Download Optimal Capacity and Two-Part Pricing for Natural Gas

Optimal Capacity and Two-Part Pricing for Natural Gas Pipelines under
Alternative Regulatory Constraints
Matthew E. Oliver, David Finnoff, Charles F. Mason*
February, 2014
Abstract
The market for natural gas pipeline transportation is comprised of two distinct tiers. The
primary market, in which pipelines sell ‘firm’ capacity contracts using a two-part tariff
structure, is subject to rate-of-return regulation.
In the secondary market for
transportation services, owners of firm contracts may either utilize or release their
contracted capacity. Both activities are transacted at decentralized market-based prices,
potentially earning firm contract owners scarcity rents. This paper extends a rich
literature on optimal capacity and pricing to account for these features of the natural gas
pipeline market, deriving optimization rules for pipeline pricing and capacity when
demand for the shipping service is stochastic and stationary. For comparison, the
analytical model presents three alternative regulatory regimes – unregulated monopoly, a
Ramsey second-best solution, and rate-of-return. As the optimality conditions for each
case are too complex to solve analytically, we parameterize and numerically solve each
set of conditions for different distributional scenarios. Results indicate that optimal
capacity under rate-of-return regulation is less than what would occur under a Ramsey
second-best solution. An important aspect of the problem is that the latter accounts for
the external effect of capacity on the consumer and producer surpluses at the markets
connected by the pipeline, whereas the former does not. Furthermore, when uncertainty
in the secondary market is high, the pipeline’s optimal capacity is scarcely greater than in
the unregulated monopoly optimum. Our results are consistent with the classic AverchJohnson hypothesis that a rate-of-return regulated firm will employ a greater capital stock
relative to the unregulated optimum. However, the result that the pipeline’s optimal
capacity under ROR is less than the Ramsey second-best socially optimal level implies
that under-investment in pipeline capacity may exacerbate congestion issues.
Calculations of social welfare under each regulatory regime show that overall economic
welfare is sub-optimal under rate-of-return regulation in each distributional scenario.
*Oliver: Georgia Institute of Technology, School of Economics; Finnoff: University of
Wyoming, Department of Economics & Finance; Mason: H.A. “Dave” True Professor of
Oil & Natural Gas Economics, University of Wyoming, Department of Economics &
Finance.
1
1
Introduction
Natural gas continues to play an increasingly prominent role as a primary energy
resource, particularly in the United States.
Domestic supplies have increased
dramatically over recent years due to advances in extraction technology, and demand has
steadily risen as electrical plant managers shift toward natural gas in response to
increased public concern over carbon emissions from coal-fired electricity generation.
However, the ability of the natural gas market to link supply and demand centers is
fundamentally limited by the capacity of the natural gas pipeline transmission network.
Insufficient capacity over certain routes results in the emergence of bottlenecks and
network congestion, which are known to have systematic and measurable effects on
transportation costs. Increased transportation cost drives apart the natural gas spot prices
at any two nodes on the network, indicating reduced market integration and, more
importantly, potential negative welfare effects.
Furthermore, federal regulation of
interstate natural gas pipelines, while having moved considerably toward a more
liberalized restructuring over the past two decades, maintains some important controls
over rate-setting behavior. This paper’s broad intent is to illuminate potential interactions
between this regulatory framework, the pipeline capacity and transportation markets, and
the natural gas spot market. Our results suggest that these interactions may result in
suppressed investment in pipeline capacity—a situation that exacerbates congestion
issues and undermines efficiency.
The natural gas pipeline industry is unique in that the market for pipeline
transportation services is comprised of two distinct tiers. Pipelines sell ‘firm’ transport
capacity contracts to gas traders, local distribution companies (LDCs), and other parties
in a primary market. The Federal Energy Regulatory Commission (FERC) regulates the
two-part tariff paid by primary customers – both the capacity reservation and usage
charges – by way of a rate-of-return (ROR) mechanism based on the pipeline’s cost-ofservice. Owners of firm capacity contracts are free to recover the market value of their
reserved capacity via unregulated secondary markets. 1 Recovery of this value can occur
through the direct mediation of gas transactions, in which the market value of capacity is
1
See Oliver (2013) for a more thorough discussion of the primary and secondary markets.
2
built into the commodity price spread, or by releasing unused capacity to others at the
market rate in a capacity-release market. Both these secondary market relationships are
marked by considerable uncertainty. As such, the specific focus of this paper is the effect
of stochastic secondary market transportation demand conditions on primary market
reservation demand decisions, which (i) depend in part on the regulatory system in place,
and (ii) ultimately affect the pipeline’s optimal capacity and two-part tariff structure.
Previous research (Marmer et al., 2007; Brown and Yücel, 2008) has asserted that
under the current regulatory framework the incentives for a pipeline to invest in greater
capacity are weakened, and are associated with potential market distortions in the form of
wealth transfers from the pipeline to the owners of capacity contracts.
Such transfers
may arise in cases where persistently constrained transport capacity regularly causes the
secondary market transportation charges levied by firm capacity contract holders to
exceed the regulated primary market tariff. As these revenue streams are diverted away
from the pipelines and toward firm capacity holders, pipelines’ incentives to invest in
greater capacity are reduced, exacerbating congestion issues over time. Our results
provide compelling evidence in support of these conjectures.
We investigate optimal capacity and two-part tariff pricing structures for a natural
gas pipeline when demand for the shipping service is stochastic but non-increasing over
time. We assume the pipeline to be a local monopoly over the route in question, and that
it exhibits increasing returns-to-scale technology. Because the stochasticity of shipping
demand occurs in the secondary market, we model its effect on firm contract owners’
capacity reservation decisions. 2 Intuitively, firm capacity is a factor of production for gas
traders. We thus derive an aggregate capacity reservation demand function, which we
then employ in the pipeline’s optimization problem. Transport demand uncertainty in the
secondary market feeds into the pipeline’s capacity and pricing decisions through its
effects on firm reservations (see Figure 1). For comparison, we consider three regulatory
alternatives: an unregulated monopoly optimum, a Ramsey second-best solution, and an
ROR regulated optimum based on FERC’s rate-design mechanism.
2
We consider unregulated gas traders only. LDCs are regulated at the state and federal levels, so to avoid
any complications we assume that all primary market capacity reservations are made by gas traders. See
Secomandi and Wang (2012) for a general overview of the standard operational activities of natural gas
traders (referred to as ‘merchants’ in that paper).
3
Figure 1. Schematic design of the optimal capacity and two-part tariff pipeline problem.
Because the optimization rules derived in each case are too complex for analytical
comparison, we provide numerical solutions to compare optimal capacities and prices
across the three scenarios. Our results are consistent with the classic Averch and Johnson
(1962) effect—ROR regulation increases the pipeline’s optimal maximum capacity
relative to the unregulated monopoly optimum. This increase is welfare-improving. A
key contribution of this paper, however, is the effect of uncertainty on the degree to
which the Averch-Johnson (A-J) effect is manifested.
With low secondary market
uncertainty, the A-J effect results in an optimal capacity that, although it falls well short
of the Ramsey second-best solution, is relatively closer to the second-best solution than it
is to the monopoly solution. Yet when uncertainty is increased, the extent by which the
ROR optimal capacity exceeds the monopoly optimal capacity becomes smaller, and the
degree to which it falls short of the second-best solution increases. Uncertainty leads to
an attenuation of the A-J effect, and causes ROR to perform more poorly relative to the
second-best solution. Numerical results show that significant wealth transfers from the
pipeline to the capacity contract holders occur under ROR regulation that do not occur in
the second-best solution. 3 We conclude that ROR pricing may be a poor instrument for
regulating pipelines serving routes marked by high transportation demand uncertainty. In
lieu of a transition to some other form of regulation for natural gas pipelines (i.e.
incentive-based), this implies that policies designed to reduce secondary market
uncertainty are likely be beneficial in terms of both technical and economic efficiency.
3
This lends support to the assertions of the Marmer et al. (2007) and Brown and Yücel (2008) analyses.
4
2
Two-Part Tariffs and Optimal Capacity: A Review
Two-part tariffs improve pricing efficiency for public utilities with large fixed costs. In
such cases, standard marginal cost pricing results in a deficit to the firm that requires
substantial tax/subsidy transfers to cover costs. Coase (1946) argued for a two-part tariff
in the presence of increasing returns as an efficient means of covering this loss when
transfers are not possible. With a uniform access fee, the profit-maximizing two-part
tariff consists of a usage price that exceeds marginal cost and an access fee that extracts
the entire consumer surplus of the customer with the least demand for the commodity
(Oi, 1971). Feldstein (1972) examined the welfare loss of the monopolistically optimal
two-part tariff, and solved for a uniform structure that balances efficiency and
distributional equity across households with different incomes. When income elasticities
of demand are non-zero, infra-marginal demand effects can offset the exclusion of
marginal customers induced by an increase in the access charge (Ng and Weisser, 1974;
Schmalensee, 1981). Under ROR regulation, a monopolistic two-part tariff creates a
trade-off between reductions in access and usage fees depending on whether increasing
the customer base or increasing output requires a larger marginal increase in capital
(Sherman and Visscher, 1982). Vogelsang (1989) developed an incentive-based two-part
tariff mechanism in which the firm is subject to an iteratively regulated access price, but
is allowed to freely set the usage price. Over time the firm finds it optimal to set the
usage price such that it converges to the second-best Ramsey price in steady state. If the
regulator sets the two-part tariff as a price-cap index, the firm is able to trade off
congestion costs against capacity expansion costs. This incentive mechanism would
provide the firm with sufficient motive to expand capacity whenever the average costs of
congestion exceed expansion costs (Vogelsang, 2001).
Most analyses of pricing and capacity under ROR regulation investigate the
propositions of Averch and Johnson (1962): first, that an ROR-regulated firm will
employ a higher amount of capital than it would if unregulated; and second, that a
“regulatory bias” will cause the firm to operate “inefficiently in the sense that (social)
cost is not minimized at the output it selects.” These results have come to be known in
the literature as the ‘A-J effect’. The regulated firm finds it advantageous to overcapitalize because its cost of capital is effectively less than the market cost. An early
5
empirical test of ROR-regulated electric utilities found evidence in support of the A-J
effect (Spann, 1974).
When demand conditions are stochastic, however, the first
proposition cannot be generalized analytically (Perrakis, 1976), whereas the second
proposition has been shown to hold (Das, 1980). Price caps may be superior to ROR
regulation in the presence of uncertainty, in that they avoid A-J effect inefficiencies
(Braeutigam and Panzar, 1993; Liston, 1993). A recent study by Cambini and Rondi
(2010) finds that among a sample of European energy utilities, those subject to incentivebased regulation had a higher investment rate than those under ROR regulation. These
authors argue that because expansion and modernization of energy infrastructure is
crucial to the efficient pricing and allocation of energy resources over the long run,
delayed investment can be associated with large social costs.
In all cases, uncertainty regarding demand for a service such as natural gas
transmission affects the optimal scale of investment in capacity. Capacity and pricing
decisions must be made before the actual quantity demanded in any period is known.
This uncertainty requires the service provider to invest in capacity based on his expected
quantity demanded and the characteristics of its distribution. The problem was originally
cast as one of efficient peak-load pricing and capacity (Boiteaux, 1949). 4 Laffont and
Tirole (1993, p. 20-21) made the point that many industries utilize facilities (i.e. electric
power plants, pipelines, or railroads) to produce the same physical good at different
times, during which demand may or may not exceed production capacity. The marginal
cost of production when capacity is fully utilized is clearly greater than when it is not, as
capacity must be expanded to meet marginal demand when existing capacity is fully
utilized. Uncertainty in the load profile creates problems for traditional methods of
marginal cost pricing when demand fluctuations both above and below maximum
capacity occur regularly. Williamson (1966) mathematically formalized the peak-load
problem by generalizing Boiteaux’s result under indivisibilities in plant expansion and
constant returns-to-scale for peak- and off-peak-load periods of unequal length. Bailey
(1972) later examined peak-load pricing under various forms of regulatory constraint, the
most standard being ROR regulation. Meyer (1975) offered an extension of monopoly
pricing and capacity investment that accounts for uncertainty by allowing demand to vary
4
For a complete historical survey of the peak-load pricing literature, see Crew et al. (1995).
6
stochastically, demonstrating that an optimal pricing-capacity choice is one in which
capacity regularly exceeds demand. 5 This countered the normative suggestion of Averch
and Johnson that excessive investments under ROR regulation are made out of motives of
purely self-interested profit maximization.
Rather, an ROR regulated firm may be
optimally managing risk by meeting “reliability standards for the service it provides.”
FERC regulations require pipelines to manage risk, in part by requiring capacity
investments to be underwritten by firm contracts. In this way, some risk is effectively
transferred to the firm capacity owners, an arrangement which may in fact reduce the
incentive for the pipeline to over-invest in capacity as Meyer’s work suggests.
The key paper for our analysis is that of Sherman and Visscher (1978, hereafter SV), who examine second-best pricing schemes with stochastic demand. S-V account for
the probability that in any period quantity demanded may exceed maximum capacity by
including in the operator’s optimization problem the minimization of “expected forgone
profits” resulting from demand in excess of maximum capacity. Forgone profits can thus
be thought of as the opportunity cost of not investing in greater capacity. Our primary
adaptation of the S-V model is the inclusion of a secondary market, which allows for
more efficient allocation of transportation services when the capacity constraint is
binding. Without the secondary market, the possibility arises that prices may not always
clear the market efficiently, requiring some inefficient non-price rationing system. The
existence of a secondary market in which the implicit price of scarce available capacity
reflects its market value overcomes the need for non-price rationing.
With the above literature in mind, our central result is that the optimal capacity of
an ROR-regulated pipeline falls short of the Ramsey second-best socially optimal level,
and as uncertainty in the secondary market increases, the ROR optimal capacity
decreases relative to the Ramsey level.
