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An optimal timing model of water reallocation and reservoir construction
European Journal of Operational Research 145 (2003) 165–174 www.elsevier.com/locate/dsw
O.R. Applications
An optimal timing model of water reallocation and reservoir construction Fathali Firoozi *, John Merrifield Department of Economics, The University of Texas at San Antonio, San Antonio, TX 78249, USA
Abstract Growth of the users base often reduces the efficiency of existing water allocation policies and causes new water supplies to cost more than the value of water in existing use. Historically, water management authorities have shown a bias toward undertaking of new supply projects and delay of necessary reallocations. This study provides an analysis of the timing and adjustments associated with construction of new reservoirs and reallocation of existing supplies. A dynamic model yields simultaneously the optimal construction timing of new projects and guidelines for reallocation of existing supplies. A number of sensitivity results provide additional insights. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Optimization theory; Systems dynamics; Facilities planning and design; Natural resources
1. Introduction Water management authorities provide strict policy guidelines regarding allocation of pumping rights and the transfer of such rights between high value and low value users. 1 Such policy guidelines are socially and politically sensitive issues and play a dominant role in determining overall availability of water and its price to the consumer. The process implies that designing an allocation policy based on commonly accepted criteria is one of the crucial
* Corresponding author. Tel.: +1-210-458-5395; fax: +1-210458-5837. E-mail address: ffi[email protected] (F. Firoozi). 1 Classifications of water users include commercial, industrial, agricultural, residential, and those with property rights that have access to underground or nearby surface reservoirs. References include Smith (1989), Blomquist (1992), and Spencer (1992).
tasks of the authorities. Meanwhile, economic growth often brings expansion and redistribution of various user groups and thus reduces the efficiency of existing water allocation policies and creates the problem of identifying an optimal balance between reallocation of existing supplies and construction of new supply projects. The issue is complicated by the fact that water from new projects often costs more than the value of water in many existing uses. In addition, policy-makers have many competing concerns and redistribution of existing water supplies appears to be one of the lesser priorities. A reallocation of a basic commodity like water may cause numerous social, legal, cultural, and equity changes with significant economic and political adjustment costs to water management authorities. Those costs are often deferred by incurring the cost of additional water supply projects. If water allocation policies are changed so that users bear a larger fraction of the
0377-2217/03/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 2 ) 0 0 1 6 0 - 1
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social marginal cost of water use, society incurs adjustment costs sooner and construction costs later. The dilemma is to find the delicate balance in that trade-off. If the demand grows across the board at a constant rate, then a reallocation may not be needed but the implications for the construction timing will remain. The efficiency of existing allocation systems and the timing of new supply projects have recently received a closer but still sporadic attention in the literature. Many analysts have cited construction of noneconomical and premature water projects and called for reallocations of existing supplies before undertaking new supply projects. 2 In addition, recent increases in construction and utilization costs and heightened environmental consciousness have provided stimuli for a more serious consideration of reallocation. However, state and local authorities continue to propose new water supply projects with unit costs well above the value of water in major existing uses. This is despite the fact that economic growth will raise the dilemma again in the near future. Economic efficiency was not a major issue even in the few recent cases where public referenda forced abandonment of proposed water projects like CaliforniaÕs Peripheral Canal and TexasÕ Applewhite Reservoirs, or when the Environmental Protection Agency blocked DenverÕs Tow Forks reservoir. Only recently the federal government has begun to pay a closer attention to reallocation as an alternative to new construction (Davis, 1992). Meanwhile, many costly and premature projects have been built (Davis, 1992; Sander, 1985). It is clear that the problems of construction timing and reallocation are closely interrelated and any decision regarding one must incorporate action toward the other. The stated issues point to the need for a formal analysis of policy toward construction tuning based on a socially optimal criterion when reallocation is also an endogenous policy choice. The present study provides such an analysis.
2 The references include Anderson (1983), Sander (1985), Welsh (1985), Gardner (1987), Langfur (1989), Schoolmaster (1991), Blomquist (1992), Griffin and Boadu (1992), Spencer (1992), and Simpson (1994).
A question may arise as to why there has been such a strong and costly preference in part of policy-makers for construction of new supply projects over reallocation of existing supplies, even when a reallocation could be on a willing buyer– willing seller or market basis. Many causes have been cited, including votersÕ insufficient information, poor economic analysis skills at regional and local home offices of government agencies (Sander, 1985), pursuit of legislative pork and log-rolling (Gardner, 1987), rent-seeking by empire-building officials (Sander, 1985; Langfur, 1989), and various adjustment costs. Gardner (1987) and Griffin and Boadu (1992) suggested that changes in the pattern of water use often threaten the wealth positions of existing water users and indirect beneficiaries. Such adjustment costs, including political concerns, must be taken into account in determination of optimal efforts to reallocate existing supplies and/or construct new supply sources (Nelson, 1987; Wehmhoefer, 1989). The focus of this study is on optimal construction timing (deferral) of new water supply projects and simultaneous reallocation of existing supplies in a dynamic setting where economic growth reduces the efficiency of existing distributive policies. The optimality will be with respect to net social benefits derived from consumption and costs. The existing dynamic models in the resources literature contain two shortcomings in relation to the stated problem of reallocation and construction; specifically, the existing models (i) do not address the issue of construction timing in conjunction with the reallocation issue and (ii) do not explicitly incorporate the adverse effect of a fixed policy in the presence of growth and the policy adjustment costs into the analysis. Section 2 develops an expository model that resolves the stated shortcomings and incorporates the key factors related to the issue. The solution and its implications for allocative policy and pricing adjustments are discussed in Section 3. The focus of Section 4 is on optimal construction timing in the presence of variable allocative policy. The simultaneous solution and its sensitivity results highlight some interesting aspects of the issue and provide certain guidelines for optimal construction timing of new supply projects as well as optimal reallocations of existing
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supplies. A numerical example in Section 5 shows an application, and some concluding remarks appear in Section 6.
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by economic growth, e.g., the higher cost of implementing a fixed allocative policy to a larger consumer base. 4 2.1. Supply and user cost
2. The model This section develops an expository formulation of the problem. The key factors are incorporated in a dynamic setting so that the timing of construction and policy toward reallocation emerge as the primary decision variables. Suppose that the water authority (the planner) has a number of allocation policy options to choose from. A measure G of these policy options is a positive number such that a higher value for G reflects a policy option that is more costly for the water authority to implement but increases the availability and access for the consumers. For instance, a flexible allocation policy (G1 ) that continuously reallocates the existing water in response to changing demand from various user groups is more costly to implement than a fixed allocation policy (G2 ), thus G2 < G1 . The relative value of G reflects the nature of allocative policy adopted by the water authority. Let Z represent the administrative costs incurred by the water authority. This cost is mainly a function of two variables, namely, the existing allocative policy and the economic growth that imposes additional cost of implementing the existing allocative policy to a growing consumer base. Thus the administrative or policy adjustment costs at time t is estimated by ZðtÞ ¼ c0 þ c1 G þ c2 t
ð1Þ
where c0 , c1 and c2 are the positive parameters of Z. The units of c1 and G are defined such that c1 G reflects the administrative cost of policy option G. 3 Thus, any arbitrary adjustment in the units of G will adjust the units of the coefficient c1 so that the role of c1 G remains unchanged. Also, c2 > 0 reflects the rise in the administrative cost imposed
3 Linear approximations often allow tractability and analytical solutions. Even with nonlinear initial functions, numerical solutions utilize linear approximations (Chiang, 1992; Kamien and Schwartz, 1991).
