Price limits and corporate investment: The consumers' perspective

Economic Modelling 50 (2015) 168–178

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Price limits and corporate investment: The consumers' perspective☆ Sudipto Sarkar ⁎ DSB 302, McMaster University, Hamilton, ON L8S 4K4, Canada

a r t i c l e

i n f o

Article history: Accepted 20 June 2015 Available online xxxx Keywords: Real option Investment Price cap Consumer welfare

a b s t r a c t This paper uses a real-option model to examine how a price cap affects a regulated firm's investment timing/capacity decision and the resulting consumer welfare. The model is too complex to allow for closed-form solutions, hence the results are derived numerically. We show that optimal investment size is an increasing function of price cap, and optimal investment trigger is initially decreasing and subsequently increasing in price cap, hence there is a unique price cap that minimizes investment trigger. The resulting consumer welfare is initially increasing and subsequently decreasing in price cap, thus there is also a unique price cap that maximizes consumer welfare. These two price caps are generally different, and can be substantially different in certain cases; therefore, accelerating investment will not necessarily make consumers better off. Also, increasing (decreasing) the price cap results in a slight (large) reduction in consumer welfare, which implies that consumers are paradoxically better off with a too-high price cap than a too-low price cap. The central message of the paper is that consumers and consumer advocates should be interested in the price-cap setting process, since the price cap impacts consumer welfare, sometimes in unexpected ways. © 2015 Elsevier B.V. All rights reserved.

1. Introduction A number of monopolistic industries are subject to governmental price limits, to prevent corporations from exploiting their monopoly power (Baldwin et al., 2012).1 Examples of such industries are electric utilities, water, gas, telecommunications, and insurance. While these may not comprise the majority of industries in the corporate sector, they are nevertheless an important segment. As pointed out by Spiegel and Spulber (1994, p. 424), the utility industry alone accounted for about 6% of the GNP of the USA and over 18.8% of the total business expenditure for new plant and equipment in 1989. This paper examines how a price limit or cap affects an unlevered firm's investment decision, and the resulting impact on consumer welfare. Using a real-option model of investment, we show that a price cap

☆ I acknowledge helpful comments and suggestions from three anonymous reviewers, which helped improve the paper. I also thank the Editor S. Mallick for his guidance and suggestions. ⁎ Tel.: +1 905 525 9140x23959; fax: +1 905 521 8995. E-mail address: [email protected]. 1 Price control can be viewed as a redistribution of wealth from corporations to consumers. Micheli and Schmidt (2015) show that price control (rent control in their example) dominates other forms of wealth redistribution such as transfer payments. It is therefore not surprising that price control is a popular means of limiting excessive profits of monopolies.

http://dx.doi.org/10.1016/j.econmod.2015.06.014 0264-9993/© 2015 Elsevier B.V. All rights reserved.

can have a significant effect on both the timing and size of a company's investment. Since price caps are to be found in monopolistic industries, the firm's investment decision can in turn have a significant effect on consumer welfare. Our model generates the following main results. First, the optimal investment size is an increasing function of price cap, but it increases at a declining rate until it flattens out to a constant size. Second, the optimal investment trigger is initially decreasing and subsequently increasing in price cap, hence there is a unique price cap that minimizes the investment trigger. Third, consumer welfare is initially increasing and subsequently decreasing in price cap, hence there is a unique price cap that maximizes consumer welfare. However, these two unique price caps are generally different (possibly very different, depending on the parameter values); thus, encouraging investment will not necessarily be best for consumers, contrary to some earlier papers. Finally, the effect of price cap on consumer welfare is generally asymmetric; that is, a lower price cap will result in a sharp fall in consumer welfare, but a higher price cap will result in a gradual fall. The practical implication is that a too-low price cap is likely to hurt consumers more than a toohigh price cap. The rest of the paper is organized as follows. Section 2 reviews the existing literature and clarifies the contribution of this paper. Section 3 analyzes the price-cap-regulated firm's investment (size and timing) decision, and Section 4 derives an appropriate consumer welfare function. Section 5 presents the results of the model, and Section 6 concludes.

S. Sarkar / Economic Modelling 50 (2015) 168–178

2. Literature review From the company's perspective, the important issue is how to modify its investment policy in the presence of a price cap.2 A few papers use the real option model to determine the optimal investment policy under a price cap. Dixit (1991) shows that, for a competitive firm, a price cap raises the investment trigger and thus has a negative effect on investment. Roques and Savva (2009) and Dobbs (2004) show, for an oligopoly and a monopoly respectively, that setting the price cap equal to the competitive entry trigger price results in the lowest investment trigger and thereby maximizes investment. None of these papers, however, look at investment size or capacity. From a consumer's perspective, the important issue is how a price cap will affect consumer welfare. The direct effect of a price cap will be to increase consumer welfare, since it limits how much consumers have to pay per unit of the good. However, there could be an indirect negative effect via the investment effect, since price caps are used in monopolistic industries. This indirect effect is explicitly incorporated in our model. Evans and Guthrie (2012) examine the effect of price cap on total welfare (consumer welfare plus producer welfare) when the firm makes incremental investments and the regulator adjusts the price cap continuously. Our paper differs from the existing literature in the following ways. First, existing real-option models (Dixit, 1991; Dobbs, 2004; Roques and Savva, 2009, etc) not consider investment capacity or consumer welfare, unlike our paper. Second, while Evans and Guthrie (2012) consider economic welfare, their model maximizes total welfare (or an equally-weighted combination of consumer and producer welfare), which is not consistent with observed regulator behavior (Dasgupta and Nanda, 1993; Evans et al., 2008; Florio, 2013, etc).3 Third, none of the above-mentioned papers consider the firm's capacity choice when investing. While some investments are indeed incremental in nature, there are many that are “lumpy” one-time decisions, where incremental additions to capacity are not possible (Bar-Ilan and Strange, 1999; Dangl, 1999). This “lumpy” investment is incorporated in our paper, where the size or capacity of the investment, and its effect on consumer well-being, are explicitly taken into account. To summarize, the existing price cap literature examines the effect of price cap on investment timing, and on total welfare (consumer plus producer) with incremental investment and continuous price cap adjustment. There is no research on the effect of price cap on the joint investment timing/capacity decision or on consumer welfare for lumpy investment. This is the gap our study aims to fill. 3. A model of investment timing and capacity 3.1. Basic model assumptions The monopolistic firm consists of a plant which produces q units of the output, which are sold at a price of $p per unit. As in Roques and 2

Price limit is a form of wealth expropriation (from corporations), and it has been shown that wealth expropriation can have a significant negative effect on corporate investment (Ochoa et al., 2015). Therefore, it is important to account for the effect of price limit on the firm's investment policy. 3 The reason price caps are used in the first place is to ensure that consumer interests are protected, hence maximizing consumer welfare should be the primary objective of the regulator (Baldwin et al., 2012; Iozzi, et al., 2002). However, corporate or producer welfare is also important because investors require adequate returns, without which future investment will be negatively impacted. Thus, the literature generally views the regulator as maximizing a weighted combination of consumer welfare and corporate welfare (Florio, 2013; Dasgupta and Nanda, 1993; Spiegel and Spulber, 1994, etc). But these weights vary widely and are determined largely by political and other non-economic factors, such as regulatory capture, lobbying, societal attitudes towards wealth redistribution, elected versus appointed regulators, degree of regulatory capture, etc (Baldwin et al., 2012, Besley and Coate, 2003; Florio, 2013). Also, a proper study of the regulator's price cap decision would need to take into account the role played by corporate leverage decisions in influencing the regulator (Bortolotti et al., 2007; Dasgupta and Nanda, 1993). Hence the regulator's objective function is beyond the scope of this paper, and we focus on the magnitude of consumer welfare, and how it is affected by a price cap.

