Why Have Voluntary Time‐of‐Use Tariffs Fallen Short in the Residential Sector?

Introduction

In the last decade, the utilization rate of power generation capacity in the United States has been less than 50% (U.S. EIA 2018). Simply put, power plants across the nation are sitting idle much of the time throughout the year. The residential sector is a significant contributor to this mismatch between supply and demand due to the high intra‐day variation in household electricity consumption patterns. The primary reason for this under‐utilization of power generation capabilities is due to the fact that the grid is built to satisfy consumer demand at its peak; that is, power generation capacity must be sufficient to meet demand even during the most brutal few days of summer and winter. What utilities pay for power during these occasional demand spikes drives up the average electricity price that consumers pay throughout the year.

One effective way to curtail such spikes in electricity demand is to induce consumers’ demand response through time‐based rate programs such as a time‐of‐use (TOU) tariff. Through the use of such incentive mechanisms, the utilities can induce a shift in electricity consumption by raising the price during the times when demand typically peaks while offering discounts during the off‐peaks. The benefit of demand response is clear to all involved parties. Consumers can save on their bills by moving some of their electricity consumption from peak to off‐peak hours. In turn, the market—utilities and generators—also benefits from the reduced need for peak‐time capacity that helps streamline electricity supply planning and reduces the overall cost. From the late 1970s, time‐of‐use tariffs have been experimentally deployed in the commercial and industrial sectors in which they have now become mandatory in many regions in the United States.

Experts envision great potential in deploying time‐based rate programs in the residential sector as well. Electricity consumption in the residential sector has been growing steadily over the last 40 years in OECD countries, from 19% of overall consumption in 1973 to 34% in 2018. The residential sector accounts for more than one‐third of total electricity consumption in the United States (IEA 2018), and its share of peak demand is constantly increasing (Faruqui and Sergici 2010). Even though the price responsiveness of residential customers can vary, recent studies point out that price elasticity in the residential sector is comparable to and may even exceed that in the commercial and industrial sectors (Badri 1992, Cappers et al. 2016, King 2001, U.S. DOE 2006, Wang and Mogi 2017). Residential consumers’ capability to shift demand is also expected to increase as more smart appliances are deployed in the near future. ¹ Hence, more states in the United States are beginning to implement time‐based electricity rate programs in the residential sector since the early 2000’s.

Unfortunately, despite the clear potential and success in the commercial and industrial sectors, TOU tariffs have not been well‐received in the residential sector. While TOU tariffs have been offered to nearly half the US population (U.S. EIA 2017), and some regions have been offered these tariffs for over 40 years, the current overall residential market penetration rate of voluntary TOU tariffs in the United States is less than 2% (Cappers et al. 2016). So, the question arises: why are time‐based electricity rate programs falling short in the residential sector?

Joskow and Wolfram (2012) summarized some of the issues left unsolved for time‐based electricity rate programs in the residential sector. Among them, technological barriers such as the need to deploy a smart metering system were once believed to be a major challenge for the effectiveness of TOU tariffs in the residential sector. This is no longer a critical obstacle, as more than 70 million (advanced) smart meters have been installed in homes across the U.S. reaching about 50% of all households in the last couple of decades (U.S. EIA 2017). Instead, many now claim that the main culprit is the economics underlying TOU tariffs and the resulting implications for customer billing. There have been studies revealing that consumers are concerned whether TOU tariffs will actually reduce their bills (Alexander 2010, Faruqui 2010, Hogan 2010). Moreover, residential customers exhibit a considerable degree of heterogeneity in electricity consumption patterns, where some households have a relatively “flat” consumption pattern while others have a “peaky” pattern and prefer to consume most of their electricity in the peak periods. As pointed out by Faruqui (2010), the traditional fixed flat rate (FFR) tariff creates an undesirable cross‐subsidy between the flat and peaky consumers, and is hence unfair to flatter consumers in the residential market.

While time‐based rate programs such as TOU tariffs can potentially alleviate the degree of cross‐subsidization, utility companies may not benefit from their deployment. Due to the difficulty in implementing mandatory or default TOU tariffs, the majority of TOU tariffs introduced in the residential sector remain voluntary opt‐in programs (Borenstein 2013). It is typically those consumers with the flattest consumption profiles that opt‐in to these programs since they can benefit from the off‐peak discounted rates without making significant changes to their consumption patterns. As a result, utility companies face potential revenue shortfalls without the upside of large cost‐savings from demand response. Hence, such consumer self‐selection behavior (also referred to as the “free‐riding effect”) makes utilities lukewarm towards marketing the TOU tariff (U.S. FERC 2012); the lack of incentive in turn can fail to entice peakier consumers who could potentially contribute more to demand response.

Our primary research goal is to enhance our understanding of why voluntary TOU tariffs have struggled in the residential sector thus far and to obtain managerial insights to overcome such challenges. To that end, we investigate the profitability, equity, and welfare implications of voluntary TOU tariffs in the residential electricity market. Specifically, we aim to: (i) investigate the optimal design of voluntary TOU tariffs and the resulting market equilibrium, (ii) identify the gap between the socially desirable level of TOU adoption and the market equilibrium with voluntary TOU tariffs, (iii) explore the consequences of voluntary TOU tariffs for cross‐subsidization across heterogeneous consumers, and (iv) identify the practical potential for TOU tariffs using a case study of the US residential electricity market.

The main contributions of our study are as follows.

First, we identify the key factors that determine the viability of voluntary TOU tariff deployment and adoption in the residential sector. On the demand side, we find that heterogeneity in consumption preferences within a target set of adopting consumers is the main determinant of how far voluntary TOU tariffs can penetrate the residential market. Residential markets that are skewed toward consumers with peakier consumption profiles are more conducive to high levels of TOU adoption than markets where consumption profiles are more evenly distributed. On the supply side, the gap between the wholesale prices of peak and off‐peak electricity is the main determinant of the success of TOU tariffs—without a sufficiently large wholesale price gap, the utility firm lacks the incentive to deploy effective TOU tariffs because it does not stand to gain as much from consumers’ demand response.

We note that our perspective as well as the managerial implications of our work are substantially different from the literature; for example, energy economists mainly focus on estimations of demand (and demand response) or mechanism design for optimal pricing, while electrical engineers focus on operations of electricity dispatching/scheduling or management of capacity/grid resilience.

In what follows, we first review related literature in section 2. In section 3, we present a game‐theoretic model to characterize the behavior of a utility firm as well as consumers with heterogeneous consumption patterns under the FFR tariff. We examine the optimal structure of a voluntary TOU tariff along with its social and equity implications in section 4. We then apply our theoretical results to obtain practical insights using data acquired from the US residential electricity market in section 5. We conclude the paper in section 6 by providing relevant managerial insights and identifying directions for future research. We study two extensions to our base model in Appendix A, and proofs of technical results are provided in Appendix B.

Literature Review

From the early 1980s, most studies on TOU tariffs in the residential sector have focused on estimating the price responsiveness of consumer demand using an econometric approach. Faruqui and Malko (1983) examine residential consumers’ demand response during peak periods from 12 pricing experiments involving about 7000 consumers in the United States. Keane and Goett (1988) verify that TOU tariffs have great potential for demand response from an experiment in California. Based on several experimental TOU tariff designs deployed in northern California, Train and Mehrez (1994) develop an econometric model that characterizes TOU adoption and consumption choices. The study finds TOU tariffs to be ineffective with a modest level of participation, and not necessarily leading to a Pareto improvement. Baladi et al. (1998) examine both the participation decision for voluntary TOU tariffs and load pattern changes of residential consumers using data from an experiment in Iowa. Faruqui and Sergici (2010) address whether consumers respond to higher peak prices by lowering their demand based on 15 recent experiments in the United States. They find that consumers do so, and provide an estimate of the magnitude of price response which depends on several factors including price increase, availability of central air conditioning, and other enabling technologies. Based on electricity consumption data in Ireland, Ata et al. (2018) empirically examine the effects of time‐based tariffs on the electricity market as well as implications for consumers, retailers, and the environment. While all the above consider voluntary TOU tariffs, some recent studies evaluate the impact of mandatory TOU pricing. Jessoe et al. (2013) study an experimental rollout of mandatory TOU tariffs targeted at large‐scale residential consumers in the northeastern United States. In addition, Maggiore et al. (2013) and Torriti (2012) examine the changes in electricity demand, price savings, and peak load shifting behavior of residential consumers in Italy where mandatory TOU tariffs have been implemented since 2010. Despite the concerns regarding the increase in consumer expenditure and electricity bills, they show that bill volatility is minimal even when the TOU program is mandated to all customers.

