The Price of Water Under the Lesotho Highlands Water Project

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<ul><li><p> Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of EconomicResearch 1998. Published by Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF,UK and 350 Main Street, Malden, MA 02148, USA.</p><p>Bulletin of Economic Research 50: 2, 1998, 0307-3378</p><p>THE PRICE OF WATER UNDER THELESOTHO HIGHLANDS WATER PROJECT*</p><p>C. R. Barrett and M. P. Senaoana</p><p>ABSTRACT</p><p>The papers main objective is to examine, in the light of contracttheory, the agreement between Lesotho and South Africa on thetransfer of water under the Lesotho Highlands Water Project. Asecondary objective is to analyse the shareout involved, using theNash bargaining solution.</p><p>I. INTRODUCTION</p><p>In 1986, Lesotho and the Republic of South Africa (from now onrespectively referred to as L and S) signed a long-term contract (Treaty,1987) for the transfer of water from L to S, the contract constituting anecessary element of a joint water resource project called the LesothoHighlands Water Project (LHWP). This contract is to run from thecompletion in 1995 of the first stage of construction, until 2045. With thehelp of World Bank loans, L is to bear the construction and operationalcosts of the project, making L the upstream producer and S the down-stream buyer. L, however, will provide a competing demand for waterfor use in irrigation, and will also use the water to generate hydroelectricpower (although this will not significantly reduce the quantity of wateravailable for other purposes). Thus, the interesting aspects of thecontract concern the rules governing the division between L and S ofuncertain quantities of water (after their use for hydroelectric power),and the payments from S to L.</p><p>The treaty distinguishes two kinds of water: water whose delivery isreliable (98 per cent sure), and excess water; the former constitutes thereserved supply. Throughout Sections I and II the focus here is on agiven hydrological year any year in the 50-year period spanned by the</p><p>* We are grateful for the helpful comments of an anonymous referee.</p><p>93</p></li><li><p>treaty. For the given year, let W denote the total supply of water, and xthe reserved supply.</p><p>In simplified form,1 the contract which was agreed specifies (for thegiven year) a price p and a royalty R, and:</p><p>(I.1) If WRx, W is transferred to S.(I.2) If WEx, at least x is transferred to S, and any additional amounts</p><p>transferred to S require the agreement of both parties.(I.3) S pays R to L.(I.4) L pays p per unit to S for each unit that W falls short of x, and S</p><p>pays p per unit to L for each unit in excess of x that L transfersto S.</p><p>Thus, water up to an amount which can be achieved with 98 per centreliability, the reserved supply, is to be transferred to S; any residualwater, excess water, is to be divided between L and S. Denoting theexcess water by w, we have w\max [Wx, 0].</p><p>It may be noted that the contract is complete: x, R and p weredetermined at the time of the agreement. One speculates that thereason was a desire to minimize potential disputes between the twocountries (bargaining costs); or it may have been that L would sign onlya complete contract in the circumstance that L was going to incur all thecosts of the project. Adjustments were, however, built in: x is to beupdated as forecasts are revised; R, which is designed to control theshareout, will be updated as new information on the projects costsbecome available; and p too is indexed.2</p><p>The main object of our paper is to analyse, in terms of a model, theform of contract for the transfer of excess water from L to S. Weexplain, in terms of the model, how p (featuring in (I.4)) is determined.As a subsidiary matter, we discuss the shareout between L and S.</p><p>II. EFFICIENCY OF THE CONTRACT</p><p>It will be supposed that any contract for the transfer of water from L toS can take account only of public knowledge. Thus the terms of thecontract can relate to events whose occurrence is public knowledge</p><p>1 The 1987 treaty refers to a hydrological year (October to September), and to monthlydeliveries and associated payments within the year. Within the hydrological year, and intosix months of the next year, it is required that an actual or anticipated shortfall in onemonth (i.e. less than the amount x/12 delivered to S) is compensated if possible by extradeliveries (above x/12) in other months.</p><p>2 Starting from 57 million cubic metres per annum in 1995, x rises gradually until, from2021 onwards, it is 2208 million cubic metres per annum. R consists of both a fixedelement relating to construction costs, and variable elements which in turn relate to theelectricity costs of pumping, and to other operating and maintenance costs. The latter arevariable because they vary with x. Throughout, p is 1.657 cents per cubic metre at 1985prices.</p><p> Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of Economic Research 1998.</p><p>94 BULLETIN OF ECONOMIC RESEARCH</p></li><li><p>(verifiable by a third party), at the time of execution of the contract, butnot to events whose occurrence is private information. Assume:</p><p>(II.1) The contract is ex ante efficient in relation to information whichis public knowledge (Coase, 1960).