169

Savva (2009), Dobbs (2004), and Evans and Guthrie (2012), we assume there are no operating costs. The output quantity cannot exceed the plant capacity Q, while the output price cannot exceed the price cap p; hence both price and quantity are constrained, p ≤ p and q ≤ Q. The firm determines both p and q optimally, subject to the above constraints, based on the realization of the demand function below. As is common in the real-option literature (Aguerrevere, 2003; He and Pindyck, 1992; Kandel and Pearson, 2002), the demand for the product is given by a linear inverse demand function: pt ¼ yt −θqt

ð1Þ

where p is the output price per unit, q is the output level, y is a continuously-varying stochastic exogenous parameter that represents the strength of demand. The state variable is y, which introduces uncertainty in the model and can be interpreted as the relative strength of the demand side of the market. When y increases, demand is stronger and price p is higher for a given q. Therefore, revenues and profits are both increasing functions of y. The parameter θ is a non-negative constant representing the slope of the linear demand function (or price sensitivity to output quantity). It can be viewed as “price responsiveness” or elasticity coefficient,4 or a measure of monopolistic market power. For θ = 0, the demand is infinitely elastic and if the company raises the price at all, sales will fall to zero; thus, θ = 0 means the firm has no market power and the output price is exogenous. For large θ, demand is inelastic and the company can raise price without a significant drop in sales, hence a large θ signifies substantial market power. We also make the standard assumption (Aguerrevere, 2003; He and Pindyck, 1992; Kandel and Pearson, 2002, etc) that y evolves continuously as a geometric Brownian motion: dy=y ¼ μdt þ σdz

ð2Þ

where μ and σ are the expected growth rate and volatility, respectively, of y; and z is a standard Wiener process. We assume r N (2 μ + σ 2); this condition is required to ensure meaningful project values (see Section 3.2). The cost of investing in the project is an increasing function of plant capacity Q, and is given by cQη (where η ≥ 1), as in Bar-Ilan and Strange (1999). The investment is irreversible, in that the company cannot recoup any part of the sunk investment cost, even though it can terminate operations and exit the industry if business conditions deteriorate sufficiently. Also, depreciation is ignored in the model. The price cap of $p per unit of output is assumed to be exogenously set by the regulator, and the firm takes it as given. All cash flows are discounted at the constant risk-free rate of r, consistent with the realoption literature (Aguerrevere, 2003; Dixit, 1991; Dobbs, 2004; Evans and Guthrie, 2012; Roques and Savva, 2009, etc). 3.1.1. Operating levels Absent constraints, operating level will be chosen to maximize the instantaneous profit, which (since there are no operating costs) is given by: π(q) = pq = (y − θq)q. Setting dπ/dq = 0, we get the optimal output level as a function of the state variable y: qt ¼ yt =2θ

ð3Þ

and the corresponding price from Eq. (1): pt ¼ yt =2:

ð4Þ

Thus, the unconstrained optimal profit stream is qtpt = (yt)2 / 4θ per unit time. However, when y rises sufficiently, the price cap will become binding (at y = 2p, from Eq. (4)) or the capacity constraint will become 4

The elasticity of demand for the demand Eq. (1) is p / (θq).

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S. Sarkar / Economic Modelling 50 (2015) 168–178

binding (at y = 2θQ, from Eq. (3)); if y exceeds both 2p and 2θQ, both price and capacity constraints will be binding. Whether price or capacity constraint becomes binding first depends on which is smaller, p or θQ. If p b θQ (p ≥ θQ), the price (capacity) constraint becomes binding first, at y = 2p (y = 2θQ). Therefore, we look at the two cases separately. When p b θQ: The price constraint becomes binding first, at y = 2p; as y rises further, price will remain at p = p, and the output will be given by Eq. (1): q = (y – p)/θ. As y keeps rising further, the output will reach the capacity limit Q at y = (p + θQ); for any higher y, the output will remain at q = Q and both price and output constraints will be binding. Thus, there will be three operating regions: Region I: small y (y b 2p), neither constraint is binding; p = y / 2, q = y / 2θ; Region II: intermediate y (2p ≤ y ≤ p + θQ), only price constraint binding; p = p, q = (y – p) / θ; Region III: large y (y N p + θQ), both constraints binding; p = p, q = Q. When p ≥ θQ: The capacity constraint becomes binding first, at y = 2θQ; as y rises further, output will remain at q = Q, while price will be given by Eq. (1): p = (y − θQ). As y keeps rising further, the price will reach the limit p at y = (p + θQ); for any higher y both price and capacity constraints will be binding. Again, there are three operating regions: Region I: small y (y b 2θQ), neither constraint is binding; p = y / 2, q = y / 2θ; Region II: intermediate y (2θQ ≤ y ≤ p + θQ), output constraint binding; q = Q, p = (y − θQ); Region III: large y (y N p + θQ), both constraints binding; p = p, q = Q.

shows the effect of the capacity constraint becoming binding, and Cyγ2 shows the effect of returning to a state with no binding constraint. In Eq. (7), ½pQ=r is the value of operating forever with both price and capacity constraints binding, and Dyγ2 shows the effect of moving to a state where only the price constraint is binding. Eq. (8) captures the fact that cash flow from the project is not a constant stream but a growing random stream. Recall that [y2 / (4θψ)] is the value of operating forever with no constraints, and that the cash flow stream in this scenario is y2 / (2θ). For a constant cash flow stream the value would be y2 / (2θr); however, in this case it will be higher. This is captured by the denominator term ψ = (r–2 μ–σ2); the higher the growth rate, the smaller the term ψ and the higher the value of the cash flow stream. Finally, γ1 (N 1) and γ2 (b0) of Eqs. (9) and (10) are the solutions of the fundamental quadratic equation: 0.5σ 2γ(γ − 1) + μγ − r = 0, which arises from the ordinary differential Eq. (A1) in the Appendix. This is an important factor in the real-option literature (see, for instance, Dixit and Pindyck, 1994), the economic significance of which lies in the fact that the multiple γ1 / (γ1 − 1) is the wedge between benefits and costs in real-option models that makes investment optimal. That is, it is optimal to invest when the benefit cost ratio (BCR) exceeds the multiple γ1 / (γ1 − 1) rather than 1 (as prescribed by the traditional literature). This multiple γ1 / (γ1 − 1) (N1) accounts for the presence of uncertainty and irreversibility (Di Corato et al., 2014). B. When p ≥ θQ. As shown in Appendix B, the project values are as follows. Region I ðy b 2θQ Þ : project value UðyÞ ¼ y2 =4θψ þ Ayγ1

ð11Þ

3.2. Project valuation

Region II ð2θQ ≤y≤p þ θQ Þ : project value VðyÞ ¼ Q y=ðr−μ Þ−θQ 2 =r þ Byγ1 þ Cyγ2

ð12Þ

In the following, we provide the derivation of the project value for both cases (p b θQ and p ≥ θQ).

Region III ðyNp þ θQ Þ : project value WðyÞ ¼ pQ=r þ Dyγ2

ð13Þ

A. When p b θQ. As shown in Appendix A, project values in the three operating regions are given by:

The constants A, B, C and D for this case are given in Appendix B, and have an interpretation similar to the above.