Ultimately, suppressed capacity investment
increases congestion, inflating the transportation charge, and reducing overall social
welfare. We now turn to an analytical model of optimal capacity and two-part pricing.
5
In peak-load problems, the availability of storage has been shown to reduce the price differential between
peak and off-peak periods, and to reduce the need to bring high-cost plants into production. In such cases,
optimal capacity occurs where the shadow value of additional capacity is exactly equal to its long-run
marginal cost net of marginal operating and storage costs (Gravelle, 1976; Nguyen, 1976). We do not
include storage in the following model, although it could be added in future extensions. Presumably, the
addition of storage would serve to mitigate the external effects of constrained capacity on spot prices, due
to its known effects on peak and off-peak prices.
7
Figure 2. Simple two-hub, one-pipeline network.
3
The Model
Our template for this problem is the simple two-hub, one-pipeline network of Cremer and
Laffont (2002), depicted in Figure 2. Flow from Hub 1 to Hub 2, denoted by 𝑦𝑡 (𝑡 =
1, … , 𝑇), is strictly unidirectional. Production, 𝑞𝑗 (𝑗 = 1, 2), is inelastically supplied to
Hub 𝑗 in period 𝑡. Consumption at Hub 𝑗 in period 𝑡 (i.e. the quantity of gas demanded
locally) is 𝑑𝑗,𝑡 .
For simplicity, we assume storage is not available in this system. Net balance of
the system implies that the two flow-balance identities define local demand at each hub.
𝑑1,𝑡 = 𝑞1 − 𝑦𝑡
𝑑2,𝑡 = 𝑞2 + 𝑦𝑡
(1a)
(1b)
In period 𝑡 available capacity on the pipeline linking Hubs 1 and 2, 𝑘𝑡𝑎 , is the difference
between the pipeline’s maximum capacity, 𝐾, and the flow of gas.
𝑘𝑡𝑎 = 𝐾 − 𝑦𝑡
(2)
The transportation charge from Hub 1 to Hub 2, 𝜏𝑡 , is a decreasing function of available
capacity: 𝜏𝑡 (𝑘𝑡𝑎 ), 𝜏𝑡 ′ (𝑘𝑡𝑎 ) < 0 (Oliver, 2013). Accordingly, one could interpret 𝜏𝑡 (𝑘𝑡𝑎 ) as
a derived demand function for available capacity—as available capacity becomes scarce,
its price (the transportation charge) rises. We assume a set of spot prices at the two hubs
�𝑝1,𝑡 , 𝑝2,𝑡 � is ‘feasible’ if 𝑝1,𝑡 ≤ 𝑝2,𝑡 .
The arbitrage condition states that the basis
differential between the two equilibrium spot prices at Hubs 1 and 2 must be equal to the
per-unit transportation charge (DeVany and Walls, 1995):
𝑝1,𝑡 = 𝑝2,𝑡 − 𝜏𝑡 (𝑘𝑡𝑎 ).
8
(3)
This condition implies that the transportation charge and the two spot prices must adjust
simultaneously to clear the secondary market and the spot markets at the two hubs.
3.1
The Individual Gas Trader’s Problem
S-V model a monopoly provider who chooses optimal pricing and capacity when demand
for the service is stochastic. Here, we model the individual gas trader as operating in a
competitive secondary market for shipping services in which price (i.e. the transportation
charge) and expected quantity demanded are exogenous.
Thus, our two main
modifications of the S-V framework as applied to the gas trader’s problem are 1) price is
not a choice variable in the trader’s problem, and 2) S-V model the monopoly provider’s
choice of the optimal capacity to build, whereas the competitive gas trader chooses the
optimal capacity to reserve. Given the pipeline’s reservation and usage charges, the
expected transportation charge, and the distribution of secondary market demand, by
maximization of expected profits the individual gas trader chooses an amount of capacity
to reserve. Despite the competitive structure of the secondary market, it is still possible
for the trader to capture economic rents. This is because the pipeline’s finite maximum
capacity creates a natural barrier to entry to the primary market once capacity is fully
reserved. Available capacity at any given moment is constrained, which through the
transportation charge creates potential scarcity rents for traders holding firm contracts.
A total of 𝑁 gas traders, 6 indexed by 𝑖, (𝑖 = 1, … , 𝑁), reserve pipeline capacity in
the primary market, and then utilize it to complete gas transactions in the secondary
market. In any period 𝑡, the quantity of gas transacted by trader 𝑖 cannot exceed his
contracted capacity, 𝑘𝑖,𝑡 . Each unit of capacity reserved incurs a reservation charge of 𝑃𝑟
per unit per period, 7 whereas each unit utilized incurs a usage charge of 𝑃𝑢 . Trader 𝑖
faces a stochastic, exogenous quantity demanded for shipping in period 𝑡, given by the
𝑁 represents the number of traders willing to commit to firm capacity contracts on the pipeline. For the
purpose of demonstration, we make the simplifying assumption that 𝑁 is fixed and exogenous. A more
complex model would endogenize 𝑁, where the number of traders entering the market is defined by a
∗
, yielded negative expected profits,
participation constraint, such that if the optimal reservation demand, 𝑘𝑖,𝑡
the trader would opt not to enter the market. This implies some value, 𝑘�𝑖 , such that profit is zero, and we
∗
thus require 𝑘𝑖,𝑡
≤ 𝑘�𝑖 . We implicitly assume that the participation constraint is satisfied for all 𝑁 traders.
7
In practice, a pipeline’s reservation charge is set at monthly intervals. According to FERC (1999),
because the average number of days in a month is 30.4, the daily reservation charge is set such that
𝑃𝑟 = 𝑃𝑚 /30.4, where 𝑃𝑚 is the monthly reservation charge.
6
9
random variable 𝑦𝑖,𝑡 , based on exogenous supply and demand factors in gas commodity
markets at nodes on the pipeline network. Hence, the shocks faced by any one trader are
faced by all other traders. Assume a log-normal distribution of 𝑦𝑖,𝑡 , represented by the
density function 𝑓𝑖 (𝑦𝑖,𝑡 ). 8 The cumulative distribution function (c.d.f.) of 𝑦𝑖,𝑡 is 𝐹𝑖 (𝑦𝑖,𝑡 ),
such that
𝐹𝑖 �𝑦�𝑖,𝑡 � = �
𝑦�𝑖,𝑡
0
𝑓𝑖 �𝑦𝑖,𝑡 �𝑑𝑦𝑖,𝑡 .
(4)
Assume that the distribution of 𝑦𝑖,𝑡 is stationary. 9
The distribution of the aggregate quantity demanded is 𝑔(𝑦𝑡 ), where 𝑦𝑡 = ∑𝑖 𝑦𝑖,𝑡 .
Similar assumptions made about the distribution of the individual demand quantities
apply to the aggregate. That is,
𝑦�𝑡
𝐺(𝑦�𝑡 ) = � 𝑔(𝑦𝑡 )𝑑𝑦𝑡
(5)
0
where 𝐺(𝑦𝑡 ) is the c.d.f. of 𝑦𝑡 . 10 The expected aggregate quantity demanded is 𝑦 𝑒 .
Given maximum capacity and the distribution of aggregate transportation
demand over (0, ∞), the expected transportation charge is given by
∞
𝜏 𝑒 ≡ � 𝜏𝑡 (𝑘𝑡𝑎 )𝑔(𝑦𝑡 )𝑑𝑦𝑡 .
(6)
0
Because determination of 𝜏 is very difficult in practice, we facilitate further discussion by
assuming linear demand and supply in the commodity spot markets, implying a linear
8
S-V model quantity demanded as an expected value plus an error term that is randomly distributed over
(−∞, ∞), and then make additional restrictions on the error term to ensure that the quantity demanded is
never negative. We avoid this awkward structure by dispensing with the error term and simply assuming
that 𝑦𝑖,𝑡 is distributed over (0, ∞). We have chosen a log-normal distribution for mathematical expedience,
although in reality other non-negative distributions may also be plausible.
9
We make this assumption for analytical convenience, acknowledging that in reality the distribution may
change over time. Such changes may occur due to weather-related and/or seasonal variation in pipeline
transportation demand, or due to structural shifts in natural gas demand and supply such as the move from
coal to natural gas in electricity generation or advances in extraction technology (i.e. hydraulic fracturing).
10
For any sum of random variables, the mean of the sum is equal to the sum of the means, 𝑦 𝑒 = ∑𝑖 𝑦𝑖𝑒 . We
implicitly assume some degree of correlation between the 𝑁 traders’ quantities demanded—when overall
demand is high, all the traders’ quantities demanded will be high, and vice-versa. For a sum of random
variables that are not independently distributed, the variance of the sum is equal to the sum of the variances
and covariances, 𝜎 2 = ∑𝑖 𝜎𝑖2 + ∑𝑖≠𝑗 𝜎𝑖 𝜎𝑗 . Thus the aggregate standard deviation is 𝜎 = √𝜎 2 . The sum of
log-normally distributed random variables is not log-normal, and derivation of an exact closed-form
representation is impossible (Krekel et al., 2004). However, Milevsky and Posner (1998) have shown that
when log-normally distributed variables are correlated, the distribution of the sum converges to the inverse
gamma distribution as 𝑛 ⟶ ∞.
10
form for 𝜏𝑡 (𝑘𝑡𝑎 ). This assumption, while likely to be unrealistic, greatly simplifies the
problem in that 𝜏 being linear in 𝑘 𝑎 implies that 𝜏 𝑒 = 𝜏(𝐾 − 𝑦 𝑒 ).
Define trader 𝑖’s planning horizon as 𝑇𝑖 . If each period is of equal length 11 and
the distribution of 𝑦𝑖,𝑡 does not change over time, we can cast the problem in terms of the
expected profit of a single period, assuming no discounting. 12 In a given period, trader
𝑖’s profit is equal to the transportation charge net of the usage charge, 𝜏𝑡 (𝑘𝑡𝑎 ) − 𝑃𝑢 , times
the quantity shipped ( 𝑦𝑖,𝑡 if 𝑦𝑖,𝑡 ≤ 𝑘𝑖,𝑡 , and 𝑘𝑖,𝑡 if 𝑦𝑖,𝑡 > 𝑘𝑖,𝑡 ), less total reservation
charges, 𝑃𝑟 𝑘𝑖,𝑡 . Because 𝑦𝑖,𝑡 and 𝜏𝑡 (𝑘𝑡𝑎 ) are uncertain, the trader must choose 𝑘𝑖,𝑡 so as
to maximize expected profit, accounting for the fact that for all 𝑦𝑖,𝑡 > 𝑘𝑖,𝑡 he will be
unable to meet 𝑦𝑖,𝑡 − 𝑘𝑖,𝑡 units of shipping demand.
Following S-V, the trader’s
objective function is thus defined by the expression,
𝜋𝑖𝑒
∞
= (𝜏 𝑒 − 𝑃𝑢 ) �𝑦𝑖𝑒 − � �𝑦𝑖,𝑡 − 𝑘𝑖,𝑡 �𝑓𝑖 �𝑦𝑖,𝑡 �𝑑𝑦𝑖,𝑡 � − 𝑃𝑟 𝑘𝑖,𝑡 .
𝑘𝑖,𝑡
(7)
In each period, trader 𝑖’s expected quantity demanded is 𝑦𝑖𝑒 and the expected variable
profit margin on each unit shipped is 𝜏 𝑒 − 𝑃𝑢 . He minimizes the opportunity cost of
expected forgone variable profits when demand cannot be satisfied because 𝑦𝑖,𝑡 > 𝑘𝑖,𝑡 .
This feature is represented by the integral, multiplied by 𝜏 𝑒 − 𝑃𝑢 . We henceforth refer to
this term as “expected opportunity cost”. The last term is the total per-period reservation
charge. 13
S-V model a planning horizon that is subdivided into 𝑇 periods of unequal length, where each period
lasts a fraction 𝛼𝑡 of the total planning horizon. This requires summation over the 𝑇 unequal periods.
12
S-V do not employ a discount rate, and we maintain this assumption purely to reduce the mathematical
complexity of the problem. Incorporating a non-zero discount rate would require summation over 𝑇𝑖
periods and multiplication of each period’s profit function by the discrete time discount factor, 𝛿𝑖 =
1/(1 + 𝑟)𝑡 . But because the profit function and its parameters do not change over time, we have no reason
to suspect that it would alter the qualitative results of the model.
13
We maintain the S-V assumption of risk neutrality. Risk preference in production has received
significant attention in the existing economic and finance literature, along with an equally significant
treatment of risk management. In expected utility maximization, ‘approximate’ risk-neutrality holds even
when the stakes are large and economically important, and the expected utility framework does not always
provide an accurate measure of risk aversion (Rabin, 2000; Rabin and Thaler, 2001). In natural gas
markets, primary capacity reservation contracts imply some degree of risk-sharing between the pipeline and
its firm customers, but in contract theory there is little evidence to support risk aversion in contract design.
Rather, transaction cost models based on the assumption of risk neutrality have found empirical support.
Mulherin (1986) finds evidence of risk neutral, transaction cost based design in long-term natural gas
contracts. Allen and Lueck (1995) find general support for the risk neutral transaction cost approach,
arguing that “risk aversion is not useful in explaining contracts”, but warn that this “does not necessarily
suggest widespread risk neutrality”. In competitive energy markets the use of futures and forward contracts
11
11
Trader 𝑖 ’s sole decision variable is an amount of capacity to reserve in each
period. The first-order condition of the expected profit function with respect to 𝑘𝑖,𝑡 yields
∗ 14
the rule for optimal capacity reservation, 𝑘𝑖,𝑡
.
