Peak load pricing under periodic and stochastic supply

European Economic Review 20 (1983) 13-21. North-Holland

PEAK LOAD

PRICING

Publishing Company

UNDER PERIODIC SUPPLY

AND STOCHASTIC

Spyros K. LIOUKAS* London

Graduate

School

of Business

Studies,

London

N WI

4SA,

UK

Received October 1979. tinal version received June 1981 The paper extends to the supply side previous work on peak load pricing embodying periodic and stochastic variations in demand. In a lirst step it introduces into the problem of periodic demand the additional problem of periodic capacity availability. Then it considers the general case where stochastic fluctuations enter both the demand and capacity availability sides. Welfare-maximising results suggest that or-peak consumers should be charged with capacity costs according to the loss-of-load probability in any period. This probability depends on periodic and random fluctuations in capacity availability. The optimal level of capacity is also alTected by such fluctuations.

1. Introduction The theory of peak load pricing has recently focused on deriving optimal pricing and capacity choice rules for a welfare maximising utility in the case of periodic and uncertain demand. i By contrast the treatment of the supply side has been relatively simple in these models because they have assumed that: (1) capacity is available at a uniform rate of Q for every period, and (2) there is no uncertainty on the capacity side. It is the purpose of this paper to examine the pricing implications of periodic and random fluctuations in capacity availability. This objective is accomplished in two stages. First, we extend the deterministic peak load problem to include the case where the capacity made available to meet demand is not the same for every period. Second, we add stochastic components into the capacity availability levels and derive new normative pricing and capacity choice rules in the general stochastic case. Before departing on formal analysis, it is important to mention the empirical justification for these generalisations. First, examination of public *I am grateful to S. Littlechild for useful comments on a previous draft, and to the many people from the U.K. Central Electricity Generating Board who spent a great deal of time providing information on their practices as regards capacity availability targeting. The author of course is solely responsible for any shortcomings. ‘Brown and Johnson (1969, 1970, 1973) and Meyer (1975) have provided principles of optimal peak load pricing under uncertain demand. The present work is an extension of these studies.

0014-2921/83/000-000/$03.00

0 1983 North-Holland

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S.K.

Lioukas,

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load pricing

utility settings (especially for electricity generation) reveals that time invariant capacity is practically untenable.2 The capacity available for meeting demand varies seasonally mainly because of maintenance requirements. Provided that the range of seasonal fluctuations in demand is wide enough and maintenance can be carried out in off-peak periods, then capacity constraints are only likely to act in peak periods. This case reflects the traditional assumption in the peak load pricing literature which maintains that marginal capacity costs are purely peak related. However, many public utilities face annual demand curves which leave narrow margins for maintenance and so there is a risk of capacity shortage in off-peak periods. Off-peak customers may therefore call for an increase in installed capacity, and so must be charged with capacity costs.3 Second, in addition to periodic fluctuations, capacity availability levels are subject to random fluctuations due to the stochastic nature of plant breakdowns and component failures. Indeed many defects are discovered only after the plant is taken out of service for inspection and maintenance. Moreover in public utility operations capacity availability targets and prices are usually set ex-ante, that is before the actual capacity available and actual demand are known. Planned availability levels are therefore subject to additional uncertainty (e.g. unforeseen delays in overhaul, breakdowns and other deviations from plans) which may affect pricing. 2. The deterministic problem We assume that the public utility is maximising the difference between willingness to pay and costs over n equal-length sub-periods of a demand cycle (e.g., corresponding to each week of the year). Capacity is treated as homogeneous4 and, in any sub-period i, it is available at a rate Zi lower than ‘Field investigation of the operational planning processes in the U.K. Central Electricity Generating Board [Lioukas (1978)] has indicated that capacity availability targets exhibit strong seasonality. Maintenance programmes are arranged so’that in every week of the year ahead enough capacity will be available to meet demand at an ‘acceptable’ risk level. Thus the capacity made available for loading tends to follow the shape of the annual demand curves. “Turvey (1968) has warned us against the dangers of over-simplitication when periodic elements of capacity availability are ignored. However he did not carry further the analysis. “This treatment has practical relevance. For example, the CEGB has found that aggregate capacity exhibits a mass behaviour with stable statistical distributions. For this reason, in the build-up of capacity availability targets, it treats capacity as homogeneous, concentrating on probabilistic deviations from mean-risk availability levels. The reader may note that discrete components of capacity are at most about 1% of the maximum capacity Q” (for the biggest power stations of 660MW). Our marginal analysis below may well be thought to deal with increments of this range. A discrete analysis of capacity categories in the form suggested by Crew and Kleindorfer (1976) would be of interest; however the complexity of full stochastic analysis with discrete technology is such that one doubts its usefulness [see eq. (10) for the simpler case].