Region I ðy b 2pÞ : project value UðyÞ ¼ y2 =4θψ þ Ayγ1

ð5Þ

Region II ð2p≤y≤p þ θQ Þ : project value VðyÞ ¼ py=ðθðr−μ ÞÞ−p2 =ðθrÞ þ Byγ1 þ Cyγ2

ð6Þ

Region III ðyNp þ θQ Þ : project value WðyÞ ¼ pQ =r þ Dyγ2 :

ð7Þ

Here, A, B, C and D are constants, and ψ ¼ r–2μ–σ 2

ð8Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð0:5−μ=σ 2 Þ þ 2r=σ 2 :

ð9Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 γ2 ¼ 0:5−μ=σ 2 − ð0:5−μ=σ 2 Þ þ 2r=σ 2 :

ð10Þ

γ1 ¼ 0:5−μ=σ 2 þ

The constants A, B, C and D are determined from boundary conditions at the two boundaries (y = 2p and y = p + θQ), and are given in Appendix A. The equations can be interpreted as follows. In Eq. (5), [y2/(4θψ)] is the value of operating forever with no constraints,5 and Ayγ1 represents the effect of price cap possibly becoming binding in the future. In Eq. (6), ½py=ðθðr−μÞÞ−p2 =ðθrÞ is the value of operating forever with only the price cap binding, Byγ1

5 For this term to be meaningful, it must be positive, hence ψ N 0, which requires that r N (2 μ + σ2).

3.3. The firm's investment decision In making its investment decision, the firm has to optimize on two dimensions – investment timing (or investment trigger, say y⁎) and capacity (say Q⁎). The investment decision is modeled in two steps. First, the optimal investment trigger is determined for a given plant capacity, y⁎(Q); next, the optimal capacity Q⁎ is determined, taking into account the optimal investment trigger for each capacity level, as in Bar-Ilan and Strange (1999) and Dangl (1999). 3.3.1. Optimal investment trigger for a given capacity The investment timing decision basically answers the question: when to invest? The objective is to maximize the net value, taking into account investment cost and time value of money (with cost of installing capacity = $cQη). This is a standard American option exercise problem (Dixit and Pindyck, 1994; Roques and Savva, 2009; Dobbs, 2004). The investment should be made when the state variable y rises to a sufficiently high level, say y⁎, which we call the investment trigger. In this section, we compute the optimal investment trigger for a given capacity, or y⁎(Q). 3.3.1.1. The option to invest. Before the firm invests in the project, the company's only asset is the option to invest. As is standard in the realoption literature, we assume it is a perpetual option. Then its value will depend only on the state variable y. Suppose the option value is F(y); then the expected return from holding the option over the next

S. Sarkar / Economic Modelling 50 (2015) 168–178

instant should be equal to the discount rate. Equating the two and simplifying, we get the ordinary differential equation or ODE (derived in Bar-Ilan and Strange, 1999): 0:5σ 2 y2 F″ðyÞ þ μyF0ðyÞ−rFðyÞ ¼ 0

ð14Þ

the general solution to which is: FðyÞ ¼ F1 yγ1 þ F2 yγ2 , where F1 and F2 are constants to be determined from boundary conditions. When y → 0, the option is worthless, i.e., F(y) → 0. This implies F2 = 0; hence we write the option value as: FðyÞ ¼ F1 yγ1 :

ð15Þ

The next step is to determine the constant F1 and the optimal trigger y⁎. These can be determined from the value-matching and smoothpasting boundary conditions (Dixit, 1993; Dixit and Pindyck, 1994). The former is a continuity condition and requires that, at option exercise (i.e., when the investment is made), the option value be equal to the project value less the option exercise cost (or cost of investing). The latter is an optimality condition and requires that the first derivative of the option value equal the first derivative of the net payoff at option exercise. The value-matching and smooth-pasting conditions are:

F1 ðy Þγ1

8 < Uðy Þ−cQ η ¼ Vðy Þ−cQ η : Wðy Þ−cQ η

if y is in Region I if y is in Region II if y is in Region III

ð16Þ

8 < U0ðy Þ ¼ V0 ðy Þ : 0  W ðy Þ

if y is in Region I if y is in Region II : if y is in Region III

ð17Þ

 γ1 −1

F1 γ1 ðy Þ



3.3.1.2. The optimal investment trigger when p b θQ. Eqs. (16) and (17) can be solved for the optimal investment trigger y⁎. Suppose y⁎ is in Region I, i.e., y⁎ b 2p. Substituting for U(y) and U′(y) in Eqs. (16) and (17), we get: y ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4θψcQ η =ð1−2=γ1 Þ:

ð18Þ

Substituting this y⁎ into condition y⁎ b 2p, we get the equivalent condition for y⁎ to be in Region I: Q η b ðpÞ2 ð1−2=γ1 Þ=ðθψcÞ:

1=γ2 cQ η −pQ =r : Dð1−γ2 =γ1 Þ



ð20Þ

Substituting this into the condition y⁎ N (p + θQ) gives the equivalent condition: ðp þ θQ Þγ2 N

cQ η −pQ =r : Dð1−γ2 =γ1 Þ

ð21Þ

Thus, if condition (21) is satisfied, the optimal trigger falls in Region III and is given by Eq. (20). Finally, if y⁎ is in Region II, i.e., (p + θQ) ≥ y⁎ ≥ 2p, then from Eqs. (16) and (17), y⁎ is the implicit solution of the nonlinear equation: py ð1−1=γ1 Þ ðpÞ2 þ Cðy Þγ2 ð1−γ2 =γ1 Þ−cQ η ¼ 0; − θr θðr−μ Þ

which has to be solved numerically since no analytical solution is available. 3.3.1.3. The optimal investment trigger when p ≥ θQ. Using the same procedure as above, the optimal investment trigger y⁎ is given by the following: ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ψc 4θψcQ η  (i) if Q η N θð1−2=γ Þ, then y ¼ ð1−2=γ Þ, 1

1

η

1=γ2

η

cQ −pQ =r cQ −pQ =r  (ii) if ðp þ θQ Þγ2 N Dð1−γ =γ Þ, then y ¼ ðDð1−γ =γ ÞÞ 2

1

2

1

, and

(iii) if neither of the above conditions is satisfied, then y⁎ is the solution to the equation

Q y ð1−1=γ1 Þ θQ 2 þ Cðy Þγ2 ð1−γ2 =γ1 Þ−cQ η ¼ 0: − r r−μ

3.3.2. Optimal project capacity The objective is to maximize the net value of the project, hence the firm's capacity-choice problem can be described as follows:  8 99 8 η    < < EPV Uðy Þ−cQ  if y is in Region I == η   : EPV Vðy Þ−cQ Max Max if y is in Region II   ;; Q : y ðQ Þ : EPV Wðy Þ−cQ η if y is in Region III

ð23Þ

From the real options literature, we know that this problem is equivalent to maximizing the ex-ante value of the option to invest, F1 yγ1 , for all y (Bar-Ilan and Strange, 1999; Dangl, 1999). Since y is the state variable (independent of Q), this is equivalent to maximizing the value of the constant F1. Thus the optimal capacity is given by: Q  ¼ arg max F1 ðQ Þ

ð24Þ

Q

where F1 is given by (from Eq. (16)): 8 η   −γ1  < Uðy Þ−cQ  ðy Þ η  F1 ðQ Þ ¼ Vðy Þ−cQ ðy Þ−γ1  : Wðy Þ−cQ η ðy Þ−γ1

if y is in Region I if y is in Region II : if y is in Region III

ð25Þ

ð19Þ

Thus, if condition (19) is satisfied, the optimal trigger is in Region I and given by Eq. (18). If the optimal trigger is in Region III, i.e., y⁎ N (p + θQ), conditions (16) and (17) give: y ¼

171

ð22Þ

As capacity Q is increased, there are two opposing effects on the value of the option to invest. The benefits associated with a larger capacity arise from the ability to raise output when demand rises (i.e., a reduction in the opportunity cost of insufficient capacity), which results in higher potential profits. On the other hand, the cost of installing capacity increases with capacity, which reduces value. For a small Q, the marginal benefits are larger since any additional capacity is likely to be utilized; for large Q, the marginal benefits are smaller since it is more likely that additional capacity will not be fully utilized. Thus, the marginal benefits are decreasing in Q, while marginal costs are increasing in Q; at some point the marginal benefit will equal marginal cost, giving the optimal Q. The value of the option to invest is therefore initially an increasing function and subsequently a decreasing function of Q, producing an inverted-U shaped relationship which allows us to identify the optimal capacity Q⁎. The optimal capacity Q⁎ will maximize, ex-ante, the value of the option to invest, F(y). This is the outer maximization problem in expression (23). The solution procedure is as follows: for each Q, the firm chooses the optimal investment trigger y⁎(Q) so as to (locally) maximize the option value parameter F1. The firm then chooses the value of Q that (globally) maximizes F1. The parameter F1 can be viewed as a measure of the company's value or the company's welfare.