(𝜏 𝑒
−𝑃
This condition can be rewritten as
𝑢)
∞
� 𝑓𝑖 �𝑦𝑖,𝑡 �𝑑𝑦𝑖,𝑡 = 𝑃𝑟
∗
𝑘𝑖,𝑡
∗
(𝜏 𝑒 − 𝑃𝑢 )�1 − 𝐹𝑖 (𝑘𝑖,𝑡
)� = 𝑃𝑟
(8)
(9)
∗
where 1 − 𝐹𝑖 (𝑘𝑖,𝑡
) is the probability of excess demand. 15 This rule states that the trader
will reserve an amount of capacity such that the marginal increase in expected variable
profit per period (gained by having access to additional capacity and thus the ability to
ship an additional unit) is equal to the marginal reservation charge per period.
The gas trader’s capacity reservation decision as given by the solution to the
above first-order condition is representative of an expected factor demand function. Firm
capacity is the essential input needed to produce the output of gas shipping services.
Thus, an individual trader’s demand for firm capacity is a function of the reservation
charge, the usage charge, the maximum capacity of the pipeline, his expected quantity
demanded of secondary market transportation services, 𝑦𝑖𝑒 , and its standard deviation, 𝜎𝑖 .
∗
𝑘𝑖,𝑡
= 𝜓𝑖,𝑡 (𝑃𝑟 , 𝑃𝑢 ; 𝐾, 𝑦𝑖𝑒 , 𝜎𝑖 )
(10)
∗
It is straightforward to show that 𝑘𝑖,𝑡
is decreasing in 𝑃𝑟 , 𝑃𝑢 , 𝐾, and 𝜎𝑖 , and increasing in
𝑦𝑖𝑒 (see Appendix 1). The first two are representative of the Law of Demand. Given the
expected aggregate quantity demanded, an increase in the maximum capacity of the
pipeline reduces the individual trader’s reservation demand because it reduces the
to hedge against price risk is widespread, and risk preferences have been shown to affect behavior in these
markets. For example, electricity cannot be stored, and greater risk-aversion significantly increases agents’
choices of hedge position, reducing the value at risk (Vehviläinen and Keppo, 2003), and also affects the
market risk premium (Benth et al., 2008). We abstract away from including hedging behavior in our
analysis in order to avoid considerable additional complexity. Given that we focus our model on the
pipeline’s optimal choice of maximum capacity under different regulatory scenarios, the relative direction
of change should be similar in spirit regardless of the risk preference assumption, although the magnitudes
might differ.
14
Note that in taking the derivative of (7) with respect to 𝑘𝑖,𝑡 , there is also a marginal effect resulting from
a change in the lower limit of the integral. However, because the integral is evaluated at 𝑦𝑖,𝑡 = 𝑘𝑖,𝑡 , this
effect cancels out in Equation (8).
15
The second-order sufficient condition for a maximum is satisfied. As long as 𝜏 𝑒 > 𝑃𝑢 , concavity is
∗
∗
apparent because −𝐹𝑖′ �𝑘𝑖,𝑡
� = −𝑓𝑖 (𝑘𝑖,𝑡
), which must be negative.
12
expected transportation charge.
A central question concerns how the structure of
uncertainty affects the capacity reservation decision. An increase in uncertainty, as
represented by an increase in the standard deviation of quantity demanded (i.e. an
increase in the mean-preserving spread), decreases the optimal capacity reservation.
Intuitively, as the probability mass of log-normal (and other non-negative) distributions is
skewed toward values below the mean, a higher standard deviation translates to a higher
probability that reserved capacity will exceed demand in any given period. Finally, it is
unnecessary to explain why an increase in the expected shipping quantity demanded
should raise the trader’s optimal capacity reservation.
The market capacity reservation demand function is simply the aggregate of
individual traders’ capacity reservation demand functions:
𝜓( 𝑃𝑟 , 𝑃𝑢 , 𝐾; 𝑦1𝑒 , … , 𝑦𝑁𝑒 , 𝜎1 , … , 𝜎𝑁 )
= � 𝜓𝑖 (𝑃𝑟 , 𝑃𝑢 ; 𝐾, 𝑦𝑖𝑒 , 𝜎𝑖 ).
(11)
𝑖
Suppressing notation, we denote the aggregate reservation demand function as
𝜓( 𝑃𝑟 , 𝑃𝑢 , 𝐾). Our next step is to utilize this primary market demand function in the
pipeline’s capacity investment and pricing decisions.
3.2
The Unconstrained Monopoly Pipeline’s Problem
We assume the pipeline has a local monopoly over the transport route in question. We
again follow the S-V model, but there are two important differences. First, the pipeline
uses a two-part tariff system. The two charges generate two distinct sources of revenue:
reservation revenue and usage revenue. Second, the maximum capacity of the pipeline
must be sufficient to satisfy the demand for capacity reservations, 𝐾 ≥ 𝜓( 𝑃𝑟 , 𝑃𝑢 , 𝐾).
Intuitively, insufficient capacity relative to market demand would entail additional costs
associated with non-price rationing. We expect this constraint to hold with equality—
maximum capacity in excess of market demand would imply that costly capacity goes
unsold. 16
16
This seems to be the likely case. According to FERC (1999, p. 36), total firm capacity reservation is
typically equivalent to maximum capacity. Upon constructing any new facility, a pipeline is also required
to provide evidence in its FERC application that all additional system capacity is fully reserved, typically
for at least ten years (Black and Veatch LLC, 2012).
13
The pipeline jointly chooses its reservation and usage charges, along with
𝑒
maximum capacity, so as to maximize its expected periodic profit function, 𝜋𝑝𝑙
, subject
to the reservation demand constraint.
Here, we combine the S-V approach with a
uniform two-part tariff profit function (Oi, 1971; Vogelsang, 1989). 17
𝑒
𝜋𝑝𝑙
𝑢
𝑒
∞
= �𝑃 − 𝑣(𝐾)� �𝑦 − � (𝑦𝑡 − 𝐾)𝑔(𝑦𝑡 )𝑑𝑦𝑡 � + 𝑃𝑟 𝜓(𝑃𝑟 , 𝑃𝑢 , 𝐾)
− 𝐶(𝐾)
𝐾
(12)
The first term is expected variable profit net of the expected opportunity cost to the
pipeline of insufficient capacity, where 𝑣(𝐾) is the pipeline’s variable cost of shipping a
unit of gas. Note that we are implicitly assuming 1) interruptible transportation (IT)
demand is zero, 18 and 2) the pipeline takes the expected quantity demanded of the
shipping service, 𝑦 𝑒 , to be exogenous. 19 We assume variable cost per unit shipped to be
constant for a given maximum capacity, but Yépez (2008) has shown that variable cost
declines at a diminishing rate as maximum capacity increases.
Therefore we have
𝑣 ′ (𝐾) < 0 and 𝑣 ′′ (𝐾) > 0. The second term in the pipeline’s expected profit function is
capacity reservation revenue. The third term, 𝐶(𝐾), is the per period cost of capacity,
where 𝐶 ′ (𝐾) > 0 and 𝐶 ′′ (𝐾) < 0. 20 We assume all other fixed costs are zero. 21
17
𝑒
Maximization of 𝜋𝑝𝑙
is subject to the reservation demand constraint.
The basic structure of a uniform two-part tariff is one in which the firm has two sources of revenue:
consumers pay a lump sum access or admission fee, as well as a price per unit of output consumed. Oi
(1971) models the optimal two-part pricing structure for an amusement park (i.e. Disneyland) based on
consumers’ demand for rides and the variable cost per ride, but does not consider the optimal scale of the
park or cost of its construction. Vogelsang (1989) considers the firm’s overall production cost to be a
function of output only, and places no restrictions on the shape of the average cost curve.
18
This assumption is to retain some parsimony in the model. IT, by definition, requires no firm claim to
capacity, and thus does not carry a reservation charge per se (McGrew, 2009, p. 109-110). However, under
FERC’s ROR framework, IT rates are set such that they are equivalent to the daily reservation charge plus
the marginal cost of shipment, 𝑃𝑟 + 𝑣(𝐾) (FERC, 1999). In the unconstrained monopoly case, there is no
reason to assume the pipeline would choose this pricing rule for IT, and allowing IT demand to be positive
would require an additional maximization rule for 𝑃𝐼𝑇 in the pipeline’s optimization problem.
19
In the S-V model, the monopoly firm’s output is a function of the price. In our model, however, because
the shipping service occurs via the competitive secondary market, it is not directly affected by the usage
charge.
20
Cremer, Gasmi, and Laffont (2003) point out the likely presence of economies of scale for natural gas
pipelines, the both in capital cost structure and from technological factors. Yépez (2008) also numerically
estimates long-run average cost (LRAC) and long-run marginal cost (LRMC) curves, but does not fully
derive a total cost curve. LRAC and LRMC are each decreasing as capacity expands, and the former
exceeds the latter, suggesting economies of scale resulting from the fact that “output can be expanded with
a less-than-proportionate increase in total cost”.
21
In practice, labor is considered a fixed cost (FERC, 1999). This is because in a given period the amount
of labor employed by the pipeline does not depend on the volume of gas shipped.
14
𝑒
max 𝜋𝑝𝑙
,
𝒔. 𝒕.
{𝑃 𝑟 ,𝑃 𝑢 ,𝐾}
Forming the Lagrangian, we have
𝑢
𝑒
𝐾 ≥ 𝜓( 𝑃𝑟 , 𝑃𝑢 , 𝐾)
∞
ℒ = �𝑃 − 𝑣(𝐾)� �𝑦 − � (𝑦𝑡 − 𝐾)𝑔(𝑦𝑡 )𝑑𝑦𝑡 � + 𝑃𝑟 𝜓( 𝑃𝑟 , 𝑃𝑢 , 𝐾)
𝐾
(13)
(14)
− 𝐶(𝐾) + λ�𝐾 − 𝜓(𝑃𝑟 , 𝑃𝑢 , 𝐾)�
where λ > 0 is the multiplier for the reservation demand constraint, and by the envelope
theorem can be interpreted as the shadow value of meeting reservation demand. The
Kuhn-Tucker conditions are given by the following expressions, where 𝜓 ∗ (∙) ≡
𝜓(𝑃𝑟 ∗ , 𝑃𝑢 ∗ , 𝐾 ∗ ) is the aggregate demand function evaluated at the optimal reservation
charge, usage charge, and maximum capacity.
𝜕ℒ
𝜕𝜓 ∗ (∙)
𝑟∗
∗)
(𝑃
=
−
λ
�
� + 𝜓 ∗ (∙) ≥ 0,
𝜕𝑃𝑟
𝜕𝑃𝑟
𝑃𝑟 ∗ > 0 →
𝜕ℒ
=0
𝜕𝑃𝑟
𝑃𝑢 ∗ > 0 →
𝜕ℒ
=0
𝜕𝑃𝑢
∞
𝜕ℒ
𝜕𝜓 ∗ (∙)
𝑒
∗ )𝑔(𝑦 )𝑑𝑦
𝑟∗
∗)
(𝑦
(𝑃
=
�𝑦
−
�
−
𝐾
�
+
−
λ
≥ 0,
𝑡
𝑡
𝑡
𝜕𝑃𝑢
𝜕𝑃𝑢
𝐾∗
∞
𝜕ℒ
= �𝑃𝑢 ∗ − 𝑣(𝐾 ∗ )�[1 − 𝐺(𝐾 ∗ )] − 𝑣 ′ (𝐾 ∗ ) �𝑦 𝑒 − � (𝑦𝑡 − 𝐾 ∗ )𝑔(𝑦𝑡 )𝑑𝑦𝑡 �
𝜕𝐾
𝐾∗
𝜕𝜓 ∗ (∙)
+ (𝑃𝑟 ∗ − λ∗ )
− 𝐶 ′ (𝐾 ∗ ) + λ∗ ≥ 0,
𝜕𝐾
𝜕ℒ
𝐾∗ > 0 →
=0
𝜕𝐾
𝜕ℒ
= 𝐾 ∗ − 𝜓 ∗ (∙) ≥ 0
𝜕λ
𝜕ℒ
λ∗ > 0 →
=0
𝜕λ
(15)
(16)
(17)
(18)
The reader will notice that the signs on conditions (15) – (17) are reversed from what
they would be in a set of typical Kuhn-Tucker maximization conditions. Intuitively, this
follows from the fact that collectively, the 𝑁 traders enjoy monopsony power over the
pipeline. In a standard monopsony model, marginal costs and marginal benefits to the
supplier (in our case, the pipeline) of increasing prices (and capacity, here) are reversed
15
compared to the way we traditionally think of them—that is, for each of our three choice
variables, 𝑃𝑟 , 𝑃𝑢 , and 𝐾, marginal benefits are upward-sloping (or flat) and marginal
costs are downward sloping. The intuitive explanation for this is that an increase in any
one of these variables results in an increase in revenues (MB), whereas a decrease leads
to a reduction in reservation demand (MC). 22 So for example, in equation (16) if MB >
MC for all 𝑃𝑢 > 0, then the corner solution 𝑃𝑢 = 0 would be optimal. Hence, the
reversal of the Kuhn-Tucker signs. Appendix 2 provides a more formal explanation in
relation to the standard monopsony model.
