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Efficient tariffs to market district heat
0360-544219053.00+ 0.00
&uqy Vol. 15. No. 9, pp. 773-719.1990
Copyright0 1990PcrgamonPressplc
Printedin Great Britain. All rights tescrvcd
EFFICIENT TARIFFS TO MARKET DISTRICT HEAT FRANZWIRL Institute for Energy Economics, Technical University of Vienna, GuShausstrassc27-29,
A-1040 Wien, Austria (Received 28 Sune 1989; received for publication 18 January 1990)
Ahstraet-The pricing of district heat is an efficient marketing instrument, in addition to a coercive measure. We derive an efficient contract, i.e., a mix of charges (for access, fixed charges and for heat) such that all costs are covered and that the consumer’s expenses are minimal. Price differentiation need not be unfair if everyone gains. Moreover, price differentiation may help to attract new classes of customers who cannot be reached with uniform tari&.
1. INTRODUCTION
an important aspect of marketing district heat, namely, efficient tariff arrangements, i.e., a proposal to a potential consumer to pay for access to the network, for heat and for a fixed charge. Numerous features have been suggested as favouring the expansion of district heat, most notably its perceived environmental advantage over conventional use of fuels. Unfortunately, elementary economic considerations are often not included. This lack of examination of economic constraints is among possible explanations for slow market penetration of district heat. In Sec. 2, we study the optimal (normative and positive) contract, i.e., the efficient mix of charges for integrating a customer into the network. This anlaysis is fairly general and applicable to other network-related services such as electricity, telecommunication, cable television, etc. Section 3 is devoted to the advantages of price differentiation on the basis of Ramsey prices. We emphasize that price differentiation need not be unfair but may actually lead to improvements for all (Pareto improvements).
This paper deals with
2. THE
EFFICIENT
CONTRACT
We consider a company which supplies and distributes district heat. The firm incurs the following costs if it wants to expand the current network: investment outlays to access a new customer (I) and costs per unit of heat (c). The investment outlays may be constant, declining with respect to sales (due to network externalities) or increasing (if expansion occurs to less densely populated regions). To simplify the exposition, we assume that I is a positive constant and neglect maintenance costs and capital depreciation. This description stresses the fact that the bulk of the costs are not only fixed but sunk. The present value of costs K for the expansion of district heat becomes K =I +
IOD 0
exp( -ti)cE
elf,
(1)
where r denotes the discount rate employed by the company to discount future costs and revenues and E is the amount of heat delivered to a typical consumer in a region. The infinite planning horizon is compatible with no capital depreciation. This assumption further simplifies the problem because no assumptions about scrap values (or other and partly arbitrary terminal conditions) bias the analysis. This assumption, although implausible at first sight, is nevertheless acceptable because integrating over a reasonable planning horizon of 25 or 50 yr is substantially equivalent to integrating to infinity if future expenses are sufficiently discounted. As an example, for an interest rate of 10% (r = 0. lo), integration of Eq. (1) over the next 25 yr yields 9&E, which is close to the value of the improper integral 10cE. 773
774
FRANZWlRL
The company offers a consumer a contract to purchase district heat. A typical district-heating contract involves the following components: payment for access to the district-heat network A, a fixed and continual charge f, price p per unit heat. Contracts of this type are typical for network-related and often publicly-distributed services such as electricity, telecommunications, etc. The vector (A, F, p) specifies a particular contractual arrangement. This contract (A, f, p) is more complicated but also more flexible than a linear contract (0, 0, p), which is typical for many commodities. More precisely, varying three variables instead of one allows us, at least in principle, to improve efficiency. A given contract (A, f, p) leads us to the following present value of revenues from an additional customer: m R(A, f,p): =A + exp(-Mf + PE(P)I dt. (2) I0 Here, E(p) denotes a conventional demand price relation such that higher marginal fuel prices p will reduce demand, E’ G 0. E. = E(3) is the ideal comfort if (additional) heat is free, i.e., when p = 0. Furthermore, the revenues are supposed to be concave, which implicitly bounds the demand-price relation, which must not be too convex; formally, (pE” - 2~5’) < 0. For a fixed tariff arrangement (A, f, p), the decision whether to expand or not depends on the inequality KaR.
(3)
We will now view the problem from the consumer’s point of view. The consumer incurs, in addition to the contractual payments, comfort sacrifices for any deviation from the ideal indoor temperature in exchange for lowering fuel costs. The convex function D quantifies this disutility D(E), D’ < 0 for E < Eo, D” > 0, D(Eo) = 0. Thus, a particular tariff (A, f, p) would induce the following present value of costs (including disutility) to a new customer: X(4
f, P) =A + ~mexp(-WWE(p))
+f + PE(P)I d.