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S. Sarkar / Economic Modelling 50 (2015) 168–178

4. The appropriate measure of consumer welfare The conventional measure of consumer welfare is consumers' surplus, given by the expression (Salant and Woroch, 1992): q

∫ ½pðqÞ−p dq, where p(q) is the price corresponding to output q

0

(from the demand function), and p ⁎ and q⁎ are the actual price charged and output sold, respectively. Since p is a function of the state variable y, consumers' surplus will also be a function of y. Further, since p⁎ and q⁎ are subject to price and capacity constraints respectively, consumers' surplus will be affected by price and capacity limits p and Q respectively. However, consumers' surplus is a continuous flow which indicates consumer well-being at a particular instant in time, not the long-term overall well-being. For that, we use the term “consumer welfare,” which basically aggregates the stream of consumers' surplus. Since distant (rather than immediate) consumers' surplus is less desirable, the surplus stream must be discounted in the same way as a cash flow stream. Thus, the appropriate measure would be the discounted sum of the consumers' surplus stream, discounted at the same rate as the cash flow stream, as in Salant and Woroch (1992) and Evans and Guthrie (2012). Further, since instantaneous consumers' surplus in our model depends on (the stochastic) y, we need to consider the expected present value of the consumers' surplus stream. Finally, consumers' surplus depends on whether the price/capacity constraints are binding, hence consumer welfare will depend on the operating region. The appropriate expression for consumer welfare must therefore take into account the current operating region as well as the possibility of moving to another operating region. Based on the above discussion, our measure of consumer welfare is the expected present value of the consumers' surplus stream from the company's output, taking into account the possibility of changes in the operating region. This can be evaluated as a contingent claim on y, similar to the valuation procedure in Appendix A.

(moves from operating Regions I to II). In Region I, consumers' surplus is (y2 / 8θ), and in Region II it is (y – p)2 / 2θ. It is easy to show that when y ≥ 2p (as in Region II), the latter exceeds the former, i.e., the price cap makes the consumers better off. On the other hand, the consumer is hurt when the capacity constraint binds, as we can see by examining the firm moving from Region II to Region III [consumers' surplus going from (y – p)2 / 2θ to fðy−pÞQ −θQ 2 =2g]. Since y N (p + θQ) in Region III, it is easy to show that the consumers' surplus is smaller when the capacity constraint binds. Thus, a lower price cap will help consumer welfare but a smaller capacity will hurt consumer welfare, everything else remaining unchanged. B. When p ≥ θQ. Region I (y b 2θQ): p⁎ = y / 2, q⁎ = y / 2θ; from Eq. (26), consumers' surplus = (y2 / 8θ) per unit time. Region II (2θQ ≤ y ≤ p + θQ): p⁎ = (y–θQ) and q⁎ = Q; consumers' surplus = θQ2 / 2 per unit time. Region III (y N p + θQ): p⁎ = p and q⁎ = Q; consumers' surplus = fðy−pÞQ −θQ 2 =2g per unit time.

4.2. Consumer welfare function As discussed in Section 4.1, consumer welfare is measured by the expected discounted value of the consumers' surplus stream over time, incorporating possible movements from one operating region to another. This measure will be state-contingent, hence we can it as follows:

8 < SI ðyÞ SðyÞ ¼ SII ðyÞ : SIII ðyÞ

if y is in Region I if y is in Region II : if y is in Region III

ð27Þ

4.1. Consumers surplus First we derive the appropriate function for instantaneous consumers' surplus in each operating region. For the demand function in our model (Eq. (1)), consumers' surplus (per unit time) is given by:

The functions SI(y), SII(y) and SIII(y) can be computed using the procedure of Appendix A, giving the following welfare functions. A. When p b θQ.



Zq

ðy−θq−p Þdq;

ð26Þ

Region I (y b 2p): @consumer welfare function SI ðyÞ ¼ y2 =8θψ þ Z1 yγ1 . 2

2

p py y Region II (2p ≤ y ≤ p + θQ): consumer welfare SII ðyÞ ¼ 2θr − θðr−μÞ þ 2θψ

0

where the actual price and quantity are p⁎ and q⁎ respectively. Thus, consumers' surplus at any point in time will depend on the realization of y, as well as the price and output level chosen by the firm (hence the price cap as well, since p⁎ and q⁎ are affected by the price cap). From Section 3, we know that price and output depend on operating regions, which depend on whether p ≥ θQ or p b θQ. We therefore look at each situation in turn. A. When p b θQ. Region I (y b 2p): From Section 3, we have p⁎ = y/2 and q⁎ = y / 2θ; substituting in Eq. (26), we get consumers' surplus = (y2 / 8θ) per unit time. Region II (2p ≤ y ≤ p + θQ): p⁎ = p and q⁎ = (y – p) / θ; from Eq. (26), consumers' surplus = (y – p)2 / 2θ. Region III (y N p+θQ): p⁎ = p and q⁎ = Q; from Eq. (26), consumers' surplus = fðy−pÞQ−θQ 2 =2g. When the price cap is lowered, consumers' surplus will rise; this can be shown by examining consumers' surplus when the price cap binds

þZ2 yγ1 þ Z3 yγ2 .

Qy Region III (y N p + θQ): consumer welfare SIII ðyÞ ¼ ðr−μÞ − Qr ðp þ θQ =2

Þ þ Z4 yγ2 . The four constants (Z1, Z2, Z3, Z4) can be derived from the four boundary conditions below. Boundary 1 (y = 2p): value-matching SI ð2pÞ ¼ SII ð2pÞ, smoothpasting SI 0ð2pÞ ¼ SII 0ð2pÞ Boundary 2 (y = p + θQ): V-M SII ðp þ θQ Þ ¼ SIII ðp þ θQ Þ , S–P SII 0ðp þ θQ Þ ¼ SIII 0ðp þ θQ Þ The four constants are not reported, but are available upon request from the author. They can be interpreted as follows. The term Z 1 yγ1 represents the effect of the price cap possibly becoming binding in future, Z 2 yγ1 shows the effect of the capacity constraint becoming binding, Z 3 yγ2 shows the effect of returning to a state where no constraint is binding, and Z 4 yγ2 shows the effect of moving from a state where both price and capacity constraints are binding to one without binding constraints. B. When p ≥ θQ.