We are unable to say whether the second-order sufficient Kuhn-Tucker conditions
are satisfied without computing a numerical solution. As such, we parameterize the
model and solve for the constrained optimum later in this paper, confirming the existence
of a unique solution. What is interesting is that it is possible in this problem to have a
unique solution even in the presence of increasing returns-to-scale. In a more basic
profit-maximization specification, increasing returns would prohibit the existence of a
unique solution. 23
3.3
The Second-Best Solution: Welfare Maximization with a Break-Even Constraint
It has long been understood that unconstrained welfare maximization under economies of
scale leads to a marginal cost pricing rule that results in significant deficit to the
monopoly firm. When transfers are not available to achieve cost coverage, a standard
economic approach has been to derive the second-best (Ramsey) pricing rule, in which
welfare maximization is subject to a break-even constraint for the monopoly firm (see
22
The overall MC component for the capacity Kuhn-Tucker condition (17) would not be monotonically
decreasing over the entire range of 𝐾. However, due to the concave shape of the of the capacity cost
function, 𝐶(𝐾), the overall MC to the pipeline of increasing capacity would eventually become strictly
decreasing in 𝐾. In other words, the negative marginal impact of greater capacity on reservation demand
eventually overtakes the marginal increase in capacity construction costs as 𝐶(𝐾) flattens out for higher
ranges of 𝐾. In any case, the overall MB component of (17) cuts the overall MC component from below,
implying the validity of the ‘greater-than-or-equal-to’ sign on the Kuhn-Tucker condition.
23
More generally, Beato (1982) provides a full analysis of the non-existence of competitive equilibria
where there are non-convexities in production technology. Following Cremer, Gasmi, and Laffont (2003),
we assume that the problem is “sufficiently concave” for the second-order sufficient conditions to hold.
This assumption is consistent with the structure of the profit function. As the capacity cost function, 𝐶(𝐾),
flattens out, the negative effect of maximum capacity on reservation demand (see Appendix 1, Equation
A1.3) implies that the profit function is globally concave in 𝐾.
16
Figure 3. Consumer and producer surpluses at Hub 1 (right panel) and Hub 2 (left panel)
with linear demand and supply curves.
Berg and Tschirhart, 1988, Ch. 3, for general reference). Ramsey pricing has been
extensively applied, most notably for our purpose by Cremer and Laffont (2002) and
Cremer, Gasmi, and Laffont (2003), who derive optimal usage charges for a pipeline
network of a given maximum capacity and without a secondary market. We extend these
authors’ methodology to account for (i) uncertainty over the quantity demanded for
shipping services, 24 (ii) the two-part tariff pricing structure, (iii) endogenous capacity
choice, and (iv) the unregulated secondary market and the expected profits of the gas
traders. 25
The capacity of the pipeline affects the secondary market transportation charge,
which in turn affects the equilibrium spot prices at the two hubs. As such, in the welfare
maximization problem the planner must account for the economic surpluses of the
producers and consumers at each hub (see Figure 3) in addition to the traders’ and the
24
Cremer, Gasmi, and Laffont (2003, Section 5) provide a simplified model of socially optimal capacity
choice under uncertainty, both for a risk averse and a risk neutral planner. Proposition 7 of their analysis
states that “when demand is uncertain and capacity has to be set ex ante, the optimal capacity level is larger
under risk aversion than under risk neutrality.” They compare higher capacity to an “insurance
mechanism”, the “premium” for which “is paid ex ante through a higher expenditure on capacity.” We
expect that the same result would hold for our model.
25
Cremer, Gasmi, and Laffont (2003, Section 4) provide a three-period model of two-part pricing with a
secondary market of exactly two traders on an even simpler pipeline network. They derive a two-part
Ramsey pricing scheme that decentralizes optimal capacity and throughput on the pipeline, given that the
pipeline operates competitively. According to these authors, however, “there is no reason why the network
operator should be expected to behave competitively.” It is the intent of the present analysis to derive an
optimal pricing arrangement with 𝑁 traders, under the assumption that the pipeline operator does not
behave competitively.
17
pipeline’s profits. Notice that as long as 𝑦 𝑒 > 0, then it must be that 𝑞1 > 𝑑1𝑒 and
𝑞2 < 𝑑2𝑒 , and economic surplus at each hub is not defined by the intersection of the local
demand and local supply schedules (given some feasible set of expected spot prices).
The left panel in Figure 3 demonstrates that at Hub 1 local demand (dashed) is not
equal to overall demand (solid), whereas the right panel shows that at Hub 2 local supply
(dashed) is not equal to overall supply (solid). The difference in each case is the amount
shipped from Hub 1 to Hub 2, which serves to link the equilibria in these markets.
Equilibrium prices are defined by the intersection of the overall demand and supply
curves—the price at Hub 1 is greater than it would be without the link, and conversely
the price at Hub 2 is lower than it would be. The secondary market transportation charge
and the two spot prices simultaneously adjust to clear both spot markets and the
secondary transportation market.
The result is that local consumer and producer
surpluses are altered. Total economic surplus at each hub increases as gas is shipped
between them, implying gains from trade.
Expected consumer and producer surpluses depend upon the local demand and
supply schedules. We assume that local demand at each hub is reasonably elastic,
implying that we must define demand functions 𝑑𝑗𝑒 = 𝐷𝑗 (𝑝𝑗𝑒 ), where 𝑝𝑗𝑒 is the expected
spot price at Hub 𝑗 and 𝐷𝑗′ (𝑝𝑗𝑒 ) < 0. Cremer and Laffont (2002) and Cremer, Gasmi, and
Laffont (2003) use inverse demand and supply functions, such that they are able to model
total economic surplus at each hub as gross consumer surplus net of total economic cost.
However, given the choice variables appropriate to our problem and the fact that we need
to use standard demand and supply functions (as opposed to inverse), it will be more
convenient for us to model total economic surplus at each hub as the sum of producer
surplus and net expected consumer surplus.
First, since we have implicitly assumed in Figure 3 that the marginal cost of
producing a unit of gas is zero, at the expected price 𝑝𝑗𝑒 producer surplus at each hub is
Expected consumer surplus is
𝑃𝑆𝑗 �𝑝𝑗𝑒 � = 𝑞𝑗 𝑝𝑗𝑒 ,
∞
𝑗 = 1,2.
𝐶𝑆𝑗 �𝑝𝑗𝑒 � = � 𝐷𝑗 (𝑝𝑗 ) 𝑑𝑝𝑗 .
𝑝𝑗𝑒
18
(19)
(20)
We now have all the necessary components for constructing the planner’s
expected social welfare function, 𝑆𝑊 𝑒 : expected consumer and producer surpluses at
each hub, the gas traders’ aggregate expected profits, and the pipeline’s expected profit.
Maximization of 𝑆𝑊 𝑒 is subject to five constraints: the pipeline’s break even constraint,
the reservation demand constraint, the two expected flow-balance identities, and the
arbitrage condition. The planner’s problem is to choose the socially optimal reservation
charge, usage charge, and maximum capacity.
max 𝑆𝑊 𝑒 = 𝐶𝑆1 (𝑝1𝑒 ) + 𝐶𝑆2 (𝑝2𝑒 ) + 𝑃𝑆1 (𝑝1𝑒 ) + 𝑃𝑆2 (𝑝2𝑒 ) + � 𝜋𝑖𝑒
{𝑃 𝑟 ,𝑃𝑢 ,𝐾}
+
𝑖
𝑒
𝜋𝑝𝑙
(21)
𝑒
=0
𝒔. 𝒕. 𝜋𝑝𝑙
𝐾 ≥ 𝜓( 𝑃𝑟 , 𝑃𝑢 , 𝐾)
𝐷1 (𝑝1𝑒 ) = 𝑞1 − 𝑦 𝑒
𝐷2 (𝑝2𝑒 ) = 𝑞2 + 𝑦 𝑒
𝑝1𝑒 = 𝑝2𝑒 − 𝜏 𝑒
After substituting the final three constraints directly into the Lagrangian, it reduces to
ℒ=�
∞
𝑝2𝑒 −𝜏𝑒
𝐷1 (𝑝1 ) 𝑑𝑝1 + �
∞
𝑝1𝑒 +𝜏𝑒
𝐷2 (𝑝2 ) 𝑑𝑝2
+ (𝐷1 (𝑝2𝑒 − 𝜏 𝑒 ) + 𝑦 𝑒 )(𝑝2𝑒 − 𝜏 𝑒 ) + (𝐷2 (𝑝1𝑒 + 𝜏 𝑒 ) − 𝑦 𝑒 )(𝑝1𝑒 + 𝜏 𝑒 )
∞
+ �𝜏 𝑒 − 𝑣(𝐾)�𝑦 𝑒 − (𝜏 𝑒 − 𝑃𝑢 ) � �𝑦𝑡 − 𝜓(𝑃𝑟 , 𝑃𝑢 , 𝐾)�𝑔(𝑦𝑡 )𝑑𝑦𝑡
∞
𝜓(∙)
− �𝑃𝑢 − 𝑣(𝐾)� � (𝑦𝑡 − 𝐾)𝑔(𝑦𝑡 )𝑑𝑦𝑡 − 𝐶(𝐾)
∞
𝐾
+ λ1 ��𝑃𝑢 − 𝑣(𝐾)� �𝑦 𝑒 − � (𝑦𝑡 − 𝐾)𝑔(𝑦𝑡 )𝑑𝑦𝑡 � + 𝑃𝑟 𝜓(𝑃𝑟 , 𝑃𝑢 , 𝐾) − 𝐶(𝐾)�
𝐾
+ λ2 �𝐾 − 𝜓( 𝑃𝑟 , 𝑃𝑢 , 𝐾)�.
(22)
The Lagrangian multipliers, λ1 ≥ 0 and λ2 ≥ 0, represent the shadow values of public
funds (Berg and Tschirhart, 1988) and of meeting reservation demand. 26
26
Some terms in the social welfare function cancel out. The gas traders’ aggregate expected profits are
∞
equal to (𝜏 𝑒 − 𝑃𝑢 ) �𝑦 𝑒 − ∫𝜓(𝑃𝑟 ,𝑃𝑢 ,𝐾)�𝑦𝑡 − 𝜓(𝑃𝑟 , 𝑃𝑢 , 𝐾)�𝑔(𝑦𝑡 )𝑑𝑦𝑡 � − 𝑃𝑟 𝜓(𝑃𝑟 , 𝑃𝑢 , 𝐾), which has two
terms that are identical to (but negative of) two terms in the pipeline’s expected profit function. Intuitively,
this makes perfect sense: expenditures for the traders are identical to revenues for the pipeline. Thus the
19
Denote the socially optimal values of the reservation charge, usage charge, and
� . To keep our notation straight, save space, and
maximum capacity as 𝑃�𝑟 , 𝑃� 𝑢 , and 𝐾
� − 𝑦 𝑒 � , 𝜏̃ 𝑒′ ≡ 𝜏 ′ �𝐾
� − 𝑦 𝑒 � , and 𝜓�(∙) ≡ 𝜓� 𝑃�𝑟 , 𝑃�𝑢 ; 𝐾
�� .
avoid clutter: 𝜏̃ 𝑒 ≡ 𝜏�𝐾
For
� are each positive and
brevity, we consider here a particular case in which 𝑃�𝑟 , 𝑃�𝑢 , and 𝐾
� = 𝜓�(∙). 27 Rearranging the first-order
the reservation demand constraint is binding, 𝐾
condition for capacity choice such that all marginal benefits are on the left-hand side and
all marginal costs are on the right-hand side, we have:
𝑒′
− 𝜏̃ �
� )𝑦 𝑒
[𝑝2𝑒 𝐷2′ (𝑝2𝑒 ) − 𝐷2 (𝑝2𝑒 )]𝜏̃ 𝑒′ − 𝑞1 𝜏̃ 𝑒′ − �1 + λ�1 �𝑣 ′ (𝐾
∞
� (∙)
𝜓
� �� �1 − 𝐺(𝐾
� )�
�𝑦𝑡 − 𝜓�(∙)� 𝑔(𝑦𝑡 )𝑑𝑦𝑡 + �1 + λ�1 � �𝑃� 𝑢 − 𝑣�𝐾
+ λ�2 �1 −
− �𝜏̃ 𝑒 − 𝑃�𝑢 �
𝜕𝜓�(∙)
� = [𝑝1𝑒 𝐷1′ (𝑝1𝑒 ) − 𝐷1 (𝑝1𝑒 )]𝜏̃ 𝑒′ − 𝑞2 𝜏̃ 𝑒′ − 𝜏̃ 𝑒′ 𝑦 𝑒
𝜕𝐾
∞
𝜕𝜓�(∙)
� ) � �𝑦𝑡 − 𝐾
� �𝑔(𝑦𝑡 )𝑑𝑦𝑡
�1 − 𝐺 �𝜓�(∙)�� − �1 + λ�1 �𝑣 ′ (𝐾
𝜕𝐾
�
𝐾
� � − λ�1 𝑃�𝑟
+ �1 + λ�1 �𝐶 ′ �𝐾
𝜕𝜓�(∙)
.
𝜕𝐾
(23)
A marginal increase in maximum capacity has six effects on marginal benefits:
1. Increase in expected consumer surplus at Hub 2.
2. Increase in expected producer surplus at Hub 1.
3. Reduction in the pipeline’s expected variable costs.
4. Reduction in the traders’ expected opportunity costs from the marginal reduction
in the expected transportation charge.
5. Reduction in the pipeline’s expected opportunity cost from a marginal decrease in
the probability of excess demand.
6. Increase in the total value of meeting reservation demand.
The net effect on marginal benefits from an increment in capacity is balanced with the net
effect on marginal costs of an increment of capacity. The marginal cost effects (righthand side of Equation 23) follow from a marginal:
third and fourth lines of the Lagrangian function embody the combined profits (net of opportunity costs) of
the traders and the pipeline.
27
There is little reason to believe this would be the true solution to a fully parameterized system, but it
allows us to significantly reduce the notational space needed for each condition and focus more clearly on
the marginal welfare effects associated with incremental changes in each endogenous variable.
20
7. Reduction in expected consumer surplus at Hub 1.
8. Reduction in expected producer surplus at Hub 2.
9. Reduction in the traders’ expected revenues.
10. Increase in the traders’ expected opportunity cost from the marginal reduction in
capacity reservations.
11. Increase in the pipeline’s expected opportunity cost from the marginal reduction
in variable costs.