</p><p>Assume too:</p><p>(II.2) L and S are risk-neutral.</p><p>As the discussion in this section concerns excess water (rather than themuch more significant reserved supply), this seems to be a reasonableassumption. Next assume:</p><p>(II.3) Costs, and the supply of water, are independent of the terms ofthe contract.</p><p>That is, once the project is undertaken, costs are given, and W for theyear in prospect is determined by just the stage of construction reachedand the vagaries of rainfall. Assumption (II.3) is a simplifying mechan-ism to enable us to focus on just the benefits from an exogenous (thoughrandom) supply of water. Finally assume, for the given year:</p><p>(II.4) The benefits from excess water are, in monetary terms:</p><p>BL\bLwL(bLa0)</p><p>BS\bSwS(bSa0)</p><p>wL and wS are the amounts of excess water retained by L and deliveredto S, respectively (wL+wS\w), and bL and bS are random parameters. Atthe beginning of the year, realizations of bL and bS are observed asprivate information, by L and S, respectively.</p><p>The assumption that the marginal benefits are independent of theallocation of excess water is quite strong. It may nevertheless be areasonable assumption on the grounds that the excess water is much lesssignificant than the reserved supply, while the two kinds of water arelikely to be close substitutes.3</p><p>Assumptions (II.1)(II.4) imply that in drawing up the contract L andS will choose to maximize their expected joint benefits from theconsumption of excess water (see the Appendix). For the given year, thismaximization will take account of the fact that, as far as each exercisescontrol, L and S will be governed by their private information aboutbenefits. Denote their joint benefits from the consumption of excesswater (for the year in prospect) by B, and the expectations operator byE. From (II.4), expected joint benefits are</p><p>3 The 1987 treaty anticipates large increases in LHWP capacity over time, full capacitynot attained until 2021 (see footnote 2). Despite this, the agreement specifies a price forexcess water which is constant in real terms over the 50-year period it spans. Downward-sloping demand curves for excess water would suggest a falling price as supply increases.</p><p> Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of Economic Research 1998.</p><p>95PRICE OF WATER UNDER LESOTHO HIGHLANDS WATER PROJECT</p></li><li><p>EB\E(bLwL)+E(bSwS) (1)</p><p>where wL+wS\w. We discuss next how the division of w between wLand wS depends on the particular contract adopted.</p><p>As indicated in the introduction, the price p of excess water is deter-mined in advance (subject to indexing). Thus we are able to formulatethe decision tree for the given year as: stage 1, p is determined; stage2, bL and bS are realized (as private information); stage 3, a decision ismade as to how, conditional on its realized value, w will be divided;stage 4, w is realized.</p><p>How the stage 3 decision is made is determined by the form ofcontract, and we now distinguish five such forms. Subject to price, onesuch form of contract (seller control) allows L to choose how w isdivided; another form (buyer control) gives this choice to S; a third formof contract (nonintegration) allows L to choose how much of w is offeredto S, and S to choose how much of the amount offered to accept;4 afourth form (reverse) interchanges the roles of L and S in nonintegration.A fifth form (command) determines directly the division of w between Land S. While in theory there are many forms of contract available, thesefive, the third of which, nonintegration, corresponds to the regimeactually adopted, are particularly simple, and will yield insights.</p><p>Assume, for simplicity:</p><p>(II.5) bL, bS and w are mutually independent random variables; bL isuniformly distributed over the interval [aL, zL]; and bS isuniformly distributed over the interval [aS, zS].</p><p>As a preliminary to examining specific regimes, we consider restric-tions on the price p of excess water, first noting that no price operatesunder the command regime. For the other regimes we can deduce thatp, if it is to be effective i.e. if the regime concerned is to bedistinguishable from command lies in both the intervals, [aL, zL] and[aS, zS]. This deduction makes use of (II.1), the assumption of efficiency.We illustrate the argument in the case of seller control.</p><p>Given seller control, p is ineffective if pRaL because with probabilityone L retains all of the excess water marginal benefit is above theprice for which L can sell the water. Analogously, p is ineffective if pEzL in this case, with probability one L delivers all of the excess water toS. Thus, effectivity implies aLspszL. It follows that p is inefficient ifpsaS. If psaS, marginally raising p improves allocation, since excesswater which L retained before, but which now because of the higher</p><p>4 The terms seller control, buyer control and nonintegration derive from Grossman andHart (1986, 1987). They consider a contract between a buyer and a seller, who may atsome fixed point in the future trade an indivisible good. Future valuations will be privateinformation. They find that, optimally, the right to enforce trade should be allocated (if toeither party) to the party whose valuation of the good is the more uncertain. (The costsof future negotiations are assumed prohibitive.)