S. Sarkar / Economic Modelling 50 (2015) 168–178

Region I (y b 2θQ): SI ðyÞ ¼ y2 =8θψ þ Z1 yγ1 Region II (2θQ ≤ y ≤ p + θQ): SII ðyÞ ¼ θQ 2 =2r þ Z2 yγ1 þ Z3 yγ2 Qy Region III (y N p + θQ): SIII ðyÞ ¼ ðr−μÞ − Qr ðp þ θQ =2Þ þ Z4 yγ2 :

The four constants can again be solved from the four boundary conditions: Boundary 1 (y = 2θQ): SI(2θQ) = SII(2θQ), SI ′ (2θQ) = SII ′ (2θQ) Boundary 2 (y = p + θQ): SII ðp þ θQ Þ ¼ SIII ðp þ θQ Þ, SII 0ðp þ θQÞ ¼ SIII 0ðp þ θQÞ:

Once again, the four constants are not reported, but are available upon request from the author. Their interpretation is similar to Section A above.

4.3. Pre-investment consumer welfare At the beginning of Section 4, we explained the difference between consumers' surplus and consumer welfare, and why the Consumer Welfare Function S(y) is a better measure of long-term consumer well-being. Since consumers' welfare depends on the price cap, we can write the function as Sðp; yÞ. This function describes the consumers' welfare from a project in operation, i.e., after the investment has been made. Now, if we look at the problem prior to investment, we know that investment will be made (and the consumers' surplus stream will begin) at some point in the future, when the state variable rises to y⁎. At that time, the consumer welfare will be given by Sðp; y Þ. Prior to the investment being made, there is no consumer surplus stream, but the expected present value of the future consumers' surplus stream is given by the expression Sðp; y Þðy=y Þγ1 , where y is the current value of the state variable. This is the appropriate pre-investment measure of consumer well-being, and this is what consumers would like to maximize, for all values of y. Since γ1 N 0, maximizing Sðp; y Þðy=y Þγ1 for all y is equivalent to maximizing ΩðpÞ ¼ Sðp; y Þ=ðy Þγ1 . Prior to the firm's investment, the appropriate measure of consumer welfare is ΩðpÞ. In the analysis that follows, ΩðpÞ is the consumer welfare function we focus on. 5. Results The complexity of the model rules out closed-form solutions, hence the results are derived numerically. We start with a base-case set of parameter values, and repeat the exercise with a wide range of values to ensure robustness of the results. The base case parameter values are as follows: expected growth rate of demand (μ) = 0, demand volatility (σ) = 20%, discount rate (r) = 10%, price sensitivity to output (θ) = 0.01, cost of installing capacity (c) = $2 per unit, and investment cost exponent (η) = 1. The fundamental characteristics of our results are not affected by varying these parameters within realistic ranges.

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5.2. Price-cap regulated firm The results for a regulated firm will depend on the price cap, hence we need to specify p. With a low price cap of p = 0.21, the results are as follows: Q⁎ = 7.54, y⁎ = 1.0188 and ΩðpÞ = 55.6234. When price cap is raised to p = 0.3, we get: Q⁎ = 10.77, y⁎ = 0.4911 and ΩðpÞ = 154.4564; when further raised to p = 0.5, the output is: Q⁎ = 16.77, y⁎ = 0.5528 and ΩðpÞ = 139.3450, and when raised to p = 1, the output is: Q⁎ = 23.22, y⁎ = 0.6344 and ΩðpÞ = 121.1770. These figures are summarized in Table 1. 5.2.1. The investment decision (Q⁎ and y⁎) The effect of price cap on firm's investment decision (capacity and investment trigger) and the resulting consumer welfare are illustrated in Fig. 1 for the base-case parameter values. It is clear that the optimal capacity Q⁎ is an increasing function of the price cap p, which is not surprising since a tighter (looser) price constraint makes additional capacity less (more) attractive. However, Q⁎ increases at a decreasing rate, until it eventually becomes flat at the “no-price-cap” level (which is 34.49 in the base case, as stated in Section 5.1). The optimal investment trigger y⁎ is initially a decreasing function, and subsequently an increasing function, of price cap p. (Note, however, that when y⁎ is increasing it does so at a decreasing rate until it flattens out at the no-cap level, y⁎ = 0.7646). This U-shaped relationship results from two opposing effects of p on y⁎: (i) a higher p increases the attractiveness of the project (by loosening the constraint), which encourages investment and thus reduces y⁎; and (ii) however, a higher p also results in a larger capacity, as seen above, and a larger capacity results in a higher y⁎ or delayed investment. Thus, when p is increased, there is a direct effect which reduces y⁎, and an indirect effect (via capacity) that increases y⁎. When the capacity is large enough, the second effect dominates and y⁎ becomes an increasing function of p. Because of the U-shaped relationship, there is a unique price cap that minimizes the investment trigger; let this price cap be pIT. Thus, a price cap of pIT will ensure the earliest possible investment by the regulated firm. For the base case, we get pIT = 0.31. If the objective is to speed up investment as much as possible (Dobbs, 2004; Roques and Savva, 2009), the optimal price cap would be pIT . 5.2.2. Consumer welfare Fig. 1 also shows how consumer welfare ΩðpÞ varies with price cap p. It is initially increasing and subsequently decreasing in p, giving an inverted-U shaped relationship. The inverted-U shaped relationship implies that there is a unique price cap at which consumer welfare reaches its highest value. For the base-case parameter values, consumer welfare is maximized when price cap is set at p = 0.32. The intuition behind this inverted-U shaped relationship is as follows. When the price cap is varied, there is a direct effect on consumer welfare because the consumers' surplus is a function of price cap (Section 4.1). In addition, there are two indirect effects – via the firm's capacity decision and investment timing decision. These do not all have the same directional effect on consumer welfare, hence the overall effect of the price cap is non-monotonic.

5.1. Benchmark: unregulated firm We first analyze the “no-price-cap” case (p→∞) as a benchmark. This case is solved by setting p high enough to ensure convergence, i.e., such that any further increase in p does not change the output values. For the above base-case parameter values, we get the following outputs: optimal capacity Q⁎ = 34.49 units, optimal investment trigger y ⁎ = 0.7646 (and corresponding entry price, from Eq. (1), of p ⁎ = y ⁎ − θQ ⁎ = 0.4197, i.e., an “unregulated” entry price of $0.4197). The resulting consumer welfare measure is Ωðp→∞Þ = 92.1951.

Table 1 Optimal investment trigger and capacity, and the resulting consumer welfare. Base-case parameter values: μ = 0, σ = 20%, r = 10%, θ = 0.01, c = 2, and η = 1. Price cap (p)

Optimal capacity (Q⁎)

Optimal trigger (y⁎)

Consumer welfare (ΩðpÞ)

0.21 0.3 0.5 1.0 ∞

7.54 10.77 16.77 23.22 34.49

1.0188 0.4911 0.5528 0.6344 0.7646

55.6234 154.4564 139.3450 121.1770 92.1951

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S. Sarkar / Economic Modelling 50 (2015) 168–178

c) consumer welfare is initially an increasing function and subsequently a decreasing function of price cap.

Investment and Consumer welfare

2.5

2

The practical implication for corporations (consumers) is that price cap can have a significant impact on investment decisions (consumer welfare).

1.5

5.3. Consumer welfare maximization versus investment trigger minimization 1

0.5

0 0

0.5

1

1.5

Price cap y*

Q*/10

Consumer welfare/100

Fig. 1. Optimal investment trigger y⁎(p), optimal capacity Q⁎(p) divided by 10, and consumer welfare function Ω(p) divided by 100, as functions of price cap p. The basecase parameter values are used: μ = 0, σ = 20%, r = 10%, θ = 0.01, c = 2 and η = 1. The price cap that minimizes investment trigger is pIT = 0.31, and the price cap that maximizes consumer welfare is pCW = 0.32.