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land pricing

15

the maximum installed capacity Q”. A constant level of maximum capacity throughout the cycle is assumed, a hypothesis which implies that investment and dis-investment (i.e., plant commissioning and decommissioning) take place only at the start of the demand cycle. The firm has to set prices pi for each sub-period i, and to choose total capacity Q” and capacity availability levels Zi at the start of each demand cycle. The welfare criterion to be maximised is max Wziil

(pr--b)Di(pJ+i$l

Dr’ [D;‘(~)-pJdx-pQ~

- WI, z,, **a,-a

(1)

where /I represents the constant annual capacity costs per unit of capacity (fixed costs) and A(Z,, Z,, . . ., Z,) a non-negative, convex and continuously differentiable function which represents the total costs of achieving an availability programme (Z,, Z,, . . ., Z,) throughout the cycle. This general cost function is not given any precise form because of the complexity of the underlying maintenance programme, which involves interdependencies among various sub-periods. For example increased availability in a period i may be achieved at the expense of availability in other periods to which plant overhaul may have to be transferred. The rest of the notation is kept similar to that of the literature, as far as possible. Welfare is maximised under the following constraints: Di(Pi)

S zi9

Zi 6 Q", PizO,

ZizO,

Q”zO,

i=l ,--*, n ,

(2)

i=l >***, n,

(3)

i=l,...,

(4)

n.

If Ii and Wi are the Lagrange multipliers for (2) and (3), the marginal conditions for optimisation provide the following pricing and capacity investment rules (assuming positive values for pi, Zi and Q”): Pi=bi+Zip

i=l,...,n,

aA az. w.=/ I i -!Aazi jgiazj c -I azi,

i=L...,n,

(5) j=L...,n,

16

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Peak load pricing

The first condition (5) requires that optimal price in each sub-period should be equated to the constant marginal operating cost plus an imputed marginal curtailment cost or quasi-rent which will restrict demand to the capacity available in each sub-period. In periods with excess available capacity, the constraint (2) will hold with inequality and the multiplier becomes zero, so that Pi=b. These results re-establish the deterministic peak load pricing principles in the new context; in sub-periods with capacity surplus consumers must pay only the operating cost, while in sub-periods with capacity shortage (not necessarily peak demand periods) consumers must pay the operating cost plus the marginal curtailment cost. Eqs. (6) and (7) together imply the following optimal rule for capacity choice:

Denoting dA/&Zi=ci the cost of an incremental increase in availability in period i and &Zj/aZi= -sji the cross-effect or substitution effect which is caused on the availability in period j by a unit increase in the availability in period i, we get the following simplified formula: 8+

~ i=l

Ci-

~

C

i=l

jti

CjSji=

~

Ii.

i=l

This rule implies that capacity should be employed up to the point where the marginal benefit it provides equals its total marginal cost. The right-hand side of the condition (9) represents the marginal benefit from an additional unit of capacity which would occur when the availability constraints are relaxed in sub-periods of excess demand. This becomes clearer if we write

iil licit1@i-b), i.e., the marginal benefit of an extra unit of capacity is equal to the added net revenue over sub-periods restricted by capacity availability. The left-hand side of (9) represents the marginal fixed cost of a unit of capacity plus the sum of the marginal availability costs, net of cross-effects, which are incurred by embodying this unit of capacity into the production system. Ci includes costs necessary to increase availability in sub-period i given that all other availabilities are kept constant, i.e., resources spent to increase maximum output capacity of existing plant or resources spent to invest in an additional unit of capacity which would increase availability in sub-period i. Sji may

S. K. Lioukas,

Peak

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17

take positive values when an increase in availability in i is achieved at the expense of availability in j, and negative values when an increase in availability in i entails an increase in the capacity available in j.