Let the aggregate of existing water sources be denoted by A, and let QA denote the market availability of water from source A to consumers, i.e., the amount of water available to consumers under a given allocative policy G and a given economic state. 5 Under the stated setting, economic growth reduces the efficiency and a given allocative policy so that at a given G, the availability QA falls over time. The value of QA is thus approximated by QA ðtÞ ¼ a0 þ a1 G a2 t
ð2Þ
where a0 , a1 and a2 are the positive parameters of QA . Note that a rise in G increases the administrative cost Z via (1) but will also increase in the water availability QA to consumers via (2). It is clear that a1 G measures the impact of a given allocative policy G on the water quantity QA from the fixed source A available to the consumers. Thus, any arbitrary change in the units of G will adjust the units of the coefficient a1 so that the role of a1 G remains unchanged. The parameter a2 > 0 with negative sign in (2) reflects the adverse effect of economic growth on the availability of water to consumers at a given allocative policy G. The cost of market availability of water from source A to consumers has two components, the usual production cost and the allocative cost. This function is assumed to take the form CA ðtÞ ¼ b0 þ b1 QA ðtÞ þ ZðtÞ
4
ð3Þ
Instead of time, one may use in (1) indicators such as GDP(t) to reflect economic growth. If one assumes that GDP grows over time at an average rate of c2 , then (1) reflects this approximation. (1) allows flexibility to incorporate any indicator with the growth rate of c2 . 5 A generalization of the model to the case of multiple existing sources can be given as follows. Replace QA with P P QA ¼ ai Qi and G ¼ ai Gi . The analysis will apply exactly with the additive terms for QA and G treated as vector variables.
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where b0 > 0 and b1 > 0 are the usual cost parameters (assuming all costs are in real terms). Eq. (3) reflects the fact that under a fixed allocative policy (fixed G), the average cost of water from source A, denoted by ACA , rises over time as the economy grows. This is shown by the division of (3) by QA :
where:
‘‘optimal’’ construction time T, we assume that all future benefits and costs of the new source beyond time T are approximated and reflected on the estimated quantities QB and k. This assumption also allows a finite-horizon statement of the problem with the following optimality criterion. The authorityÕs objective is to determine the construction time T and the allocative policy GðtÞ such that the net social benefits emerging from aggregate consumption and cost over the period [0; T ] are maximized. Although the problem could be formulated in a control-theoretic setting, for the purpose of identifying certain analytical characteristics of the solution it is helpful to choose the classical setting and treat the problem as one in variational calculus. 6 To provide a formal statement of the problem within the classical setting, assume that the market demand for water is approximated by the simple form
r0 ðkÞ :¼ b0 þ b1 ðkÞa0 þ c0 ;
P ðQÞ ¼ a bQ
r1 ðkÞ :¼ b1 ðkÞa1 þ c1 ;
where a > 0 and b > 0 are the demand parameters, P is the market price, and QðtÞ is the consumption path. The net social benefit (welfare) in period t is estimated by the total consumer surplus net of the total costs: Z QA N ðtÞ ¼ P ðxÞ dx CA ðtÞ 0 b 2 ¼ aQA ðtÞ ½QA ðtÞ CA ðtÞ 2
ACA ðtÞ ¼ b1 þ
b0 þ ZðtÞ QA ðtÞ
where, from Eqs. (1) and (2), Z rises and QA falls over time (under a fixed G). Any exogenous rise in the average cost of water can be introduced by a rise in b1 . Hence, write: b1 ðkÞ ¼ b1 þ k, where k P 0 represents an exogenous rise in the average cost. Substituting (1) and (2) for ZðtÞ and QA ðtÞ, Eq. (3) can be written as CA ðtÞ ¼ r0 ðkÞ þ r1 ðkÞG þ r2 ðkÞt
ð4Þ
r2 ðkÞ :¼ c2 b1 ðkÞa2 : As stated above, in the presence of positive exogenous increment in the average cost ðk > 0Þ, b1 ðkÞ ¼ b1 þ k. For simplicity (or when k ¼ 0) the argument k is made an implicit argument of b1 until the discussions in Sections 3 and 4 where the introduction of new supplies and the crucial role of k will be discussed. As for the signs of the parameters in (4), it is clear that r0 and r1 are positive. Also, assuming that economic growth increases the user cost of water from the fixed source A at a given allocative policy G, the condition r2 > 0 is adopted. 2.2. Problem statement At a future time T (unknown) a new water source B can be constructed, which is expected to provide the water quantity QB . The construction, however, is expected to raise the average cost of water by an estimated k > 0. To focus on the essence of our argument regarding the ‘‘optimal’’ allocative policy GðtÞ over the period [0; T ] and the
where QA ¼ QA ðG; tÞ and CA ¼ CA ðG; tÞ are those specified in (2) and (4). The problem is stated formally as Z T F ðG; tÞ dt ð5Þ MaxV ¼ G;T
0
subject to the terminal constraints QðT Þ ¼ QA ðG; T Þ þ QB : At time t ¼ T : b1 ðkÞ ¼ b1 þ k;
ð6Þ for k > 0
ð7Þ
where
6 The standard references include Miller (1979), Kamien and Schwartz (1991), and Chiang (1992).
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F ðG; tÞ ¼ N ðtÞeqt b 2 ¼ aQðG; tÞ ½QðG; tÞ CðG; tÞ eqt : 2 ð8Þ Q and C are specified in (2) and (4), and q is the time discount rate. The next two sections discuss the simultaneous solutions for the optimal allocative policy path, pricing, and construction timing of the new supply project.
3. Optimal allocation and pricing The solution and its policy implications regarding optimal allocation and pricing of existing supplies are examined in this section. The stated problem is a variable-terminal point problem in variational calculus. The necessary conditions for an optimum consist of the Euler equation and the transversality condition: FG
d FG0 ¼ 0; dt
½F G0 FG0 ¼ 0; where FG ¼ oF =oG, G0 ¼ dG=dt, and FG0 ¼ oF = oG0 . Since FG0 ¼ 0, the two conditions reduce to FG ¼ 0;
ð9Þ
½F t¼T ¼ 0:
ð10Þ
The procedure is to solve (9) and (10) simultaneously for G and T. Note that FG ¼ aðdQ=dGÞ bQðG; tÞðdQ=dGÞ ðdC=dGÞ ¼ aa1 bða0 þ a1 G a2 tÞa1 r1 ðkÞ: Utilizing the above statement in (9) and solving for G yield the optimal path of the allocative policy to be implemented by the water authority: aa1 ba0 a1 r1 ðkÞ a2 G ðk; tÞ ¼ þ t: a1 ba21
Then G ðk; tÞ ¼ DðkÞ þ
a2 t: a1
169
ð12Þ
As indicated earlier, G ðk; tÞ provides a guide for the time path of allocative policy adjustments over the time period [0; T ] such that the net social benefits with respect to consumption and costs are maximized. Four important policy implications follow from the results: (i) The stated solution G ðtÞ in (12) is independent of initial point, i.e., the initial optimal policy measure G ð0Þ may not satisfy the observed initial value Gð0Þ ¼ G0 . This aspect of the solution emerges from the fact that FG0 ¼ 0; neither the Euler Eq. (9) nor the transversality condition (10) involve an arbitrary constant to be determined by the initial condition Gð0Þ ¼ G0 . 7 According to (12), the initial allocative policy must be such that it generates the policy level G ðk; 0Þ ¼ DðkÞ. If this is not the case, then a policy adjustment that yields the level D for G must be implemented in the initial period. (ii) The optimality result in (12) implies that the policy variable G must rise over tune at the rate a2 =a1 . In other words, dynamic allocative policy adjustments are needed to accommodate economic growth. (iii) By the statements before and after (4), dr1 =dk ¼ a1 ðdb1 =dkÞ ¼ a1 . Therefore, dDðkÞ 1 dr1 ðkÞ 1 ¼ 2 < 0: ð13Þ ¼ dk ba1 dk ba1 Hence, oG =ok ¼ dD=dk < 0. This implies an interesting result: the economy can tolerate a less costly allocative design and lower water availability to users (smaller G ) before the construction time when the expected increment in average water cost emerging from the construction rises (larger k). The effect of a rise in k on the optimal construction time will be discussed in the next section.