When the price cap is increased, the direct effect is to reduce consumer welfare. The indirect effects come through the firm's capacity and timing decisions. A higher price cap results in a larger capacity, which consumers benefit from; hence the “capacity effect” is positive. The “timing effect” depends on the price cap level. When the price cap is low (high), an increase in the cap results in earlier (later) investment. Consumers benefit from earlier investment, and are hurt by delayed investment; thus, the “investment effect” will be positive (negative) when the price cap level is low (high). When the price cap is increased, the direct effect reduces the consumer welfare while the two indirect effects depend on the existing price cap – for low price cap, both are positive; for high price cap, one is positive and the other is negative. When the price cap is low, the direct effect is dominated by the indirect effects, and the consumer welfare rises with the price cap. When the price cap is large, the negative effects dominate, hence consumer welfare is a decreasing function of the price cap. The result is an inverted-U shaped relationship between ΩðpÞ and p. We also note from Fig. 1 that the choice of price cap can make a significant difference to consumer well-being. Without a price cap, the consumer welfare measure is 92.1951; with a price cap of p = 0.3, consumer welfare is significantly higher at 154.4564. On the other hand, an inappropriate choice of price cap can significantly reduce consumer welfare, e.g., with p = 0.21, consumer welfare is much lower at 55.6234; thus, even with a low price cap (which should ostensibly protect consumer interests), consumer welfare is significantly lower than in an unregulated environment. The above computations were repeated with a wide range of parameter values, and the effect of price cap on the firm's investment decision (both timing and size) and consumer welfare was found to be very similar in all cases. To summarize the results of this section:

5.2.2.1. Result 1. The price cap (p) affects the firm's investment decision (Q⁎ and y⁎) and the consumer welfare (ΩðpÞ) as follows: a) Q⁎ increases with p, but at a declining rate until it reaches a constant level (the “no price-cap” level); b) y⁎ is a U-shaped function of p, initially falling and subsequently rising (at a declining rate, leveling off at the “no-cap” level); and

Suppose the price cap that maximizes consumer welfare is pCW , and the price cap that minimizes the investment trigger is pIT . In both Roques and Savva (2009) and Dobbs (2004), the optimal price cap was defined as the one that minimized the investment trigger, or pIT . This would accelerate investment timing (which would be good for consumers). However, the overall effect on consumer welfare is not clear because other decisions (investment capacity or timing) will be affected in a manner that might have a negative effect on consumers. In order to address this issue, we compare below the investment-triggerminimizing price cap pIT with the consumer-welfare-maximizing price cap pCW . For the base-case parameter values, we get: pCW = 0.32 and pIT = 0.31. Thus, the price cap that minimizes the investment trigger does not result in the highest level of consumer welfare. As discussed in Section 2, we do not look at what should be maximized or what should be the regulator's objective. Rather, we take the consumers' perspective, and examine how the price cap affects consumer welfare. From the consumers' perspective, the desirable price cap is one that maximizes consumer welfare, not necessarily one that accelerates investment. If the price cap is set below pIT , there are three effects: (i) a lower price cap means higher consumers' surplus, hence consumer is better off; (ii) smaller capacity, hence consumer is worse off; and (iii) delayed investment, hence consumer is worse off. Similarly, setting the price cap above pIT will result in lower consumers' surplus, larger capacity, and delayed investment. In both cases, the net effect on consumer welfare is not clear – the consumer could be better off or worse off, depending on the magnitudes of these individual effects. Whether the consumer is better off with price cap above or below pIT (i.e., whether pCW is above or below pIT ) depends primarily on the importance of the installed capacity to consumer welfare. When the capacity has a significant impact on consumer welfare, the capacity effect is more important, and pCW will be higher than pIT (in order to have larger capacity investment). For instance, if the demand is very volatile (σ is high), then it becomes more important to have larger capacity to absorb the large demand shocks; thus, capacity is more desirable for consumer welfare. Therefore, when σ is high, we expect pCW to be larger than pIT . For low σ, a large capacity is less important, and it is more important to look after consumer welfare by reducing the price cap; hence pCW would be lower than pIT for low σ. Also, a higher discount rate r, keeping all other parameter values unchanged, implies that future consumer surplus will be less valuable relative to current consumer surplus, hence a large capacity is not so desirable for consumer well-being (since large capacity is mainly to accommodate increased demand in future). For large r, therefore, pCW will be lower than pIT . Table 2 shows the consumer-welfare-maximizing price cap and the investment-trigger-minimizing price cap for different parameter values. We note that, for the base case, the former exceeds the latter, or pCW N pIT . The other comparative static results are consistent with the above discussion. For small demand volatility (σ), we see that pCW is lower than pIT; for large σ, pCW is higher than pIT ; these are consistent with the above discussion. Importantly, the difference betweenpCW andpIT can be substantial, e.g., with σ = 24%, we have pCW = 0.42 and pIT = 0.34. As demand growth rate (μ) is increased, pCW rises while pIT falls; thus, the difference between pCW and pIT increases with μ. This is

S. Sarkar / Economic Modelling 50 (2015) 168–178 Table 2 Comparative static results. Base-case parameter values: μ = 0, σ = 20%, r = 10%, θ = 0.01, c = 2, and η = 1. Base-case results: Q⁎ = 11.48, y⁎ = 0.4904, pCW = .32, pIT = 0.31. σ

Q⁎

y⁎

pCW

pIT

c

Q⁎

y⁎

pCW

pIT

0.16 0.18 0.20 0.22 0.24

7.70 9.28 11.48 14.12 19.14

0.4081 0.4458 0.4904 0.5447 0.6272

0.27 0.29 0.32 0.35 0.42

0.29 0.30 0.31 0.33 0.34

1.0 1.5 2.0 2.5 3.0

5.74 8.61 11.48 14.36 17.23

0.2452 0.3678 0.4904 0.6131 0.7357

0.16 0.24 0.32 0.40 0.48

0.16 0.23 0.31 0.39 0.47

μ

Q⁎

y⁎

pCW

pIT

r

Q⁎

y⁎

pCW

pIT

0.0 0.005 0.01 0.015 0.02

11.48 12.82 15.32 18.43 22.91

0.4904 0.4990 0.5182 0.5477 0.5939

0.32 0.32 0.34 0.36 0.39

0.31 0.31 0.30 0.30 0.29

0.08 0.09 0.10 0.11 0.12

11.95 11.85 11.48 11.23 11.37

0.4460 0.4685 0.4904 0.5156 0.5415

0.29 0.31 0.32 0.33 0.35

0.26 0.29 0.31 0.34 0.36

Note: Q⁎ y⁎ pCW pIT

optimal capacity, optimal investment trigger, price cap that maximizes consumer welfare Ω(p), and price cap that minimizes investment trigger y⁎.

because capacity becomes more desirable when the demand growth μ is higher (see argument above). For high growth rate, the difference can be substantial, e.g., with μ = 2%, we have pCW = 0.39 and pIT = 0.29. For low discount rate (r), we have pCW N pIT , and the relationship is reversed for large r. This is also consistent with our above discussion. Finally, as investment cost (c) is increased, both pCW and pIT rise, but the spread between the two remains virtually unchanged. In earlier studies (Dobbs, 2004; Roques and Savva, 2009), the recommendation was to set price cap at pIT so as to minimize the investment trigger. However, as shown above, such a price cap would generally not maximize consumer welfare. Following the prescription of these papers would mostly result in a price cap that was too low, since pCW N pIT in most cases (the exceptions being low demand volatility and high interest rate). In many cases, the difference between pCW and pIT is small, hence pIT can be a reasonable proxy for pCW . However, there are some cases (e.g., high demand volatility and high demand growth rate) when the difference is substantial, hence the price cap that minimizes the investment trigger will be quite sub-optimal for consumers. The practical implication for regulators, policymakers and consumers is that is possible to choose the price cap to encourage investment as soon as possible, but that price cap is not necessarily the best for consumers. The results of this section are summarized below. 5.3.1. Result 2 If the price cap is set so as to minimize the firm's investment trigger, consumer welfare will generally not be maximized; for some parameter values (e.g., high demand volatility and high demand growth rate) the difference is substantial. We also note from Table 2 that the price cap that maximizes consumer welfare is an increasing function of demand volatility (σ), demand growth rate (μ), per-unit investment cost (c), and discount rate (r). These relationships should be of interest to regulators and certainly to consumers and consumer advocates. 5.4. Asymmetry of the price-cap consumer-welfare relationship In Fig. 1, we note that, as the price cap is increased from low levels, consumer welfare first rises sharply until the maximum is reached, and then declines gradually as price cap is raised further. Thus, the

