trans-boundary water projects and political uncertainty

16

TRANS-BOUNDARY WATER PROJECTS AND POLITICAL UNCERTAINTY Yacov Tsur Hebrew University of Jerusalem, Israel and University of Minnesota, USA Amos Zemel Ben Gurion University of the Negev, Israel

We investigate effects of political uncertainty on economic viability and time profile of international water development projects. The political uncertainty considered is manifested in the form of sudden, discrete events that upon occurrence may irreversibly damage or terminate the project. Occurrence risk is treated as an exogenous hazard, increasing the effective discount rate. One class of events may occur before or after the project completion date. Other projects are exposed to risk only until completion, following which the benefits are guaranteed. This distinction bears important consequences regarding the project’s time profile (duration) and whether or not it should be undertaken.

1. INTRODUCTION Joint exploitation of a stock of natural resource, such as a water basin, by several nations (states or provinces) typically involves considerations that are not restricted to the economic and technical aspects of the project. If all the parties involved are rational, i.e., base their decisions on economic costs-benefit criteria, and when no uncertainty prevails, an optimal development strategy meeting the needs and interests of the parties can be worked out in a rather straightforward (though far from simple) fashion. The situation is different if the normal development of the project may be interrupted at any given time as sudden changes in the political atmosphere bring non-economic considerations to the focus of attention. Often, the interruption decision of one party involves uncertainty, e.g., when it is instigated by political instability internal to the interrupting state. If development of a resource requires little or no initial investment, the presence of interruption risk may not be of much importance as not much is at stake. However, a development program typically involves considerable initial investment,

2

EXPANDING SCOPE OF TRANS-BOUNDARY PROBLEMS

and the interruption risk can easily tilt the decision against the project. It is this latter situation that we investigate in this work. Trans-boundary water projects appear to be the natural approach when political and natural resource boundaries cross each other. For a large number of arid and semi-arid countries, whose in-house resources are nearly exhausted, international water bodies are the only major new source of water which can yet be economically developed (Biswas, 1994). The Middle East is abundant with examples of potential trans-boundary development projects, including the integrated development of the Nile basin (Mageed, 1994) and the resolution of the long standing Iraqi-Syrian-Turkish water dispute (Kolars, 1994; Bilen, 1994). The development of joint Jordanian-Israeli water projects currently appears promising, since these issues are an important part of the peace treaty recently signed between the two nations. At different times, the construction of large-scale desalination plants along the Arava valley as well as the development of a desalination-hydroelectricity co-generation plant in the Dead Sea basin have been proposed (Murakami and Musiake, 1994). Clearly, an acceptable sharing of the scarce water supplies will become a major issue on the Israeli-Palestinian agenda. Indeed, the various plans addressing exploitation of the Jordan River water (including Lebanon and Syria, as well as Jordan, Israel and the West Bank) have marked the development of Israeli-Arab relations from their early stages (Wolf, 1994; Murakami and Musiake, 1994). Many examples from other regions around the Globe are listed in the works compiled by Dinar and Loehman (1995), while the Center for Natural Resources, Energy and Transportation (CNRET) Register lists as many as 214 international river and lake basins (CNRET, 1978). In most cases, joint projects are not favorably viewed by some sectors of the population and of the political establishments. Rather, water resources are seen as national assets, not to be shared with neighbors/rivals. Just as the mutual benefit derived from joint projects is believed to push forward the political processes of peace and reconciliation, hampering these projects can play to the hands of those who disapprove of these processes. It is this tension that underlies the uncertainty we wish to analyze. These potential projects share common features in that they (i) involve two or more states, i.e., are trans-boundary, (ii) require considerable initial investment, (iii) require construction over a prolonged period of time (before exploitation begins), (iv) can be interrupted due to conditions which are exogenous to the project’s costs and benefits, and (v) can be interrupted by events that are largely uncertain at the time the decision to undertake the project is made. Not surprisingly, interruption uncertainty often casts doubt on prospects for trans-boundary development of natural resources which otherwise pass the cost-benefit criterion. While the reduction in a project’s valuation appears intuitively obvious, we wish to draw attention also to the temporal aspects of the problem. How should the investment scheduling respond to the interruption risk? On the one hand, delayed investments decrease the damage in case the project is abandoned. On the other hand, the benefits are also delayed and, under some scenarios, early completion signals the end of the risky period. It is our purpose here to describe precisely the manifestation of these constraints. Understanding

TRANS-BOUNDARY PROJECTS UNDER POLITICAL UNCERTAINTY

3

them is useful for the design of arrangements that mitigate the impeding effects of political uncertainty.

