Hydropower pricing for smelters

Paul van Moeseke

Hydropower, in contrast to hydroenergy, is shown to be governed by depletion pricing: its real price is theoretically subject to a secular rise at the real social rate of discount. The alternative is to charge the equivalent constant cost for long term contracts with major consumers such as local authorities and smelters. Contract prices will there- fore be above current marginal power cost in the initial phase and below it in the final phase of the contract.

Reactions to a section on long run electricity prices in my previous article in this journal’ indicate that hydroelectric- ity is generally considered to be a renewable resource and therefore exempt from depletion pricing, unlike fossil fuels, which are depletable and subject to a real long run price increase determined by the social rate of discount. This long run price increase is known as the Hotelling rule? While the matter is of general interest, it is of most immediate concern to major users such as smelters and local authorities, as well as to the host countries or power boards supplying the power.

In fact, hydroelectricity is a twin resource, consisting of hydroenergy (kWh) - which is evidently renewable as long as it rains - and hydropower (kW). The latter is depletable, since there are only so many valleys to be dammed. Hydropower and hydroenergy are roughly related like the capacity of, respectively, the traffic on, a canal.

But it is precisely hydropower, blocks of several hundred thousand kW constantly available, that smelters pay for since their cost of hydroenergy, in practice the distribution cost of electricity per unit (kWh). is very small. It is shown below that the real price of hydropower should rise over time at the social rate of discount; in other words that the Hotelling rule, or a variant thereof, applies. If so, efficiency requires that long run hydropower contracts, in real constant prices, for smelters include prices initially above, and later below, the current margin-

al cost of power. In particular, the mere fact that the

The author is Professor and Head, Department of Economics, Massey University, Palmerston North, New Zealand.

Research was initiated during the author’s tenure of the Professorial Fellowship in Economic Policy of the Reserve Bank of Australia. The author is indebted to Anthony Bird Associates (Kingston-upon-Thames) and to Dr F. Vande Ginste (Brussels) for research data on aluminium smelters.

current marginal cost exceeds current charges resulting from a longstanding contract is not necessarily inefficient.

An intuitive rationale behind a secular rise in the real price of hydropower is, of course, that more accessible (hence cheaper) sites, such as more readily accessible wells, are evidently exploited first. Again, the rising capital costs sunk in deeper wells, offshore exploration and shale development imply, by substitution, a corres- ponding rise in the marginal value of investment in, inter ah, power dams as well as coal mines.

The secular rise of the hydropower price at the social rate of discount is a theoretical result derived at the end and holds as long as the number and capacity of potentially dammable sites is finite.

It is of particular interest that depletion pricing can be shown to hold, as we shall see, whether the power board maximizes the present value of net returns or, more realistically, just minimizes the present value of the cost of systematically damming potential sites over time, mean- while auctioning off to the highest bidder new blocks of power as they become available.

At realistic real discount rates of 2% and 3% the implications for major baseloaders such as smelters are very significant: eg for 25year contracts efficiency re- quires that constant prices should be higher (by 27.1% and 42.1%) than current power prices at the beginning of the contract but considerably lower (by 22.9% and 32.9%) at the end. Although there are wide variations, current new hydroelectricity contracts typically charge smelters 1.5-2~ per unit.

Every one cent difference in electricity price to a 300 000 t/y (tonnes per year) aluminium smelter repre- sents an annual payment of some $45 million to the power authority or host country. Conversely, it also represents about a $150 cost difference per tonne (6&/Ib) to the company or between 13% and 15% of recent LME spot prices (mid-1984 to mid-1986). Anthony Bird’s” estimate for the current (April 1986) world average cost of production in the non-socialist world is 51c/lb, excluding interest and depreciation. According to the same author,” at recent prices a 5c/lb decrease of the aluminium price reduces by more than 2 million tonnes the aluminium capacity that can profitably stay in production (in t/y: 10.6 million at 6Oc, 8.4 million at 55c, 6.3 million at 50~). The same holds, one assumes, for a 5c/lb increase in produc- tion cost.