175

consumer welfare function has an asymmetric response to changes in the price cap. The asymmetric response can be explained as follows. Suppose the price cap is set to maximize consumer welfare, i.e., p = pCW . Now, if the price cap is reduced below pCW , it will affect the firm's investment decision in two ways (from Section 5.2): (i) capacity will be smaller, which will hurt consumers, and (ii) investment trigger will rise, which will also hurt consumers (although it might initially fall slightly and then start rising, if pCW N pIT ). Since both effects will hurt consumers, consumer welfare will fall sharply. On the other hand, if the price cap is raised above pCW , it will affect the firm's investment decision as follows (Section 5.2): (i) capacity will be larger, which will help consumers, and (ii) investment trigger will rise, which will hurt consumers (although it might initially fall slightly and then start rising, if pCW b pIT ), Because there is some canceling out of the effects in this latter case, the net effect will be weaker; as a result, consumer welfare will fall more gradually. This results in an asymmetric relationship between price cap and consumer welfare, whereby consumer welfare falls steeply when price cap is reduced, but falls much more slowly when price cap is raised. To test for the generality of this asymmetric relationship, we repeated the numerical computations for a wide range of parameter values. The results are summarized in Fig. 2(a)–(f). In virtually all the cases, we note that the asymmetric relationship persists: if price cap is reduced below pCW consumer welfare falls sharply, but if raised above pCW consumer welfare falls gradually. The only exception is for low demand volatility (σ = 15%); in this case, consumer welfare falls sharply when price cap is lowered, and when price cap is raised it first falls sharply and then falls gradually. The asymmetric relationship between price cap and consumer welfare has an important practical implication for regulators. When the regulator is not sure about the exact value of the optimal price cap (say, because of uncertainty regarding input parameter estimates), consumers will generally be better off if the price cap is set too high rather than too low. While this might seem contradictory to conventional wisdom, it reflects the importance of investment capacity and timing in determining consumer welfare. As pointed out above, the exception is for low volatility, in which case the regulator must be careful in both directions, since consumer welfare falls significantly when price cap is moved away from pCW in either direction. The results of this section are summarized below. 5.4.1. Result 3 The effect of price cap on consumer welfare is generally asymmetric – if price cap is raised consumer welfare ΩðpÞ falls gradually, but if it is lowered, consumer welfare falls rapidly. 6. Conclusion This paper uses a real-option model to examine the effect of a price limit on a firm's investment timing and size decision, and the resulting effect on consumer welfare. The complexity of the model precludes closed-form solutions, hence the results are illustrated numerically. We show that investment trigger is a U-shaped function of price cap, investment size is an increasing function of price cap, and consumer welfare is an inverted-U-shaped function of price cap. Thus there is a unique price cap that minimizes investment trigger and another that maximizes consumer welfare. These two price caps are generally different, in some cases substantially so. The practical implication is that speeding up investment (as recommended by the existing literature) will not necessarily increase consumer welfare. Further, the effect of price cap on consumer welfare is generally asymmetric – when price cap is lowered, consumer welfare falls precipitously, but when it is raised, consumer welfare falls gradually. Thus, a too-low price cap will paradoxically hurt consumers more than a toohigh price cap.

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S. Sarkar / Economic Modelling 50 (2015) 168–178

b

300

Consumer welfare

Consumer welfare

a

250 200 150 100

300 250 200 150 100 50

50

0

0 0

0.5

1

0

1.5

0.5

Price cap Sigma=15%

Sigma=20%

Mu=0

Sigma=25%

Mu=1%

200 150 100 50

300 250 200 150 100 50 0

0.5

1.0

1.5

0

0.5

r=8%

1

1.5

Price cap

Price cap

Theta=0.01

Theta=0.005

r=10%

f

200

Consumer welfare

Consumer welfare

Mu=2%

350

0 0.0

e

1.5

d

250

Consumer welfare

Consumer welfare

c

1

Price cap

150

100

50

Theta=0.015

200

150

100

50

0

0 0

0.5

1

1.5

c = 1.9

0

0.5

1

1.5

Price cap

Price cap c=2

Eta = 1

Eta = 1.1

Fig. 2. (a). Consumer welfare Ω(p) as a function of price cap p, for three different levels of demand volatility, σ = 15%, 20% and 25%. Base-case parameter values: μ = 0, r = 10%, θ = 0.01, c = 2 and η = 1. (b). Consumer welfare Ω(p) as a function of price cap p, for three different demand growth rates, μ = 0, 1% and 2%. Base-case parameter values: σ = 20%, r = 10%, θ = 0.01, c = 2 and η = 1. (c). Consumer welfare Ω(p) as a function of price cap p, for two different discount rates, r = 8% and 10%. Base-case parameter values: μ = 0, σ = 20%, θ = 0.01, c = 2 and η = 1. (d). Consumer welfare Ω(p) as a function of price cap p, for three different levels of price sensitivity, θ = 0.005, 0.01 and 0.015. Base-case parameter values: μ = 0, σ = 20%, r = 10%, c = 2 and η = 1. (e). Consumer welfare Ω(p) as a function of price cap p, for two different levels of per-unit investment cost, c = 1.9 and 2. Base-case parameter values: μ = 0, σ = 20%, r = 10%, θ = 0.01 and η = 1. (f). Consumer welfare Ω(p) as a function of price cap p, for two different levels of investment cost convexity, η = 1 and 1.1. Base-case parameter values: μ = 0, σ = 20%, r = 10%, θ = 0.01 and c = 2.

From a practical standpoint, there are three important implications of our results. One, consumer welfare can be substantially affected by price cap, hence consumers and consumer advocates should be deeply interested in the price-cap-setting process. To cite an example, Besley and Coate (2003) report that electricity prices in states with elected regulators are significantly lower than states with Governor-appointed regulators. Thus, the selection process for regulators will matter a great deal to consumers, since it affects the price cap and thereby consumer welfare.