12. Increase in the pipeline’s capacity cost.
13. Reduction in the pipeline’s capacity reservation revenue.
This rule is by nature far more complex than the unconstrained monopoly capacity rule,
owing to the planner’s consideration of all affected parties and to the break-even
constraint. Without defining explicit functional forms and parameters, we are unable to
say with certainty whether the socially optimal capacity is greater than the unconstrained
monopolistic optimum.
However, our numerical analysis confirms the standard
economic result that an unregulated profit-maximizing monopoly will constrain output
below the socially optimal level. The unregulated monopolist is able to take advantage of
market power by constraining capacity, which constrains output and pushes prices above
marginal cost. Further deviations from the second-best solution occur because the profitmaximizing pipeline does not account for the external effects of its choice of maximum
capacity on the consumer and producer surpluses at the hubs.
The necessary conditions for the socially optimal reservation and usage charges
are
𝜕ℒ
𝜕𝜓�(∙)
𝜕𝜓�(∙)
𝜕𝜓�(∙)
𝑒
� (∙)�� + λ�1 �𝑃�𝑟
� (∙)� − λ�2
� 𝑢�
=
�𝜏̃
−
𝑃
�1
−
𝐺
�𝜓
+
𝜓
𝜕𝑃𝑟
𝜕𝑃𝑟
𝜕𝑃𝑟
𝜕𝑃𝑟
(24)
= 0,
∞
𝜕ℒ
𝜕𝜓�(∙)
�
� �𝑔(𝑦𝑡 )𝑑𝑦𝑡
=−
�1 − 𝐺 �𝜓(∙)�� − � �𝑦𝑡 − 𝐾
𝜕𝑃𝑢
𝜕𝑃𝑢
�
𝐾
∞
� �𝑔(𝑦𝑡 )𝑑𝑦𝑡 � + 𝑃� 𝑟
+ λ�1 ��𝑦 𝑒 − � �𝑦𝑡 − 𝐾
�
𝐾
𝜕𝜓�(∙)
𝜕𝜓�(∙)
�2
�
−
λ
= 0.
𝜕𝑃𝑢
𝜕𝑃𝑢
(25)
In the same way as (23) balances the marginal social benefits and marginal social benefits
of increased capacity, so do these rules for the reservation and usage charges. However,
21
it is rather clear that the welfare effects of changes in the two-part tariff charges occur
primarily via their impacts on aggregate reservation demand.
The Kuhn-Tucker
conditions for the multipliers λ1 and λ2 are also necessary for computing the solution of a
fully parameterized mixed-complementarity problem.
3.4
Rate-of-Return Regulation
Under ROR regulation the specific allowable rates of return are determined by
calculating a pipeline’s cost of service (also referred to as a “revenue requirement”).
First, various cost components are parsed into distinct categories such as gathering,
transmission, or storage. Costs are then further identified as either “fixed” or “variable”.
Fixed costs are those incurred regardless of whether service is provided: for example
office rent, depreciation, or interest payments. 28 Variable costs are mostly made up of
compressor fuel usage, which varies with the provision of service. All fixed costs are
allotted to the “reservation” component of transportation rates, and all variable costs to
the “usage” component.
Once the total cost of service for the pipeline has been
determined, it is allocated among the pipeline’s various classes of customers, such that
each class of customer is designated with a specific portion of the total revenue
requirement. Using this cost-sharing mechanism, unit rates are set for each class of
service (McGrew, 2009, p.97-99).
FERC (1999) outlines five steps for calculating a reasonable ROR for a natural
gas pipeline based on cost-of-service. (1) Establish a revenue requirement, i.e. cost-ofservice. (2) Functionalize the cost-of-service. 29 (3) Classify costs. 30 (4) Allocate costs.31
(5) Design the applicable rates. The basic cost-of-service formula is
where
Rate Base × Overall Rate of Return = Total Cost-of-Service
Total Cost-of-Service = Return + Operation & Maintenance Expenses
+ Administrative & General Expenses + Depreciation Expense
28
Recall that labor is classified as a fixed cost as well.
This is the process of categorizing costs as operating & maintenance, administrative & general,
depreciation, etc.
30
Functionalized costs are then classified as fixed or variable.
31
Cost allocation apportions functionalized and classified costs between geographic zones, and between
‘jurisdictional’ services. Jurisdictional services are basically firm and interruptible transportation services.
29
22
+ Non-income Taxes + Income Taxes – Revenue Credits.
The rate base is computed as
Gross Plant – Accumulated Depreciation = Net Plant
Net Plant – Accumulated Deferred Income Taxes + Working Capital = Rate Base.
For tractability, let depreciation, all other costs and expenses (including labor), taxes,
credits, and working capital all be zero. Thus, we have simply
Gross Plant × Overall Rate of Return = Return + Operating Expenses.
The overall rate-of-return is a weighted average of the cost of capital (WACC), and is
based on three components: the pipeline’s capitalization ratio, the pipeline’s cost of debt,
and the allowed rate of return on equity. To illustrate, if the pipeline’s capitalization ratio
is 75% debt to 25% equity, the cost of debt is 8%, and the allowed ROR on equity 12%,
then the overall ROR is (0.75 × 0.08) + (0.25 × 0.12) = 0.09, or 9%. Denoting the
overall ROR as 𝑟, we define the ROR constraint on the pipeline’s profits:
𝑒
𝜋𝑝𝑙
= 𝑟𝐶(𝐾) = [𝑟𝐷 𝜌 + 𝑟𝐸 (1 − 𝜌)]𝐶(𝐾),
(26)
where 𝑟𝐷 is the cost of debt, 𝑟𝐸 is the allowed ROR on equity, and 𝜌 is the fraction of
gross plant that is debt financed. We assume that 𝐾, 𝑟𝐷 , 𝑟𝐸 , and 𝜌 are each taken as
exogenous parameters by the regulator.
Under the FERC’s ROR regulation, the usage charge is equal to the expected
average usage cost of shipping a unit of gas. The equivalent variable here is the marginal
cost of shipping, 𝑣(𝐾) , implying that the pipeline’s variable profits (and expected
opportunity cost) are always equal to zero. 32 The pipeline’s allowed profits are realized
solely through capacity reservation:
𝑒
𝜋𝑝𝑙
= 𝑃𝑟 𝜓(∙) − 𝐶(𝐾).
(27)
We assume that both the pipeline and regulator have complete information, and the
pipeline has a corresponding optimization problem of:
32
FERC (1999) states that “…the firm usage rate is computed by dividing the total usage costs by the
projected annual firm and IT volumes.” 100% of fixed costs are allocated to the reservation charge, and
100% of variable costs to the usage charge, where “variable costs represent the non-labor… portion of the
O&M accounts related to compressor and meter stations.” That is, the fuel necessary for running the
compressor and meter stations. Thus, in our model, in which the marginal cost of shipping is constant for a
given maximum capacity, 𝑃� 𝑢 = average expected usage cost = 𝑣(𝐾)𝑦 𝑒 /𝑦 𝑒 = 𝑣(𝐾) = marginal usage
cost.
23
𝑒
max 𝜋𝑝𝑙
(28)
{𝑃 𝑟 ,𝐾}
𝑒
𝒔. 𝒕. 𝜋𝑝𝑙
≤ 𝑟𝐶(𝐾)
The Lagrangian function is
𝐾 ≥ 𝜓(∙).
ℒ = 𝑃𝑟 𝜓(∙) − 𝐶(𝐾) + λ1 �𝑃𝑟 𝜓(∙) − (1 + 𝑟)𝐶(𝐾)� + λ2 �𝐾 − 𝜓(∙)�.
� ≥ 0, λ�1 ≤ 0, and λ�2 ≥ 0 are given by:
The Kuhn-Tucker conditions for 𝑃� 𝑟 ≥ 0, 𝐾
𝜕ℒ
𝜕𝜓�(∙)
𝜕𝜓�(∙)
�1 � �𝜓�(∙) + 𝑃�𝑟
�2
=
�1
+
λ
�
−
λ
≥ 0,
𝜕𝑃𝑟
𝜕𝑃𝑟
𝜕𝑃𝑟
𝑃�𝑟 > 0 →
(30)
𝜕ℒ
= 0,
𝜕𝑃
𝜕ℒ
𝜕𝜓�(∙)
𝜕𝜓�(∙)
𝜕𝜓�(∙)
� � + λ�1 �𝑃�𝑟
� �� + λ�2 �1 −
= 𝑃�𝑟
− 𝐶 ′ �𝐾
− (1 + 𝑟)𝐶 ′ �𝐾
�
𝜕𝐾
𝜕𝐾
𝜕𝐾
𝜕𝐾
≥ 0,
(29)
(31)
�>0 →
𝐾
𝜕ℒ
= 0,
𝜕𝐾
(32)
λ�1 < 0 →
𝜕ℒ
= 0,
𝜕λ2
(33)
λ�2 > 0 →
𝜕ℒ
= 0.
𝜕λ2
𝜕ℒ
� � ≤ 0,
= 𝑃� 𝑟 𝜓�(∙) − (1 + 𝑟)𝐶�𝐾
𝜕λ1
𝜕ℒ
� − 𝜓�(∙) ≥ 0,
=𝐾
𝜕λ2
The optimality conditions on the reservation charge and maximum capacity, (31)
and (32), are interpreted in the usual way. They each balance marginal benefits and
marginal costs, but as we have already discussed, a corner solution for either variable
would obtain if MB > MC (see Appendix 2). The most important qualitative difference
between ROR regulation and the Ramsey second-best is that in the ROR case there is no
account of the economic welfare of producers and consumers at the two hubs. Without
accounting for the external effect of the pipeline’s capacity on these agents’ surpluses,
ROR regulation fails to achieve welfare maximization.
In the next section, we
parameterize our model, and for comparison of outcomes numerically compute solutions
24
for each of the three cases: unregulated monopoly, second-best solution, and ROR
regulation. ROR increases the pipeline’s optimal capacity relative to the unregulated
monopoly case, but constrains it to a level that is lower than the optimal maximum
capacity in the Ramsey second-best solution.
4
Numerical Implementation
To demonstrate the implications of the model, we numerically solve the optimality
conditions for the three regulatory cases. We consider 𝑁 = 10 relatively identical traders
with log-normally distributed individual demands.
33
The artificially generated
distributional parameters for the 10 traders’ individual shipping demands are positively
correlated. The intuition is that we would expect in periods of high demand that most (if
not all) of the traders’ individual demands should be relatively high, and vice versa for
periods of low demand. From the 10 traders’ individual distributional parameters, we
compute the distributional parameters of aggregate shipping demand. 34 Appendix 3
provides a detailed description of the parameter generation process.
We examine four combinations of the 10 traders’ distributional parameters. Each
individual trader’s distributional parameters may have 1) a low variance and a low
correlation with the other traders (henceforth denoted as LL), 2) a low variance and a
high correlation with the other traders (henceforth LH), 3) a high variance and a low
correlation with the other traders (henceforth HL), and 4) a high variance and a high
correlation with the other traders (henceforth HH). For each of the four distributional
scenarios, we compute the solutions for each of the three regulatory cases. Table 1
provides the randomly generated distributional parameters used in our computation.
volumetric parameters are in units of 100,000 MMBtu.
All
Note that the alternative
distributions are such that the expected aggregate shipping demand stays relatively
constant across the four scenarios, allowing us to pinpoint the effect of uncertainty on the
optimal solution for a given regulatory regime. Increasing the correlation between the
traders’ shipping demands increases the aggregate standard deviation without affecting
33
We say ‘relatively’ because the 10 traders each have slightly different distributional parameters within a
tight range (see Table 1).
34 The distribution of the sum of 𝑁 > 1 log-normally distributed random variables converges to an inverse
gamma distribution as the number of observations approaches infinity. We thus assume an inverse gamma
distribution for aggregate shipping demand.
25
their individual standard deviations. The rest of the model parameters, described below,
remain constant across all computed solutions.
To ease the computational burden, we hold the equilibrium spot price at Hub 2
constant. The intuition of the analysis, however, is not lost. The only implication is that
the change in the transportation charge (which by the arbitrage condition is equivalent to
the spot price differential) affects only the equilibrium spot price at Hub 1. An increase
in the expected transportation charge depresses the spot price at Hub 1, and a decrease in
the expected transportation charge raises it. To create this effect, we hold constant the
linear slopes of the local and overall demand curves at Hub 1 (see Figure 3), but allow the
intercept to vary such that these curves shift up and down in response to changes in the
basis differential. 35
Other parameters we extrapolate from the dataset and empirical estimation in
Oliver (2013). First, we construct a linear form for 𝜏(𝐾 − 𝑦) with a slope of −0.064
(see Appendix 4 for further details). 36 Using the proportions of average aggregate flows
on each segment of the pipeline network in that dataset, we have 𝑑1𝑒 = 𝑦 𝑒 /0.685 and
𝑑2𝑒 = 𝑦 𝑒 /0.735, from which we can back out the production quantities, 𝑞1 and 𝑞2 , using
the flow balance identities (1a) and (1b). The final parameter needed is the allowed
ROR, which we set at 𝑟 = 0.116. This is the average FERC-allowed ROR, as computed
from a set of 56 interstate pipelines (Loeffler, 2004; von Hirschhausen, 2008).
Appendix 4 provides a detailed description of the explicit functional forms used to
compute the solutions.
We present the following results for each solution: reservation
charge, usage charge, maximum capacity, aggregate reserved capacity, expected
equilibrium Hub 1 spot price, expected spot price differential (i.e. expected transportation
charge), and expected social welfare. Tables 2a through 2d present the solutions for the
four distributional scenarios. The primary results of interest are that in all four distribu-
Referring to Figure 3, we arbitrarily set the slope of the Hub 1 local inverse demand curve to −0.05, and
the slope of the export inverse demand curve to −0.05 (that is, the demand for Hub 1 gas by Hub 2). This
gives us a slope for the Hub 1 overall inverse demand curve of −0.025. We set the equilibrium spot price
at Hub 2 equal to $5 per MMBtu, and the slope of the Hub 2 local inverse demand curve to −0.05. We
allow the intercept of the Hub 1 local/overall inverse demand curves and the slope of the Hub 2 overall
inverse supply curve to be endogenously determined by the model.