There is also a stream of research that analytically examines optimal pricing strategies within time‐based tariffs and the resulting interactions between utility firms and consumers. These studies connect time‐based tariffs in the electricity markets to the Operations Management (OM) literature (e.g., Banal‐Estanol and Micola 2009, Garcia et al. 2005, Kok et al. 2018, Mays and Klabjan 2017, Sioshansi 2012). Mays and Klabjan (2017) examine the optimal design and configuration of various time‐based tariffs, such as a time‐of‐use (TOU) tariff, critical peak pricing (CPP), and real‐time pricing (RTP). More recently, there is an emerging stream of work that focuses on the demand side of electricity management, typically in conjunction with a renewable energy source. Papier (2016) studies a control mechanism to manage electricity peak loads in manufacturing lines. Qi et al. (2017) propose a revenue sharing mechanism which ensures voluntary participation of prosumers (e.g., an owner of a solar panel who actively consumes and produces electricity) while enhancing resource utilization efficiency. Gärttner et al. (2018) examine a demand aggregator’s problem that includes designing and dispatching a portfolio of electricity supply (volatile renewable and conventional generators) and demand (inflexible bases and shiftable load) under various market scenarios. Liang et al. (2019) explore the demand side energy management problem taking into consideration energy storage and trading decisions. The problem is modeled as a finite markov decision process and an approximate dynamic programming approach is presented as a solution method. Qi and Shen (2018) point out the needs and opportunities in exploring the pricing and transaction structure of demand side energy management.

Most of the studies cited above do not consider a consumer’s demand response between the peak and off‐peak period. A few exceptions are described below. Yang et al. (2013a) is one of the first to highlight the importance of considering consumer response in designing and implementing the TOU tariff program. Using a game‐theoretic framework with two consumption periods, peak and off‐peak, they find that a well‐designed TOU tariff can induce a win–win situation for both the utility and the consumers. In addition, Yang et al. (2013b) consider a game‐theoretic model consisting of multiple types of consumers (residential, commercial, and industrial) with differing price responsiveness. These studies however focus on the case in which the TOU tariff is mandatory, and not a voluntary option for consumers. Dong et al. (2017) examine a utility firm’s problem in which consumers can voluntarily choose between TOU and FFR tariffs. They show that under realistic scenarios, the required power generation capacity can go down with the TOU tariff compared to the status quo, that is, the fixed flat rate. Webb et al. (2017) study the value of coordinating energy efficiency and demand response incentives provided to industrial consumers. We extend this literature by studying the design of voluntary TOU tariffs while considering heterogeneity in consumption patterns among residential consumers, as well as taking a social perspective on demand response programs. One distinct contribution of our study is that we are able to identify the misalignment between the incentives of a utility firm and a social planner that illustrates the ongoing struggle for TOU penetration in residential markets.

Finally, there is a stream of work that focuses on the economics underlying TOU tariffs and the resulting redistribution of customer bills given a certain consumer’s electricity consumption patterns. Using a stylized economic model, Mackie‐Mason (1990) illustrates that a voluntary TOU tariff can be Pareto superior or Pareto inferior depending on consumers’ consumption patterns. Woo et al. (1995) and Horowitz and Woo (2006) show that the TOU tariff with a cost‐sharing structure can be Pareto superior regardless of consumers’ consumption patterns. Due to heterogeneity among consumers in their electricity consumption patterns, Ericson (2011) highlights the importance of incorporating the free‐riding effect in TOU design; that is, voluntary TOU tariffs will only attract those consumers whose consumption patterns are already favorable under the TOU pricing scheme. Consequently, a discrete choice model that forecasts residential consumers’ self‐selection into TOU tariffs is presented. Finally, Borenstein (2013) singles out the concern about increases in electricity bills as one of the key reasons for resistance to time‐based pricing (including TOU tariffs) in the residential sector, and proposes improvements to rate designs based on an empirical analysis. Our paper extends this literature by analytically investigating the interaction between the utility firm and consumers, while quantifying the cross‐subsidization across consumers with heterogeneous consumption patterns in our case study of the US market.

Baseline Model

Model Setting and Assumptions

We consider a parsimonious stylized model setting to capture the essential characteristics of the utility’s (henceforth, the firm) and consumers’ optimization problems. In particular, we consider two electricity consumption periods, peak and base (off‐peak), denoted by j ∈ J = {p, b}. Normalizing the length of a time period to 1, and representing the beginning and end of the peak period by

t ̲

and

t ¯

respectively, we denote the fraction of the peak period as

τ = t ¯ − t ̲

where

0 ≤ t ̲ < t ¯ ≤ 1

. We assume that the electricity market is deregulated—the firm is only responsible for the retail provision of electricity and is a price‐taker in the wholesale market. We assume that consumption patterns and load shifting due to demand response programs do not affect the prices in the wholesale electricity market, since the load of an individual firm is small relative to the size of the entire wholesale market; that is, we focus on the short‐run effect of consumers’ demand response to TOU deployment. We denote

w b

as the wholesale market price during the base period and

w p

as the wholesale market price during the peak period, where

w b < w p

.

Not all consumers have the same consumption preferences. To capture consumer heterogeneity, we assume that there are N distinct consumer types based on their electricity consumption preferences. In particular, we denote the consumer types by i ∈ I,= {1, 2, …, N} where the consumption patterns are ordered by the index from the flattest, i = 1, to the peakiest, i = N. That is, flat consumers exhibit the least difference in consumption preferences between peak and base periods, whereas peaky consumers have the highest preference for peak period consumption relative to the base period. ² Therefore, a type i consumer’s consumption of (or demands for) electricity (in MWh) in each period can be expressed as

Q i b = ∫ 0 t ̲ q i b ( t ) d t + ∫ t ¯ 1 q i b ( t ) d t

and

Q i p = ∫ t ̲ t ¯ q i p ( t ) d t

, respectively. We assume that each type of consumer i ∈ I consists of

n i ∈ ( 0 , 1 )

fraction of the entire consumer mass where

∑ i ∈ I n i = 1

. Figure1a illustrates prototypical consumption profiles for three types of consumers under the current fixed flat rate (FFR) tariff. Figure 1b illustrates an example of demand response under the time‐of‐use (TOU) tariff with two tier‐priced periods.

OPEN IN VIEWER

Figure 1

Illustration of Consumer Types and Consumption Behavior under Fixed Flat Rate and Time‐of‐Use Tariffs. (a) Electricity consumption profiles, (b) Consumption after demand response

We define the following consumer utility functions under a given tariff scheme depending on consumption levels in the peak and base periods and payments to the firm (Gilbert and Jonnalagedda 2011, Mohsenian‐Rad et al. 2010):

U i = ∑ j ∈ { p , b } v i j Q i j − ( Q i j ) 2 2 − ∑ j ∈ { p , b } p i j Q i j ∀ i ∈ I ,

where

v i j > 0

represents consumer i’s preference for electricity consumption in period j and

p i j > 0

is the corresponding price charged to consumer i in period j. This specification reflects diminishing (but non‐negative) marginal utility from incremental electricity consumption during each of the peak and base periods while accounting for the disutility from paying larger electricity bills. Note that the prices charged by the firm are not consumer specific; rather, the prices paid in any period by a given consumer depends on the tariff structure that is chosen by the consumer.

We express consumer type i’s consumption preference in period j as

v i j = v + α i j

where v is the common valuation while

α i j

is a type‐ and time‐specific value of electricity consumption. We assume that a consumer’s valuation difference between peak and base periods is symmetric for the ensuing analysis; that is,

α i = α i p = − α i b

such that

v i p = v + α i

and

v i b = v − α i

. Therefore, a consumer’s additional preference for electricity consumption during peak periods (reduced preference for consumption during base periods) is captured by

α i ∈ 0 , v

; as

α i

increases, the consumer’s preferences are skewed further toward consumption during the peak period. Since consumer types are indexed in increasing order of their peakiness,

α i ≤ α i + 1

(i.e.,

α i p ≤ α i + 1 p

and

α i b ≥ α i + 1 b

) hold for all i ∈ I. The assumption of symmetric differences in electricity consumption preferences between the peak and base periods is considered for tractability. We note that relaxing this stipulation does not affect the key findings of this study. For analytical convenience, we further stipulate that

∑ j ∈ J Q i j = 1

for i ∈ I, as in Faruqui (2010) and Yang et al. (2013a). This implies that the aggregate electricity consumption is identical and invariant across consumer types and tariff structures, that is, our base model is a consumption scheduling model and not a consumption choice model. ³

We do not presuppose any relationship between customers’ income level and their electricity consumption preferences, as no conclusive evidence of any such relationship is found in the literature; for example, see Faruqui and Sergici (2010) and George et al. (2018). Therefore, we restrict our attention to differences in consumption characteristics in the residential market and remain agnostic to the relationships these characteristics might share with other demographic factors.