(4)
This formulation assumes a utility-maximizing choice of heat. More precisely, a consumer can only respond with heat purchases once a contract has been signed. The consumer determines his individual heat requirements E from the following calculus: minimize disutility plus fuel expenses, i.e., arithmetically find min D(E) +pE. The demand relation E(p) denotes the corresponding solution from the implicit relation D’ + p = 0. The consumer applies a discount rate 6 to calculate the present value. It is assumed that a consumer is less patient and S > r. The expenditures X denote the present value of the costs (including disutility) obtainable from the existing space-heating system or from subsitution for any alternative fuel except district-heat. In other words, X denotes the reservation price, i.e., the best a consumer can attain without considering d.h. Obviously, the d.h. company must beat this threshold in order to win this customer, i.e., X(A, f, p) d X. A competitive supply or a regulated public monopoly will offer a contract such that costs public enterprise (or a competitive equal revenues: K = R. Moreover, a welfare-oriented company) will design the contract such that the total expenditures from a consumer’s point of view (i.e., including disutility) become minimal, subject to the constraint that all costs are covered. Technically speaking, the d.h. company acts as a Stackelberg leader, i.e., the company adjoins the optimal response of the customer. This procedure leads to the following optimization problem: minX(A,f,p)sothat[R(A,f,p)-K]>O0,A30,f30,ps0.
(5)
A.f.P The following proposition relegated to the Appendix.
describes
the optimal
contract.
The
analytical
derivation
is
Proposition 1. A friendly consumer offer should not charge for access to the network and A = 0. Instead, a fied and continual charge, f > 0 and f = rl, should generate the corresponding
Efficient tariffs to market district heat
775
stream of revenues to compensate for the initial investment and for interest. Marginal costs determine the price for district-heat, p = c. Summarizing, (0, rl, c) characterizes the optimal contract. This proposition is intuitively plausible. Due to their impatiencet relative to the d.h. company consumers prefer an annual payment of a fixed charge f = rl to a single initial investment I. The variable part of the tariff to a single initial investment I. The variable part of the tariff for consumed heat equals the marginal costs, which is a standard result in welfare analysis. The possibility to split tariffs into fixed and variable components allows for marginal-cost pricing and simultaneously covers total costs. This flexibility provides an edge over those competing fuels that must include fixed costs in the linear tariff. Linear tariffs, (A, f, p) = (0,0,p), create a dead-weight loss if the associated price p must cover variable plus lixed costs. The reason is that prices exceeding marginal costs lead to comfort sacrifices of the consumers. This contract shows some resemblance to but also important differences from real-world arrangements. Typically, all elements of (A, f, p) are positive. As in Proposition 1, fixed costs skim off the bulk of the revenues and fuel costs are typically low. This fact complies with the preceding result because the marginal costs of heat, especially from cogeneration, are small. The difference is that many d.h. companies charge for access to the network. The motivation from the company’s point of view is fairly obvious: to recover the sunk investments as rapidly as possible. However, this policy increases the consumers’ costs and thus slows the market penetration of district-heat. The basic structure of the contract specified in Proposition 1 is not only optimal with respect to welfare but also applies to a private monopoly. More precisely, the following proposition holds (the proof is given in the Appendix): Proposition 2. A profit maximizing d.h. company will offer the consumer the contract (0, f, c). In other words, the monopoly applies a two-part tariff and charges marginal costs. However, the jked charge f > ri provides the profit. This result may be of little relevance for the current organization of the d.h. industry, which is characterized by social ownership and/or heavy regulation. However, Proposition 2 applies once d.h. utilities are deregulated.
3. PRICE
DIFFERENTIATION
(RAMSEY
PRICES)
An important characteristic of marketing district-heat is the large portion of fixed costs required to extend the network. Moreover, these costs are not only fixed but also sunk. Thus, wrong decisions cannot be corrected, nor can the invested money be recovered. Therefore, a subtle strategy is necessary to market d.h. successfully. Proposition 1 stresses one way to attract new customers. Another possibility is to differentiate the contracts according to the consumers’ willingness to pay. This differentiation is restricted to the fixed charges f because an optimal contract is completely specified by fixed charges and heat prices and the latter must always balance the marginal costs c. In the following exposition, we will employ the concept of second-best pricing, called Ramsey prices. ’ To simplify the exposition, we suppose that the fixed and sunk investments I,, are necessary to serve a region. Additionally, the investments i are required to connect a particular customer to the grid. Two classes of consumers consider the acquisition of district-heat. Both groups are equally large and each group consists of n members. The only difference is that the first group’s threshold &, is larger than the second group’s, i.e., Y1 > X2. For example, the first group uses an inefficient and overall costly fuel, e.g., coal, and thus has a higher willingness to pay for district-heat because they would accept any contract satisfying X 6 X1. On the other hand, the second group’s willingness to pay for d.h. is lower for various reasons, e.g., they use a more convenient fuel such as oil. The opposite applies, if a consumer is less impatient than the d.h. company
(r > 6),
then A = I,f = 0.
776
FRANZWtRL
The total investments to hook up an entire region to the d.h. network amount to (4 + hi). a uniform tariff, identified by the superscript U, must result in the same tixed charge
Thus,
f” = r{[?J(2n)]
+ i}
(6)
to each consumer. No marketing problem arises if the associated consumer expenses X” are less than X, because then both consumer groups would switch to d.h. and share the common costs &, equally. We suppose that the second group rejects this contract while the first group accepts the offer. In this case, X(O,r(e+i),
c)
OCeCiJ2n.