36
This is the empirically estimated change in the spot price differential resulting from a 100,000 MMBtu
increase in available capacity between the two hubs analyzed in Oliver (2013).
35
26
Table 1. Distributional parameters.
Trader
Distributional Scenario
Low Variance
Low Variance
High Variance
High Variance
High
Correlation
Low
Correlation
High
Correlation
Low
Correlation
Parameter
ye
std. dev.
μ
σ
ye
std. dev.
μ
σ
ye
std. dev.
μ
σ
ye
std. dev.
μ
σ
ye
std. dev.
μ
σ
1.6979
2.1720
0.0101
1.0279
1.5957
2.0014
-0.0218
0.9941
1.6579
2.2768
0.0143
0.9821
1.6043
2.2895
-0.0413
0.9954
1.5985
1.9315
0.0018
0.9701
1.6248
2.2005
-0.0468
1.0325
1.5571
1.7924
-0.0275
1.0060
1.5738
1.8724
-0.0388
1.0174
1.5927
1.8154
-0.0145
1.0090
1.6252
2.1213
-0.0337
1.0308
3.2498
12.5717
-0.646
1.8487
1.8920
5.8170
-0.8798
1.8071
1.6480
4.4098
-0.9716
1.7773
1.3402
4.2771
-1.1988
1.8024
1.2764
2.9100
-1.1965
1.8213
2.3406
6.8041
-0.6441
1.7712
1.9334
5.5708
-0.8651
1.7917
1.7876
5.9870
-1.0529
1.7968
1.8151
7.5411
-1.0913
1.8123
1.5846
5.4475
-1.1492
1.8191
6
ye
std. dev.
μ
σ
1.5949
1.8486
-0.0077
0.9913
1.5608
1.9290
-0.0401
1.0000
1.4293
5.5033
-1.2725
1.8107
1.3759
3.7142
-1.1654
1.7475
7
ye
std. dev.
μ
σ
1.6045
1.9947
-0.0250
1.0141
1.6678
2.2163
-0.0135
1.0359
1.5804
6.0982
-1.3115
1.8838
1.3308
3.8794
-1.2779
1.7890
8
ye
std. dev.
μ
σ
1.5796
1.9224
-0.0278
0.9913
1.6079
1.9727
-0.0172
1.0105
1.3401
5.1531
-1.4194
1.8858
1.4288
4.7246
-1.3100
1.7823
9
ye
std. dev.
μ
σ
1.5683
2.1319
-0.0287
0.9592
1.6089
1.9761
-0.0303
1.0305
1.1975
5.1104
-1.4981
1.7954
1.3478
3.9180
-1.3630
1.8262
10
ye
std. dev.
μ
σ
1.5603
2.2133
-0.0586
0.9816
1.6383
2.1291
-0.0217
1.0485
1.1097
3.4420
-1.5251
1.8682
1.1159
3.3660
-1.4084
1.7882
1
2
3
4
5
ye
16.062
16.057
16.063
16.061
std. dev.
10.495
12.742
26.977
29.465
α
4.342
3.588
2.355
2.297
β
53.683
41.557
21.759
20.832
Notes: Trader's distributions are log-normal with the parameters μ and σ , which are the mean
and standard deviation of the normally distributed random variable ln( y ). Aggregate distribution
is inverse gamma with shape parameter α and scale parameter β (see Appendix 5).
Aggregate
27
tional scenarios, the maximum capacity of the pipeline and the resulting expected social
welfare are greater in the ROR solution than in the monopoly solution, but less than in
the socially optimal solution. This result is consistent with the A-J effect. But because
the ROR framework does not account for the external effect of pipeline capacity on the
economic welfare of producers and consumers at the hubs, the optimal maximum
capacity under ROR falls short of the second-best solution. More importantly, there is a
net expected welfare loss associated with ROR regulation when compared to the secondbest solution. ROR increases the profits of the traders and pipeline, but decreases the
economic surplus at Hub 1 (economic surplus at Hub 2 is constant because we pegged its
expected equilibrium spot price). The reduction in economic surplus at Hub 1 is greater
in magnitude than the combined increase in the traders’ and pipeline’s profits, resulting in
a net loss of expected social welfare. But how are these results related to the key
parameters of the model?
Upon closer inspection, we find that it is the individual variances of the traders’
shipping quantities that are the centrally important parameters in determining the optimal
solution.
To see this, compare the capacity reservation choices of two traders whose
expected shipping quantities are identical, but whose variances are vastly different.
Figure 4a shows the distribution and optimal capacity reservation across regulatory
regimes for Trader 8 in the LL scenario, who has an expected shipping demand of 1.58
and standard deviation of 1.92. Figure 4b shows this same information for Trader 7 of
the HL scenario, who also has an expected shipping demand of 1.58, but a much higher
standard deviation of 6.10. Granted, the equilibrium reservation and usage charges are
slightly higher for the latter, but intuitively it is rather clear that the marked difference in
optimal capacity reservations is more strongly affected by the different demand
distributions. When a trader faces greater uncertainty, i.e. a higher variance, more of the
probability mass of the log-normal distribution is contained within the low range of
values, implying that it is optimal to reserve less capacity. This is consistent with the
comparative static expression derived in Appendix 1. We find further evidence for the
central importance of the traders’ individual variances by examining the aggregate
distributions and capacity reservations (equal to the maximum capacity of the pipeline for
28
Table 2a. Solution results: LL scenario.
Unregulated
Monopoly
Regulatory Case
Rate-of-Return
Regulation
Second-Best
Social Optimum
0.447
0.444
0.758
0.741
0.913
0.887
0.465
0.435
0.771
0.727
0.921
0.87
Trader 5 Capacity Reservation, k 5
Trader 6 Capacity Reservation, k 6
0.464
0.452
0.764
0.752
0.91
Trader 7 Capacity Reservation, k 7
Trader 8 Capacity Reservation, k 8
Trader 9 Capacity Reservation, k 9
Trader 10 Capacity Reservation, k 10
0.436
0.443
0.882
0.882
0.454
0.735
0.737
0.743
0.433
0.717
0.884
0.856
Aggregate Capacity Reservation, � 𝑘𝑖
4.471
7.446
8.904
Maximum Capacity of Pipeline
Reservation Charge ($/MMBtu)
Usage Charge ($/MMBtu)
4.471
0.84
0
7.446
0.523
0.02
8.904
0.423
0
3.932
1.068
205.738
4.123
0.877
211.768
4.216
0.784
214.743
Variable
Trader 1 Capacity Reservation, k 1
Trader 2 Capacity Reservation, k 2
Trader 3 Capacity Reservation, k 3
Trader 4 Capacity Reservation, k 4
𝑖
Hub 1 Exp. Equilibrium Spot Price ($/MMBtu)
Exp. Basis Differential ($/MMBtu)
Exp. Social Welfare ($100,000's)
0.9
Table 2b. Solution results: LH scenario.
Unregulated
Monopoly
0.422
Variable
Trader 1 Capacity Reservation, k 1
Trader 2 Capacity Reservation, k 2
Trader 3 Capacity Reservation, k 3
Trader 4 Capacity Reservation, k 4
Trader 5 Capacity Reservation,
Trader 6 Capacity Reservation,
Trader 7 Capacity Reservation,
Trader 8 Capacity Reservation,
k5
k6
k7
k8
Trader 9 Capacity Reservation, k 9
Trader 10 Capacity Reservation, k 10
Regulatory Case
Rate-of-Return
Regulation
Second-Best
Social Optimum
0.706
0.859
0.44
0.726
0.431
0.444
0.715
0.734
0.429
0.436
0.435
0.443
0.716
0.718
0.729
0.878
0.867
0.889
0.871
0.868
0.888
0.887
0.43
0.428
0.732
0.718
0.721
Aggregate Capacity Reservation, � 𝑘𝑖
4.338
7.216
8.761
Maximum Capacity of Pipeline
Reservation Charge ($/MMBtu)
Usage Charge ($/MMBtu)
4.338
0.845
0
7.216
0.535
0.021
8.761
0.429
0
3.924
1.076
204.713
4.108
0.892
210.495
4.207
0.793
213.639
𝑖
Hub 1 Exp. Equilibrium Spot Price ($/MMBtu)
Exp. Basis Differential ($/MMBtu)
Exp. Social Welfare ($100,000's)
29
0.874
0.88
Table 2c. Solution results: HL scenario.
0.23
0.185
0.171
Regulatory Case
Rate-of-Return
Regulation
0.266
0.214
0.197
0.135
0.134
0.156
0.155
0.34
0.342
0.125
0.116
0.104
0.1
0.095
0.144
0.135
0.121
0.116
0.11
0.316
0.306
0.274
0.252
0.247
Aggregate Capacity Reservation, � 𝑘𝑖
1.396
1.615
3.566
Maximum Capacity of Pipeline
Reservation Charge ($/MMBtu)
Usage Charge ($/MMBtu)
1.396
0.85
0
1.615
0.678
0.195
3.566
0.313
0.465
3.735
1.265
189.786
3.749
1.251
191.113
3.874
1.126
194.874
Unregulated
Monopoly
Variable
Trader 1 Capacity Reservation, k 1
Trader 2 Capacity Reservation, k 2
Trader 3 Capacity Reservation, k 3
Trader 4 Capacity Reservation, k 4
Trader 5 Capacity Reservation, k 5
Trader 6 Capacity Reservation, k 6
Trader 7 Capacity Reservation, k 7
Trader 8 Capacity Reservation, k 8
Trader 9 Capacity Reservation, k 9
Trader 10 Capacity Reservation, k 10
𝑖
Hub 1 Exp. Equilibrium Spot Price ($/MMBtu)
Exp. Basis Differential ($/MMBtu)
Exp. Social Welfare ($100,000's)
Second-Best
Social Optimum
0.593
0.468
0.426
Table 2d. Solution results: HH scenario.
Unregulated
Monopoly
0.227
0.18
0.149
0.142
0.134
0.136
0.119
0.116
0.108
0.105
Regulatory Case
Rate-of-Return
Regulation
0.266
0.212
0.175
0.168
0.158
0.159
0.14
0.136
0.127
0.123
Second-Best
Social Optimum
0.578
0.464
0.385
0.371
0.35
0.343
0.307
0.297
0.283
0.27
Aggregate Capacity Reservation, � 𝑘𝑖
1.414
1.664
3.648
Maximum Capacity of Pipeline
Reservation Charge ($/MMBtu)
Usage Charge ($/MMBtu)
1.414
0.862
0
1.664
0.689
0.187
3.648
0.31
0.473
Hub 1 Exp. Equilibrium Spot Price ($/MMBtu)
Exp. Basis Differential ($/MMBtu)
Exp. Social Welfare ($100,000's)
3.737
1.263
190.91
3.753
1.247
192.001
3.88
1.12
195.991
Variable
Trader 1 Capacity Reservation, k 1
Trader 2 Capacity Reservation, k 2
Trader 3 Capacity Reservation, k 3
Trader 4 Capacity Reservation, k 4
Trader 5 Capacity Reservation, k 5
Trader 6 Capacity Reservation, k 6
Trader 7 Capacity Reservation, k 7
Trader 8 Capacity Reservation, k 8
Trader 9 Capacity Reservation, k 9
Trader 10 Capacity Reservation, k 10
𝑖
30
all solutions). Comparing the result of the LL scenario (Figure 5a) to that of the LH
scenario (Figure 5b), where the latter maintains the individual variances but increases the
aggregate variance, we see very little qualitative difference between them. The same can
be seen by comparing the HL scenario (Figure 6a) to the HH scenario (Figure 6b). An
increase in the aggregate variance has little to no effect on the outcomes of the three
regulatory regimes when the traders’ individual variances are held relatively constant.
However, comparing Figures 5a-b to Figures 6a-b, there are considerable
differences in outcomes.
Most notably, we find that when the traders’ individual
variances are low, ROR regulation performs reasonably well as compared to the secondbest solution (although with ROR capacity is still lower). On the other hand, when the
traders’ individual variances are high, the ROR regulation performs quite poorly, and the
capacity of the pipeline is constrained to a very low level that is in fact almost as low as
the monopoly outcome. The reason for this is related to both the pricing structure of the
two-part tariff and the exclusion of economic welfare at the two hubs under ROR
regulation. Notice that in the high variance scenarios (HL and HH), the reservation
charge is over two times greater in the ROR solution than in the socially optimal solution.
With ROR regulation, the usage charge is constrained to being equal to the marginal
variable cost of shipping a unit of gas on the pipeline. As a result, the pipeline’s return
must be made-up through a much higher reservation charge. The implication is that
when secondary market uncertainty is high, firm capacity demand cannot be stimulated
through a reduction in the reservation charge. 37 In the constrained welfare-maximizing
case, when the individual traders face greater uncertainty the planner finds it optimal to
lower the reservation charge while raising the usage charge above the marginal variable
cost of shipping.
The reduction in the reservation charge stimulates firm capacity
demand, and the pipeline’s break-even constraint is instead achieved through an increase
in the usage charge.
Traders pay less to reserve capacity, but more to actually utilize it.
37
Intuitively, one might expect that if we recast the ROR solution in such a way as to allow the usage
charge to exceed the marginal cost of shipping, it might alleviate this issue and bring the ROR capacity
solution closer in line with the second-best solution. Surprisingly, it does not. Re-deriving and solving the
FOC’s for the ROR optimum such that the usage charge is endogenously chosen in equilibrium, we find
that the usage charge falls to zero, the reservation charge exceeds that of the monopoly solution, and
capacity is slightly lower than in the monopoly solution! This indicates the importance of the external
effect of capacity on the economic welfare at the hubs.