2.

THE INVESTMENT PROBLEM

In this section, we consider the investment policy project owners should adopt in order to best derive the advantages offered by the resource available to them. Rather than analyze a specific project in detail, we construct the simplest model that incorporates the uncertainty and temporal aspects we wish to highlight. The model thus admits analytical solutions, yet is rich enough to illustrate the delicate effects of uncertainty on investment scheduling—in particular on project duration. The status of the project is defined in terms of the state variable Nt. The project is complete and begins to bear fruits when its state reaches the level N , at which time no further investment is required and the project owners enjoy a stream of revenues at constant rate B. The project construction progress depends on the investment rate Rt via the production function y(R) according to dN/dt = y(R) .

(1)

The function y is increasing, concave and bounded of finite initial slope:

y( 0) = 0, y' ( R) > 0, y" ( R) < 0, y( ∞ ) = y < ∞ and y' ( 0 ) < ∞ . Thus, the −1 inverse function Q( z ) ≡ y ' ( z ) is well defined and decreasing over [0, y ' ( 0 )], with Q(y' ( 0 )) = 0 and Q( 0 ) = ∞.

When it is certain that the project can proceed uninterrupted until fruition, project planning involves determining the investment profile Rt , t ∈ [0, T ] , and the completion date T according to T

V c (ψ , N ) = Max { Rt },T {∫ − Rt e − ρt dt + e − ρT ψ }

(2)

0

Rt ≥ 0, N 0 = 0, N T = N (or Rt ≡ 0 and T = ∞). In (2), ρ is the discount rate and ψ = B / ρ is the project’s value at completion time. The choice ( Rt = 0 for all t and T = ∞) corresponds to the decision not to undertake

subject to (1),

the project which is adopted when ψ is too low to justify the investment required to complete the project. This simple structure permits solving the problem in a closed form, yielding an explicit expression for the optimal investment rate. This task is carried out elsewhere (Tsur and Zemel, 1997) and the derivation is appended for convenience (Appendix A). We list the salient properties required for our discussion:

ρ be defined over [0, y ' ( 0 )] and let zψ be the unique root of the equation W ( zψ ) = ψ [or ρW ( zψ ) = B ].

(P1) Let W ( z ) ≡ [ y ( Q( z )) / z − Q( z )] /

4


Then, the benefit rate B affects the solution through the parameter zψ, which

N. (P2) The completion level N determines the project duration T according to the is independent of T

end-constraint

∫ y(Q(z

ψ

e ρt ))dt = N .

0

(P3) The project value is given by

V c (ψ , N ) = W ( zψ e ρT ).

V c (ψ , N ) increases with ψ and decreases with N . (P5) The project duration T increases with N and decreases with ψ. (P6) With any completion level N there is associated a critical value ψ1 such that (P4) The value

the project should not be undertaken for ψ < ψ. The corresponding critical duration is given by Tmax = log[ y '( 0 ) / zψ ] / ρ. For fixed values of ψ2

(and varying values of profitable project.

N ), Tmax is the maximum possible duration for a

3. INTERRUPTION RISK Suppose now that the region is under a constant threat of some hostilities which force project abandonment and effectively render all previous investments worthless. We refer to this kind of interruption as the event. The event occurrence conditions are exogenous and can be viewed, from the planner’s point of view, as random. In general, the occurrence hazard can depend on the status of the project. It is expedient, however, to restrict this dependence to two extreme scenarios. In the permanent-risk scenario, the hazard is completely blind to the project’s status and the event can occur both before and after the project’s completion—with the same devastating result of terminating the project construction (if it has not been completed yet) and eliminating all future benefits. At the other extreme, occurrence hazard holds only during the construction period and, once completed, the project’s benefit ψ is guaranteed. In this case, the event is influential only as long as N < N . This situation is denoted as temporary-risk. Under both scenarios, the planners must take into account the hazard of an untimely irreversible termination of the investment problem. Yet, the distinction between durations of the risk periods turns out to be important to the effect of occurrence uncertainty on optimal investment policies. Let us denote by λt the occurrence hazard rate, representing the probability that the event will immediately occur following time t given that it has not occurred at or before time t. When λ is constant (independent of t), the event occurrence time Te s