(7)
The first inequality involves the assumption that the second group will contribute a small factor E to the fIxed costs b but not their appropriate share (second inequality). The last inequality in Eq. (7) expresses the fact that the first group would accept the offer in Eq. (6). The average costs rise for the first group when the second group refuses to join the d.h. network, because the sunk costs & must then be raised from n instead of 2n consumers. Indeed, the costs associated with this uniform contract may exceed the higher reservation price X2. As a consequence, nobody acquires district-heat and both consumer groups use their initial fuel and pay X1 and X2, respectively. However, price differentiation allows reductions in costs for both groups if the “willingness to pay” exceeds the total cost. We define ni as the maximal willingness of consumer i to contribute to the common costs b, which is the difference between the opportunity costs X, and the costs associated with a contract without coverage for &, i.e. 0 < [x2 - X(0, ri, c)]: =
ar2 < xl:
=
[Z1 - X(0, ri, c)].
(8)
The assumption of a sufficient willingness to pay ensures that n(rci + ark) > (r&,/6). Hence, it is possible to share the common costs I,, such that both consumer groups gain as compared to their initial expenses Xi (i = 1,2). We propose to differentiate according to the individual willingness to pay by writing 4 = r{i + [Jci/(nl+
~dl(b/n)),
j= 1,2.
(9)
These contracts (0,4, c) with Z = 1, 2, ensure that everybody acquires district-heat. Moreover, everybody gains compared with the uniform contract specified by Eq. (6) because the second group will pay less than their reservation price X2 and less revenues have to be raised from the first group to cover lo. In technical terms, this arrangement constitutes a Pareto improvement. Prices that use willingness to pay to split unattributable costs are called Ramsey-prices.? Of course, discriminatory Ramsey pricing leads to a Pareto improvement only if additional consumers can be attracted, i.e., group 1 in the example hooks up to the network. Otherwise (e.g., when X” < X2 < X1), discriminatory pricing cannot reduce the costs to one customer by raising the payments for another consumer. Ramsey-prices are second-best in the sense that they maximize the social welfare if marginal-cost pricing does not cover the costs and if a deficit is infeasible for various reasons.’ Moreover, this arrangment may be considered to be fair because each group has to spend the same fraction, i.e. X1 - X1(0, ri, c)
r
Xl
=iin,=
&In
X2-
X2(0,
6
c) (10)
x2
Here, we refer to maximal willingness to pay for IO, Ed,, which becomes a proxy for the potential gain. The specified arrangement does not lead to cross subsidizations$ because each consumer covers the costs attributable to his or her behaviour. Indeed, cross subsidization is unlikely for tRamsey prices for a multi-product industry confronted by a given system of demand are determined by the inverse elasticity rule, i.e., the relative mark-up over marginal costs is proportional to the reciprocal value of the (absolute) price elasticity. A commodity with a low price elasticity corresponds to a high willingness to pay, i.e., the value to the consumer exceeds the price by far. $Cross subsidies are not only inefficient but also unfair.’
Efficient tariffs to market district heat
777
a competitive company. Suppose that the company engages in cross subsidizations and uses the revenues from the second group to win also the first group that would not join even for fi = i. Then the second group of consumers could reject the offer, or another company could make a lower bid to serve the second market only. In fact, Ramsey prices may help to protect a local monopoly from the entry of competitors. Baumol et al3 prove that such a regime is sustainable, i.e., does not provoke entry, if the cost function satisfies well defined conditions. This analysis addresses the examples of Faulhaber4 who showed that Ramsey prices need not be sustainable. At least three groups (e.g., oil, gas, and coal heating systems) and a U-shaped average cost curve are necessary to construct an example parallel to that of Faulhaber. Our methodology implies monotonically declining average-cost curves and thus Ramsey prices are sustainable even if additional groups are considered.
4. SUMMARY
We have considered the design of efficient contracts to market district-heat. More precisely, the question is which system of charges for access, a tixed fee or payment for each unit of heat, covers the expenses of the company and is simultaneously attractive for new potential customers. The optimal welfare contract would not charge for access if consumers are impatient, as is most likely. Instead, a tixed charge should generate a revenue stream to compensate for the initial outlays and interest. Heat should be priced according to marginal costs. Surprisingly, even a monopolist would offer heat at marginal costs and would also refrain from access charges, but he would charge a larger fixed fee. The possibility to split tariffs provides an edge to d.h. vis a vis competitive fuels because the d.h. consumer receives heat at the marginal costs. This saving reduces the comfort sacrifice that is associated with the use of other fuels, for which a charge must be included to cover fixed and marginal costs. It is possible to imagine consumers who choose d.h. heat rationally despite its greater cost compared to liquid fuels because they can afford higher levels of comfort with lower marginal costs. The ratio of fixed to marginal costs is much lower for conventional fuels and, therefore, gains from a two-part tariff will presumably not outweigh the institutional costs associated with such an arrangement. The arguments of Sec. 3 encourage price differentiation. This policy may attract new customers with a lower willingness to pay, which allows them to share the common costs within a wider group of consumers. Price differentiation may appear to be discriminatory and potentially unfair at first sight. Actually, price differentiation does not lead to cross subsidizations because everybody has to cover at least the costs attributable to his or her consumption. Furthermore, this policy will not be unfair because everybody gains in the form of lower costs. The reason is simple. Accepting all consumers with a positive, albeit below average, willingness to contribute to the common costs lowers the fraction to be covered by each individual who ultimately chooses district heat.