31
(a)
(b)
Figure 4. (a) Individual capacity reservation and distribution: 𝑦𝑖𝑒 = 1.58, 𝑠𝑡𝑑𝑒𝑣𝑖 =
1.92. (b) Individual capacity reservation and distribution: 𝑦𝑖𝑒 = 1.58, 𝑠𝑡𝑑𝑒𝑣𝑖 = 6.01.
(a)
(b)
Figure 5. (a) Optimal capacities and aggregate distribution, LL scenario. (b) Optimal
capacities and aggregate distribution, LH scenario.
Because the ROR framework lacks this flexibility, the firm capacity reservations of the
traders are strongly negatively affected by greater uncertainty. 38
38
A key question is whether these results would hold in similar fashion in a deterministic setting.
Increased uncertainty reduces the traders’ optimal capacity reservations (see Appendix 1, Equation A1.5),
which then reduces the optimal maximum capacity of the pipeline. Thus, for a deterministic setting, we
could reasonably expect that for each regulatory alternative, the traders’ capacity reservations would
increase relative to the model presented here, increasing the pipeline’s optimal capacity. By the first
Averch-Johnson proposition, optimal capacity under ROR should still exceed optimal capacity for the
unregulated monopoly in a deterministic setting. Although likely due to the structural differences between
the ROR and welfare-maximizing problems, it is unclear whether the ROR optimal capacity would be less
than the socially optimal capacity in a deterministic setting.
32
Figure 6. (a) Optimal capacities and aggregate distribution, HL scenario. (b) Optimal
capacities and aggregate distribution, HH scenario.
The other issue of importance is the notion of wealth transfers from the pipeline to
the traders due to scarcity rents generated by constrained capacity. Equations (7) and
(17) are the economic profit functions of the traders and pipeline, respectively. Recall
that the integral term in each of these expressions represents the opportunity cost of
insufficient capacity relative to shipping demand. These terms do not represent actual
monetary costs to either the traders or the pipeline. Thus, if we omit the integral term
from each expression, we are left with the traders’ and pipeline’s accounting profit
functions, which can also be interpreted as the rents accruing to scarce pipeline capacity.
Comparing accounting profits (rents) across regulatory regimes and distributional
scenarios in Table 3, we find evidence of large transfers of accounting profits from the
pipeline to the traders, but only in the high variance distributional scenarios. With a high
degree of uncertainty (HL and HH), the pipeline’s maximum capacity is strongly
constrained in ROR relative to the second-best solution. The pipeline captures significant
rents in the socially optimal solution, yet with ROR its rents are constrained to the
allowed return, and the traders’ rents increase by an amount roughly equal to the
reduction in the pipeline’s rents. In the low variance distributional scenarios (LL and
LH), we find that the accounting profits of both the traders and the pipeline increase
when the regulatory regime is switched from Ramsey second-best to ROR.
The
implication is that these rents are captured at the expense of economic welfare at Hub 1.
The remaining reduction in economic surplus at Hub 1 is pure deadweight loss.
33
Table 3. Economic profit vs. accounting profit ($100,000’s).
Regulatory Case
Unregulated Monopoly
Low Variance - Low Correlation
Trader Aggregate
Pipeline
Low Variance - High Correlation
Trader Aggregate
Pipeline
High Variance - Low Correlation
Trader Aggregate
Pipeline
High Variance - High Correlation
Trader Aggregate
Pipeline
Rate-of-Return Regulation
Social Optimum
Economic
Profit
Accounting
Profit
Economic
Profit
Accounting
Profit
Economic
Profit
Accounting
Profit
-2.809
0.712
13.396
0.217
-3.995
0.405
9.879
0.405
-4.302
0
8.829
-0.106
-3.4
0.669
13.614
0.146
-4.627
0.402
10.123
0.402
-4.985
0
8.978
-0.112
-10.357
0.066
19.128
-3.493
-9.631
0.114
15.857
0.114
-10.688
0
9.5
5.173
-9.262
0.084
19.073
-3.404
-8.839
0.119
15.888
0.119
-9.746
0
9.275
5.297
The policy implications of these results are quite clear. When the individual
variances of the traders’ shipping demands are high, the outcome is that the optimal
maximum capacity of an ROR regulated pipeline is severely constricted relative to the
second-best solution.
Even when uncertainty is relatively low, ROR regulation
suppresses the optimal capacity, resulting in a loss of welfare. In terms of relative
capacity, the ROR framework performs more poorly in the high variance scenarios.
Finally, Tables 2a-2d show that the magnitude of the welfare loss is comparable in the
low uncertainty scenarios to those of the high uncertainty scenarios, but the proportion of
the welfare loss relative to the second-best solution is greater with high uncertainty.
ROR regulation induces a 1.4% welfare loss in the low uncertainty cases, as compared to
a 2.0% welfare loss in the high uncertainty cases. As such, an ROR pricing framework
is likely to be an unwise regulatory mechanism for pipelines serving transport routes
characterized by considerable stochasticity.
This result lends particular weight to the attractiveness of incentive-based
regulatory mechanisms, in which the firm not only finds it optimal to converge to the
Ramsey result over time (Vogelsang, 1989; 2001), but also is provided with sufficient
incentive to increase capacity to a more socially efficient level (Cambini and Rondi,
2010).
Incentive-based
regulations, most prominently price- and
34
revenue-cap
mechanisms, motivate operators to improve efficiency (in the Averch-Johnson sense).39
Guthrie (2006), in similar spirit to the intuition presented above, notes that ROR
regulation provides the infrastructure firm with the least flexibility in setting prices,
which adversely affects capital investment.
In contrast, price-cap regulation over a
basket of the firm’s goods and services allows the firm to respond to efficiency gains
related to capital investment patterns and price structure. Guthrie argues that because
investment in infrastructure is crucial to both prices and quantities in the long-run (of the
commodity itself, i.e. natural gas or electricity), insufficient (or delayed) investment in
infrastructure implies substantial welfare costs.
Our numerical results demonstrate
Guthrie’s assessment of ROR regulation in a concrete way. A key opportunity for
ongoing research is to derive and calculate the optimal solution under a price-cap
framework, to determine whether the optimal capacity and two-part tariff of a pipeline
subject to price-cap regulation is welfare-improving relative to ROR.
One caveat,
however, is that implementation of a price-cap mechanism is generally more complicated
than ROR (Joskow, 2008). If the costs of transitioning U.S. pipeline regulation to an
incentive-based scheme are too high, our results imply that ROR regulations may instead
need to be augmented by policies designed to reduce uncertainty in the secondary market.
5
Conclusion
In the markets for natural gas pipeline capacity (primary) and transportation (secondary),
uncertainty in the secondary market plays a central role in the capacity reservation
decisions of primary firm customers. The key implication of rate-of-return pricing in
natural gas transport is the real potential of severely distorted optimality conditions that
determine the maximum capacity of a pipeline. Constrained capacity results in negative
external effects on the economic welfare of producers and consumers at the hubs
connected by the pipeline. Even when uncertainty in the secondary market is relatively
low, we find that ROR regulation constricts maximum capacity relative to what would
occur under a Ramsey second-best socially optimal rule. This finding emanates primarily
from the fact that an ROR pricing rule for pipelines does not force the pipeline to
internalize the aforementioned external costs. Furthermore, our analysis has shown that,
39
Joskow (2008) provides a survey of incentive-based regulation as applied to electricity networks.
35
under ROR regulation, a high degree of uncertainty regarding daily fluctuations in
secondary market transportation demand strongly suppresses reservation demand in the
primary capacity market relative to the second-best outcome. When individual primary
contract holders face high uncertainty in the secondary market, their capacity reservation
decisions are significantly restrained. There are two reasons for this. One is related to
the distribution of demand for shipping services—a higher variance implies a higher
probability on any given day that reserved capacity will go unused. It is thus optimal to
reserve less capacity. The second reason, however, is subtler, and is related to the
rigidity of the ROR pricing rule. The ROR usage charge, which in our model is based on
FERC’s rate-setting guidelines, is required to be equal to the marginal cost of shipping a
unit of gas.
Thus, the entirety of the allowed return must be realized through the
reservation charge. When primary market purchasers of capacity face high uncertainty
regarding the utilization of reserved capacity, the Ramsey two-part tariff reduces the
reservation charge and sets the utilization charge far above the marginal cost of shipping
a unit of gas. This provides the primary purchasers with sufficient incentive to reserve
more capacity, while allocating the pipeline’s cost coverage requirement more evenly
between reservation and usage revenues. Conversely, the ROR two-part tariff does not
allow for such flexibility, maintaining a low, marginal cost based usage charge and a
high, ROR based reservation charge that together imply greater expected expenditures for
the primary contract holder.
The pipeline has incentive not to over-invest in capacity relative to primary
market reservation demand—i.e., it is optimal to build capacity such that it just meets
primary market reservation demand in all cases. Accordingly, the key implication of our
work is that the tandem effects of market externalities, secondary market uncertainty, and
tariff structure rigidity on reservation demand in the primary market can cause the
maximum installed capacity of the pipeline under ROR to be only a slight improvement
over the unregulated monopoly optimum capacity under certain conditions. This, in turn,
has the potential to generate large wealth transfers from the pipeline to the primary
contract holders, who are able to capture significant scarcity rents accruing to the
excessively constrained transport capacity. Previous researchers have suspected that such
wealth transfers reduce a pipeline’s incentives to expand capacity where needed, thereby
36
exacerbating congestion issues. Based on our results, we argue that this is likely to be
especially true for pipeline routes characterized by a high degree of stochasticity of
demand in the secondary market for transportation services.
Acknowledgements
This paper has greatly benefited from the help and insights of David Aadland, Jason
Shogren, Alexandre Skiba, Brian Towler, and Aaron Wood. The School of Energy
Resources at the University of Wyoming provided partial financial support for this
research.
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39
Appendices
A1. Comparative Statics
The comparative statics associated with Equation (10) are given by the following
expressions:
𝜕𝑘𝑖∗
−1
= 𝑒
<0
𝑟
(𝜏 − 𝑃𝑢 )𝑓𝑖 (𝑘𝑖∗ )
𝜕𝑃
(A1.1)
𝜕𝑘𝑖∗
−[1 − 𝐹𝑖 (𝑘𝑖∗ )]
=
<0
𝜕𝑃𝑢 (𝜏 𝑒 − 𝑃𝑢 )𝑓𝑖 (𝑘𝑖∗ )
(A1.2)
𝜕𝑘𝑖∗
𝜏 𝑒′ [1 − 𝐹𝑖 (𝑘𝑖∗ )]
= 𝑒
<0
𝜕𝐾 (𝜏 − 𝑃𝑢 )𝑓𝑖 (𝑘𝑖∗ )
(A1.3)
𝜕𝑘𝑖∗ −𝜏 𝑒′ [1 − 𝐹𝑖 (𝑘𝑖∗ )] − 𝜏 𝑒 (𝜕𝐹𝑖 (𝑘𝑖∗ )/𝜕𝑦𝑖𝑒 )
=
>0
(𝜏 𝑒 − 𝑃𝑢 )𝑓𝑖 (𝑘𝑖∗ )
𝜕𝑦𝑖𝑒
(A1.4)
𝜕𝑘𝑖∗
𝜕𝐹𝑖 (𝑘𝑖∗ )/𝜕𝜎𝑖
=−
<0
𝜕𝜎𝑖
𝑓𝑖 (𝑘𝑖∗ )
(A1.5)
A2. Relation to the Monopsony Model
The standard textbook monopsony model is of a labor pool seeking employment with a
single firm, and is represented by Figure A2.1. The labor supply curve (𝑆) is upward
sloping. The firm wants to choose an amount of labor (𝐿) to maximize its profits. The
downward sloping red curve is the marginal revenue product of labor (𝑀𝑅𝑃).
The
marginal cost curve (𝑀𝐶) is above the supply curve, because the firm has to increase the
wage paid to all the workers it already employs if it wants to hire an additional worker.
The optimal amount of labor hired by the firm, 𝐿∗ , occurs where 𝑀𝑅𝑃 = 𝑀𝐶, and the
monopsony wage is 𝑤 ∗ .
Next, let us consider a case in which the supply of labor is fixed at 𝐿�. In this case,
marginal cost and supply are equal, 𝑀𝐶 = 𝑆 (see Figure A2.2). In Figure A2.2 the
equilibrium again occurs where 𝑀𝑅𝑃 = 𝑀𝐶, giving us an equilibrium wage rate of 𝑤
�. If
we rotate this diagram around the 45-degree line, we get Figure A2.3.
Now, it is
important to understand that so far we have been looking at this market from the
40
Figure A2.1. Standard monopsony model.
Figure A2.2. Monopsony model with fixed supply.
perspective of the monopsony firm. What if we look at it from the perspective of the
workers? The marginal cost to the firm of hiring labor is equivalent to the marginal
benefit to the workers of being hired. Not only that, but the firm’s MRP curve is
equivalent to the workers’ MC curve – that is, the marginal cost to the workers of
demanding a higher wage is that some of them will go unemployed (like the classic labor
union problem). Now we draw this same diagram with the pipeline being the ‘workers’
41
Figure A2.3. Inverted monopsony model with fixed supply,
from the perspective of the supplier (i.e. workers).
Figure A2.4. Marginal costs and marginal benefits to a
pipeline facing a set of traders with monopsony power.