Also, let DðkÞ :¼
aa1 ba0 a1 r1 ðkÞ : ba21
ð11Þ
7 A similar independence issue is discussed in Chiang (1992, pp. 42, 149–150).
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(iv) An important corollary to the solution in (12) is the identification of a complementary pricing policy. Applying (12) in (2) yields
Pðk; QB Þ :¼ aQðk; T Þ ðb=2Þ½Qðk; T Þ ¼ a½a0 þ a1 DðkÞ þ QB
2
ðb=2Þ½a0 þ a1 DðkÞ þ QB :
QA ðtÞ ¼ a0 þ a1 ½DðkÞ þ ða2 =a1 Þt a2 t ¼ a0 þ a1 DðkÞ: Using the above quantity in the demand function leads to the optimal price: P ðkÞ ¼ a bQA ¼ a b½a0 þ a1 DðkÞ: An application of the definition in (11) reduces the above optimal price to P ðkÞ ¼
r1 ðkÞ c ¼ b1 ðkÞ þ 1 a1 a1
ð14Þ
where the last equality emerges from the definition following Eq. (4). It is clear from (12) that the distributive policy G must change over time to adjust for the growth. However, (14) shows that the corresponding market price has a steady state for a given k. Note that a rise in the expected cost increment emerging from the new project (a rise in k) will increase the optimal price P over the period [0; T ] prior to the construction time.
ð16Þ
The equation prior to (5) shows that the statement in (16) is the consumer surplus before subtraction of the costs. The following features of P will be utilized in later analysis. Note that oP ¼ aa1 ba1 ða0 þ a1 D þ QB Þ: oD Hence, utilizing (13), oP oP dD ¼ ok oD dk ¼ ½aa1 ba1 ða0 þ a1 D þ QB Þ½1=ðba1 Þ ¼ ða0 þ a1 D þ QB Þ ða=bÞ:
ð17Þ
Similarly, oP ¼ a bða0 þ a1 D þ QB Þ: oQB
ð18Þ
The terminal cost is characterized by applying (12) in (4): Cðk; T Þ ¼ CA ðk; T Þ ¼ r0 ðkÞ þ r1 ðkÞG ðk; T Þ þ r2 ðkÞT
4. Optimal construction time The central issue of construction timing (deferral) is studied in this section. The optimal construction time and its sensitivity to the predicted output and cost of the new water supply project will be examined. Utilizing (12) and the terminal conditions (6) and (7), the transversality condition (10) can be solved for the optimal construction time (T ) of the new water source B. The terminal components of F in (8) are evaluated first. With an application of (2) the terminal condition (6) can be written as Qðk; T Þ ¼ QA ðk; T Þ þ QB
¼ ½r0 ðkÞ þ r1 ðkÞDk þ ½r1 ðkÞða2 =a1 Þ þ r2 ðkÞT :
ð19Þ
Define: RðkÞ :¼ r0 ðkÞ þ r1 ðkÞDðkÞ;
ð20Þ
CðkÞ :¼ r1 ðkÞða2 =a1 Þ þ r2 ðkÞ;
ð21Þ
where C > 0. Then, Cðk; T Þ ¼ RðkÞ þ CðkÞT . The following features of C and R will be utilized. The definitions of r1 ðkÞ and r2 ðkÞ given after (4) show that CðkÞ ¼ r1 ðkÞða2 =a1 Þ þ r2 ðkÞ ¼ ½b1 ðkÞa1 þ c1 ða2 =a1 Þ þ ½c2 b1 ðkÞa2
¼ a0 þ a1 G ðk; T Þ a2 T þ QB ¼ a0 þ a1 DðkÞ þ QB
2
¼ c1 ða2 =a1 Þ þ c2 ; ð15Þ
where the last equality follows from an application of (12). Define
hence, CðkÞ is in fact independent of k, CðkÞ ¼ C. Also, the definitions of r0 ðkÞ and r1 ðkÞ given following (4) and the statement in (13) imply
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dR dr0 dD dr1 ¼ þD þ r1 dk dk dk dk db1 1 db1 þ r1 ¼ a0 þ D a1 ba1 dk dk r1 ¼ ða0 þ Da1 Þ ba1
ð22Þ
where db1 ðkÞ=dk ¼ 1 follows from the earlier definition b1 ðkÞ ¼ b1 þ k. Applications of the above definitions in (8) lead to ½F t¼T ¼ F ðT Þ 2
¼ faQðk; T Þ ðb=2Þ½Qðk; T Þ Cðk; T ÞgeqT ¼ fPðk; QB Þ RðkÞ CT geqT : ð23Þ Utilizing (23), the transversality condition ½F t¼T ¼ 0 stated in (10) yields two solutions for the optimal construction time T , one infinite and one finite. The infinite solution T ¼ 1 (i.e., infinite postponement of the project) emerges from setting eqT ¼ 0. But only the finite solution is feasible. To see this, note that the optimal time path of the adjustment cost Z is identified by an application of (12) in (1): Z ðk; tÞ ¼ ½c0 þ c1 DðkÞ þ ½c1 ða2 =a1 Þ þ c2 t: It is clear that Z ! 1 as t ! 1. Hence, unless the authority has access to infinite budget over time, the solution T ¼ 1 is not feasible. The finite solution for the optimal construction time emerges from setting the term inside the braces in (23) equal to zero and solving for T: T ðk; QB Þ ¼
1 ½Pðk; QB Þ RðkÞ: C
ð24Þ
Eq. (24) characterizes the optimal construction time (T ) as a function of the output and cost increment (QB , k) that are expected to emerge from the new water source B. The sensitivity of the optimal construction time to the predicted output and cost increment of the new project can now be evaluated. Utilizing the relation in (18), the result in (24) yields oT 1 oP 1 ¼ ¼ ½a bða0 þ a1 D þ QB Þ C oQB C oQB ð25Þ
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where C is positive. An evaluation of the term in the brackets shows that the derivative in (25) is negative if QB > ð1=bÞ½b1 þ ðc1 =a1 Þ, and is nonnegative if ‘‘>’’ is replaced by ‘‘ 6 ’’. Thus for a sufficiently large expected output from the new source B, a rise in QB will pull the optimal construction time T forward. Next, the results in (17) and (22) are utilized in the following evaluation of (24): oT 1 oP oR ¼ C ok ok ok 1 ¼ f½a0 þ a1 D þ QB ða=bÞ C ½ða0 þ Da1 Þ r1 =ðba1 Þg 1 r1 QB þ ða=bÞ : ð26Þ ¼ C ba1 It is clear that the sign of the term in (26) depends essentially on the size of the market demand ða; bÞ relative to the expected output of the new source (QB ). In the case where the market demand is relatively large (large ‘‘a’’), the sign is negative and hence a rise in the expected increment in average cost emerging from the construction (a rise in k) will pull the construction time forward (a fall in T ). An intuitive justification can be given along the following lines. As elaborated in the last section, a rise in k will raise the optimal price in period T , which generates a reduction in the consumer surplus over that period. Due to the large size of the demand, this fall in the consumer surplus is relatively large, which justifies the pulling forward of the construction time. Similarly, when the market demand is relatively small, the sign in (26) is positive and hence a rise in k will postpone the construction time.