REFERENCES
1. W. J. Baumol and D. Bradford, Am. Econ. Rev. 60,265 (1970). 2. G. R. Faulhaber and S. B. Levinson, Am. Econ. Rev. 7l, 1083 (1981). 3. W. J. Baumol, E. E. Bailey, and R. D. Willig, Am. Econ. Rw. 67, 350 (1977). 4. G. R. Faulhaber, Am. Econ. Rev. 65,966 (1975). 5. M. D. Intriligator, Mathematical Optimization and Economic Theory, Prentice-Hall,
New York, NY
(1971). APPENDIX
Proposition 1 The constraints of the optimization problem given in Eq. (5) are concave (due to the supposition of concave revenues). However, the objective (viewed as a maximization problem)
7%
FRANZWlRL
need not be concave because the revenues enter negatively. Thus, the Kuhn-Tucker provide only necessary optimality conditions. We define the Lagrangian
L=-,+(D+F+pE)+~ f+(P-c)E+A-I s
r
+FA+~
I
1
f+/, 2
conditions
p 3
(Al)
and use the Kuhn-Tucker multiplier A.for the revenue constraint of Eq. (3) and p, (i = 1,2, 3) for the non-negativity of the instruments. Equation (Al) shows the integrals in explicit form. This procedure leads to the following necessary optimality conditions:’
auaA= auaf = -(i/q
-(l
- ApI) = 0,
+ p/r) + ji2 = 0,
aL_D’E’+pE’+E+A(p-c)E’+E 6
ap-
)L
(A2)
+p3=0, r
f+(P-c)E+A-I E 1, ~0
AsO
r
1.0 = 0, P2f
=o,
clsp=o, The joint conditions
(A3)
,
P,~O,
W)
P250,
647)
p390.
M-9
of Eqs. (A3) and (A7) and the assumption j42= (l/S) - (A/r) = [(t - M)/(&)]
3 oj
6 > r imply
(r/S) 5 A.
(A9)
Therefore, no interior solution is possible for the decision variable A. The opposite A > 0 implies p1 = 0 [from Eq. (A6)], thus, I = 1 [from Eq. (A2)] and hence a contradiction follows. Therefore, A = 0 and pl 3 0. We assume that interior solutions exist for the instruments f and p. It will be shown that these assumptions, and the implications c(~= p3 = 0 are compatible with the specified optimality conditions. From the supposition p2 = 0 follows that A = r/6 due to Eq. (A3). Therefore, the strong inequality must hold for pl, ccl > 0, because (r/6) < 1 and Eq. (A2) applies. Substitution of this result into Eq. (A4) and using p3 = 0 gives (A/r)[(p - c)E’] = (l/S)[D’E’
+pE’] = 0.
(AlO)
The term between the square brackets on the rhs of Eq. (AlO) must vanish because D’ = -p if the consumer determines the fuel demand rationally, i.e., by minimizing {D(E) -pE}. Therefore, the Ihs of Eq. (AlO) must also vanish, which requires p = c. With this solution, A = 0, f > 0 and p = c; we may compute the amount of fixed charges f from the revenue constraint, which reduces to
(f/r) = I.
(All)
Thus, f = rl. The sequence (0, ri, c) corresponds to the optimal strategy, although the Kuhn-Tucker conditions are not sufficient because the space of admissible policies is compact due to the supposition of equality between costs and revenues. The theorem of Weierstral3 therefore conditions. ensures the existence of an optimum, which must satisfy the Kuhn-Tucker However, (0, ri, c) is the unique solution of Eqs. (A2)-(AS) because different assumptions contradict at least one of the optimality conditions Eqs. (A2)-(A8). Hence, (0, ri, c) must be the optimum solution. Proposition 2 The optimal welfare policy is to accept any customer who is willing to cover the costs. However, a private monopolist might refuse to serve all of these customers if this restrictive policy can enhance profits. We suppose that there exists a distribution of reservation prices &
Efficient tariffs to market district heat
779
such that N(X) denotes the number of consumers acquiring a d.h. contract if their costs are equal to X. We assume N’ < 0 and that the implied profit function (R - K)N is concave in all instruments (A, f, p). A monopolist will find
(Al21 subject to the conventional
non-negativity
constraints.
We again define the Lagrangian
L=N(R-K)+FIA+YJ+c~~P
(A13)
and obtain the following necessary and sufficient (due to the assumption of concave revenues) optimality conditions: aLlaA = (R - K)N’aXldA
+ NaRIdA + /.J~= 0,
(AW
auaf= (R- K)N'aXlaf+NaRlaf+j4,=0, aLlap=(R-K)N~axlap+NaR~ap-NaKlap+~3=0,
(A13 6416)
plus the suppressed complimentary slackness conditions of Eqs. (A6)-(A8). We assume a policy similar to Proposition 1, i.e., ccl 3 0 and pL2= c(~= 0. Substitution partial derivativest
axiaf= 116, axiap=~Id, aRiaf=lir, aRiap=(E +pE’)lr, aKiap= cE’lr,
of the
6417) 6418) 6419)
simplifies Eq. (A15) to
[(R-K)N'/6]+Nlr=O, which implies ccl > 0 because of Eq. (A14). Substitution into Eq. (A16) and rearrangement yields
of the derivatives of Eqs. (A17)-(A19)
E{[(R - K)N’lS] + Nlr} + (p - c)NE’lr = 0. The term between the braces must vanish in order to satisfy the equality Therefore, p = c whenever the monopolist chooses to supply (N > 0).
tThe derivative aX/ap
involves again the use of (D’ + p) = 0.