The intention of the TOU program is to induce consumers to shift consumption from peak hours to off‐peak hours. The TOU tariff consists of a set of prices, a peak price

p p

and a base price

p b

. Naturally, consumers are charged a higher price for electricity consumed during peak hours; that is,

p b < p p

holds, where this price gap provides consumers with an incentive to shift demand from peak to off‐peak hours. While some jurisdictions have made TOU tariffs mandatory and several others are considering moving in that direction, ⁴ they are generally difficult to push through the regulatory process and are thus typically introduced in the form of voluntary programs wherein consumers can choose to opt‐in to these tariffs instead of the default FFR tariff (Borenstein 2013). Therefore, we consider the design and deployment of voluntary TOU tariffs to a market of heterogeneous residential consumers in section 4.1, and examine the gap between the firm’s and regulator’s preferred degree of TOU adoption in section 4.2. We then examine the equity of voluntary TOU tariffs in section 4.3.

In what follows, we first examine the market equilibrium for the baseline model that employs a default fixed flat rate program.

Status Quo: Fixed Flat Rates

We consider the following two‐stage model. In the first stage, the profit‐maximizing firm chooses the fixed flat retail electricity price

p f = p b = p p

subject to restrictions placed by regulatory agencies. In stage 2, consumers respond by choosing their optimal consumption levels in the peak and base periods, and pay the firm the corresponding tariffs. Consumers must choose a level of consumption

Q i j ∗

in each period j ∈ J to maximize their respective utilities. From the stipulated utility functions, the optimal consumption levels for consumer type i can be obtained as

Q i j ∗ ( F F R ) = v i j / 2 v

.

Next, we address the profit‐maximizing firm’s problem as follows:

maximize π ( F F R ) = ∑ i ∈ I n i ∑ j ∈ J ( p f − w j ) Q i j ∗

subject to: p f ∑ i ∈ I n i ∑ j ∈ J Q i j ∗ ≤ ( 1 + r ) ∑ j ∈ J w j ∑ i ∈ I n i Q i j ∗ ,

where r > 0 is the maximum allowed rate of return over the costs of electricity provision. The constraint 3 represents a rate of return regulation imposed by a governing or regulatory body such as a public utilities commission (PUC) as is typical in electricity markets (Lazar 2016, Liston 1993, Sappington and Weisman 2016). Rate of return regulation allows a utility to recover its operating costs while also obtaining a fixed return on its investments in generation and transmission infrastructure. Since we consider the situation in which the firm is only responsible for the retail provision of electricity in a deregulated market, we assume that the rate of return regulation is applied solely to the operating costs in the utility’s problem. In the ensuing analysis, we impose an additional condition on the wholesale prices,

w p − w b < v

, to ensure that the incremental cost of electricity provision in peak periods is never so high that it is unprofitable to meet the electricity needs of consumers at those times.

The lemma below summarizes the solution to the firm’s problem in the FFR program. In addition, the resulting social welfare is obtained which is defined as the sum of consumer and producer surplus, where consumer surplus is captured by the weighted sum of utilities enjoyed by each consumer type in equilibrium.

Lemma 1

Under the FFR program, the firm charges the price of

p f ∗ = ( 1 + r ) w ¯

and earns profit of

π ∗ ( F F R ) = r w ¯

, where

w ¯ : = 1 2 v v ( w p + w b ) + ( w p − w b ) ∑ i ∈ I α i n i

is the average cost of electricity provision under the FFR tariff. The resulting social welfare in the equilibrium is given by

S W ( F F R ) = 1 4 v 3 v 2 + ∑ i ∈ I α i 2 n i − w ¯

.

We note that the constraint 3 is binding in equilibrium, and thus the firm’s profit is always restricted by the regulator. As the status quo, this equilibrium will provide the basis for our further analyses of demand response programs in subsequent sections. To that end, we substitute the optimal price in the FFR program and obtain the consumer type i’s utility in the status quo (which will serve as their reservation utility while considering voluntary time‐based tariffs) as follows:

U i ∗ ( F F R ) = 1 4 v ( 3 v 2 + α i 2 ) − p f ∗ .

In the next section, we consider the impact of introducing TOU tariffs and examine the interactions between the utility firm and the consumers.

Analysis: Introduction of Voluntary Time‐of‐Use Tariffs

To take a comprehensive perspective of TOU tariffs, we investigate the impact of TOU tariffs on the three major parties involved in the residential electricity market: the private utility firms (Investor‐Owned Utilities), regulatory bodies (Public Utility Commissions), and consumers. In particular, we examine how the introduction of voluntary TOU tariffs affects the market and the firm’s profit in section 4.1, social welfare in section 4.2, and the distribution of benefits across the different types of consumers in section 4.3.

Utility Firm’s Perspective

To examine the market equilibrium under voluntary TOU tariffs, we consider the following three‐stage problem. In the first stage, the firm sets the respective TOU electricity prices

p b

and

p p

for the base and peak periods, where

p b < p f < p p

to ensure that consumers are incentivized to shift consumption from the peak to the base period. The requirement for

p b < p f < p p

has precedence in the literature (see Dong et al. 2017) and is also consistent with practice (Shao et al. 2010). Note that the default FFR tariff

p f ∗

is maintained as before in Lemma 1 since changing

p f ∗

would violate the voluntary nature of the TOU offering and could consequently result in regulatory issues (Braithwait et al. 2007). In the second stage, each type of consumer decides whether to opt‐in to the TOU tariff or stick with the FFR tariff. If a consumer decides to opt‐in to the TOU tariff, then the consumer makes the demand shifting decision in the third stage; that is, the consumer determines how much electricity to consume during the peak and base periods respectively in response to the incentive provided by the TOU program. The consumer who decides to stick with the FFR tariff maintains the same consumption levels as in section 3.

Solving the problem backwards, we first examine the consumers’ problem to find their best responses (i.e., consumption patterns) to the TOU tariffs if they decide to opt‐in. Recall (1) for the respective utility of each type of consumer. Given the prices under the TOU tariff, consumer i’s optimal consumption level in each period is obtained by solving the following:

maximize Q i j U i ( T O U ) = ∑ j ∈ J v i j Q i j − ( Q i j ) 2 2 − ∑ j ∈ J p j Q i j ∀ i ∈ I .

From the above, we obtain the optimal consumption of consumer type i in period j as

Q i b ∗ ( T O U ) = 1 2 v ( v − α i + ( p p − p b ) )

and

Q i p ∗ ( T O U ) = 1 2 v ( v + α i − ( p p − p b ) )

. Here, the price differential

Δ p = p p − p b

represents the price incentive that each consumer receives for shifting consumption from the peak to the off‐peak period. However, not all types of consumers will find it optimal to opt‐in to the TOU program. By substituting the consumption decisions under the TOU tariffs into their respective utility functions, we obtain consumer type i’s utility from opting in to the voluntary TOU tariff as

U i ∗ ( T O U ) = 1 4 v v + α i − ( p p − p b ) 2 + 1 2 v − α i − p b .

In the second stage of the game, each consumer type chooses whether or not to opt‐in to the TOU tariff. A consumer belonging to type i will opt‐in if and only if

U i ∗ ( T O U ) ≥ U i ∗ ( F F R )

holds. Below we outline a key result regarding the structure of consumer adoption behavior under voluntary TOU tariffs.

Lemma 2

Suppose it is optimal for consumer type i to opt‐in to the TOU program, that is,

U i ∗ ( T O U ) ≥ U i ∗ ( F F R )

. Then, it is optimal for all consumer types ℓ ≤ i to opt‐in to the TOU program as well.