5. A numerical example The numerical example in this section shows an application of the stated results. For Eqs. (1)–(3), the following parametric values are adopted: c0 ¼ 120; c1 ¼ 3; c2 ¼ 6; a0 ¼ 0; a1 ¼ 10; a2 ¼ 5; b0 ¼ 0; b1 ¼ 0:8:
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In the terminal period T (when the new source will be added), b1 ¼ 0:8 þ k with the expected average cost increment k ¼ 0:2 (k ¼ 0 for the periods prior to T ) where the expected output of the new source is QB ¼ 80. The market demand for water has the parameters a ¼ 140; b ¼ 0:6: The parameters of Eq. (4) can now be computed via those of Eqs. (1)–(3): r0 ¼ b0 þ b1 a0 þ c0 ¼ 120; r1 ¼ b1 a1 þ c1 ¼ 11; r2 ¼ c2 b1 a2 ¼ 2: The optimal price is now computed via Eq. (14): c c P ¼ b1 ðkÞ þ 1 ¼ ðb1 þ kÞ þ 1 a1 a1 ¼ f1:1 for t < T ; 1:3 for t ¼ T g: For the period prior to the optimal construction time T (when k ¼ 0) the parameter D in (11) is computed via the above values, resulting in D ¼ 23:15. Thus, the optimal allocative policy path prior to the construction timer T is computed via (12): G ðtÞ ¼ D þ ða2 =a1 Þt ¼ 23:15 þ 0:5t: In order to compute the optimal construction time, the necessary parameters are first computed via (16), (20), (21) with k ¼ 0: R ¼ r0 þ r1 D ¼ 374:65; C ¼ r1 ða2 =a1 Þ þ r2 ¼ 7:5; a0 þ a1 D þ QB ¼ 311:5; P ¼ a½a0 þ a1 D þ QB ðb=2Þ½a0 þ a1 D þ QB
2
¼ 14500:325: The optimal construction time is then computed via (24): T ¼
1 ðP RÞ ¼ 1883:42: C
The above value for T represents the optimal number of time units (time intervals, periods) to pass before the new source is added to the supply
pool. The sensitivity results in (25) and (26) are computed next: oT 1 ¼ ½a bða0 þ a1 D þ QB Þ ¼ 6:25; oQB C oT 1 r1 QB þ ¼ ða=bÞ ¼ 20:2: C ok ba1 Thus a rise in the expected output of the new source will pull the construction time forward. Also, in this example the market demand is large relative to the output of the new source, hence, a rise in the expected cost increment (k) will pull the construction time forward. The justification following Eq. (26) applies.
6. Concluding remarks Economic growth often reduces the efficiency of existing allocative policies toward pumping rights and brings many water management authorities to face the challenging problem of reallocating existing supplies and construction of new reservoirs. The problem is aggravated by increases in water cost when new reservoirs are added. Historically, there has been a strong and costly preference in part of policy-makers for construction of new supply projects over reallocation of existing supplies without enough attention to the crucial timing and reallocation issues. The expository model in this study provided a formalization of the delicate concepts and processes involved in the analysis of allocative and pricing policies, adjustment costs, and the timing (deferral) of supply augmentation projects. The dynamic optimally conditions led to a simultaneous solution for the optimal allocative policy and construction timing as well as a number of sensitivity results. Two fundamental implications follow: (i) It is clear that dynamic (periodic) adjustments in allocative policy are needed to accommodate economic growth. The nature of such adjustments is determined in part by the effect of economic growth on the existing policies. (ii) The timing of any supply augmentation project must be evaluated jointly with flexible allocative policy toward existing supplies. A result is that, when the expected cost increment emerging
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from the construction rises, the economy can tolerate a less costly allocative design that generates a lower degree of water availability prior to constructing a new supply project. In addition, to attain maximum efficiency, the optimal price over the periods prior to construction rises when there is a rise in the expected cost increment. Furthermore, the expected cost increments and output of the new project are the primary determinants of the optimal construction time. In particular, a rise in the expected output of the new project will pull the construction time forward. The impact of a rise in the expected cost increment depends mainly on the market demand size relative to the expected output of the new project. When the relative demand is large, a rise in the expected cost increment will pull the construction time forward. The study provides a basis for further formal modeling and investigation of the issues. As is typical of expository models, the present formulation may be extended or refined in a number of ways. For instance, tractability forced the use of linear approximations of some of the initial functions. It is clear that introduction of nonlinearity could generate results with more complicated forms than those generated above. For instance, if the initial functions are in exponential forms, then the corresponding logarithmic forms, which are linear, will be adopted and the stated steps will hold but the parameters would then be elasticities and the solutions would be in logarithmic form. Since logarithm is a monotonic function, the stated comparative static results continue to hold for the exponential case as well. However, there are nonlinearities that generate difficulties in arriving at analytical solutions and in those cases linear approximations and numerical solutions could be of utility. Furthermore, random components may be introduced into a number variables. For instance, randomness could be introduced into the output and demand functions via an additive random term with the mean zero and a given distribution or by a random multiplicative parameter with a given distribution. In both cases the analysis could be carried out with the expectation terms involving the properties of the underlying distributions. A more elaborate model may introduce a utility
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function for the decision-maker and work with the expected utility terms that incorporate the decision-makerÕs subjective risk aversion as well. Another feature of the present model is that its planning horizon is the pre-construction period where the construction time itself is endogenous. A possible extension may incorporate a post-construction setting. The present model, however, highlights some of the underlying issues and, within its context, provides certain guidelines for valuation and timing of supply augmentation projects. Optimal timing models could also generate valuable insight for most resources with controlled supply, e.g., fishery and forestry.
Acknowledgements The authors thank two anonymous referees for helpful comments.
References Anderson, T.L., 1983. Water Crisis: Ending the Policy Drought. Johns Hopkins University Press, Baltimore, MD. Blomquist, W., 1992. Dividing the Waters. Institute for Contemporary Studies Press, San Francisco. Chiang, A., 1992. Dynamic Optimization. McGraw-Hill, New York. Davis, P.A., 1992. An agency in transition. Congressional Quarterly Weekly Report 50 (March 7), 530. Gardner, B.D., 1987. Removing impediments to water markets. Journal of Soil and Water Conservation (November– December), 384–388. Griffin, R.C., Boadu, F.O., 1992. Water marketing in Texas opportunities for reform. Natural Resources Journal. Kamien, M., Schwartz, N., 1991. Dynamic Optimization, second ed. North-Holland, New York. Langfur, H., 1989. The two forks debate: Water in the west. Dissent Summer, 391–393. Miller, R., 1979. Dynamic Optimization and Economic Applications. McGraw-Hill, New York. Nelson, R.H., 1987. The economics profession and the making of public policy. Journal of Economic Literature March, 49–91. Sander, W., 1985. The state of economic analysis in water resources planning. American Journal of Economics and Sociology, 121–128. Schoolmaster, F.A., 1991. Water marketing and water rights transfers in the lower Rio Grande Valley, Texas. Professional Geographer, 292–304.