is distributed according to Pr{T >s} = exp{ − e

∫ λdt } 3= e 0

− λs

with the associated


5

− λs

density λ e . For a project planned to be completed by time T, the expected investment cost is T

s

T

T

0

0

0

0

− ρt − ρt − ( ρ +λ )t − λs − λT dt. ∫ λe { ∫ Rt e dt }ds + e ∫ Rt e dt = ∫ Rt e

Similarly, the expected benefit is decreased by the probability e

− λT

that the event

−( ρ+λ )T − λT − ρT will not occur until T, yielding e e ψ = e ψ . However, the relevant value of ψ differs depending on the specification of the post-completion hazard. In the permanent-risk scenario ψ = B / ( ρ + λ ), whereas in the temporary-risk case the benefit is not affected by the risk and ψ = B / ρ . For both scenarios, we see

that the optimization problem under uncertainty reduces to that under certainty, with ρ replaced by ( ρ + λ ) as the effective discount rate. Next, we investigate the implications of this difference on the optimal project duration as well as on its value.

4. PROJECT VALUE AND DURATION UNDER UNCERTAINTY Let the superscripts “c” and “uc” relate to the various quantities associated with the certainty and uncertainty problems, respectively. We formulate the uncertainty

ρ uc = ρ + λ , ψuc and N uc as a certainty equivalent c c c problem (with ρ = ρ and modified values for ψ and N ). This can be done

problem with the parameters

because increasing the discount rate is equivalent to rescaling the time variable τ = t( ρ + λ ) / ρ , T

V (ψ , N ) = Max{ Rt },T { ∫ − Rt e − ( ρ + λ )t dt + e − ( ρ + λ ) Tψ uc } = uc

uc

uc

0

Tc

c ρ − Rτc e − ρτ dτ + e − ρT ψ uc } , = 45 Max{ R c }, T c { ∫ τ ρ+λ 0

subject to (1), Rt ≥ 0,

N o = 0,

NT = N uc (or Rt ≡ 0 and T = ∞), where

Rτc = Rt ; and T c = T ( ρ + λ ) / ρ.

(3)

Observe that (1) is written in terms of the original time variable t. Transforming the time variable to τ entails a similar transformation on the state variable,

6


N c = N ( ρ + λ ) / ρ, to ensure that dN c / dτ = dN / dt = y( Rτc ). Expressed in terms of the scaled variables, the other constraints are Rτ ≥ 0, c

N 0c = 0, and

N Tc c = N c . Setting ψ c = ψ uc ( ρ + λ ) / ρ, we find V uc (ψ uc , N uc ) =

(4)

Tc

c ρ ρ V c (ψ c , N c ) Max { Rc },T c 6 { ∫ − Rτc e − ρτ dτ + e − ρT ψ c } = τ ρ + λ ρ+λ 0

The last equality follows because the latter optimization problem is formulated in terms of the “certainty” rate ρ and is therefore equivalent to the certainty problem (with the “scaled” benefit and completion level). Thus,

ρ ucV uc (ψ uc , N uc ) = ρ cV c (ψ c , N c ).

(5)

This scaling relation between the uncertainty and certainty value functions induces a similar relation between the corresponding optimal completion times. Denoting the optimal value of T by

T uc (ψ uc , N uc ), we find from (3)

ρ uc T uc (ψ uc , N uc ) = ρ c T c (ψ c , N c ).

(6)

To investigate how uncertainty affects the value V and the completion date T requires evaluating the sensitivity of these functions to the discount rate (because the effects of uncertainty are manifested through the discount rate). Equations (5) and (6) do not explicitly provide the desired changes of V and T with ρ because they relate quantities corresponding to different values of ψ and N . We seek the effect of increasing ρ, holding the latter parameters constant. Recall, from (P4) and (P5) that increasing these parameters entails conflicting trends, ∂V / ∂ψ > 0 but

∂V / ∂N < 0, Similarly, ∂T / ∂N > 0, but ∂T / ∂ψ < 0. To determine the trend entailed by increasing ρ alone, we consider first the “permanent-risk” scenario, for which the analysis is simpler. In this case,