Lemma 2 illustrates a useful property of the voluntary TOU adoption equilibrium. It implies that TOU adoption occurs in increasing order of consumers’ peakiness

α i

. That is, the firm penetrates the market in incremental order, targeting from the flattest consumers, type 1, to the peakiest consumers, type N. This trend is due to the fact that the flattest consumers have the most to gain by opting in to TOU tariffs as they consume the least during the peak period and hence have the least resistance to demand response. This phenomenon is in line with the free‐riding effect observed in practice, since the flat consumers do not consume a lot of electricity in the peak periods anyhow and thus do not have to shift much demand in order to see significant savings on their electricity bills (Hartway et al. 1999, Woo et al. 2008).

Lemma 2 also indicates that it suffices to only consider (N + 1) possible outcomes in examining the equilibrium with N consumer types. That is, starting with Case 1, we consider cases up to Case N in an incremental order. When no consumer type opts in, then the situation reduces to the FFR tariff which we refer to as Case ∅.

We now turn to the first stage wherein the firm makes its pricing decisions under the voluntary TOU tariffs. Note that the profit the firm obtains in Case ∅ is identical to that for the FFR tariff, that is,

π ∗ ( T O U ( ∅ ) ) = π ∗ ( F F R )

. Therefore, we only consider the remaining N cases in what follows. To ensure a meaningful solution, we stipulate that

p p − p b < v

, that is, the price differential between the peak and off‐peak is not larger than a consumer’s common valuation for a unit of electricity consumption. This allows us to focus on the scenarios in which the consumer’s incentive for demand response is not so large that they completely forgo consumption in the peak period. In addition, we assume that the firm sets electricity prices that are decoupled from the costs of electricity provision under the TOU tariff. Under the TOU program, the utility will always be worse off (relative to the FFR program) if its profit is coupled with the cost of electricity provision since these costs will generally decrease as more consumers participate in the demand response program. Lazar (2016) notes that such unintended consequences can also arise when energy efficiency programs are introduced. To prevent this, other regulatory schemes such as a price‐cap policy (see, e.g., Dong et al. 2017) may be considered to allow the firm to recover any lost profit resulting from the deployment of energy efficiency or demand response programs. We do not impose explicit price caps in our model; rather, we allow the utility company to choose TOU prices that maximize its profit. In equilibrium, if there is no feasible set of TOU prices (

p p > p f > p b

) that can return a higher profit than the status quo (FFR tariffs), then the utility will optimally set

p p = p b = p f

, that is, it will eschew the TOU program in favor of the FFR tariffs. [Correction added on 3 December 2019, after first online publication: the subscript values ‘p’, ‘f’, and ‘b’ in the inline equations at the last sentence of the 2nd paragraph and in equation (9) have been superscripted.]

Before we introduce the utility’s TOU pricing problem, we first define the TOU adoption equilibrium as follows: Case k equilibrium refers to the case in which the utility firm sets TOU prices optimally such that the first k flattest consumer types,

{ 1 , … , k } = I k ⊆ I

, choose to opt‐in to the TOU program. To determine the TOU prices that maximize its profit for Case k, the firm solves the following problem:

maximize p b , p p π ∗ ( T O U ( k ) ) = ∑ i ∈ I k ∑ j ∈ { b , p } n i ( p j − w j ) Q i j ∗ + ∑ i ∈ I \ I k ∑ j ∈ { b , p } n i ( p f − w j Q i j ∗ )

subject to: U i ∗ ( T O U ) ≥ U i ∗ ( F F R ) ∀ i ∈ I k = { 1 , … , k } ,

U i ∗ ( T O U ) ≤ U i ∗ ( F F R ) ∀ i ∈ I \ I k = { k + 1 , … , N } ,

p p ≥ p f ≥ p b .

In the objective function 6, the first term represents the profit obtained from consumers adopting the TOU tariff and the second term represents the profit from the remaining consumers that stick with the FFR tariff. Due to the incremental inclusion property in Lemma 2, the firm only needs to ensure that consumer type k opts in to TOU program. Thus, it follows that constraint 7 will be binding in equilibrium for i = k, that is,

U k ∗ ( T O U ) = U k ∗ ( F F R )

in equilibrium.

We now obtain the resulting outcome under each equilibrium case. For expositional convenience, we define the heterogeneity within the adoption set for Case k as

H ( k ) : = ∑ i ∈ I k n i ∑ ℓ ∈ I k n ℓ ( α k − α i ) .

The above measure can be interpreted as the weighted absolute deviation in consumption preferences within adoption set k which captures the heterogeneity among the consumers who are included in the TOU adoption set relative to the peakiest consumer in the set, that is, the type k consumer. Barring highly disproportionate characteristics across consumer segments, H(k) is generally increasing in k. Also, if only one type of consumer opts in, that is, Case 1, then H(1) = 0 since the consumption preferences of adopting consumers is homogeneous.

Lemma 3

In the presence of TOU tariffs,

H ( k ) < w p − w b

is a necessary condition for Case k to be feasible in equilibrium.

If TOU Case k emerges in equilibrium, then the firm charges prices

p b ∗ < p f ∗

and

p p ∗ > p f ∗

, and earns a profit of

π ∗ ( T O U ( k ) ) = π ∗ ( F F R ) + ∑ i ∈ I k n i 4 v ( w p − w b − H ( k ) ) 2 > π ∗ ( F F R )

. The corresponding social welfare is

S W ( T O U ( k ) ) = S W ( F F R ) + ∑ i ∈ I k n i 4 v ( ( w p − w b ) 2 − H ( k ) 2 )

. Complete expressions for

p b ∗

and

p p ∗

are provided in Appendix B.

Lemma 3(i) presents the necessary condition for each equilibrium case to be feasible. If the condition does not hold, that is, if

H ( k ) ≥ w p − w b

, then the firm charges

p b ∗ = p p ∗ = p f ∗

and earns profit of

π ∗ ( T O U ( k ) ) = π ∗ ( F F R )

; that is, the firm resorts to the FFR tariff as in the status quo since there is no set of prices

p p ∗ > p f ∗ > p b ∗

that is more profitable to the firm. This shows that the deviation in consumption preferences of the k flattest consumers must not be too high compared to the wholesale price differential for Case k to be feasible.

If the necessary condition holds, then Lemma 3(ii) demonstrates the tariff scheme that induces the first k consumer types to opt‐in to TOU while yielding

π ∗ ( T O U ( k ) ) > π ∗ ( F F R )

. Note that since constraint 7 for i = k is binding in the Case k, the firm is able to extract all surplus from consumer type k, leaving that consumer type indifferent between opting in to the TOU program and sticking with the FFR program. However, all consumer types i < k obtain a positive surplus from opting in to the TOU program in Case k. The surplus accruing to a consumer type i is larger when that consumer’s consumption preferences are flatter relative to the k’th consumer type. Therefore, a large deviation in consumption preferences of the adopting consumer types relative to consumer type k forces the firm to leave a large amount of surplus to these consumers. Hence, it is profitable for the firm to offer the TOU tariff only if the deviation in consumption preferences relative to the peakiest adopting consumer (type k) is sufficiently low—this is also evident from Lemma 3 where

π ∗ ( T O U ( k ) )

is decreasing in H(k).

Lemma 3 also suggests that the TOU tariff dominates the FFR tariff from the firm’s perspective. That is, the Case 1 TOU equilibrium always yields a greater profit than FFR, since

π ∗ ( T O U ( 1 ) ) − π ∗ ( F F R ) = n 1 4 v ( w p − w b − H ( 1 ) ) 2 > 0

holds since

w p − w b > 0

by stipulation and H(1) = 0 by definition. Note also that Case 1 is always feasible per Lemma 3(i). However, inducing only the flattest type of consumer to opt‐in to TOU tariffs is of limited value to both the firm and society, as the flattest consumers have very little resistance to demand shifting and do not impose large electricity provision costs on the firm in the first place. Ideally, the firm would like to induce peakier consumers to also opt‐in to the TOU program. The degree of penetration into the consumer market, that is, precisely which of the remaining N cases of the TOU tariff emerge in equilibrium remains to be seen. Building on Lemma 3, we next characterize the optimality conditions for the voluntary TOU tariff.

Proposition 1

Case k emerges as the equilibrium under the voluntary TOU tariff if and only if the following two conditions hold:

H ( k ) < w p − w b

and

for all

i ∈ I k

(i.e., i ≤ k),

H ( k ) − H ( i ) ≤ max 0 , ( w p − w b − H ( k ) ) 2 ∑ ℓ ∈ I k \ I i n ℓ 2 ( w p − w b − H ( k ) + H ( i ) 2 ) ∑ ℓ ∈ I i n ℓ

; for all

i ∉ I k

(i.e., i > k),

H ( i ) − H ( k ) ≥ ( w p − w b − H ( i ) ) 2 ∑ ℓ ∈ I i \ I k n ℓ 2 ( w p − w b − H ( k ) + H ( i ) 2 ) ∑ ℓ ∈ I k n ℓ

.