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Simpson, L.D., 1994. Are water markets a viable option. Finance and Development (June), 30–32. Smith, Z.A., 1989. Groundwater in the West. Academic Press, San Diego, CA. Spencer, L., 1992. Water: The WestÕs most misallocated resource. Forbes April 27, 68–74.
Wehmhoefer, R.A., 1989. Water law in the Southwest. In: Smith, Z. (Ed.), Water and the Future of the Southwest. University of New Mexico Press, Albuquerque, pp. 19– 46. Welsh, F., 1985. How to Create a Water Crisis. Johnson Publishing, Boulder, CO.
O.R. Applications
An optimal timing model of water reallocation and reservoir construction Fathali Firoozi *, John Merrifield Department of Economics, The University of Texas at San Antonio, San Antonio, TX 78249, USA
Abstract Growth of the users base often reduces the efficiency of existing water allocation policies and causes new water supplies to cost more than the value of water in existing use. Historically, water management authorities have shown a bias toward undertaking of new supply projects and delay of necessary reallocations. This study provides an analysis of the timing and adjustments associated with construction of new reservoirs and reallocation of existing supplies. A dynamic model yields simultaneously the optimal construction timing of new projects and guidelines for reallocation of existing supplies. A number of sensitivity results provide additional insights. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Optimization theory; Systems dynamics; Facilities planning and design; Natural resources
1. Introduction Water management authorities provide strict policy guidelines regarding allocation of pumping rights and the transfer of such rights between high value and low value users. 1 Such policy guidelines are socially and politically sensitive issues and play a dominant role in determining overall availability of water and its price to the consumer. The process implies that designing an allocation policy based on commonly accepted criteria is one of the crucial
* Corresponding author. Tel.: +1-210-458-5395; fax: +1-210458-5837. E-mail address: ffi[email protected] (F. Firoozi). 1 Classifications of water users include commercial, industrial, agricultural, residential, and those with property rights that have access to underground or nearby surface reservoirs. References include Smith (1989), Blomquist (1992), and Spencer (1992).
tasks of the authorities. Meanwhile, economic growth often brings expansion and redistribution of various user groups and thus reduces the efficiency of existing water allocation policies and creates the problem of identifying an optimal balance between reallocation of existing supplies and construction of new supply projects. The issue is complicated by the fact that water from new projects often costs more than the value of water in many existing uses. In addition, policy-makers have many competing concerns and redistribution of existing water supplies appears to be one of the lesser priorities. A reallocation of a basic commodity like water may cause numerous social, legal, cultural, and equity changes with significant economic and political adjustment costs to water management authorities. Those costs are often deferred by incurring the cost of additional water supply projects. If water allocation policies are changed so that users bear a larger fraction of the
0377-2217/03/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 2 ) 0 0 1 6 0 - 1
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social marginal cost of water use, society incurs adjustment costs sooner and construction costs later. The dilemma is to find the delicate balance in that trade-off. If the demand grows across the board at a constant rate, then a reallocation may not be needed but the implications for the construction timing will remain. The efficiency of existing allocation systems and the timing of new supply projects have recently received a closer but still sporadic attention in the literature. Many analysts have cited construction of noneconomical and premature water projects and called for reallocations of existing supplies before undertaking new supply projects. 2 In addition, recent increases in construction and utilization costs and heightened environmental consciousness have provided stimuli for a more serious consideration of reallocation. However, state and local authorities continue to propose new water supply projects with unit costs well above the value of water in major existing uses. This is despite the fact that economic growth will raise the dilemma again in the near future. Economic efficiency was not a major issue even in the few recent cases where public referenda forced abandonment of proposed water projects like CaliforniaÕs Peripheral Canal and TexasÕ Applewhite Reservoirs, or when the Environmental Protection Agency blocked DenverÕs Tow Forks reservoir. Only recently the federal government has begun to pay a closer attention to reallocation as an alternative to new construction (Davis, 1992). Meanwhile, many costly and premature projects have been built (Davis, 1992; Sander, 1985). It is clear that the problems of construction timing and reallocation are closely interrelated and any decision regarding one must incorporate action toward the other. The stated issues point to the need for a formal analysis of policy toward construction tuning based on a socially optimal criterion when reallocation is also an endogenous policy choice. The present study provides such an analysis.
2 The references include Anderson (1983), Sander (1985), Welsh (1985), Gardner (1987), Langfur (1989), Schoolmaster (1991), Blomquist (1992), Griffin and Boadu (1992), Spencer (1992), and Simpson (1994).
A question may arise as to why there has been such a strong and costly preference in part of policy-makers for construction of new supply projects over reallocation of existing supplies, even when a reallocation could be on a willing buyer– willing seller or market basis. Many causes have been cited, including votersÕ insufficient information, poor economic analysis skills at regional and local home offices of government agencies (Sander, 1985), pursuit of legislative pork and log-rolling (Gardner, 1987), rent-seeking by empire-building officials (Sander, 1985; Langfur, 1989), and various adjustment costs. Gardner (1987) and Griffin and Boadu (1992) suggested that changes in the pattern of water use often threaten the wealth positions of existing water users and indirect beneficiaries. Such adjustment costs, including political concerns, must be taken into account in determination of optimal efforts to reallocate existing supplies and/or construct new supply sources (Nelson, 1987; Wehmhoefer, 1989). The focus of this study is on optimal construction timing (deferral) of new water supply projects and simultaneous reallocation of existing supplies in a dynamic setting where economic growth reduces the efficiency of existing distributive policies. The optimality will be with respect to net social benefits derived from consumption and costs. The existing dynamic models in the resources literature contain two shortcomings in relation to the stated problem of reallocation and construction; specifically, the existing models (i) do not address the issue of construction timing in conjunction with the reallocation issue and (ii) do not explicitly incorporate the adverse effect of a fixed policy in the presence of growth and the policy adjustment costs into the analysis. Section 2 develops an expository model that resolves the stated shortcomings and incorporates the key factors related to the issue. The solution and its implications for allocative policy and pricing adjustments are discussed in Section 3. The focus of Section 4 is on optimal construction timing in the presence of variable allocative policy. The simultaneous solution and its sensitivity results highlight some interesting aspects of the issue and provide certain guidelines for optimal construction timing of new supply projects as well as optimal reallocations of existing
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supplies. A numerical example in Section 5 shows an application, and some concluding remarks appear in Section 6.