ψ uc = B / ( ρ + λ ) = B / ρ uc . Hence, ψ c = ψ uc ( ρ + λ ) / ρ = B / ρ c , and rescaling does not affect the value of B. Consider now solutions of the optimization problems with the two different discount rates, but the same values of B and N . According to (P1), the parameter zψ depends on B but is independent of ρ in this scenario. The completion time T is

ρ to ρ uc uc in the uncertainty problem and keeping N constant implies a larger value of T .

determined by (P2). Since y is increasing and Q is decreasing, increasing


7

The increase in the effective discount rate encourages delayed investments and extends the project duration. The project value should decrease when interruption risk is introduced. To verify this, use (P3) noting that the product

ρ uc T uc is larger than the certainty value of

ρT while zψ remains unchanged. Since W(z) is a decreasing function, the uncertainty value must be lower than its certainty counterpart. These results are illustrated in Figure 1, in which the project duration, T, and value, V, are plotted against the effective discount rate, ρ + λ , under permanent risk. The form y ( R ) = aR / ( R + b ), where a and b are given positive parameters is assumed for the production function. The project duration increases from about eight years to over eleven years as the project value decreases from $2 billion at certainty (with ρ = 5 percent per annum) down to zero. While it is easy to verify that the latter conclusion concerning the decrease in value associated with uncertainty is robust under scenario variations, the effect on project duration is more subtle. For the temporary-risk scenario we can identify situations in which uncertainty actually implies reduced project durations. Under this scenario, it is the post-project completion value, ψ, which remains invariant

ρ uc = ρ + λ . We holding ψ and N constant. The

when the discount rate is increased to the uncertainty rate of compare, therefore, the values of uncertainty duration,

V c and V uc

T uc , is suboptimal for the certainty problem. T uc

V c ≥ Max{ Rt } { ∫ − Rt e − ρt dt + e − ρT ψ } = uc

0

T uc

Max{ Rt } { ∫ − Rt e − ρ t e λt dt + e − ρ uc

uc

T uc

e λT ψ } > uc

0

T uc

( 7)

Max{ Rt } e

λT uc

{ ∫ − Rt e − ρ t dt + e − ρ uc

0

uc

T uc

ψ } = e λT V uc > V uc . uc

Hence,

8

Figure 1.


Project Duration T (right scale) and Value (left scale) versus the Effective Discount Rate ρ+λ under Permanent Risk.1

Thus, introducing uncertainty or increasing the discount rate decreases the value under temporary risk as well. In contrast, project duration cannot be a monotonic function of the discount rate. Consider first projects of very short duration, as in the case of infinitesimally small

N = 0 then T = 0 and V = ψ , regardless of ρ . Alternatively, let N = δ > 0 be very small and expand all quantities to o(δ ). where o(δ ). satisfies ______________. Then (see Appendix B) the optimal investment rate is given by R = Q( zψ ) + o(δ ). The corresponding project duration is of the same order, T = δ / y ( Q( zψ )). Unlike the case of N.

Evidently, if

permanent risk, the parameter zψ now depends on ρ because W(z) depends on ρ, and this dependence determines the variation of T with ρ,

dT = ( T / ρ )ψ [ y ' ( R )]3 / [ y 2 ( R ) y"( R )] < 0. dρ

7 (8)

In this limit, T decreases with ρ. For short project duration, uncertainty tends to increase R and decrease T. At the other extreme, when ψ and T are close to their critical values [see (P6)], the behavior is reversed. Assume the project is barely profitable at the discount rate ρ and that an infinitesimal hazard rate λ = δ is enough to drive the value to zero. Using the scaling laws (5) and (6), we find in Appendix B that

T uc (ψ , N ) − T c (ψ , N ) = z (T c )V c (ψ , N ) / y( R0 ) + O(δ ),

(9)

where the first right-hand term of (9) is positive and of order O(δ ). Thus, project duration increases with uncertainty towards its critical value with an ever 1/ 2

increasing slope. Contrasting this result with small 1

The form

y(R) = aR / (R + b)

N behavior we see that, unlike

is assumed for the production function.