6420)
WI) in Eq. (A20).
&uqy Vol. 15. No. 9, pp. 773-719.1990
Copyright0 1990PcrgamonPressplc
Printedin Great Britain. All rights tescrvcd
EFFICIENT TARIFFS TO MARKET DISTRICT HEAT FRANZWIRL Institute for Energy Economics, Technical University of Vienna, GuShausstrassc27-29,
A-1040 Wien, Austria (Received 28 Sune 1989; received for publication 18 January 1990)
Ahstraet-The pricing of district heat is an efficient marketing instrument, in addition to a coercive measure. We derive an efficient contract, i.e., a mix of charges (for access, fixed charges and for heat) such that all costs are covered and that the consumer’s expenses are minimal. Price differentiation need not be unfair if everyone gains. Moreover, price differentiation may help to attract new classes of customers who cannot be reached with uniform tari&.
1. INTRODUCTION
an important aspect of marketing district heat, namely, efficient tariff arrangements, i.e., a proposal to a potential consumer to pay for access to the network, for heat and for a fixed charge. Numerous features have been suggested as favouring the expansion of district heat, most notably its perceived environmental advantage over conventional use of fuels. Unfortunately, elementary economic considerations are often not included. This lack of examination of economic constraints is among possible explanations for slow market penetration of district heat. In Sec. 2, we study the optimal (normative and positive) contract, i.e., the efficient mix of charges for integrating a customer into the network. This anlaysis is fairly general and applicable to other network-related services such as electricity, telecommunication, cable television, etc. Section 3 is devoted to the advantages of price differentiation on the basis of Ramsey prices. We emphasize that price differentiation need not be unfair but may actually lead to improvements for all (Pareto improvements).
This paper deals with
2. THE
EFFICIENT
CONTRACT
We consider a company which supplies and distributes district heat. The firm incurs the following costs if it wants to expand the current network: investment outlays to access a new customer (I) and costs per unit of heat (c). The investment outlays may be constant, declining with respect to sales (due to network externalities) or increasing (if expansion occurs to less densely populated regions). To simplify the exposition, we assume that I is a positive constant and neglect maintenance costs and capital depreciation. This description stresses the fact that the bulk of the costs are not only fixed but sunk. The present value of costs K for the expansion of district heat becomes K =I +
IOD 0
exp( -ti)cE
elf,
(1)
where r denotes the discount rate employed by the company to discount future costs and revenues and E is the amount of heat delivered to a typical consumer in a region. The infinite planning horizon is compatible with no capital depreciation. This assumption further simplifies the problem because no assumptions about scrap values (or other and partly arbitrary terminal conditions) bias the analysis. This assumption, although implausible at first sight, is nevertheless acceptable because integrating over a reasonable planning horizon of 25 or 50 yr is substantially equivalent to integrating to infinity if future expenses are sufficiently discounted. As an example, for an interest rate of 10% (r = 0. lo), integration of Eq. (1) over the next 25 yr yields 9&E, which is close to the value of the improper integral 10cE. 773
774
FRANZWlRL
The company offers a consumer a contract to purchase district heat. A typical district-heating contract involves the following components: payment for access to the district-heat network A, a fixed and continual charge f, price p per unit heat. Contracts of this type are typical for network-related and often publicly-distributed services such as electricity, telecommunications, etc. The vector (A, F, p) specifies a particular contractual arrangement. This contract (A, f, p) is more complicated but also more flexible than a linear contract (0, 0, p), which is typical for many commodities. More precisely, varying three variables instead of one allows us, at least in principle, to improve efficiency. A given contract (A, f, p) leads us to the following present value of revenues from an additional customer: m R(A, f,p): =A + exp(-Mf + PE(P)I dt. (2) I0 Here, E(p) denotes a conventional demand price relation such that higher marginal fuel prices p will reduce demand, E’ G 0. E. = E(3) is the ideal comfort if (additional) heat is free, i.e., when p = 0. Furthermore, the revenues are supposed to be concave, which implicitly bounds the demand-price relation, which must not be too convex; formally, (pE” - 2~5’) < 0. For a fixed tariff arrangement (A, f, p), the decision whether to expand or not depends on the inequality KaR.
(3)
We will now view the problem from the consumer’s point of view. The consumer incurs, in addition to the contractual payments, comfort sacrifices for any deviation from the ideal indoor temperature in exchange for lowering fuel costs. The convex function D quantifies this disutility D(E), D’ < 0 for E < Eo, D” > 0, D(Eo) = 0. Thus, a particular tariff (A, f, p) would induce the following present value of costs (including disutility) to a new customer: X(4
f, P) =A + ~mexp(-WWE(p))
+f + PE(P)I d.