</p><p> Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of Economic Research 1998.</p><p>96 BULLETIN OF ECONOMIC RESEARCH</p></li><li><p>price L delivers to S, is transferred to where it yields a higher marginalbenefit. Analogously, p is inefficient if pazS. Efficiency therefore impliesaSRpRzS.</p><p>We examine each of the five regimes in turn, relegating details of thecalculations involved to the Appendix. For notational convenience,define: rL\zLaL; rS\zSaS; mL\EbL; and mS\EbS.</p><p>Seller control: L retains the excess water if bLap (wL\w) and delivers itto S if bLsp (wL\0). The probability of the former is (zLp)/rL, and ofthe latter (paL)/rL. Thus, from (1) and (II.5):</p><p>EB\zLp</p><p>rL</p><p>p+zL2</p><p>Ew+paL</p><p>rLmS Ew. (2)</p><p>Maximize EB. For the efficient level of p we obtain</p><p>p\mS. (3)</p><p>On substituting (3) into (2), we obtain</p><p>EB\C(zLmS)2</p><p>2rL+mSD Ew. (4)</p><p>Buyer control: Interchanging the roles of L and S, we obtain</p><p>p\mL (5)</p><p>EB\C(zSmL)2</p><p>2rS+mLD Ew. (6)</p><p>Nonintegration: L retains the excess water if bLap (wL\w), and offers itto S if bLsp. The probability of the former is again (zLp)/rL, and ofthe latter (paL)/rL. S accepts if bSap (wS\w). The probability that Saccepts is (zSp)/rS, and that S rejects (paS)/rS. Thus, from (1) and(II.5):</p><p>EB\zLp</p><p>rL</p><p>p+zL2</p><p>Ew+paL</p><p>rL</p><p>zSprS</p><p>p+zS2</p><p>Ew</p><p>+paL</p><p>rL</p><p>paSrS</p><p>aL+p2</p><p>Ew. (7)</p><p>Maximize EB. For the efficient level of p we obtain</p><p>p\aL+zS</p><p>2. (8)</p><p> Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of Economic Research 1998.</p><p>97PRICE OF WATER UNDER LESOTHO HIGHLANDS WATER PROJECT</p></li><li><p>On substituting (8) into (7), we obtain</p><p>EB\C(zSaL)3</p><p>8rL rS+mLD Ew. (9)</p><p>Reverse: Interchanging the roles of L and S, we obtain</p><p>p\aS+zL</p><p>2(10)</p><p>EB\C(zLaS)3</p><p>8rL rS+mSD Ew. (11)</p><p>Command: In this case, maximizing EB we obtain</p><p>EB\max (mL, mS)Ew. (12)</p><p>Let S denote seller control, B buyer control, N nonintegration, R reverse,and C command. It is immediate from (4), (6) and (12) that C isdominated by (i.e. is less efficient than) either S or B, and so can berejected. Noting that aL\mL12 rL, etc., it is easy to calculate that theremaining four regimes all give the same value for EB when mL\mS\mand rL\rS\r, namely (r/8+m)Ew. (The corresponding value for EBwhen bL and bS are public knowledge is (r/6+m)Ew proof omitted.)Let NaS denote that N dominates S, etc. We explore the cases: differ-ing expected marginal benefits from excess water but the same variance;the same expected marginal benefit but differing variances. The follow-ing are proved in the Appendix:5</p><p>(A) If mL\m+ar (0sas12), mS\m and rL\rS\r, then</p><p>NaS\BaR.6</p><p>(B) If mL\mS\m, rL\br (1sbR4) and rS\r, then</p><p>SaN\RaB.</p><p>Two further results can be obtained by use of symmetry:</p><p>(Ap) If mL\m, mS\m+ar (0sas12) and rL\rS\r, then5 For the four regimes in turn, the partial derivatives with respect to mL, mS, rL and rS,</p><p>evaluated at the point mL\mS\m, rL\rS\r and Ew\1, are, respectively:</p><p>S: 1/2 1/2 1/8 0</p><p>B: 1/2 1/2 0 1/8</p><p>N: 5/8 3/8 1/16 1/16</p><p>R: 3/8 5/8 1/16 1/16.6 Under buyer control or seller control, for p to be effective and efficient, a is restricted</p><p>to (12, 12).</p><p> Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of Economic Research 1998.</p><p>98 BULLETIN OF ECONOMIC RESEARCH</p></li><li><p>RaS\BaN.</p><p>(Bp) If mL\mS\m, rL\r and rS\br (1sbr4), then</p><p>BaN\RaS.</p><p>These results are at least suggestive. We tentatively conclude thatregime N was selected because mLamS, and either rLRrS or, if rLarS, thedifference is insufficient to reverse the implied ranking of N and S.</p><p>How can this conclusion, which may at first seem paradoxical, beinterpreted? We infer that, as all of the reserved supply is allocated toS, the benefits from the reserved supply are greater in S than they wouldbe in L, at the margin. (As L is richly endowed with water resources, thisis not surprising.) Given this inference, and assuming the two kinds ofwater are close substitutes, we might anticipate that the benefits fromexcess water will also be greater in S than in L. As we have seen, ourconclusion has been the opposite, i.e. mLamS.</p><p>An interpretation is, however, not hard to find. A key consideration is,we feel, Ls geographic proximity to the LHWP. This has a specificeffect in raising Ls benefits from the excess water in comparison withthose of S. This is because costly installations, required to transfer waterlarge distances, are likely to be a more important factor in reducing thebenefits from excess water than in the case of the reserved supply. Thereserved supply (98 per cent sure and suitable for municipal uses) isboth more highly valued and likely to be more manageable than excesswater.</p><p>III. THE SHAREOUT</p><p>We now comment briefly on the shareout from the LHWP. In thiscontext it is...</p></li></ul>

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