9

the value V, optimal project duration is not a monotonic function of the discount rate. For temporary risk with fixed values of ψ and N , it obtains some minimum value. Let the discount rate of ρmin correspond to this minimum duration. Then

the effect of uncertainty depends on whether ρ lies below or above ρmin . In the former case, uncertainty entails shorter duration, while in the latter case it implies longer projects. Of course, if the hazard rate is large enough to bring the effective discount rate above the critical value, the project is no longer profitable and T is infinite. As an example, we illustrate these findings in Figure 2 using again y ( R ) = aR / ( R + b ) for the production function and the certainty discount rate of 5 percent per annum. While the project value is seen again to decrease monotonically (albeit at a lower rate relative to permanent risk), the project duration is found to decrease by nearly a year until it begins to rise again at the effective discount rate of 12 percent per annum, at which point the project would have lost some 75 percent of its certainty value. It is easy to understand the robustness of the results concerning the value function. Occurrence is never profitable. Hence, the hazard associated with it can only decrease the expected value. How should one interpret the sensitivity of the project duration to variations in the parameters? In the temporary-risk scenario, two conflicting considerations call for a trade off. Increased early investments can bring about early completion and ensure a safe stream of benefits so the overall risk is reduced. On the other hand, the expected loss due to interruption prior to completion should encourage prudence and delay investments. For highly profitable short projects, the first consideration dominates while, for long projects of insignificant value the prudent policy of reduced investment rates is more advantageous. In contrast, under permanent risk even an early completion cannot guarantee the profit. Hence uncertainty is always associated with the more prudent investment policy.

10

Figure 2.


Project Duration T (right scale) and Value V (left scale) versus the Effective Discount Rate ρ+λ under Temporary Risk.2

5. CONCLUDING COMMENTS The water crisis approaching many countries in the Middle East may increase competition and hostility among nations sharing an international water body. But it may also be an impetus towards cooperation in trans-boundary projects that better exploit the common resource (Naff and Matson, 1984). Internal debates on such projects tend to be emotionally loaded and compounded with arguments that do not relate directly to water issues alone. The resolution of such debates often depends on a delicate balance of political powers which introduces uncertainty regarding the will and power of governments to carry on development projects until completion. The political uncertainty considered in this work takes the form of irreversible interrupting events for which the occurrence hazard is exogenous to the planning task. As is well known, event uncertainty of this type is equivalent to an increased effective discount rate. In the present context, this observation gives rise to simple scaling laws for both the project value as well as its duration. Expressing the optimization problem under uncertainty as an equivalent certainty problem (with

2

The form

y(R) = aR / (R + b)

is assumed for the production function.


11

scaled parameters), one is able to derive the loss of value associated with uncertainty in a straightforward manner. The loss of value comes as no surprise. The interrupting events represent total investment loss. Hence, the threat of their occurrence can only reduce value. In contrast, the effect of uncertainty on the optimal project duration is more subtle. This observation may be of practical importance. Realizing the losses associated with the interruption hazard, project owners will naturally consider incentive schemes to encourage development firms to expedite their operations. As is shown above, this strategy is not always optimal and certain scenarios call for a more prudent investment policy under political uncertainty. It is clear, however, that the resolution of problems associated with this type of uncertainty cannot be restricted to economic measures. Perhaps the most important message stemming from studies like the one presented here should be directed at political decision makers. A clear presentation of the prohibitive costs of political uncertainty on development programs and how they dramatically effect social well being might provide the drive for effective measures towards the elimination of its political sources.

12


APPENDIX A: SOLUTION OF THE INVESTMENT PROBLEM In this

Appendix we

evaluate the project duration T (ψ , N ) and the

corresponding value function V (ψ , N ) for the investment problem under certainty. The investors seek to maximize the benefit from the project, c

T  V (ψ , N ) = Max{ Rt },T ∫ − Rt e − ρt dt + e − ρTψ  , 0  c

(A1)

subject to

dN / dt = y( Rt ), Rt ≥ 0, N o = 0, N T = N (or Rt ≡ 0 and T = ∞). (A2) The Hamiltonian for the optimization problem (A1) is

H t = − Rt e − ρt + e − ρt pt y ( Rt ) where pt is the current value costate variable. (Leonard and Long 1992, pp. 334-335)

(A3)

The first order conditions are

∂ H / ∂ R = 0 ⇒ y ' ( Rt ) = 1 / p t ,

(A4)

p& t − ρ pt = −∂ H t / ∂ N t = 0 ⇒ pt = p0 e ρt .