0301-4207/87/010085-03$03.00 0 1987 Butterworth & Co (Publishers) Ltd 85

Hydmpower pricing for smrlkvs

Equivalent constant costs

Before stating the reasoning behind the necessary secular rise of the cost of hydropower we compute below the magnitudes of the cost differentials involved. Normally, the electricity price charged to huge baseloaders requiring uninterrupted access to blocks of hundreds of MW of power 8 760 hours a year hardly exceeds the price of the power: this is reasonable since the cost of energy distribution per unit (kWh) ‘. 1s a small fraction of that to the average consumer or business. We can therefore, by way of illustration, simply identify electricity price with power

price. Power contracts with smelters are normally long run (25

to 50 years) and intended to be constant in real terms, eg by tying them to the aluminium price (whose secular trend has been more or less constant in real terms) or some such index as the OECD GNP deflator. If, however, as argued below, depletion pricing applies, the initial power price should rise at the real social rate of discount with a secular trend of, say, 2% or 3%. A constant real price contract must therefore be based on the so-called equivalent constant cost which, of course, will be higher than the rising price initially and lower later on, as illustrated in Table 1.

The first line of figures in the table, for instance, says that an initial cost of one cent rising at 2% pa to 1.649~ after 25 years, overtakes the equivalent constant cost of 1.271~ after 11.98 years (break even time). The equivalent constant cost can be found in any annuity table as the annuity with a present value of 2.5~: a 25year cost of one cent rising at 2% pa is equivalent to (has the same present value as) a constant cost of 1.271~ over the same period.

The table assumes the usual case of a constant block of power; otherwise the annual costs should, of course, be weighted by power supply. To fix the ideas: Anthony Bird Associates are quoted in Metal Bulletin5 as stating that the weighted average power cost to Western world aluminium smelters in 1983 was ‘the very low figure of 18.8 mills’ (1.88~) which ‘compares with recent reports that most large industrial users with a steady load face costs in the range of 3&70 mills for their electricity’ (3-7 cents). They forecast that the aluminium industry will be paying an average 2.81~ per kWh (in constant 1983 dollars) by 1991.

Average hydropower prices, while still lower, are being

Table 1. Equivalent constant cost.

Power price (cents/unit)

Rate of Equivalent Break even Contract discount Initial Final constant cost time (years)

25 years 2% 1 1.649 1.271 11.98 3% 1 2.117 1.421 11.72 4% 1 2.718 1.582 11.47

40 years 2% 1 2.226 1.453 18.67 3% 1 3.320 1.717 18.02 4% 1 4.953 2.005 17.39

50 years 2% 1 2.718 1.582 22.93

3% 1 4.482 1.931 21.93 4% 1 7.389 2.313 20.96

revised upward almost everywhere, as a perusal of the Light Metals sections of Metal BuMin 108486 will make clear. Be that as it may, if by way of illustration one takes 2c as a representative initial figure one can derive the corresponding final prices and equivalent costs simply by doubling the figures of Table 1. Break-even times are unaffected.

If the current marginal cost is 2c, new 25.year contracts imply either a constant real price of 2 x 1.271~ (at 2%) or il periodic adjustment of 2% pa. Hence a smelter approaching the end of a 25-year constant price contract should be paying 2 X 1.271~ towards the end or only 77% of the terminal 2 x 1.649~. Again, a smelter presently operating in the second half of a fixed contract should be paying less than the current marginal cost of new power and only about 77% if the contract is nearly up.

The rising cost of hydropower: rationale

In practice power boards in small and developing host countries seem to dam remaining potential hydrosites systematically and to more or less auction off to the highest bidder new blocks of power as they come on stream. The power boards’ objective is then to minimize present discounted cost of damming potential sites over time.” We now indicate why this model yields the equivalent of a Hotelling rule. A more complex model in terms of net returns is mentioned at the end.