Two, the price cap that minimizes investment trigger is generally different from that which maximizes consumer welfare. In certain cases (e.g., high demand volatility and high demand growth rate), the difference between the two can be significant. Thus, setting the rice cap to minimize investment trigger could result in substantial reduction in consumer welfare. Three, because of the asymmetric effect of price cap on consumer welfare, consumers will be better off if the regulator sets the price cap too high rather than too low. This is often an important factor, since

S. Sarkar / Economic Modelling 50 (2015) 168–178

regulators and policymakers are generally unsure of the exact values of input parameters (because of estimation errors, unanticipated economic changes, etc). Finally, as in any theoretical model, we make certain assumptions for tractability. The demand parameter y is assumed to follow a geometric Brownian Motion, consistent with most real-option models. The other plausible process considered in the literature is the mean-reverting process, examined by Wong and Yi (2013), who show that it results in a lower investment trigger. Thus, a mean-reverting y in our model would result in earlier investment. However, although the mathematics would undoubtedly be more complex, the effect of price limit on the various outcomes should not be qualitatively different. We also assume there is no depreciation in the model. If we include depreciation, the output and the cash flow stream to the firm would be eroded over time; the smaller cash flows would affect the firm's investment decision, while the lower output level would affect consumer welfare. However, the basic trade-offs in the model would not be affected by this change, hence the results should be qualitatively similar. The model also assumes the investment is irreversible (i.e., no disinvestment). Most projects in real life are substantially, but not completely, irreversible (Dixit and Pindyck, 1994). Partial reversibility can be incorporated as in Wong (2010), who finds that it lowers investment trigger but does not affect capacity. In our model, therefore, partial reversibility would result in earlier investment, and consumer welfare would be higher. However, once again the basic trade-offs remain the same, hence the effect of price limit on investment decision and consumer welfare should be qualitatively the same. Appendix A. Project valuation when p b θQ The project value will be a function of the state of the world (y); let us say it is Z(y). Then it can be shown, using standard arguments (e.g., Bar-Ilan and Strange, 1999; Dobbs, 2004; Dixit and Pindyck, 1994), that Z(y) satisfies the ordinary differential equation (ODE): 0:5σ 2 y2 Z″ðyÞ þ μyZ0ðyÞ−rZðyÞ þ ςðyÞ ¼ 0

ðA1Þ

where ς(y) is the cash flow per unit time. Since ς(y) depends on the operating region, so will the project value. Thus, the project value must be derived separately for each region. The general solution to ODE (Eq. (A1) is ZðyÞ ¼ Z1 yγ1 þ Z2 yγ2 , where Z1 and Z2 are constants to be determined from boundary conditions. The particular solution will depend on the exact form of the cash flow term ς(y). Region I (no binding constraint, or y b 2p): The unconstrained cash flow is y2 / (4θ) from Section 3.1, hence ς(y) = y2 / (4θ). This gives the particular solution [y2 / (4 ψ θ)], where ψ = (r–2 μ–σ 2 ). The project value U(y) is the complete solution to ODE (A1), given by: UðyÞ ¼ y2 =4θψ þ Ayγ1 þ A0 yγ2 , where A and A0 are constants to be determined from boundary conditions, and γ1 (N0) and γ2 (b 0) are the solutions of the quadratic equation: 0.5σ 2γ(γ − 1) + μγ − r = 0, and are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 γ1 ¼ 0:5−μ=σ 2 þ ð0:5−μ=σ 2 Þ þ 2r=σ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 γ2 ¼ 0:5−μ=σ 2 − ð0:5−μ=σ 2 Þ þ 2r=σ 2 :

γ1

UðyÞ ¼ y2 =4θψ þ Ay :

½py=ðθðr−μÞÞ−p2 =ðθrÞ. The project value in this region, V(y), must also satisfy ODE (A1), giving the complete solution: VðyÞ ¼ py=ðθðr−μ ÞÞ−p2 =ðθrÞ þ Byγ1 þ Cyγ2 :

ðA2Þ

Region II (only price constraint binding, or 2p ≤ y ≤ p + θQ): in this case, we have ς(y) = p(y – p) / θ, which gives the particular solution

ðA3Þ

Region III (both price and capacity constraints binding, or y N p + θQ): In this case, we have ς(y) = pQ, which gives the particular solution [pQ / r]. The project value W(y) must satisfy ODE (A1), giving the complete solution: WðyÞ ¼ pQ=r þ Dyγ2 þ D0 yγ1 . Next, note that when y is very large (y → ∞) the probability of returning to Region II is negligible, hence WðyÞ→pQ=r. This is possible only if D0 = 0 in the above expression for W(y); we therefore write the project value in this region as: ðA4Þ WðyÞ ¼ pQ=r þ Dyγ2 : In the above projects values, A, B, C and D are constants to be determined from boundary conditions. Boundary Conditions: When y rises to 2p, the price cap becomes binding. This gives the first boundary y = 2p (between Regions I and II). When y rises further to y = (p + θQ), both constraints become binding; this gives the second boundary y = (p + θQ) between Regions II and III. Two conditions must be satisfied at each boundary (Dixit and Pindyck, 1994): (a) the value matching condition (V–M), which requires that the two values be equal at the boundary, to ensure continuity at the boundary; and (b) the smooth pasting condition (S–P), which requires that the two slopes be equal at the boundary, to rule out arbitrage opportunities at the boundary. These are given below. • Boundary 1: V–M: U(2p) = V(2p), S–P: U′(2p) = V′(2p); • Boundary 2: V–M: V(p + θQ) = W(p + θQ), S–P: V′(p + θQ) = W′ðp þ θQÞ: The four equations can be solved for the four unknowns A, B, C and D, which are given below. pQ ðpÞ2 pðp þ θQ Þð1−1=γ2 Þ þ − r θðr−μ Þ θr B¼ C¼ ð1−γ1 =γ2 Þðp þ θQ Þγ1

p2 θ

! 1−2=γ1 þ 1 − 2ð1−1=γ1 Þ r ðr−μ Þ ψ ð1−γ2 =γ1 Þð2pÞγ2

2p2 p2 p2 pðp þ θQ Þ γ − þ Bð2pÞγ1 þ Cð2pÞγ2 − þ B 1 ðp þ θQ Þγ1 θγ2 ðr−μ Þ θðr−μ Þ θr θψ γ2 A¼ D¼ Cþ ð2pÞγ1 ðp þ θQ Þγ2

Appendix B. Project valuation when p ≥ θQ In this case, the valuation ODE is the same as in Section A above. From Section 3.1, the cash flow areς(y) = y2 / (4θ) in Region I, ς(y) = Q(y–θQ) in Region II, and ς(y) = pQ in Region III. The project values in the different regions are as follows.

Region I ð y b2θQ Þ :

Next, note that when y is very small (y → 0) the project value approaches zero since y = 0 is an absorbing boundary of the lognormal process y; that is, when y → 0, U(y) → 0. This implies A0 = 0 in the above expression for U(y); we therefore write the project value in this region as:

177

UðyÞ ¼ y2 =4θψ þ Ayγ1

ðA5Þ

Region II ð2θQ ≤ y≤p þ θQ Þ : VðyÞ ¼ Q y=ðr−μ Þ−θQ 2 =r þ Byγ1 þ Cyγ2

ðA6Þ Region III ðyNp þ θQ Þ : WðyÞ ¼ pQ =r þ Dyγ2 :

ðA7Þ

The two boundaries are: y = 2θQ and y = (p + θQ), giving the following boundary conditions: • First boundary (y = 2θQ): U(2θQ) = V(2θQ) & U′(2θQ) = V′(2θQ); • Second boundary (y = (p + θQ)): V(p + θQ) = W(p + θQ) & V′ (p + θQ) = W′(p + θQ).

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S. Sarkar / Economic Modelling 50 (2015) 168–178

As before, the four boundary conditions provide the four unknowns A, B, C and D: p þ θQ ðp þ θQ Þð1−1=γ2 Þ 1 2ð1−1=γ1 Þ 1−2=γ1 −   − ðr−μ Þ þ r ðr−μ Þ ψ 2 r B¼Q C ¼ θQ ð1−γ1 =γ2 Þðp þ θQ Þγ1 ð1−γ2 =γ1 Þð2θQ Þγ2   2 1 1 θQ 2 − − þ Bð2θQ Þγ1 þ Cð2θQ Þγ2 ðr−μ Þ r ψ A¼ ð2θQ Þγ1 Q ðp þ θQ Þ þ Bγ1 ðp þ θQ Þγ1 þ Cγ2 ðp þ θQ Þγ2 ðr−μ Þ D¼ : γ2 ðp þ θQ Þγ2

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