(A5)

Combining (A4) and (A5) gives the investment rate

Rt = Q(e − ρt / p0 ) with

(A6)

1 / p0 ≤ y' ( 0 ) to ensure that Rt ≥ 0 for all t. The transversality condition

corresponding to free T is

HT + ∂ (e − ρTψ ) / ∂T = 0 , giving − RT + pT y( RT ) = ρψ or − Q(1 / pT ) + pT y ( Q(1 / pT )) = ρψ unless T = ∞. Finally, for a finite T, the terminal condition

N t = N implies

(A7)

TRANS-BOUNDARY PROJECTS UNDER POLITICAL UNCERTAINTY T

13

T

∫ y( R )dt = ∫ y( Q( e

− ρt

t

0

/ p0 ))dt = N .

(A8)8

0

Since the solution in (A6) is always available, one needs only to determine the parameters p0 and T from (A7) and (A8) without violating (A2). Define the function

W ( z ) = [ y ( Q( z )) / z − Q( z )] / ρ ,

z ∈[ 0, y ' ( 0)]

(A9)

with

W ' ( z ) = − y(Q(z )) / ρz 2 < 0, W" ( z ) = − Q' ( z ) / ρz + 2 y(Q(z )) / ρz 3 > 0 It follows that

(A10)

W ( y '(0)) = 0 and W(0)=∞. Thus, the equation W ( z ) = ψ must

have a single positive root for every ψ, denoted zψ, and (A7) implies that pT = 1 / zψ. Thus, pT is independent of N . Evidently, zψ decreases continuously with ψ [because z'ψ = 1 / W ' ( zψ ) < 0 ] and z 0 = y ' ( 0) . For a feasible solution with Rt ≥ 0 we must ensure that the argument of Q is always less than y′(0) (see A6). In particular, for t = 0,

zψ e ρT = 1 / p0 ≤ y '(0)

and, unless T = ∞ and N∞ = 0, T such that

T ≤ log( y '(0) / zψ ) / ρ ≡ Tmax (ψ )

,

(A11)

express the final level NT in the form T

NT ≡ K (ψ , T ) ≡ ∫ y(Q(e − ρt / p0 ))dt = 0

T

∫ y(Q(z

ψ

0

T

(A12)

e ρ ( T −t ) ))dt = ∫ y(Q(zψ e ρt )) dt 0

Evidently, K(ψ,0) = 0 and, for all T < Tmax(ψ),

∂K = y(Q(zψ e ρT )) > 0; ∂T

∂ 2K = ρzψ2 e 2 ρT Q' ( zψ e ρT ) < 0 . (A13) ∂T 2

N and ψ, equation (A8) [recast as K (ψ , T ) = N ] must have a unique feasible solution T (ψ , N ) ≤ Tmax (ψ ) if It follows that for

14


N ≤ K (ψ , Tmax (ψ )) ,

(A14)

and no solution otherwise. The latter case implies that (A4)-(A8) cannot be solved consistently to provide a feasible plan, and the policy [Rt ≡ 0,

T (ψ , N ) = ∞ ]

V (ψ , N ) = 0 . Indeed, since K (ψ , Tmax (ψ )) is a continuous and strictly increasing function of ψ with K ( 0, Tmax ( 0)) = 0 , equality in (A14) determines the minimal benefit ψ9 for any finite N . Clearly, the higher the completion level N , the higher must be the minimum benefit ψ10 that would render the project profitable. Having determined T (ψ , N ) , the

forms the optimal plan, yielding

c

characterization of the optimization problem (A1) is complete, because the parameter p0 is readily obtained via

1 / p0 = zψ e ρT (ψ , N ) .

We now turn our attention to the value function, cumulative investment present value, T

∫Re t

0

11 Set

V c (ψ , N ) . Consider first the

T

− ρt

dt = ∫ Q( zψ e

T

ρ ( T −t )

)e

− ρt

dt = e

0

− ρT

∫ Q( z

ψ

e ρt ) e ρt dt .