3. The stochastic problem We express stochastic components in an additive form for both capacity and demand.’ The public utility is assumed to maximise the expected welfare which consists of the following components: E(W)= E(willingness to pay) - E(average variable costs x sales) - E(lixed capacity costs) - E(capacity availability costs) - E(rationing costs). The last term (i.e., rationing costs) reflects the costs required to distinguish between potential claimants when demand exceeds capacity and serve only those with the highest willingness to pw6 The random components are assumed to be independent of the availability and demand levels. The capacity expected to be available for a sub-period i is Qi=Zi+ei, where Zi is the ‘deterministic’ part (that is the mean expectation) and e, is an independently distributed stochastic term. e, takes values SO that 05 Qi~Q”, that is it varies within the range -Z. I-le.51Q”‘- Zi. We denote J(ei) the probability distribution function of e, and Fi(ei) the corresponding cumulative distribution. It is assumed that E(e, ej)=O (i#j), OlE($< CC, E(ei,Zj)=O and E(ei,Dj)=O. In a similar way demand is assumed to include two components: Dr(pr,Ui) =X&J+ Ui, where Xi(pi) is the deterministic part or mean expectation (X’(pi) CO), and Ui an independently distributed random part (- Xi(pi) 5 Ui < a). Random components for various sub-periods are assumed to be independent (i.e., cross-elasticities zero) and also independent of the capacity components as well as the deterministic part of the demand. We denote g{(u,) the probability distribution function of Ui and G,(Ui) the cumulative distribution. As Crew and Kleindorfer (1976) suggested rationing costs could be represented by a monotonically increasing convex function of the capacity deficit, i.e., Ri = Ri(Di - Qi) = Ri[Xi(pi) + ui- Zi-ei]. For expository purposes below we assume that the differential of Ri with respect of oi-Qi is a constant Ti (i.e., Ri=ri and R; =O). Under these conditions we can estimate the expected value of the objective ‘A multiplicative form could also be used. However, we expect it to yield similar results [see Brown and Johnson (1969)]. $Rationing of excess demand is assumed to be allocated to consumers having the lowest marginal valuations (i.e., willingness to pay). This scheme was used by Brown and Johnson (1969, 1973) and Meyer (1975). Note, however, that alternative rationing schemes could also be employed [see Visscher (1973) and Sherman and Visscher (1978)]. The rationing cost formulation was suggested by Crew and Kleindorfer (1976).

SK

18

Lioukos,

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load pricing

function component by component. Analytically we have E(W)=I$I

Q~~z’J(eJz’~~C:‘~’ gi(ui)(pi-b)[Xi(pi)+ui]duidei i i I n Q'"-zr + 1 j fi(ei)(pi-b)(Zi+ei)z,+.,-x(p.)gi(U3duidei I i=l -Z, I

I

It

-/IQ”’ - A(Z,, . . ., Z,) n +&

Q”-Zi

zr+er-XI(PI)

-S,

S -X&p,)

I

h@iI

xi-+-ur)

gi(uJ

f

[Xi(X) + Ui]dxdui

Pi

+z,+e~x,~p,gi(uJ [“‘I}-“” CXi(x)+uildx I i Ii PI X;‘(Zr+ei~“i)

-

J Pi

[Xi(x)+ui-Zi-ei]dx

+.I, (p,)

1I

dni dei

Ri[Xi(pi) + Ui - Zi - eJgi(uJduidei.

f I

(10)

it

The first term of the welfare function (10) is the expected net revenue for the case when output is not constrained by capacity at the preset (pi,Zi, Q”) values (chosen at the start of the cycle). The second term is the expected revenue when demand exceeds capacity. The third and fourth terms are the fixed for the cycle capacity costs and the costs required to attain an availability programme (Z,, Z,, . . ., Z,). The next expression is the expected consumers’ surplus consisting.of (i) the expected area under the demand curve all the way up to the quantity demanded when output is unconstrained, and (ii) the expected area all the way up to the quantity demanded in cases of constrained output minus the expected loss of surplus because of constrained supplies (i.e., area L in fig. 1). The welfare expression (10) will be maximised subject to the constraints Zi 5 Q",

i=l,...,n.

Setting the differential of the Lagrangian following condition for optimal pricing:

function equal to zero, we get the

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Lioukas,

Peak

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19

x;’ (-lJi)

p;= x;‘Kli-ui) Pi

Fig. 1. Components of welfare when demand exceeds capacity.

Provided that the differential condition is simplified to pi=b+ri

LLpi

1 - LLP,’

Xy@i) is not zero and Ri=ri

is constant this

i= 1,...,n,

(11)

where LLPi is the loss-of-load probability in period i, that is pm-Zi

LLP,= Prob. [Xi(pi) + pi > Zi + ei] =

jz

h(ei) I

7

gi(ui)durdei.

z, + e, - X,(P,)

The pricing rule (11) includes respective results of the literature as subcases. For example without rationing costs (ri=O) and for certain capacity (ei=O) the optimal price becomes equal to the marginal operating cost b, a result which has been derived by Brown and Johnson for the single-period (n= 1) case and by Meyer for the multiperiod case.’ It is evident from (11) that optima1 prices are related to the loss-of-load probability in any sub-period, peak and off-peak. The surcharge above operational costs is zero only when the probability of excess demand is zero (with LLPi = 0 we have pi = b). An intuitive interpretation of (11) can be made if we write (pi-b)(l-LLPJ=riLLP,

i= 1,. . .,n.