The argument below is valid given a limited supply of potential hydrosites. The fact that the exact upper limit to potential supply is unknown is, as is well known, mathe- matically irrelevant.

Denote the cost of building an amount X, of power in period t by the function c,(x,) and call marginal cost c: in period t the cost of building one last, or one extra, unit (or generator) of power. Further call the present discounted marginal cost PMC, of power in period t:

PMC, = c,‘/a’ (1)

where eg a = 1.02 if the rate of discount is 2%. The standard optimality rule is: build in each period up to the point where the PMC, is the same for all periods. The rule is intuitively clear. For if, say, PMC, > PMCz total present discounted cost would not be minimal: it could be reduced by delaying construction of the last unit of period 1 by one period. Since therefore PMC, = PMC,,, it follows from (1) that

PMC, = cilu’ = PMC,, , = c:, ,/a’+’

so that

c:+, = ac:

ie the marginal cost, which, as is well known, is the amount that, on average, should be charged for efficient allocation, goes up by the rate of discount (Hotelling rule).

An alternative, theoretically optimal if politically less realistic, model is for the host country, or its power board, to maximize the present value of net returns, ie sales minus power cost.

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Hydropower pricing for smelters

the rate of social discount, exactly like the prices of

depletable resources. In the case of constant price long run contracts efficient

allocation requires that charges reflect the equivalent constant cost: this will initially be higher than the marginal cost (current price) of power, break even sometime during the first half of the contract and be considerably lower towards the end.

Similarly, a major user now in the second half of an earlier constant price contract, should be paying less than current marginal cost. The precise price ratios underlying these conclusions are set out in Table 1.

This approach is more complex in that income from electricity generated by a turbine constructed in period t accrues to later periods as well, while building costs pertain to t only. Reasoning analogous to the above model (detail and proof in Moeseke)’ leads to the requirement that the present discounted value of net marginal revenue in period t:

PMR, = f: (l/a’ + l/u’+ + .) - ~;/a’

be the same in every period; where f,(x,) is revenue from electricity generated by X, in every period from t onward and f: is the corresponding marginal revenue.

It is easy to show that the requirement is satisfied if both marginal revenue and marginal cost, hence price, increase over time at the rate of discount, again a rule of the

Hotelling type.

Conclusions

Hydroelectricity is a twin resource consisting of hyd- roenergy, which is renewable, and hydropower, which is depletable. The price of power is subject to depletion pricing, that is to say, efficient allocation implies a secular rise at the social rate of discount. This conclusion holds whether power boards, or host countries, maximize present value of net revenue or simply minimize damming costs over time.

Hydropower prices, if periodically adjusted, may there- fore be expected to be subject to a secular rise, roughly at

‘Paul van Moeseke, ‘Export efficiency of large projects: the case of aluminium smelters’, Resources Policy, Vol 11, No 2, June 1985, pp 110-118. ‘Harold Hotelling, ‘The economics of exhaustible resources’, Journal of fo/itica/ Economy, Vol 39, No 2, 1931, pp 137-175. 3Anthony Bird Associates, Aluminium Analysis, No 29, April 1986. 4Anthony Bird, ‘Aluminium production costs’, Unpublished paper presented at Metal Bulletin’s Third International Aluminium Congress, Munich, September 1984 and Anthony Bird Associ- ates, Aluminium Analysis, No 24, January 1985. %deta/ Bulletin, ‘Moderate aluminium growth predicted’, 10 April 1984, pp 13-l 4. 6PauI van Moeseke, ‘Aluminium and power: the host nation’s export efficiency’, Metal Bulletin, Proceedings of the Second International Aluminium Congress, 1983, pp 81-7 and op tit, Ref 1. 7PauI van Moeseke, ‘Hydropower as a depletable resource’, Massey Economic Papers, Vol 3, No A8502, February 1985, pp 34-42.

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