0

ρt

z (t ) = zψ e to write the latter integral as z(T )

(e − ρT / ρzψ ) ∫ Q(z )dz = (e − ρT / zψ )[ zW ( z )]zz((T0)) = ψe − ρT − W ( z (T )). z(0)

[The latter equality follows because

z (0) = zψ and W ( zψ ) = ψ . ] Thus,

V c (ψ , N ) = W ( z(T )) = W ( zψ e ρT (ψ , N ) ) .

(A15)

In the critical case with equality in (A11) and (A14), z(T) = y’(0) and W[z(T)] vanishes, consistent with the continuity of the value function at ψ12. Otherwise, (A11) implies that z(T) < y’(0), and W[z(T)] > 0, justifying identification of ψ13 as the root of (A14). We now investigate sensitivity of project value and duration to parameters of the problem. Note that T is determined by (A12). Hence,

∂T (ψ , N ) ∂K / ∂ψ z'ψ [ y(Q(zψ )) − y(Q(z(T ))))] =− = < 0 . (A16) ∂ψ ∂K / ∂T y(Q(z(T ))) Using (A15), we write

z'ψ z(T ) ∂V c = W ' ( z (T ))[ + ρz(T )∂T / ∂ψ ] = zψ ∂ψ


W ' ( z (T ))z'ψ z (T ) zψ

[1 +

y(Q( zψ )) - y(Q(z(T))) ]= y(Q(z(T)))

W ' ( z (T ))z'ψ z (T ) y(Q(zψ )) zψ y(Q(z (T ))) 14However,

15

.

z'ψ = 1 / W ' ( zψ ). Using (A10) to evaluate W ′(z), we find zψ ∂V c (ψ , N ) = = e − ρT ( ψ , N ) > 0 ∂ψ z(T )

(A17)

∂T ∂ 2V c (ψ , N ) = − ρ e − ρT ( ψ , N ) > 0. 2 ∂ψ ∂ψ

(A18)

and

So far we have been interested in the dependence of the duration T and the value Vc on ψ. It is also illuminating to consider the variation of these parameters with N. Using (A8) and (A13) we find

∂T  ∂K  =  ∂N  ∂T 

−1

=

1 >0, y(T )

 ∂ 2 K   ∂K  − ρz 2 ( T )Q' (T ) ∂ 2T = > 0. = −    2  ∂T   ∂T  ∂N 2 y 3 (T )

(A19)

3

Note that

(A20)

y (T ) = y (Q( z (T (ψ , N )))), Q(T ) = Q( z (T (ψ , N ))) etc.

Similarly, using (A10), (A13), (A15), and (A19), we obtain

∂T ∂Vc 1 = ρz( T )W ' ( z(T )) =− 0. 2 ∂N z(T ) ∂ N z(T ) y(T )

(A22)

and

APPENDIX B: PROJECT DURATION UNDER TEMPORARY RISK

16


In this Appendix, we derive the dependence of project duration on the hazard rate under the temporary-risk scenario. Consider first projects of very short duration, when N = δ is very small, and expand all quantities to o(δ). Neglecting the variation of R in time we write T

δ ≈ ∫ y( R)dt = y( R)T . 0

T

Hence T = δ / y( R) + o(δ ) and V =

∫ − Re

− ρt

dt + e − ρTψ =

0

− RT + ( 1 − ρT )ψ + o(δ ) = ψ − (ψρ + R)T + o(δ ) = ψ − δ (ψρ + R) / y( R) + o(δ ). Maximizing V with respect to R, we find (ψρ+R)y’(R)/y(R) = 1.

(B1)

Transforming to the variable z = y’(R) [so that R = Q(z)], We write (B1) as W(z) = ψ .

(B2)

It follows that z = zψ and the optimal investment rate is given by R = Q(zψ). The corresponding project duration is, to o(δ), T = δ/y[Q(zψ)]. In this scenario, the parameter zψ depends on ρ because W(z) depends on ρ, implying a variation of T with ρ,

dzψ δzψ Q' ( −∂W / ∂ρ ) dT δ = − 2 y' (Q(zψ ))Q' ( zψ ) =− 2 = dρ y dρ y ∂W / ∂z

δzψ Q' W ( zψ ) / ρ 2 − 2 = δ [ zψ / y( R)] 3 (ψ / ρ )Q' ( zψ ) = y − y / ( ρzψ2 )

(B3)