Proposition 1 characterizes the necessary and sufficient conditions, (a) and (b), respectively, under which each equilibrium case emerges under the voluntary TOU tariff. These conditions, when taken together, imply that the optimal degree of TOU adoption is achieved when the heterogeneity in consumption preferences H(k) within a given adoption set

I k

is not too large. Specifically, the heterogeneity of the adoption set H(k) must be less than both the wholesale price differential

w p − w b

and a threshold in relation to other H(i)’s. Recall from our earlier discussion following Lemma 3 that this is a result of the firm wanting to leave less surplus to flatter consumers who will opt‐in to the TOU tariffs along with the peakiest consumer being targeted by the TOU tariffs.

Next, we examine the impact of consumer characteristics on the benefits of deploying voluntary TOU tariffs.

Proposition 2

Suppose TOU(k) is optimal to the utility. Then, the incremental profit to the firm,

Δ π k = π ∗ ( T O U ( k ) ) − π ∗ ( F F R )

, and the incremental social welfare,

Δ S W k = S W ( T O U ( k ) ) − S W ( F F R )

, are increasing in

n k

, but decreasing in

α k

.

Proposition 2 indicates that the benefit to the firm and society of inducing the next peakiest consumer segment into the TOU adoption set is increasing in the size of that consumer segment but decreasing in the peakiness of that consumer segment. Intuitively, an increase in

n k

increases the number of consumers participating in demand response. However, inducing a very peaky consumer segment (high

α k

) to join the TOU program may not necessarily be beneficial for two reasons. First, from the firm’s perspective, targeting voluntary TOU tariffs at a relatively peaky consumer segment will cause the firm to leave a large amount of surplus to the flatter consumers that will also opt‐in. Second, targeting TOU tariffs at a relatively peaky consumer segment requires the firm to ensure that this consumer segment does not suffer from giving up its peaky (and costly) consumption preferences, that is, the peak price set by the firm should be sufficiently low to induce participation from this consumer segment; this in turn reduces every flatter consumer type’s incentive for demand response, and consequently undermines the value of TOU pricing to the firm as well as other participating consumers. Therefore, there is a clear trade‐off between inducing more consumers to participate in the TOU program and generating sufficient demand response from those consumers that are already willing participants in the program. The following formalizes the relationship between the magnitude of the incentive for demand response and the number of TOU adopters in equilibrium.

Proposition 3

For any k and

k ′

such that

H ( k ) < w p − w b

and

H ( k ′ ) < w p − w b

, it follows that

p k p ∗ − p k b ∗ ≤ p k ′ p ∗ − p k ′ b ∗

holds if and only if

H ( k ′ ) < H ( k )

.

Proposition 3 indicates that an equilibrium with a more homogeneous adoption set (lower H(k)) will result in a larger price gap, and hence, a larger incentive for demand response from consumers. In other words, the magnitude of the demand response incentive,

p k p ∗ − p k b ∗

, is decreasing in H(k). It is worth noting that H(k) is generally increasing in k due to its additive nature, and therefore, the incentive for demand response is likely to decrease as the number of adopting consumer types increases. ⁵ Therefore, while we may be concerned with low TOU enrollment rates as a sign of the failure of demand response programs in the residential market, it is also important to consider the level of participation among enrolled consumers as measured by the amount of demand shifted from the peak to the off‐peak periods by each consumer.

Social Welfare Perspective

We now extend our analysis by taking the perspective of regulatory bodies such as Public Utility Commissions. In particular, we examine the social‐welfare maximizing equilibrium under the voluntary TOU tariff, and obtain relevant policy implications. The following result is the counterpart of Proposition 1 from the social welfare perspective.

Proposition 4

Under the TOU tariff, Case k achieves the greatest social welfare if and only if the following two conditions hold:

H ( k ) < w p − w b

and

for all

i ∈ I k

(i.e., i ≤ k),

H ( k ) − H ( i ) ≤ max 0 , ( ( w p − w b ) 2 − H ( k ) 2 ) ∑ ℓ ∈ I k \ I i n ℓ ( H ( k ) + H ( i ) ) ∑ ℓ ∈ I i n ℓ

; for all

i ∉ I k

(i.e., i > k),

H ( i ) − H ( k ) ≥ ( ( w p − w b ) 2 − H ( i ) 2 ) ∑ ℓ ∈ I i \ I k n ℓ ( H ( k ) + H ( i ) ) ∑ ℓ ∈ I k n ℓ

.

As in the firm’s perspective on TOU tariffs, condition (a) in Proposition 4 provides the necessary condition for the feasibility of TOU tariffs targeted to an adoption set

I k

in Case k. Condition (b) characterizes the sufficient condition. Altogether, as in the case of the firm’s optimal tariffs, Proposition 4 suggests that the optimal level of TOU adoption corresponds to a consumption set

I k

that exhibits a relatively low degree of heterogeneity. As it was the case from the firm’s perspective, we find that the TOU tariff dominates the FFR tariff from a social welfare perspective as well. That is, TOU tariffs represent a win–win for both the firm as well as society overall. However, we note from Proposition 4(b) that the sufficient condition for the social optimality of Case k is different from the analogous condition for the firm’s perspective in Proposition 1. Hence, while both perspectives may result in the deployment of TOU tariffs over FFR, it is not clear what degree of TOU adoption is optimal from either perspective. We outline the differences in the equilibria arising out of either perspective in the following result. For notational convenience, we denote Case k’s sufficient condition from the firm’s perspective, Proposition 1(b), by

ω k π

and its social welfare counterpart, Proposition 4(b), by

ω k SW

.

Proposition 5

Suppose TOU(k) is optimal from the social welfare perspective and

T O U ( k ′ )

is optimal from the firm’s perspective. Then,

k ≥ k ′

holds for all

k , k ′ ∈ I

; that is,

Ω k SW ⊆ Ω k π

holds for all k ∈ I where

Ω k π = ∪ ℓ = 1 k ω ℓ π

and

Ω k SW = ∪ ℓ = 1 k ω ℓ SW

.

Proposition 5 highlights the misalignment between the firm and social welfare perspectives on TOU tariffs. In particular, we find that the deployment of TOU tariffs by the firm will always be lagging relative to the socially optimal degree of TOU adoption. While a greater degree of TOU adoption benefits more customers by allowing them to lower their bills, it also cuts into the firm’s revenue due to the free‐riding behavior of flatter consumer types. Figure 2 illustrates the misalignment between the optimal deployment of TOU tariffs according to the two perspectives when there are three consumer segments (N = 3): flat, average, and peaky.

OPEN IN VIEWER

Figure 2

Time‐of‐Use (TOU) Adoption Equilibria from Firm’s Perspective, Regulator’s Perspective, and the Contrast between the Two as a Function of the Number of Flat (

n 1

) and Peaky (

n 3

) Consumers.Note: We set v = 1,

w p = 0.5

,

w b = 0

, r = 0.05,

α 1 = 0.25

,

α 2 = 0.5

,

α 3 = 0.75

. Case k ∈ {1, 2, 3} represents the equilibrium in which only the first k consumer segments opt‐in to the TOU tariff. In the rightmost graph, case {i, j} represents the case wherein only the first i (j) consumer segments opt‐in from the firm’s (regulator’s) perspective.

The first two graphs in Figure 2 illustrate the equilibrium TOU adoption cases from the firm’s perspective and the regulator’s perspective. The third graph illustrates the misalignment between the two perspectives, where the unshaded region indicates a difference in the firm’s and regulator’s preferred degrees of TOU adoption. Alignment between the firm’s and social welfare perspectives occurs in the top and bottom of the rightmost graph in Figure 2 (shaded gray) when there is either a large or small number of extreme consumers. When the number of peaky consumers is sufficiently large (small), then the optimal number of adopting consumers is identical from both perspectives because there is a (no) clear benefit to the firm and society to inducing (not inducing) demand response from the peakiest consumer segment. It is when the peaky consumer segment is intermediate in size that there arises a gap between the firm’s and regulator’s preferred levels of TOU adoption, wherein the firm does not perceive the added benefit of inducing demand response from the peakiest consumer segment while the regulator may continue to do so. Therefore, misalignment between the firm and the regulator’s perspectives occurs when consumption preferences are relatively uniform.