(i.e. the supplier) and the 𝑁 traders collectively having monopsony power. Also, let us
simplify away from the two-part tariff and just say there is one price, 𝑝, that the traders
� units of pipeline capacity. The
have to pay to the pipeline to reserve and utilize 𝐾
optimal price that the pipeline should charge, 𝑝∗ , is shown in Figure A2.4. Notice that if
the pipeline charges more than 𝑝∗ , the traders will not reserve the entire capacity of the
pipeline. And it would not charge less than 𝑝∗ because then it could be making more
revenue per unit by raising the price. The intuition to be taken away from this is that in a
42
Figure A2.5. Marginal benefits and marginal costs of a two-part tariff – reservation
charge (right) and usage charge (left) – that would result in a corner solution for the usage
charge.
Figure A2.6. Marginal benefits and marginal costs
to the pipeline of increased maximum capacity.
model where the buyers have market power as the traders do in our pipeline model, the
marginal benefits and marginal costs to the supplier in choosing an optimal price are
reversed from the way we traditionally think of them.
Next, we could have a situation where, with a two-part tariff, we had a picture
similar to the previous one, except that 𝑀𝐵 > 𝑀𝐶 for all 𝑝𝑢 > 0. The reason for this is
that the pipeline can increase demand for reservations by reducing the usage charge, but
the revenues from the reservation charge are everywhere high enough to (more than)
compensate for the lost usage revenue. The result would be the corner solution 𝑝𝑢 = 0.
The MB-MC diagrams would look similar to those depicted in Figure A2.5.
43
Finally, when looking at the MB-MC relationships for choosing maximum
capacity, we have a marginal cost curve that, while not globally decreasing, is strictly
decreasing over the relevant range of 𝐾. The marginal costs to the pipeline of increasing
capacity are made up of two effects: one due to the strictly negative relationship between
capacity and reservation demand by way of the transportation charge, and the other due
to the concavity of the capacity construction cost curve. The net effect of these two
components results in a marginal cost curve for capacity similar to that depicted in Figure
A2.6. As the capacity cost curve flattens out over higher ranges of 𝐾, the negative
marginal effect of capacity on reservation demand outweighs positive marginal effect on
capacity construction cost. The marginal benefits of increasing capacity for a given
reservation and usage charge are related to increased revenues.
The main implication of this analysis is that we find critical theoretical support for
setting the signs on each of the appropriate Kuhn-Tucker conditions such that a corner
solution obtains if 𝑀𝐵 > 𝑀𝐶 for all positive values of the associated endogenous
variable.
A3. Generation of individual and aggregate distributional parameters
Using Stata, we generate shipping demand data, 𝑦𝑖,𝑡 (𝑖 = 1, … 10), for the individual
traders, each with a log-normal distribution. We then compute each log-normal mean
and standard deviation of 𝑦𝑖,𝑡 , as well as the fundamental distributional parameters, 𝜇𝑖
and 𝜎𝑖 , which are the mean and standard deviation of the normally distributed random
variable, ln(𝑦𝑖,𝑡 ). We want the ten traders’ shipping demands to be positively correlated.
We use a predefined loop program called ‘mkcorr’ (www.stata.com) that employs the
Cholesky transformation of a correlation matrix of our choosing. Setting the number of
observations at 1000, the resulting generated data are ten normally distributed random
variables that have a correlation matrix similar to (but not exactly identical to) the
specified correlation matrix. For the two low correlation cases, we specify a correlation
matrix such that every off-diagonal element is 0.5, whereas in the high correlation cases
every off-diagonal element is 0.75. We then take the exponential function of each of
those ten correlated, normally distributed random variables to get ten correlated, lognormally distributed variables. For the two low-variance cases, we generate the data with
44
a standard normal distribution, ln(𝑦𝑖,𝑡 )~𝑁(0,1). For the two high-variance cases, we
adjust 𝜇 and 𝜎 such that we preserve generally the log-normal means but increase the
variances. From those ten log-normally distributed random variables, we calculate the
aggregate mean and standard deviation. The mean of the sum is equal to the sum of the
means. Because the ten random variables are not independent of each other, the variance
of the sum is equal to the sum of the variances and covariances, the square root of which
is the aggregate standard deviation.
To generate the four different distributional
scenarios across which the aggregate mean is preserved, we take random draws of the
generated data until we have four distributions that have roughly the same aggregate
mean. Code available upon request.
A4. Explicit functional forms used in numerical computations
A4.1. Transportation charge as a decreasing function of available capacity
The transportation charge is 𝜏(𝑘 𝑎 ), where 𝜏 ′ (𝑘 𝑎 ) < 0. In Oliver (2013), the empirically
estimated a change in the spot price differential (i.e. transportation charge) of −0.064 for
a 100,000 MMBtu increase in flows at the bottleneck. At a mean of 300,000 MMBtu in
available capacity, the mean transportation charge predicted by the estimation was 0.134.
Using these numbers, we construct a simple linear form of 𝜏(𝑘 𝑎 ) , where 𝜏 ′ (𝑘 𝑎 ) =
−0.064 .
If 𝜏 = 0.134 and 𝑘 𝑎 = 3 (recall that in our numerical computation, all
capacities and flows are in units of 100,000 MMBtu’s per day), we have
0.134 = 𝑏 − 0.064(3).
(A4.1)
𝜏(𝑘 𝑎 ) ≡ 𝜏(𝐾 − 𝑦) = 0.326 − 0.064(𝐾 − 𝑦).
(A4.2)
Solving this expression for 𝑏 yields 𝑏 = 0.326, and our simple linear function for the
transportation charge is
A4.2. Log-normal p.d.f. and c.d.f.
The traders’ individual shipping quantities are assumed to be log-normally distributed.
The c.d.f. appears in the individual trader’s first-order condition, given by equation (9).
Both the p.d.f.’s and c.d.f.’s appear in the comparative static expressions (A1.1) through
(A1.3), which are then used in the pipeline’s first-order conditions, because
45
𝜕𝜓(∙)
𝜕𝜓𝑖 (∙)
𝜕𝑘𝑖∗
≡�
≡�
, 𝑖 = 1, … , 𝑁; 𝑗 = 𝑃𝑟 , 𝑃𝑢 , 𝐾.
𝜕𝑗
𝜕𝑗
𝜕𝑗
𝑖
𝑖
(A4.3)
The log-normal p.d.f. evaluated at the trader 𝑖’s capacity reservation choice, 𝑘𝑖 , is given
by
𝑓𝑖 (𝑘𝑖 ) =
1
𝑘𝑖 �2𝜋𝜎𝑖 2
exp �−
(ln(𝑘𝑖 ) − 𝜇𝑖 )2
�,
2𝜎𝑖 2
(A4.4)
where 𝜇𝑖 and 𝜎𝑖 are the mean and standard deviation of ln(𝑦𝑖,𝑡 ). The log-normal c.d.f.
evaluated at 𝑘𝑖 is given by
𝐹(𝑘𝑖 ) =
1 1
ln(𝑘𝑖 ) − 𝜇𝑖
+ erf �
�,
2 2
𝜎𝑖 √2
(A4.5)
where erf[∙] denotes the error function. We also need an explicit form of the integral
∞
� �𝑦𝑖,𝑡 − 𝑘𝑖 �𝑓𝑖 �𝑦𝑖,𝑡 �𝑑𝑦𝑖,𝑡 ,
(A4.6)
𝑘𝑖
found in the traders’ objective function. Using Mathematica to evaluate this integral, we
have
(𝜇 + 𝜎𝑖 2 − ln(𝑘𝑖 ))
𝜎2
exp �𝜇𝑖 + 2𝑖 � �1 + erf � 𝑖
��
𝜎𝑖 √2
(ln(𝑘𝑖 ) − 𝜇𝑖 )
erfc �
�
𝜎𝑖 √2
− 𝑘𝑖 ,
(A4.7)
where erfc[∙] ≡ 1 − erf[∙] is the error function complement.
A4.3. Inverse Gamma p.d.f. and c.d.f.
We have assumed that the distribution of aggregate shipping demand is inverse gamma
(Milevsky and Posner, 1998; Krekel et al., 2004). The inverse gamma c.d.f. of aggregate
shipping demand evaluated at the maximum capacity of the pipeline, 𝐾, is
𝛽
𝐺(𝐾) = 𝛾 �𝛼, �,
𝐾
𝛽
(A4.8)
where 𝛾 �𝛼, 𝐾� is the regularized gamma function, and is pre-defined in GAMS. The
parameters 𝛼 and 𝛽 are the shape and scale parameters, and are defined as functions of
the aggregate mean and variance, 𝑦 𝑒 and 𝜎.
𝛼=
2𝜎 2 + (𝑦 𝑒 )2
𝑦 𝑒 [(𝑦 𝑒 )2 + 𝜎 2 ]
,
𝛽
=
𝜎2
𝜎2
46
(A4.9)
We do not need the inverse gamma p.d.f., but we do need an explicit form of the integral
term,
∞
� (𝑦𝑡 − 𝐾)𝑔(𝑦𝑡 )𝑑𝑦𝑡 ,
(A4.10)
𝐾
which appears multiple times in the pipeline’s first-order conditions. Again evaluating
the integral using Mathematica, we have
𝛽
1
𝛽
𝛽
�−𝐾Γ[𝛼] + 𝐾Γ �𝛼, � + 𝛽 �Γ[𝛼 − 1] − Γ �𝛼 − 1, ���,
Γ[𝛼]
𝐾
𝐾
(A4.11)
𝛽
where Γ �𝛼, 𝐾� ≡ 1 − 𝛾 �𝛼, 𝐾� is the upper incomplete gamma function, and Γ[𝛼] is the
lower incomplete gamma function.
A4.4. Pipeline cost functions
The last explicit functional forms to assign are the pipeline’s capacity cost function,
𝐶(𝐾), and its variable cost function, 𝑐(𝐾). Note that we do not specify these functional
forms with any pretense that they are perfectly accurate depictions of any real pipeline’s
cost functions. To achieve that end would require estimation of true cost curves, which is
beyond the scope of this paper. We simply want these functions to have properties that
reflect the theoretically assumed economies of scale with respect to capacity.
Specifically, we want 𝐶 ′ (𝐾) > 0, 𝐶 ′′ (𝐾) < 0, and conversely 𝑐 ′ (𝐾) < 0, 𝑐 ′′ (𝐾) > 0.
We assume forms of these functions that have these characteristics, and solve for specific
parameters that yield reasonable costs using data from an actual interstate pipeline.
Rockies Express Pipeline (REX) Zone 1 extends from the Opal Hub in Southwest
Wyoming to the Cheyenne Hub along the Wyoming-Colorado border.
Its current
operating capacity is approximately 1,870,000 MMBtu’s per day. The basic FERCregulated tariff schedule for REX Zone 1 is $7.072 per MMBtu for the (monthly)
reservation charge and $0.005 per MMBtu for the usage charge. 40 Because we want to
express the reservation charge in terms of its daily increment, we divide it by 30.4 to get
$0.233 (see Footnote 4). Using these parameters (and the average FERC allowed return
of 𝑟 = 0.116), we calibrate the capacity and variable cost functions in the following way.
40
Source: FERC Gas Tariff – Third Revised Volume No. 1, Fall 2011 – Rockies Express Pipeline, LLC.
47
First, using equation (33), we know that if the ROR and reservation demand
constraints hold with equality, such that 𝑃𝐾 − 𝐶(𝐾) = 𝑟𝐶(𝐾) , we can solve this
expression for the ROR reservation charge.
𝑃=
Using the above parameters, we have
(1 + 𝑟)𝐶(𝐾)
𝐾
0.233 =
(1.116)𝐶(𝐾)
.
1,870,000
Solving this expression for 𝐶(𝐾) gives us 𝐶(𝐾) ≈ $390,421 .
following functional form for 𝐶(𝐾),
𝑏𝐾 2 − 10𝑚𝑖𝑙𝑙𝑖𝑜𝑛
𝐶(𝐾) = 2
.
𝐾 + 80𝑏𝑖𝑙𝑙𝑖𝑜𝑛
(A4.12)
(A4.13)
We then specify the
(A4.14)
Plugging in 𝐶(𝐾) = $390,421 and 𝐾 = 1,870,000, we then solve (A4.13) for 𝑏 to get
𝑏 ≈ 399,353. Our calibrated daily capacity cost function is thus
𝐶(𝐾) =
399,352 × 𝐾 2 − 10𝑚𝑖𝑙𝑙𝑖𝑜𝑛
.
𝐾 2 + 80𝑏𝑖𝑙𝑙𝑖𝑜𝑛
(A4.15)
This may seem an odd looking function at first, but graphing it (Figure A4.1) we find that
it displays the key characteristics that we wanted in our capacity cost function, 𝐶 ′ (𝐾) >
0 and
𝐶 ′′ (𝐾) < 0 , and it results in reasonable costs over the relevant range of
capacities. 41
Using a similar rationale, we create the variable cost function using ROR rule for
the usage charge, 𝑝 = 𝑐(𝐾) .
We specify the following form for the variable cost
function:
𝑐(𝐾) =
𝑏
,
(𝐾 + 1000)1.5
(A4.16)
and again solve for the parameter 𝑏 using the data from REX Zone 1. Here, we get
𝑏 = 12,796,190, so our calibrated variable cost function is
𝑐(𝐾) =
12,796,190
.
(𝐾 + 1000)1.5
(A4.17)
Figure A4.2 displays this function graphically
41
Note that in the full numerical computation, we scale capacity in units of 100,000 MMBtu, and therefore
scale the cost functions accordingly.
48
$/period
400000
𝐶(𝐾)
300000
200000
100000
500000
1.0106
1.5106
Figure A4.1. Capacity cost function, 𝐶(𝐾).
2.0106
Capacity, 𝐾
$/MMBtu
0.14
0.12
0.10
0.08
0.06
0.04
0.02
500000
1.0106
𝑐(𝐾)
1.5106
Figure A4.2. Variable cost function, 𝑐(𝐾).
49
2.0106
Capacity, 𝐾