(4)
This formulation assumes a utility-maximizing choice of heat. More precisely, a consumer can only respond with heat purchases once a contract has been signed. The consumer determines his individual heat requirements E from the following calculus: minimize disutility plus fuel expenses, i.e., arithmetically find min D(E) +pE. The demand relation E(p) denotes the corresponding solution from the implicit relation D’ + p = 0. The consumer applies a discount rate 6 to calculate the present value. It is assumed that a consumer is less patient and S > r. The expenditures X denote the present value of the costs (including disutility) obtainable from the existing space-heating system or from subsitution for any alternative fuel except district-heat. In other words, X denotes the reservation price, i.e., the best a consumer can attain without considering d.h. Obviously, the d.h. company must beat this threshold in order to win this customer, i.e., X(A, f, p) d X. A competitive supply or a regulated public monopoly will offer a contract such that costs public enterprise (or a competitive equal revenues: K = R. Moreover, a welfare-oriented company) will design the contract such that the total expenditures from a consumer’s point of view (i.e., including disutility) become minimal, subject to the constraint that all costs are covered. Technically speaking, the d.h. company acts as a Stackelberg leader, i.e., the company adjoins the optimal response of the customer. This procedure leads to the following optimization problem: minX(A,f,p)sothat[R(A,f,p)-K]>O0,A30,f30,ps0.
(5)
A.f.P The following proposition relegated to the Appendix.
describes
the optimal
contract.
The
analytical
derivation
is
Proposition 1. A friendly consumer offer should not charge for access to the network and A = 0. Instead, a fied and continual charge, f > 0 and f = rl, should generate the corresponding
Efficient tariffs to market district heat
775
stream of revenues to compensate for the initial investment and for interest. Marginal costs determine the price for district-heat, p = c. Summarizing, (0, rl, c) characterizes the optimal contract. This proposition is intuitively plausible. Due to their impatiencet relative to the d.h. company consumers prefer an annual payment of a fixed charge f = rl to a single initial investment I. The variable part of the tariff to a single initial investment I. The variable part of the tariff for consumed heat equals the marginal costs, which is a standard result in welfare analysis. The possibility to split tariffs into fixed and variable components allows for marginal-cost pricing and simultaneously covers total costs. This flexibility provides an edge over those competing fuels that must include fixed costs in the linear tariff. Linear tariffs, (A, f, p) = (0,0,p), create a dead-weight loss if the associated price p must cover variable plus lixed costs. The reason is that prices exceeding marginal costs lead to comfort sacrifices of the consumers. This contract shows some resemblance to but also important differences from real-world arrangements. Typically, all elements of (A, f, p) are positive. As in Proposition 1, fixed costs skim off the bulk of the revenues and fuel costs are typically low. This fact complies with the preceding result because the marginal costs of heat, especially from cogeneration, are small. The difference is that many d.h. companies charge for access to the network. The motivation from the company’s point of view is fairly obvious: to recover the sunk investments as rapidly as possible. However, this policy increases the consumers’ costs and thus slows the market penetration of district-heat. The basic structure of the contract specified in Proposition 1 is not only optimal with respect to welfare but also applies to a private monopoly. More precisely, the following proposition holds (the proof is given in the Appendix): Proposition 2. A profit maximizing d.h. company will offer the consumer the contract (0, f, c). In other words, the monopoly applies a two-part tariff and charges marginal costs. However, the jked charge f > ri provides the profit. This result may be of little relevance for the current organization of the d.h. industry, which is characterized by social ownership and/or heavy regulation. However, Proposition 2 applies once d.h. utilities are deregulated.
3. PRICE
DIFFERENTIATION
(RAMSEY
PRICES)
An important characteristic of marketing district-heat is the large portion of fixed costs required to extend the network. Moreover, these costs are not only fixed but also sunk. Thus, wrong decisions cannot be corrected, nor can the invested money be recovered. Therefore, a subtle strategy is necessary to market d.h. successfully. Proposition 1 stresses one way to attract new customers. Another possibility is to differentiate the contracts according to the consumers’ willingness to pay. This differentiation is restricted to the fixed charges f because an optimal contract is completely specified by fixed charges and heat prices and the latter must always balance the marginal costs c. In the following exposition, we will employ the concept of second-best pricing, called Ramsey prices. ’ To simplify the exposition, we suppose that the fixed and sunk investments I,, are necessary to serve a region. Additionally, the investments i are required to connect a particular customer to the grid. Two classes of consumers consider the acquisition of district-heat. Both groups are equally large and each group consists of n members. The only difference is that the first group’s threshold &, is larger than the second group’s, i.e., Y1 > X2. For example, the first group uses an inefficient and overall costly fuel, e.g., coal, and thus has a higher willingness to pay for district-heat because they would accept any contract satisfying X 6 X1. On the other hand, the second group’s willingness to pay for d.h. is lower for various reasons, e.g., they use a more convenient fuel such as oil. The opposite applies, if a consumer is less impatient than the d.h. company
(r > 6),
then A = I,f = 0.
776
FRANZWtRL
The total investments to hook up an entire region to the d.h. network amount to (4 + hi). a uniform tariff, identified by the superscript U, must result in the same tixed charge
Thus,
f” = r{[?J(2n)]
+ i}
(6)
to each consumer. No marketing problem arises if the associated consumer expenses X” are less than X, because then both consumer groups would switch to d.h. and share the common costs &, equally. We suppose that the second group rejects this contract while the first group accepts the offer. In this case, X(O,r(e+i),
c)
OCeCiJ2n.