Consumers’ Perspective

Time‐of‐use tariffs have been proposed as a measure to reduce the undesirable cross‐subsidization across consumers with different consumption preferences by passing on to customers the true costs of their consumption patterns (Borenstein 2013, Faruqui 2010). Faruqui (2010) notes that a lower degree of cross‐subsidization is more desirable in an electricity tariff. Accordingly, we seek to compare the cross‐subsidization resulting from voluntary TOU tariffs with those under FFR tariffs. As discussed in the literature (e.g., Faulhaber 1975,2005), we consider cross‐subsidization as the difference between the actual electricity bill paid by a consumer and the cost‐based allocation of the revenue collected by the firm from all consumers. Specifically, we define the cross‐subsidization of a single consumer of type k under a given tariff scheme T as follows:

ξ k ( T ) = ∑ j ∈ J p j ∗ ( T ) Q k j ∗ ( T ) − ∑ i ∈ I ∑ j ∈ J n i p j ∗ ( T ) Q i j ∗ ( T ) × ∑ j ∈ J w j ( T ) Q k j ∗ ( T ) ∑ i ∈ I ∑ j ∈ J n i w j ( T ) Q i j ∗ ( T ) .

The first term represents the payment made to the firm by a consumer of type k. The second term represents the proportional share of revenue that is due from consumer type k which is computed by multiplying the total revenue collected by the firm with the proportion of the firm’s electricity provision costs attributable to a single consumer of type k. Note that if

ξ k ( T )

is negative, it implies that the type k consumer is paying less than their proportional share for electricity while (some or all of) the other types are cross‐subsidizing the consumer type k. In contrast, a positive

ξ k ( T )

implies that the consumer type k is paying more than its proportional share for electricity and is cross‐subsidizing (some or all of) the other types of consumers. By construction, we have

∑ k ∈ I n k ξ k ( T ) = 0

. To determine the cross‐subsidization of a given tariff scheme across the entire residential market, we define the system‐wide cross‐subsidization under tariff T as follows:

Ξ ( T ) : = ∑ k ∈ I n k | ξ k ( T ) | ,

which aggregates the absolute degree of individual cross‐subsidization among all consumer types. We note that lower Ξ(T) represents a more equitable tariff (i.e., a tariff with less cross‐subsidization) scheme resulting from a smaller magnitude of under(over)payments by consumers.

We characterize the impact of a consumer’s type on their degree of individual cross‐subsidization in the following proposition.

Proposition 6

Under both the FFR and TOU tariffs,

ξ k

is decreasing in k and

there exists a unique

k ¯ ( T )

such that

ξ k ( T ) > 0

for all

k ≤ k ¯ ( T )

and

ξ k ( T ) < 0

for all

k > k ¯ ( T )

for T ∈ {FFR, TOU}.

Proposition 6(a) implies that flatter consumers are worse off than their peakier counterparts under both the FFR and TOU tariffs; that is, a flatter consumer will pay more in cross‐subsidies (alternatively, receive less in cross‐subsidies) than peakier consumers. Proposition 6(b) further indicates that flatter consumers (specifically

k ∈ { 1 , … , k ¯ ( T ) }

) in the market will pay more than their proportional share of electricity prices under both the FFR and TOU tariffs; that is, they will cross‐subsidize their peakier counterparts. In particular, the flattest consumer type is always a cross‐subsidizer (

ξ 1 ( T ) > 0

). Conversely, the peakier consumers in the market will pay less than their proportional share, that is, they will be cross‐subsidized by the flatter consumers in the market, and the peakiest consumer type is always cross‐subsidized (

ξ N ( T ) < 0

). Altogether, Proposition 6 indicates that peakier consumers benefit from the overpayments made by flatter consumers, and therefore, that both FFR and TOU programs can be inequitable. This implies that those consumer types that impose the most costs on the firm by virtue of their peakier consumption preferences are undercharged relative to the flatter types with less expensive consumption preferences.

TOU tariffs were introduced in part to resolve the problem of disproportionate benefit distribution inherent to the FFR tariff structure (Faruqui 2010). However, the TOU tariffs considered thus far are voluntary in nature (as is generally the case with the rollouts of TOU tariffs across the United States), and as such, they often suffer from the inability to induce the peakiest consumers to adopt TOU tariffs. In what follows, we describe the impact of voluntary TOU tariffs on the degree of cross‐subsidization of the peakiest consumers who do not opt‐in to the voluntary TOU tariffs.

Lemma 4

If

H ( k ) < w p − w b

, then

ξ i ( T O U ( k ) ) < ξ i ( F F R )

holds for all

i ∉ I k

.

Lemma 4 shows that those consumers who do not opt‐in to the TOU tariff are cross‐subsidized more heavily (alternatively, cross‐subsidize less) under the voluntary TOU tariff structure relative to the FFR tariff. This results from the fact that non‐adopting consumers continue to make the same payments to the firm as they did under the FFR tariff while imposing higher costs relative to their counterparts that opt‐in to the TOU tariffs and participate in demand response. This has significant implications for the performance of voluntary TOU tariffs from an equity perspective—if enough consumers fail to opt‐in to the TOU tariffs, that leads to a large volume and magnitude of underpayments that must be compensated by larger overpayments from the adopting consumers. Therefore, voluntary TOU tariffs are likely to be less equitable than FFR tariffs unless they induce a sufficiently high degree of adoption. We next contrast overall performance by comparing the system‐wide cross‐subsidization of FFR and TOU tariffs in the following result.

Proposition 7

For all

k ≤ k ¯ ( F F R )

, Ξ(TOU(k)) > Ξ(FFR) holds.

In addition, Ξ(TOU(N)) < Ξ(FFR) holds.

From Lemma 4, we noted that lower levels of adoption would likely lead to an inequitable tariff structure due to the relative underpayments made by the non‐adopting consumer types. Indeed, Proposition 7(a) reveals that if sufficiently few consumer types opt‐in to the TOU tariffs, the overall distribution of benefits under the TOU tariff would be less equitable than the default FFR tariff. Specifically, if the only types of consumers that opt‐in to the TOU tariff are those that were cross‐subsidizers under the FFR tariff (i.e., if

ξ i ( F F R ) > 0

for all

i ∈ I k

which holds when

k ≤ k ¯ ( F F R )

), then it follows from Lemma 4 that voluntary TOU tariffs fare worse from an equity perspective because the underpaying consumer types are even more heavily cross‐subsidized under the TOU tariffs. We note here that

k ¯ ( F F R ) = max [ ℓ : α ℓ ≤ ∑ i ∈ N n i α i ]

, which implies that TOU tariffs fare worse when the peakiest adopting consumer type is less peaky than the average consumer in the residential market.

It is worth pointing out that the converse of Proposition 7(a) does not necessarily hold. That is, it is not guaranteed that TOU tariffs will be more equitable than FFR tariffs if they achieve more adoption than the threshold indicated. However, Proposition 7(b) notes that if all consumers are induced to adopt TOU tariffs thus yielding a Case N equilibrium, then equity is always improved relative to the status quo. We illustrate the degree of cross‐subsidization through an example of a market with three consumer segments in Figure 3.

OPEN IN VIEWER

Figure 3

Individual and System‐Wide Cross‐Subsidization for Three Consumer Types with

n 1 = n 2 = n 3 = 0.33

,

α 1 = 0.25

,

α 2 = 0.5

,

α 3 = 0.75

,

w b = 0.1

,

w p = 0.4

, v = 1, r = 0.05. (a) Individual cross‐subsidization across tariff structures, (b) System‐wide cross‐subsidization as a percentage of total revenue across tariff structures

We observe from Figure 3a that regardless of the tariff structure, flatter consumers overpay relative to their peakier counterparts as noted in Proposition 6. However, while full adoption under the voluntary TOU tariff (Case 3) results in each consumer overpaying or underpaying (weakly) less than in the FFR tariff, the TOU tariffs with lower degrees of adoption, particularly TOU Case 1, result in significant over and underpayments relative to the FFR tariff as shown in Lemma 4. As a consequence, Figure 3b illustrates that only full adoption under the TOU tariff outperforms the FFR tariff from an equity perspective. While such high adoption rates are difficult to achieve in practice, several states such as California have taken the unique step to install TOU tariffs as the default tariff structure for their residential consumers (Rocky Mountain Institute 2015). While not mandatory, these default TOU tariffs benefit from the inertia faced by consumers as documented by literature on the status quo bias (Kahneman et al. 1991), and are thus likely to improve equitable distribution of benefits by approaching full adoption, particularly if no consumer is worse‐off relative to the FFR tariffs.