(T / ρ )ψ [ y' ( R)] 3 / [ y 2 ( R) y" ( R)] < 0. Thus, in this limit T decreases when the hazard rate is added to ρ. For short project duration, uncertainty tends to increase R and decrease T. Consider now the other extreme, when the project is barely profitable at discount rate ρ, and an infinitesimal hazard rate λ = δ reduces the value to zero. Using (5), this situation is described by


17

ρ V c (ψ + ψδ / ρ , N + Nδ / ρ ) = V uc (ψ , N ) = 0 . ρ +δ

(B4)

Expanding to o(δ), and using (A18) and (A21) for the partial derivatives of V , we rewrite (B4) as c

V c (ψ , N ) = (δ / ρ )( N − ψzψ ) / z (T c ) + o(δ ).

(B5)

Using (6) and (B5), as well as the explicit expression for the investment rate Rt=Q[z(Tc-t)], we follow the same procedure for T and find

T uc (ψ , N ) − T c (ψ , N ) = z ( T c )V c (ψ , N ) / y ( R0 ) + (δ / ρ )[ψzψ / y ( RT c ) − T c (ψ , N )] + o(δ ).

(B6)

The sign of T (ψ , N ) − T (ψ , N ) is seen to depend on the relative magnitudes of the terms forming the right hand side of (B6). Obviously, the second term is of uc

order o(δ), and V

c

c

in the first term is of the same order. However, y(R0) is also of

order o(1). Hence, the first term dominates implying Tuc(ψ, N ) > Tc(ψ, N ), and T c

increases with uncertainty. Indeed, because W[z(Tc)] = V = o(δ), and because W[y’(0)] = W’[y’(0)] = 0, it follows that z(Tc) = y’(0) + oδ1/2) and y(R0) = y{Q[z(Tc)]} = o(δ1/2). Thus, (B5) and (B6) imply

T uc (ψ , N ) − T c (ψ , N ) = (δ / ρ )( N − ψzψ ) / y ( R0 ) + O(δ ) = O(δ 1/ 2 ).

(B7)

18


REFERENCES Bilen, O. 1994. Prospects for Technical Cooperation in the Euphrates-Tigris Basin. In A.K. Biswas, ed., International Waters of the Middle East. Oxford University Press. Biswas, A.K. 1994. Management of International Water Resources: Some Recent Developments. In A.K. Biswas, International Waters of the Middle East. Oxford University Press. Centre for Natural Resources, Energy and Transport. 1978. Register of International Rivers. Water Supply & Management 2: 1-58. Dinar, A., and E.T. Loehman, eds. 1995. Water Quantity/Quality Management and Conflict Resolution. London: Praeger. Kolars, J. 1994. Problems of International River Management: The Case of the Euphrates. In A.K. Biswas, ed., International Waters of the Middle East. Oxford University Press. Leonard, D., and N.V. Long. 1992. Optimal Control Theory and Static Optimization in Economics. Cambridge University Press. Mageed, A.Y. 1994. The Nile Basin: Lessons from the Past. In A.K. Biswas, ed., International Waters of the Middle East. Oxford University Press. Murakami, M., and K. Musiake. 1994. The Jordan River and the Litani. In A.K. Biswas, ed., International Waters of the Middle East. Oxford University Press Naff, T., and R. Matson, eds. 1984. Water in the Middle East: Conflict or Cooperation? Westview Press. Tsur, Y., and A. Zemel. 1997. Intertemporal Regulation of Public Projects. Submitted for publication. Wolf, A.T. 1994. A Hydropolitical History of the Nile, Jordan and Euphrates River Basins. In A.K. Biswas, ed., International Waters of the Middle East. Oxford University Press.


19

Perm anent R isk 2

11

Value ($)

1.5

10 1 9 0.5 0 0.05

8

0.06

Duration (Years)

12

7 0.07

Effective D iscount R ate

FIGURE 1 - Use figure in Tsur’s original paper, as this one doesn’t have the floating boxes.

20


Temporary Risk 8.4

2 1.8

8.2 8

Value ($)

1.4 1.2

7.8

1 7.6

0.8 0.6

7.4

0.4 7.2 0.2

0.15

0.1

7 0.05

0

Effective Discount Rate FIGURE 2. Use one in Tsur’s original paper, as this one doesn’t have the floating boxes.

Duration (Years)

1.6