(7)
The first inequality involves the assumption that the second group will contribute a small factor E to the fIxed costs b but not their appropriate share (second inequality). The last inequality in Eq. (7) expresses the fact that the first group would accept the offer in Eq. (6). The average costs rise for the first group when the second group refuses to join the d.h. network, because the sunk costs & must then be raised from n instead of 2n consumers. Indeed, the costs associated with this uniform contract may exceed the higher reservation price X2. As a consequence, nobody acquires district-heat and both consumer groups use their initial fuel and pay X1 and X2, respectively. However, price differentiation allows reductions in costs for both groups if the “willingness to pay” exceeds the total cost. We define ni as the maximal willingness of consumer i to contribute to the common costs b, which is the difference between the opportunity costs X, and the costs associated with a contract without coverage for &, i.e. 0 < [x2 - X(0, ri, c)]: =
ar2 < xl:
=
[Z1 - X(0, ri, c)].
(8)
The assumption of a sufficient willingness to pay ensures that n(rci + ark) > (r&,/6). Hence, it is possible to share the common costs I,, such that both consumer groups gain as compared to their initial expenses Xi (i = 1,2). We propose to differentiate according to the individual willingness to pay by writing 4 = r{i + [Jci/(nl+
~dl(b/n)),
j= 1,2.
(9)
These contracts (0,4, c) with Z = 1, 2, ensure that everybody acquires district-heat. Moreover, everybody gains compared with the uniform contract specified by Eq. (6) because the second group will pay less than their reservation price X2 and less revenues have to be raised from the first group to cover lo. In technical terms, this arrangement constitutes a Pareto improvement. Prices that use willingness to pay to split unattributable costs are called Ramsey-prices.? Of course, discriminatory Ramsey pricing leads to a Pareto improvement only if additional consumers can be attracted, i.e., group 1 in the example hooks up to the network. Otherwise (e.g., when X” < X2 < X1), discriminatory pricing cannot reduce the costs to one customer by raising the payments for another consumer. Ramsey-prices are second-best in the sense that they maximize the social welfare if marginal-cost pricing does not cover the costs and if a deficit is infeasible for various reasons.’ Moreover, this arrangment may be considered to be fair because each group has to spend the same fraction, i.e. X1 - X1(0, ri, c)
r
Xl
=iin,=
&In
X2-
X2(0,
6
c) (10)
x2
Here, we refer to maximal willingness to pay for IO, Ed,, which becomes a proxy for the potential gain. The specified arrangement does not lead to cross subsidizations$ because each consumer covers the costs attributable to his or her behaviour. Indeed, cross subsidization is unlikely for tRamsey prices for a multi-product industry confronted by a given system of demand are determined by the inverse elasticity rule, i.e., the relative mark-up over marginal costs is proportional to the reciprocal value of the (absolute) price elasticity. A commodity with a low price elasticity corresponds to a high willingness to pay, i.e., the value to the consumer exceeds the price by far. $Cross subsidies are not only inefficient but also unfair.’
Efficient tariffs to market district heat
777
a competitive company. Suppose that the company engages in cross subsidizations and uses the revenues from the second group to win also the first group that would not join even for fi = i. Then the second group of consumers could reject the offer, or another company could make a lower bid to serve the second market only. In fact, Ramsey prices may help to protect a local monopoly from the entry of competitors. Baumol et al3 prove that such a regime is sustainable, i.e., does not provoke entry, if the cost function satisfies well defined conditions. This analysis addresses the examples of Faulhaber4 who showed that Ramsey prices need not be sustainable. At least three groups (e.g., oil, gas, and coal heating systems) and a U-shaped average cost curve are necessary to construct an example parallel to that of Faulhaber. Our methodology implies monotonically declining average-cost curves and thus Ramsey prices are sustainable even if additional groups are considered.
4. SUMMARY
We have considered the design of efficient contracts to market district-heat. More precisely, the question is which system of charges for access, a tixed fee or payment for each unit of heat, covers the expenses of the company and is simultaneously attractive for new potential customers. The optimal welfare contract would not charge for access if consumers are impatient, as is most likely. Instead, a tixed charge should generate a revenue stream to compensate for the initial outlays and interest. Heat should be priced according to marginal costs. Surprisingly, even a monopolist would offer heat at marginal costs and would also refrain from access charges, but he would charge a larger fixed fee. The possibility to split tariffs provides an edge to d.h. vis a vis competitive fuels because the d.h. consumer receives heat at the marginal costs. This saving reduces the comfort sacrifice that is associated with the use of other fuels, for which a charge must be included to cover fixed and marginal costs. It is possible to imagine consumers who choose d.h. heat rationally despite its greater cost compared to liquid fuels because they can afford higher levels of comfort with lower marginal costs. The ratio of fixed to marginal costs is much lower for conventional fuels and, therefore, gains from a two-part tariff will presumably not outweigh the institutional costs associated with such an arrangement. The arguments of Sec. 3 encourage price differentiation. This policy may attract new customers with a lower willingness to pay, which allows them to share the common costs within a wider group of consumers. Price differentiation may appear to be discriminatory and potentially unfair at first sight. Actually, price differentiation does not lead to cross subsidizations because everybody has to cover at least the costs attributable to his or her consumption. Furthermore, this policy will not be unfair because everybody gains in the form of lower costs. The reason is simple. Accepting all consumers with a positive, albeit below average, willingness to contribute to the common costs lowers the fraction to be covered by each individual who ultimately chooses district heat.