Numerical Study: US Residential Sector

In this section, we explore the application of our theoretical insights by simulating the state of TOU deployment in the US residential sector and comparing those results with the current state of TOU adoption. For the simulation, we calibrate our models using the data set and assumptions that are outlined below.

US wholesale electricity market data is obtained from the Federal Energy Regulatory Commission (U.S. FERC 2018) which publishes reports on regional wholesale electricity markets. FERC also provides market reports for each regional transmission organization (RTO) and independent system operator (ISO) that can be used to determine the average locational marginal prices (LMPs) during peak and off‐peak periods across various regions of the United States. Using this data, we calibrate

w p

w b

To illustrate some potential customer types across states in the U.S., we use the data for simulated energy profiles for US residential buildings for cities in the United States developed by the U.S. Department of Energy (U.S. DOE 2013). These profiles include three residential building types: baseline, low‐load, and high load. The baseline residential building has heating and cooling set points of 71 and 76

∘

∘

∘

For the numerical analysis, we choose one representative city from 17 different states in the United States. These 17 states are mainly served by electricity trading hubs in four different regional wholesale markets. We characterize the peakiness of electricity consumption for a city as being representative of the corresponding state. Using the load profiles of the three consumer types and the assumptions stated above, we estimate the peakiness

α i

OPEN IN VIEWER

Table 1

Summary Statistics on the Simulated Consumption Scale, Patterns and Wholesale Market Prices in 17 U.S. States

State	Representative city	ISO/RTO	Daily avg. cons. (kWh)	α 1	α 2	α 3	w b ($MWh)	w p ($MWh)
AR	Little Rock	MISO	33.5	5.7	12.5	15.5	20.7	26.4
AZ	Phoenix	Southwest	33.8	15.1	17.5	21.3	36.6	60.1
DE	Wilmington	PJM	29.7	3.2	6.0	10.0	23.1	33.5
IA	Des Moines	MISO	26.5	2.9	7.8	8.3	21.5	29.4
IL	Chicago	MISO	21.4	1.9	7.5	8.6	22.1	29.7
IN	Indianapolis	MISO	29.8	4.1	8.0	8.4	23.1	32.2
KY	Louisville	MISO	34.4	4.1	8.1	13.3	21.5	29.4
MD	Baltimore	PJM	30.7	2.3	7.9	11.7	30.6	43.9
MI	Detroit	MISO	20.6	5.6	6.0	12.2	22.8	31.4
MN	Minneapolis	MISO	24.2	4.0	4.1	9.1	16.2	23.2
NJ	Newark	PJM	21.0	5.4	9.2	11.4	20.1	29.2
OH	Columbus	PJM	27.3	6.1	6.6	14.1	20.5	29.9
PA	Philadelphia	PJM	25.1	1.8	7.5	11.7	20.0	29.1
VA	Norfolk	PJM	35.4	6.2	10.9	12.6	26.4	36.5
WA	Seattle	Northwest	33.4	7.0	7.3	7.9	14.0	35.8
WI	Milwaukee	MISO	21.4	4.5	4.9	9.0	21.5	29.4
WV	Charleston	PJM	33.5	7.1	12.0	16.6	23.8	34.1

The resulting equilibrium outcomes from the respective firm’s and regulator’s perspectives based on our model are presented in Figure 4. We summarize the current state of time‐varying tariff deployment in Table 2 based on firm‐by‐firm level data provided by the U.S. EIA (2018). While we compare the equilibrium outcomes with the actual state of the TOU adoption in the United States, we note that our simulation results are meant to provide insights on the general trend of TOU adoption rather than to estimate the size of TOU enrollment or to predict consumption volumes in each city. This is because our stylized model does not capture certain aspects of market conditions (e.g., market power, local regulation), technical details of electricity operation (e.g., transmission constraints, generator’s characteristics) as well as consumer behaviors (e.g., psychological barriers to TOU adoption).

OPEN IN VIEWER

Figure 4

Equilibrium Cases for 17 US States from the Firm’s Perspective and the Social Welfare Perspective. (a) Utility firm’s perspective, (b) Regulator’s perspective [Color figure can be viewed at wileyonlinelibrary.com]

OPEN IN VIEWER

Table 2

Summary Statistics on the Deployment of Time‐Varying Tariffs for the Residential Sector in 17 U.S. States

State	Available time‐varying tariff types*	Number of households (mil.)	Enrollment rate (%)	Total number of utility firms	Firms offering tariffs (%)
AR	TOU, VPP	1.4	7.8	30	10
AZ	TOU	2.8	32.9	33	24
DE	TOU, CPR	0.4	64.6	13	23
IA	TOU	1.4	3.8	91	10
IL	TOU, CPR, RTP	3.4	8.0	37	14
IN	TOU, CPP	2.8	1.6	63	27
KY	TOU	2.0	0.3	49	12
MD	TOU, CPR, VPP	1.9	95.6	16	25
MI	TOU, CPP, CPR	4.3	0.6	38	13
MN	TOU, CPP	2.4	0.8	112	9
NJ	TOU, VPP, RTP	3.0	0.6	17	18
OH	TOU, CPP, RTP	2.7	27.4	57	12
PA	TOU, VPP, RTP	3.6	0.0	31	16
VA	TOU	3.4	0.3	29	24
WA	TOU	3.0	0.0	49	2
WI	TOU, CPP	2.7	1.7	85	61
WV	TOU	0.9	0.3	6	33

Note

*This includes all time‐varying tariff types including TOU (time‐of‐use), CPP (critical peak pricing, CPR (critical peak rebate), VPP (variable peak pricing), and RTP (real time pricing).

Figure 4 highlights that the TOU adoption trend from the firm’s perspective is lagging relative to the socially optimal degree of TOU adoption, as shown in Proposition 5. In particular, misalignment between the two perspectives is prevalent at intermediate levels of consumer heterogeneity in peakiness and the wholesale price gap, that is, where consumers are not too dissimilar in their consumption preferences and the utility firm’s incentive to deploy TOU tariffs is moderate. Comparing Figure 4 and Table 2, we see that the three states with the highest TOU adoption—Maryland, Delaware, and Arizona—all appear in the utility firms’ Case 3 region. In addition, two of three states in the utility firms’ Case 1 region have very low TOU adoption. This adoption pattern is consistent with the model developed in this study (exceptions include Washington and Virginia that have low adoption and Arkansas that has relatively high adoption). Furthermore, the proportion of utility firms offering demand response programs to consumers in the states falling within the Case 3 equilibrium in Figure 4, with the exception of Washington, is nearly 25%, which is relatively higher than that in other states. Utility firms in several other states have not even introduced TOU tariffs in the residential sector; currently, only 5 utility firms out of 31 in Pennsylvania and 6 out of 49 in Kentucky are offering TOU programs.

We find that a considerable degree of misalignment persists in states that exhibit a moderate degree of consumer heterogeneity and wholesale price gaps. For instance, while the Northern Indiana Public Service Company identified a 36% market penetration potential for residential demand response in 2015 (Applied Energy Group 2016), TOU tariffs were not offered to customers there until 2017. Other states too including Pennsylvania (adoption rate nearly 0%) and Wisconsin (1.7%) have not realized their adoption potential. Therefore, despite public interest in demand response programs, utility companies may lack the necessary incentives to deploy TOU programs in a manner that induces meaningful large‐scale adoption while maintaining profitability.

We next turn to an analysis of cross‐subsidization under TOU tariffs in the states considered previously. Here, we rely on the interpretation of our simulation results since the relevant data for these states are not available. Using Lemma 1 along with the

w p

and

w b

obtained above, we compute the average cost of electricity provision under the FFR tariff,

w ¯

for each city. In addition, referring to the historical rate of return in California during the last 10 years (U.S. CPUC 2018) and the real cost of capital in electricity sectors of OECD countries (IRENA 2018), we stipulate the maximum allowed rate of return r as 7.5% and compute the electricity price under the FFR tariff,

p f ∗

. From Lemma 3, we compute the retail electricity prices under the voluntary TOU tariff,

p b ∗

and

p p ∗

. Finally, with the prices obtained under the TOU tariffs, we determine the amount of consumption shifted from peak to off‐peak for each consumer type when the consumer opts in to the TOU tariff. To contrast the equilibrium TOU adoption with the maximum achievable potential for demand response, we also estimate the demand response achieved in the Case N equilibrium. This is identical to a mandatory TOU tariff subject to Pareto improvement constraints such that no consumer is worse off from participating in the TOU program relative to the FFR program (henceforth referred to as mandatory TOU). Table 3 summarizes all calculated prices and the resulting amount of demand response from consumers.