167
by economic growth, e.g., the higher cost of implementing a fixed allocative policy to a larger consumer base. 4 2.1. Supply and user cost
2. The model This section develops an expository formulation of the problem. The key factors are incorporated in a dynamic setting so that the timing of construction and policy toward reallocation emerge as the primary decision variables. Suppose that the water authority (the planner) has a number of allocation policy options to choose from. A measure G of these policy options is a positive number such that a higher value for G reflects a policy option that is more costly for the water authority to implement but increases the availability and access for the consumers. For instance, a flexible allocation policy (G1 ) that continuously reallocates the existing water in response to changing demand from various user groups is more costly to implement than a fixed allocation policy (G2 ), thus G2 < G1 . The relative value of G reflects the nature of allocative policy adopted by the water authority. Let Z represent the administrative costs incurred by the water authority. This cost is mainly a function of two variables, namely, the existing allocative policy and the economic growth that imposes additional cost of implementing the existing allocative policy to a growing consumer base. Thus the administrative or policy adjustment costs at time t is estimated by ZðtÞ ¼ c0 þ c1 G þ c2 t
ð1Þ
where c0 , c1 and c2 are the positive parameters of Z. The units of c1 and G are defined such that c1 G reflects the administrative cost of policy option G. 3 Thus, any arbitrary adjustment in the units of G will adjust the units of the coefficient c1 so that the role of c1 G remains unchanged. Also, c2 > 0 reflects the rise in the administrative cost imposed
3 Linear approximations often allow tractability and analytical solutions. Even with nonlinear initial functions, numerical solutions utilize linear approximations (Chiang, 1992; Kamien and Schwartz, 1991).
Let the aggregate of existing water sources be denoted by A, and let QA denote the market availability of water from source A to consumers, i.e., the amount of water available to consumers under a given allocative policy G and a given economic state. 5 Under the stated setting, economic growth reduces the efficiency and a given allocative policy so that at a given G, the availability QA falls over time. The value of QA is thus approximated by QA ðtÞ ¼ a0 þ a1 G a2 t
ð2Þ
where a0 , a1 and a2 are the positive parameters of QA . Note that a rise in G increases the administrative cost Z via (1) but will also increase in the water availability QA to consumers via (2). It is clear that a1 G measures the impact of a given allocative policy G on the water quantity QA from the fixed source A available to the consumers. Thus, any arbitrary change in the units of G will adjust the units of the coefficient a1 so that the role of a1 G remains unchanged. The parameter a2 > 0 with negative sign in (2) reflects the adverse effect of economic growth on the availability of water to consumers at a given allocative policy G. The cost of market availability of water from source A to consumers has two components, the usual production cost and the allocative cost. This function is assumed to take the form CA ðtÞ ¼ b0 þ b1 QA ðtÞ þ ZðtÞ
4
ð3Þ
Instead of time, one may use in (1) indicators such as GDP(t) to reflect economic growth. If one assumes that GDP grows over time at an average rate of c2 , then (1) reflects this approximation. (1) allows flexibility to incorporate any indicator with the growth rate of c2 . 5 A generalization of the model to the case of multiple existing sources can be given as follows. Replace QA with P P QA ¼ ai Qi and G ¼ ai Gi . The analysis will apply exactly with the additive terms for QA and G treated as vector variables.
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where b0 > 0 and b1 > 0 are the usual cost parameters (assuming all costs are in real terms). Eq. (3) reflects the fact that under a fixed allocative policy (fixed G), the average cost of water from source A, denoted by ACA , rises over time as the economy grows. This is shown by the division of (3) by QA :
where:
‘‘optimal’’ construction time T, we assume that all future benefits and costs of the new source beyond time T are approximated and reflected on the estimated quantities QB and k. This assumption also allows a finite-horizon statement of the problem with the following optimality criterion. The authorityÕs objective is to determine the construction time T and the allocative policy GðtÞ such that the net social benefits emerging from aggregate consumption and cost over the period [0; T ] are maximized. Although the problem could be formulated in a control-theoretic setting, for the purpose of identifying certain analytical characteristics of the solution it is helpful to choose the classical setting and treat the problem as one in variational calculus. 6 To provide a formal statement of the problem within the classical setting, assume that the market demand for water is approximated by the simple form
r0 ðkÞ :¼ b0 þ b1 ðkÞa0 þ c0 ;
P ðQÞ ¼ a bQ
r1 ðkÞ :¼ b1 ðkÞa1 þ c1 ;
where a > 0 and b > 0 are the demand parameters, P is the market price, and QðtÞ is the consumption path. The net social benefit (welfare) in period t is estimated by the total consumer surplus net of the total costs: Z QA N ðtÞ ¼ P ðxÞ dx CA ðtÞ 0 b 2 ¼ aQA ðtÞ ½QA ðtÞ CA ðtÞ 2
ACA ðtÞ ¼ b1 þ
b0 þ ZðtÞ QA ðtÞ
where, from Eqs. (1) and (2), Z rises and QA falls over time (under a fixed G). Any exogenous rise in the average cost of water can be introduced by a rise in b1 . Hence, write: b1 ðkÞ ¼ b1 þ k, where k P 0 represents an exogenous rise in the average cost. Substituting (1) and (2) for ZðtÞ and QA ðtÞ, Eq. (3) can be written as CA ðtÞ ¼ r0 ðkÞ þ r1 ðkÞG þ r2 ðkÞt
ð4Þ
r2 ðkÞ :¼ c2 b1 ðkÞa2 : As stated above, in the presence of positive exogenous increment in the average cost ðk > 0Þ, b1 ðkÞ ¼ b1 þ k. For simplicity (or when k ¼ 0) the argument k is made an implicit argument of b1 until the discussions in Sections 3 and 4 where the introduction of new supplies and the crucial role of k will be discussed. As for the signs of the parameters in (4), it is clear that r0 and r1 are positive. Also, assuming that economic growth increases the user cost of water from the fixed source A at a given allocative policy G, the condition r2 > 0 is adopted. 2.2. Problem statement At a future time T (unknown) a new water source B can be constructed, which is expected to provide the water quantity QB . The construction, however, is expected to raise the average cost of water by an estimated k > 0. To focus on the essence of our argument regarding the ‘‘optimal’’ allocative policy GðtÞ over the period [0; T ] and the
where QA ¼ QA ðG; tÞ and CA ¼ CA ðG; tÞ are those specified in (2) and (4). The problem is stated formally as Z T F ðG; tÞ dt ð5Þ MaxV ¼ G;T
0
subject to the terminal constraints QðT Þ ¼ QA ðG; T Þ þ QB : At time t ¼ T : b1 ðkÞ ¼ b1 þ k;
ð6Þ for k > 0
ð7Þ
where
6 The standard references include Miller (1979), Kamien and Schwartz (1991), and Chiang (1992).
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F ðG; tÞ ¼ N ðtÞeqt b 2 ¼ aQðG; tÞ ½QðG; tÞ CðG; tÞ eqt : 2 ð8Þ Q and C are specified in (2) and (4), and q is the time discount rate. The next two sections discuss the simultaneous solutions for the optimal allocative policy path, pricing, and construction timing of the new supply project.