REFERENCES
1. W. J. Baumol and D. Bradford, Am. Econ. Rev. 60,265 (1970). 2. G. R. Faulhaber and S. B. Levinson, Am. Econ. Rev. 7l, 1083 (1981). 3. W. J. Baumol, E. E. Bailey, and R. D. Willig, Am. Econ. Rw. 67, 350 (1977). 4. G. R. Faulhaber, Am. Econ. Rev. 65,966 (1975). 5. M. D. Intriligator, Mathematical Optimization and Economic Theory, Prentice-Hall,
New York, NY
(1971). APPENDIX
Proposition 1 The constraints of the optimization problem given in Eq. (5) are concave (due to the supposition of concave revenues). However, the objective (viewed as a maximization problem)
7%
FRANZWlRL
need not be concave because the revenues enter negatively. Thus, the Kuhn-Tucker provide only necessary optimality conditions. We define the Lagrangian
L=-,+(D+F+pE)+~ f+(P-c)E+A-I s
r
+FA+~
I
1
f+/, 2
conditions
p 3
(Al)
and use the Kuhn-Tucker multiplier A.for the revenue constraint of Eq. (3) and p, (i = 1,2, 3) for the non-negativity of the instruments. Equation (Al) shows the integrals in explicit form. This procedure leads to the following necessary optimality conditions:’
auaA= auaf = -(i/q
-(l
- ApI) = 0,
+ p/r) + ji2 = 0,
aL_D’E’+pE’+E+A(p-c)E’+E 6
ap-
)L
(A2)
+p3=0, r
f+(P-c)E+A-I E 1, ~0
AsO
r
1.0 = 0, P2f
=o,
clsp=o, The joint conditions
(A3)
,
P,~O,
W)
P250,
647)
p390.
M-9
of Eqs. (A3) and (A7) and the assumption j42= (l/S) - (A/r) = [(t - M)/(&)]
3 oj
6 > r imply
(r/S) 5 A.
(A9)
Therefore, no interior solution is possible for the decision variable A. The opposite A > 0 implies p1 = 0 [from Eq. (A6)], thus, I = 1 [from Eq. (A2)] and hence a contradiction follows. Therefore, A = 0 and pl 3 0. We assume that interior solutions exist for the instruments f and p. It will be shown that these assumptions, and the implications c(~= p3 = 0 are compatible with the specified optimality conditions. From the supposition p2 = 0 follows that A = r/6 due to Eq. (A3). Therefore, the strong inequality must hold for pl, ccl > 0, because (r/6) < 1 and Eq. (A2) applies. Substitution of this result into Eq. (A4) and using p3 = 0 gives (A/r)[(p - c)E’] = (l/S)[D’E’
+pE’] = 0.
(AlO)
The term between the square brackets on the rhs of Eq. (AlO) must vanish because D’ = -p if the consumer determines the fuel demand rationally, i.e., by minimizing {D(E) -pE}. Therefore, the Ihs of Eq. (AlO) must also vanish, which requires p = c. With this solution, A = 0, f > 0 and p = c; we may compute the amount of fixed charges f from the revenue constraint, which reduces to
(f/r) = I.
(All)
Thus, f = rl. The sequence (0, ri, c) corresponds to the optimal strategy, although the Kuhn-Tucker conditions are not sufficient because the space of admissible policies is compact due to the supposition of equality between costs and revenues. The theorem of Weierstral3 therefore conditions. ensures the existence of an optimum, which must satisfy the Kuhn-Tucker However, (0, ri, c) is the unique solution of Eqs. (A2)-(AS) because different assumptions contradict at least one of the optimality conditions Eqs. (A2)-(A8). Hence, (0, ri, c) must be the optimum solution. Proposition 2 The optimal welfare policy is to accept any customer who is willing to cover the costs. However, a private monopolist might refuse to serve all of these customers if this restrictive policy can enhance profits. We suppose that there exists a distribution of reservation prices &
Efficient tariffs to market district heat
779
such that N(X) denotes the number of consumers acquiring a d.h. contract if their costs are equal to X. We assume N’ < 0 and that the implied profit function (R - K)N is concave in all instruments (A, f, p). A monopolist will find
(Al21 subject to the conventional
non-negativity
constraints.
We again define the Lagrangian
L=N(R-K)+FIA+YJ+c~~P
(A13)
and obtain the following necessary and sufficient (due to the assumption of concave revenues) optimality conditions: aLlaA = (R - K)N’aXldA
+ NaRIdA + /.J~= 0,
(AW
auaf= (R- K)N'aXlaf+NaRlaf+j4,=0, aLlap=(R-K)N~axlap+NaR~ap-NaKlap+~3=0,
(A13 6416)
plus the suppressed complimentary slackness conditions of Eqs. (A6)-(A8). We assume a policy similar to Proposition 1, i.e., ccl 3 0 and pL2= c(~= 0. Substitution partial derivativest
axiaf= 116, axiap=~Id, aRiaf=lir, aRiap=(E +pE’)lr, aKiap= cE’lr,
of the
6417) 6418) 6419)
simplifies Eq. (A15) to
[(R-K)N'/6]+Nlr=O, which implies ccl > 0 because of Eq. (A14). Substitution into Eq. (A16) and rearrangement yields
of the derivatives of Eqs. (A17)-(A19)
E{[(R - K)N’lS] + Nlr} + (p - c)NE’lr = 0. The term between the braces must vanish in order to satisfy the equality Therefore, p = c whenever the monopolist chooses to supply (N > 0).
tThe derivative aX/ap
involves again the use of (D’ + p) = 0.
6420)
WI) in Eq. (A20).