That is prices should be set so that at the margin

the expected net

‘Note also that our LLP takes into account both demand and capacity uncertainty and so it is more general than the equivalent result by Crew and Kleindorfer (i976). These authors introduce uncertainty only in the demand side.

20

S.K. Lioukas,

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contribution to revenue, weighted by its probability, is equal to the marginal cost of consumption rationing, again weighted by its probability of occurrence.’ As regards the optimal capacity, differentiating the Lagrangian function with respect to Zi and Q” and combining the results, we get the following relationship: P=i~,@i-b)LLPi-i~,~i+

i i=l

C CiSji+ii, T+i$l’iLLf’i* j#i

(12)

where T=‘JZZijJei) I

7 [X;‘(Zi+ei-ui)-pi]gi(ui)duidei z, +ei- Xi(Pi)

= D EQ, (Pi* - Pi). I- I

‘I;: is the truncated mean of the difference between the marginal willingness to pay and the actual price for the marginal disappointed consumer in subperiod i (as indicated in fig. 1, the difference Pt-pi is the vertical dimension of L). Since ‘&>O, the marginal benetit of an extra unit of capacity is now higher than in the deterministic case. This becomes clear if we re-arrange the condition (12) as follows:

The left-hand side of (13) is the extra costs incurred for installing and making available over the cycle an additional unit of capacity. The righthand side is the sum of the extra net revenue weighted by its probability, the savings on rationing costs from an additional unit of capacity again weighted by their probabilities, plus the marginal willingness of consumers to pay above the price charged when output is restricted. Taking into account (11) we can rewrite (13) as p+ i ci- i i=l

i=l

C CjSji= i (Pi-b)+ j#i

i=l

i i=l

T,

(14)

sWhether the lirm is operating at a delicit under these prices depends on the value of r,. We can imagine that r, could reflect the costs of establishing a market of rights for future use of the product whenever output will be constrained by capacity unavailability [see Brown and Johnson (1973), Crew and Kleindorfer (1976)]. These forward rights could be priced so as to recoup costs and thus break even.

S.K. Lioukas,

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21

which differs from (9) only in that it comprises the terms ?;. Since z are positive, the demand for capacity (at the margin) in the case of a complete stochastic setting is higher than in a deterministic setting, when compared for the same set of peak prices, that is prices contributing to capacity costs. 4. Conclusions We have shown that periodic and stochastic components of capacity affect the optimal pricing and capacity choice rules of a welfare maximising public utility. The solution indicates that optimal prices include a capacity surcharge above the marginal operating cost which depends on the loss-ofload probability. This probability combines uncertainties in both capacity and demand sides. The marginal rule for capacity investment suggests that the optimal level of capacity depends on both the costs of the availability programme over a demand cycle and stochastic variations in availability levels. References Brown, G. and M.B. Johnson, 1969, Public utility pricing and output under risk, American Economic Review 59, 119-128. Brown, G. and M.B. Johnson, 1970, Public utility pricing and output under risk: Reply, American Economic Review 60,489-490. Brown, G. and M.B. Johnson, 1973, Welfare-maximising price and output with stochastic demand: Reply, American Economic Review 63,230-231. Crew, M.A. and P.R. Kleindorfer, 1976, Peak load pricing with a diverse technology, Bell Journal of Economics 7,207-231. Lioukas, S., 1978, Goal-setting and performance evaluation in the operational planning processes ol the CEGB, IPSM working paper (London Business School, London). Meyer, R.A., 1975, Monopoly pricing and capacity choice under uncertainty, American Economic Review 65,326-337. Sherman, R. and M.L. Visscher, 1978, Second best pricing with stochastic demand, American Economic Review 68,41-53. Turvey, R., 1968, Peak-load pricing, Journal of Political Economy 76, 701-713. Turvey, R., 1970, Public utility pricing and output under risk: Comment, American Economic Review 60.485-486. Visscher, M.L., 1973, Welfare-maximising price and output with stochastic demand: Comment, American Economic Review 63.224-229.