [ ] β 1 <F = e q(β) β N( d β ) (D.7) [ ] β 1 >F = e q(β) β N(d β ) (D.8) 26

27 d β1 = e r E Q 0 β = β 1 q(β 1 ) = 0 ln F + ( µ + σ 2 ( β )) σ [ K β 1 ] = K β 1 N( d β1 ) (D.9) d 0 = e r E Q 0 β = 0 q(0) = r ln ( ) F + µ σ2 2 σ [ ] F = e r F r r N( d 0) (D.10) d β2 = β = β 2 q(β 2 ) = 0 ln F + ( µ + σ 2 ( β e r E Q 0 σ [ B β 2 ] = B β 2 N(d β2 ) )) (D.11) 27

28 β = 1 q(1) = µ r ln ( F + µ + σ ) 2 d 1 = σ [ ] e r E Q 0 r µ = r µ N(d 1)e (r µ) (D.12) Recall that V (, ) = V (, ) e r E Q 0 [V (, )] For < F K β 1 + F [ r K β 1 N( d β1 ) + F r e r N( d 0 ) + B β 2 N(d β2 ) + ] r µ e (r µ) N(d 1 ) N( d β ) = 1 N(d β ). Hence, for < F (D.13) K β 1 N(d β1 ) + F r (1 e r (1 N(d 0 ))) B β 2 N(d β2 ) r µ e (r µ) N(d 1 ) (D.14) For > F B β 2 + r µ [ K β 1 N( d β1 ) + F r e r N( d 0 ) + B β 2 N(d β2 ) + ] r µ e (r µ) N(d 1 ) (D.15) 28

29 K β 1 (1 N(d β1 )) F r e r (1 N(d 0 )) + B β 2 (1 N(d β2 )) + r µ (1 e (r µ) N(d 1 )) (D.16) K β 1 N(d β1 ) + F r (1 e r (1 N(d 0 ))) B β 2 N(d β2 ) r µ e (r µ) N(d 1 ) V G () = K β 1 (1 N(d β1 )) F r e r (1 N(d 0 )) +B β 2 (1 N(d β2 )) + r µ (1 e (r µ) N(d 1 )) Now, we add the art without guarantee V () = V G () + r µ e (r µ) < F > F (D.17) (D.18) 29

30 Aendix E Limits of the value of roject for a finite FI: Price taker Checking the limit for + ln ( ) lim d F + µ σ2 2 0 = lim + + σ lim d 1 = + ln ( ) F + µ + σ2 2 lim + σ µ σ2 2 < 0 = + µ σ2 2 > 0 (E.1) = + (E.2) ln ( ( lim d F + µ + σ 2 β 1 1 )) 2 β 1 = lim + + σ = + (E.3) ln ( ( lim d F + µ + σ 2 β 2 1 )) 2 β 2 = lim + + σ = (E.4) Recall that β 2 < 0 and β 1 > 1. In addition, β 1 and β 2 is the solution of the following quadratic equation: 0.5σ 2 β(β 1)+µβ r. Hence lim + d β2 is always. In addition, N(+ ) = 1 and N( ) = 0. lim + for < F [ lim K β 1 N(d β1 ) + F + r (1 e r (1 N(d 0 ))) B β 2 N(d β2 ) r µ e (r µ) N(d 1 ) + ] r µ e (r µ) = = K β 1 + F r (E.5) Hence the same result for eretual otion lim + for > F 30

31 [ lim K β 1 (1 N(d β1 )) F + r e r (1 N( d 0 )) +B β 2 (1 N(d β2 )) + r µ (1 e (r µ) N(d 1 )) + ] r µ e (r µ) = = B β 2 + r µ (E.6) Checking the limit for 0 lim 0 for < F lim d 0 = +0 < F + > F [ lim K β 1 N(d β1 ) + F 0 r (1 e r (1 N(d 0 ))) B β 2 N(d β2 ) r µ e (r µ) N(d 1 ) + ] r µ e (r µ) = = r µ (E.7) (E.8) lim 0 for > F [ lim K β 1 (1 N(d β1 )) F 0 r e r (1 N( d 0 )) +B β 2 (1 N(d β2 )) + r µ (1 e (r µ) N(d 1 )) + ] r µ e (r µ) = = r µ (E.9) Hence for both the lim 0 is the value of the roject without guarantee, in other words is the NPV. 31

32 Aendix F Investment thresholds: Price taker Aendix F.1 he value-matching condition is: First interval: < F A β 1 = K β 1 N(d β1 ) + F r (1 e r (1 N(d 0 ))) B β 2 N(d β2 ) r µ e (r µ) N(d 1 ) + r µ e (r µ) I he smooth-asting condition is: β 1 A β 1 1 = K β 1 N(d β 1 ) + β 1 K β 1 1 N(d β1 ) + F r e r N(d 0) β 2 B β2 1 N(d β2 ) B β N(d 2 β 2 ) 1 r µ e (r µ) N(d 1 ) N(d 1) r µ e (r µ) + 1 r µ e (r µ) (F.1) (F.2) As in Shackleton & Wojakowski (2007) (Aendix B), the artial derivatives of the cumulative distribution function cancel across the betas, reducing the value-matching and smooth-asting conditions to the following nonlinear equation: ( ) (β 1 β 2 )B β 2 N(d β2 ) (β 1 1) r µ e (r µ) N(d 1 ) ( ) F + β 1 r (1 e r (1 N(d 0 ))) I (F.3) Equation F.3 must be solved numerically to find otimal exercise threshold,. Aendix F.2 he value-matching condition is: Second interval: > F A 1 β 1 = K β 1 (1 N(d β1 )) F r e r (1 N(d 0 )) + B β 2 (1 N(d β2 )) + r µ (1 e (r µ) N(d 1 ) + r µ e (r µ) I (F.4) 32

33 he smooth-asting condition is: β 1 A 1 β1 1 = β 1 K β1 1 N(d β1 ) K β N(d 1 β 1 ) + e r F N(d 0 ) r + β 2 B β2 1 (1 N(d β2 )) B β N( d 2 β 2 ) + 1 r µ (1 e (r µ) N(d 1 )) N(d 1) r µ e (r µ) + 1 r µ e (r µ) (F.5) As before, the artial derivatives of the cumulative distribution function cancel across the betas, reducing the value-matching and smooth-asting conditions to the following nonlinear equation: ( ) (β 1 β 2 )B β 2 (1 N(d β2 )) + (β 1 1) r µ (1 e (r µ) N(d 1 )) ( ) F β 1 r e r (1 N(d 0 )) + I (F.6) Equation F.6 must be solved numerically to find otimal exercise threshold,. 33

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