3. Optimal allocation and pricing The solution and its policy implications regarding optimal allocation and pricing of existing supplies are examined in this section. The stated problem is a variable-terminal point problem in variational calculus. The necessary conditions for an optimum consist of the Euler equation and the transversality condition: FG
d FG0 ¼ 0; dt
½F G0 FG0 ¼ 0; where FG ¼ oF =oG, G0 ¼ dG=dt, and FG0 ¼ oF = oG0 . Since FG0 ¼ 0, the two conditions reduce to FG ¼ 0;
ð9Þ
½F t¼T ¼ 0:
ð10Þ
The procedure is to solve (9) and (10) simultaneously for G and T. Note that FG ¼ aðdQ=dGÞ bQðG; tÞðdQ=dGÞ ðdC=dGÞ ¼ aa1 bða0 þ a1 G a2 tÞa1 r1 ðkÞ: Utilizing the above statement in (9) and solving for G yield the optimal path of the allocative policy to be implemented by the water authority: aa1 ba0 a1 r1 ðkÞ a2 G ðk; tÞ ¼ þ t: a1 ba21
Then G ðk; tÞ ¼ DðkÞ þ
a2 t: a1
169
ð12Þ
As indicated earlier, G ðk; tÞ provides a guide for the time path of allocative policy adjustments over the time period [0; T ] such that the net social benefits with respect to consumption and costs are maximized. Four important policy implications follow from the results: (i) The stated solution G ðtÞ in (12) is independent of initial point, i.e., the initial optimal policy measure G ð0Þ may not satisfy the observed initial value Gð0Þ ¼ G0 . This aspect of the solution emerges from the fact that FG0 ¼ 0; neither the Euler Eq. (9) nor the transversality condition (10) involve an arbitrary constant to be determined by the initial condition Gð0Þ ¼ G0 . 7 According to (12), the initial allocative policy must be such that it generates the policy level G ðk; 0Þ ¼ DðkÞ. If this is not the case, then a policy adjustment that yields the level D for G must be implemented in the initial period. (ii) The optimality result in (12) implies that the policy variable G must rise over tune at the rate a2 =a1 . In other words, dynamic allocative policy adjustments are needed to accommodate economic growth. (iii) By the statements before and after (4), dr1 =dk ¼ a1 ðdb1 =dkÞ ¼ a1 . Therefore, dDðkÞ 1 dr1 ðkÞ 1 ¼ 2 < 0: ð13Þ ¼ dk ba1 dk ba1 Hence, oG =ok ¼ dD=dk < 0. This implies an interesting result: the economy can tolerate a less costly allocative design and lower water availability to users (smaller G ) before the construction time when the expected increment in average water cost emerging from the construction rises (larger k). The effect of a rise in k on the optimal construction time will be discussed in the next section.
Also, let DðkÞ :¼
aa1 ba0 a1 r1 ðkÞ : ba21
ð11Þ
7 A similar independence issue is discussed in Chiang (1992, pp. 42, 149–150).
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(iv) An important corollary to the solution in (12) is the identification of a complementary pricing policy. Applying (12) in (2) yields
Pðk; QB Þ :¼ aQðk; T Þ ðb=2Þ½Qðk; T Þ ¼ a½a0 þ a1 DðkÞ þ QB
2
ðb=2Þ½a0 þ a1 DðkÞ þ QB :
QA ðtÞ ¼ a0 þ a1 ½DðkÞ þ ða2 =a1 Þt a2 t ¼ a0 þ a1 DðkÞ: Using the above quantity in the demand function leads to the optimal price: P ðkÞ ¼ a bQA ¼ a b½a0 þ a1 DðkÞ: An application of the definition in (11) reduces the above optimal price to P ðkÞ ¼
r1 ðkÞ c ¼ b1 ðkÞ þ 1 a1 a1
ð14Þ
where the last equality emerges from the definition following Eq. (4). It is clear from (12) that the distributive policy G must change over time to adjust for the growth. However, (14) shows that the corresponding market price has a steady state for a given k. Note that a rise in the expected cost increment emerging from the new project (a rise in k) will increase the optimal price P over the period [0; T ] prior to the construction time.
ð16Þ
The equation prior to (5) shows that the statement in (16) is the consumer surplus before subtraction of the costs. The following features of P will be utilized in later analysis. Note that oP ¼ aa1 ba1 ða0 þ a1 D þ QB Þ: oD Hence, utilizing (13), oP oP dD ¼ ok oD dk ¼ ½aa1 ba1 ða0 þ a1 D þ QB Þ½1=ðba1 Þ ¼ ða0 þ a1 D þ QB Þ ða=bÞ:
ð17Þ
Similarly, oP ¼ a bða0 þ a1 D þ QB Þ: oQB
ð18Þ
The terminal cost is characterized by applying (12) in (4): Cðk; T Þ ¼ CA ðk; T Þ ¼ r0 ðkÞ þ r1 ðkÞG ðk; T Þ þ r2 ðkÞT
4. Optimal construction time The central issue of construction timing (deferral) is studied in this section. The optimal construction time and its sensitivity to the predicted output and cost of the new water supply project will be examined. Utilizing (12) and the terminal conditions (6) and (7), the transversality condition (10) can be solved for the optimal construction time (T ) of the new water source B. The terminal components of F in (8) are evaluated first. With an application of (2) the terminal condition (6) can be written as Qðk; T Þ ¼ QA ðk; T Þ þ QB
¼ ½r0 ðkÞ þ r1 ðkÞDk þ ½r1 ðkÞða2 =a1 Þ þ r2 ðkÞT :
ð19Þ
Define: RðkÞ :¼ r0 ðkÞ þ r1 ðkÞDðkÞ;
ð20Þ
CðkÞ :¼ r1 ðkÞða2 =a1 Þ þ r2 ðkÞ;
ð21Þ
where C > 0. Then, Cðk; T Þ ¼ RðkÞ þ CðkÞT . The following features of C and R will be utilized. The definitions of r1 ðkÞ and r2 ðkÞ given after (4) show that CðkÞ ¼ r1 ðkÞða2 =a1 Þ þ r2 ðkÞ ¼ ½b1 ðkÞa1 þ c1 ða2 =a1 Þ þ ½c2 b1 ðkÞa2
¼ a0 þ a1 G ðk; T Þ a2 T þ QB ¼ a0 þ a1 DðkÞ þ QB
2
¼ c1 ða2 =a1 Þ þ c2 ; ð15Þ
where the last equality follows from an application of (12). Define
hence, CðkÞ is in fact independent of k, CðkÞ ¼ C. Also, the definitions of r0 ðkÞ and r1 ðkÞ given following (4) and the statement in (13) imply
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dR dr0 dD dr1 ¼ þD þ r1 dk dk dk dk db1 1 db1 þ r1 ¼ a0 þ D a1 ba1 dk dk r1 ¼ ða0 þ Da1 Þ ba1
ð22Þ
where db1 ðkÞ=dk ¼ 1 follows from the earlier definition b1 ðkÞ ¼ b1 þ k. Applications of the above definitions in (8) lead to ½F t¼T ¼ F ðT Þ 2
¼ faQðk; T Þ ðb=2Þ½Qðk; T Þ Cðk; T ÞgeqT ¼ fPðk; QB Þ RðkÞ CT geqT : ð23Þ Utilizing (23), the transversality condition ½F t¼T ¼ 0 stated in (10) yields two solutions for the optimal construction time T , one infinite and one finite. The infinite solution T ¼ 1 (i.e., infinite postponement of the project) emerges from setting eqT ¼ 0. But only the finite solution is feasible. To see this, note that the optimal time path of the adjustment cost Z is identified by an application of (12) in (1): Z ðk; tÞ ¼ ½c0 þ c1 DðkÞ þ ½c1 ða2 =a1 Þ þ c2 t: It is clear that Z ! 1 as t ! 1. Hence, unless the authority has access to infinite budget over time, the solution T ¼ 1 is not feasible. The finite solution for the optimal construction time emerges from setting the term inside the braces in (23) equal to zero and solving for T: T ðk; QB Þ ¼
1 ½Pðk; QB Þ RðkÞ: C
ð24Þ
Eq. (24) characterizes the optimal construction time (T ) as a function of the output and cost increment (QB , k) that are expected to emerge from the new water source B. The sensitivity of the optimal construction time to the predicted output and cost increment of the new project can now be evaluated. Utilizing the relation in (18), the result in (24) yields oT 1 oP 1 ¼ ¼ ½a bða0 þ a1 D þ QB Þ C oQB C oQB ð25Þ
171
where C is positive. An evaluation of the term in the brackets shows that the derivative in (25) is negative if QB > ð1=bÞ½b1 þ ðc1 =a1 Þ, and is nonnegative if ‘‘>’’ is replaced by ‘‘ 6 ’’. Thus for a sufficiently large expected output from the new source B, a rise in QB will pull the optimal construction time T forward. Next, the results in (17) and (22) are utilized in the following evaluation of (24): oT 1 oP oR ¼ C ok ok ok 1 ¼ f½a0 þ a1 D þ QB ða=bÞ C ½ða0 þ Da1 Þ r1 =ðba1 Þg 1 r1 QB þ ða=bÞ : ð26Þ ¼ C ba1 It is clear that the sign of the term in (26) depends essentially on the size of the market demand ða; bÞ relative to the expected output of the new source (QB ). In the case where the market demand is relatively large (large ‘‘a’’), the sign is negative and hence a rise in the expected increment in average cost emerging from the construction (a rise in k) will pull the construction time forward (a fall in T ). An intuitive justification can be given along the following lines. As elaborated in the last section, a rise in k will raise the optimal price in period T , which generates a reduction in the consumer surplus over that period. Due to the large size of the demand, this fall in the consumer surplus is relatively large, which justifies the pulling forward of the construction time. Similarly, when the market demand is relatively small, the sign in (26) is positive and hence a rise in k will postpone the construction time.
5. A numerical example The numerical example in this section shows an application of the stated results. For Eqs. (1)–(3), the following parametric values are adopted: c0 ¼ 120; c1 ¼ 3; c2 ¼ 6; a0 ¼ 0; a1 ¼ 10; a2 ¼ 5; b0 ¼ 0; b1 ¼ 0:8:
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In the terminal period T (when the new source will be added), b1 ¼ 0:8 þ k with the expected average cost increment k ¼ 0:2 (k ¼ 0 for the periods prior to T ) where the expected output of the new source is QB ¼ 80. The market demand for water has the parameters a ¼ 140; b ¼ 0:6: The parameters of Eq. (4) can now be computed via those of Eqs. (1)–(3): r0 ¼ b0 þ b1 a0 þ c0 ¼ 120; r1 ¼ b1 a1 þ c1 ¼ 11; r2 ¼ c2 b1 a2 ¼ 2: The optimal price is now computed via Eq. (14): c c P ¼ b1 ðkÞ þ 1 ¼ ðb1 þ kÞ þ 1 a1 a1 ¼ f1:1 for t < T ; 1:3 for t ¼ T g: For the period prior to the optimal construction time T (when k ¼ 0) the parameter D in (11) is computed via the above values, resulting in D ¼ 23:15. Thus, the optimal allocative policy path prior to the construction timer T is computed via (12): G ðtÞ ¼ D þ ða2 =a1 Þt ¼ 23:15 þ 0:5t: In order to compute the optimal construction time, the necessary parameters are first computed via (16), (20), (21) with k ¼ 0: R ¼ r0 þ r1 D ¼ 374:65; C ¼ r1 ða2 =a1 Þ þ r2 ¼ 7:5; a0 þ a1 D þ QB ¼ 311:5; P ¼ a½a0 þ a1 D þ QB ðb=2Þ½a0 þ a1 D þ QB
2
¼ 14500:325: The optimal construction time is then computed via (24): T ¼
1 ðP RÞ ¼ 1883:42: C
The above value for T represents the optimal number of time units (time intervals, periods) to pass before the new source is added to the supply
pool. The sensitivity results in (25) and (26) are computed next: oT 1 ¼ ½a bða0 þ a1 D þ QB Þ ¼ 6:25; oQB C oT 1 r1 QB þ ¼ ða=bÞ ¼ 20:2: C ok ba1 Thus a rise in the expected output of the new source will pull the construction time forward. Also, in this example the market demand is large relative to the output of the new source, hence, a rise in the expected cost increment (k) will pull the construction time forward. The justification following Eq. (26) applies.
6. Concluding remarks Economic growth often reduces the efficiency of existing allocative policies toward pumping rights and brings many water management authorities to face the challenging problem of reallocating existing supplies and construction of new reservoirs. The problem is aggravated by increases in water cost when new reservoirs are added. Historically, there has been a strong and costly preference in part of policy-makers for construction of new supply projects over reallocation of existing supplies without enough attention to the crucial timing and reallocation issues. The expository model in this study provided a formalization of the delicate concepts and processes involved in the analysis of allocative and pricing policies, adjustment costs, and the timing (deferral) of supply augmentation projects. The dynamic optimally conditions led to a simultaneous solution for the optimal allocative policy and construction timing as well as a number of sensitivity results. Two fundamental implications follow: (i) It is clear that dynamic (periodic) adjustments in allocative policy are needed to accommodate economic growth. The nature of such adjustments is determined in part by the effect of economic growth on the existing policies. (ii) The timing of any supply augmentation project must be evaluated jointly with flexible allocative policy toward existing supplies. A result is that, when the expected cost increment emerging
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from the construction rises, the economy can tolerate a less costly allocative design that generates a lower degree of water availability prior to constructing a new supply project. In addition, to attain maximum efficiency, the optimal price over the periods prior to construction rises when there is a rise in the expected cost increment. Furthermore, the expected cost increments and output of the new project are the primary determinants of the optimal construction time. In particular, a rise in the expected output of the new project will pull the construction time forward. The impact of a rise in the expected cost increment depends mainly on the market demand size relative to the expected output of the new project. When the relative demand is large, a rise in the expected cost increment will pull the construction time forward. The study provides a basis for further formal modeling and investigation of the issues. As is typical of expository models, the present formulation may be extended or refined in a number of ways. For instance, tractability forced the use of linear approximations of some of the initial functions. It is clear that introduction of nonlinearity could generate results with more complicated forms than those generated above. For instance, if the initial functions are in exponential forms, then the corresponding logarithmic forms, which are linear, will be adopted and the stated steps will hold but the parameters would then be elasticities and the solutions would be in logarithmic form. Since logarithm is a monotonic function, the stated comparative static results continue to hold for the exponential case as well. However, there are nonlinearities that generate difficulties in arriving at analytical solutions and in those cases linear approximations and numerical solutions could be of utility. Furthermore, random components may be introduced into a number variables. For instance, randomness could be introduced into the output and demand functions via an additive random term with the mean zero and a given distribution or by a random multiplicative parameter with a given distribution. In both cases the analysis could be carried out with the expectation terms involving the properties of the underlying distributions. A more elaborate model may introduce a utility
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function for the decision-maker and work with the expected utility terms that incorporate the decision-makerÕs subjective risk aversion as well. Another feature of the present model is that its planning horizon is the pre-construction period where the construction time itself is endogenous. A possible extension may incorporate a post-construction setting. The present model, however, highlights some of the underlying issues and, within its context, provides certain guidelines for valuation and timing of supply augmentation projects. Optimal timing models could also generate valuable insight for most resources with controlled supply, e.g., fishery and forestry.
Acknowledgements The authors thank two anonymous referees for helpful comments.
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