OPEN IN VIEWER

Table 3

Comparison between Voluntary and Mandatory TOU Tariffs

State	Δw	FFR	Voluntary TOU			Mandatory TOU
		p f ∗	Δp	Daily avg.	% DR 2	Δp	Daily avg.	% DR 2
	($/MWh)	($/MWh)	($/MWh)	DR(kWh) 1		($/MWh)	DR(kWh) 1
AR	5.69	25.66	5.69	0.16	0.46	1.46	0.24	0.73
AZ	23.49	54.25	20.18	3.41	10.09	20.18	3.41	10.09
DE	10.47	30.97	7.24	1.08	3.62	7.24	1.08	3.62
IA	7.90	27.73	7.63	0.46	1.75	5.00	0.66	2.50
IL	7.59	28.22	7.04	0.35	1.62	5.65	0.60	2.83
IN	9.10	30.17	8.88	0.61	2.05	4.98	0.74	2.49
KY	7.90	27.87	7.90	0.22	0.63	3.80	0.65	1.90
MD	13.29	40.56	8.85	1.36	4.43	8.85	1.36	4.43
MI	8.53	29.49	8.37	0.40	1.96	4.26	0.44	2.13
MN	7.01	21.38	6.98	0.40	1.65	3.64	0.44	1.82
NJ	9.13	26.90	7.21	0.36	1.74	6.38	0.67	3.19
OH	9.46	27.55	9.22	0.59	2.16	4.30	0.59	2.15
PA	9.11	26.75	9.11	0.19	0.77	4.38	0.55	2.19
VA	10.14	34.34	7.48	1.32	3.74	7.48	1.32	3.74
WA	21.81	27.65	9.65	3.56	10.66	21.32	3.56	10.66
WI	7.91	27.59	7.70	0.39	1.82	5.04	0.54	2.52
WV	10.33	31.77	7.90	0.64	1.90	5.65	0.95	2.83

Note

¹This indicates the average amount of demand response per consumer during a day.

²This indicates the proportion of demand response relative to total electricity consumption; that is, percentage demand response = average demand response/average electricity consumption.

Table 3 compares the retail prices and demand responses achieved under three tariffs: FFR, voluntary TOU, and mandatory TOU tariffs. In all states, the gaps between retail prices,

Δ p = p p ∗ − p b ∗

, are (weakly) larger under the voluntary TOU than the mandatory TOU; nevertheless, the average levels of demand response per consumer under the voluntary TOU are (weakly) lower than those under the mandatory TOU. We note that these quantities are identical under both tariffs if the voluntary TOU tariff yields full adoption (Case 3). In the states where Cases 1 or 2 arise in equilibrium, the average amount of demand response per consumer is higher under the mandatory TOU tariff despite the greater incentive provided under the voluntary tariff because of the increased coverage of the residential market under the mandatory tariff. Next, we obtain the segment‐wide as well as the system‐wide cross‐subsidization under each tariff, as summarized in Table 4.

OPEN IN VIEWER

Table 4

Cross‐Subsidization under the FFR, Voluntary TOU, and Mandatory TOU Tariffs (mil.$/yr)

State	FFR				Voluntary TOU				Mandatory TOU
	n 3 ξ 3	n 2 ξ 2	n 1 ξ 1	Ξ	n 3 ξ 3	n 2 ξ 2	n 1 ξ 1	Ξ	n 3 ξ 3	n 2 ξ 2	n 1 ξ 1	Ξ
AR	−0.73	−0.23	0.95	1.91	−0.90	−0.40	1.31	2.61	−0.55	−0.17	0.73	1.46
AZ	−4.75	0.60	4.15	9.50	−1.06	0.13	0.93	2.13	−1.06	0.13	0.93	2.13
DE	−0.26	−0.03	0.29	0.59	−0.09	−0.01	0.11	0.21	−0.09	−0.01	0.11	0.21
IA	−0.55	0.22	0.32	1.10	−1.05	0.52	0.53	2.10	−0.23	0.09	0.13	0.45
IL	−0.70	0.15	0.55	1.41	−1.57	0.75	0.81	3.13	−0.22	0.05	0.17	0.44
IN	−2.07	0.93	1.14	4.14	−3.66	1.82	1.84	7.31	−1.02	0.46	0.56	2.04
KY	−1.44	−0.20	1.64	3.27	−1.93	−0.70	2.63	5.26	−0.80	−0.11	0.91	1.81
MD	−2.21	−0.30	2.51	5.01	−0.85	−0.11	0.97	1.94	−0.85	−0.11	0.97	1.94
MI	−2.14	0.99	1.15	4.27	−3.63	1.81	1.82	7.25	−1.15	0.53	0.62	2.29
MN	−0.90	0.44	0.46	1.79	−1.57	0.78	0.78	3.13	−0.47	0.23	0.24	0.93
NJ	−1.05	−0.21	1.26	2.52	−1.86	−0.73	1.13	3.72	−0.37	−0.07	0.45	0.89
OH	−2.35	1.06	1.28	4.69	−3.84	1.91	1.93	7.67	−1.36	0.62	0.74	2.72
PA	−2.56	−0.25	2.81	5.63	−3.46	−1.14	4.59	9.19	−1.42	−0.14	1.56	3.12
VA	−2.13	−0.80	2.93	5.86	−0.68	−0.26	0.94	1.87	−0.68	−0.26	0.94	1.87
WA	−0.27	0.05	0.21	0.53	−0.10	0.02	0.08	0.20	−0.10	0.02	0.08	0.20
WI	−0.85	0.36	0.49	1.70	−1.66	0.82	0.83	3.31	−0.35	0.15	0.20	0.70
WV	−0.91	−0.02	0.93	1.85	−1.35	0.54	0.81	2.70	−0.45	−0.01	0.46	0.92

Our simulation results for the analysis of cross‐subsidization in the US residential sector are in line with the analytical results obtained in section 4.3. We find that while flat consumers always cross‐subsidize their peakier counterparts regardless of the tariff scheme, average consumers may either provide or receive cross‐subsidies. Notably, in states where voluntary TOU tariffs do not induce enough consumers to opt‐in (states with a Case 1 or Case 2 equilibrium), cross‐subsidization increases relative to the FFR tariff; that is, the TOU tariff relatively overcompensates the peakier, non‐adopting consumers. It is only in those states with a Case 3 equilibrium that the degree of cross‐subsidization within each consumer segment as well as throughout the entire residential market is ameliorated relative to the FFR tariff. Interestingly, despite participating consumers being better off as a result of decreased electricity bills, these same consumers also end up cross‐subsidizing the others more when the TOU tariff induces low levels of adoption. This confirms that voluntary TOU programs are not guaranteed to be improving equity among residential consumers. A mandatory TOU program with Pareto improvement constraints (or a policy mechanism that promotes greater TOU adoption than the voluntary program) may be considered by policymakers to improve the equity of electricity pricing in the residential market. Figure 5 illustrates the comparison of the simulated system‐wide cross‐subsidization under the three tariffs across the 17 US states.

OPEN IN VIEWER

Figure 5

System‐Wide Cross‐Subsidization under the Fixed Flat Rate (FFR), Voluntary Time‐of‐Use (TOU), and Mandatory TOU Tariffs

Concluding Remarks

Voluntary Time‐of‐Use tariffs were conceived as a simple demand response tool to address the problem of matching electricity supply with demand—a feat which longstanding flat rate tariffs were unable to achieve. On the supply side, high wholesale prices during the peak period provide utility firms with the incentive to induce consumers to shift demand to the off‐peak periods when it is cheaper to meet demand. Further, appropriately designed voluntary TOU tariffs also have the advantage of giving consumers the opportunity to lower their electricity bills. Therefore, TOU tariffs present a win–win opportunity for the utility firms as well as consumers. Despite this potential, TOU tariffs have been sluggishly deployed across the US residential sector and adoption rates among consumers have been extremely low.

A major source of difficulty in designing appropriate TOU tariffs arises from the heterogeneity in the consumption preferences of residential consumers. We find that adoption of voluntary TOU tariffs by residential consumers follows an incremental pattern wherein it is easiest for the firm to induce those consumers with the flattest consumption preferences to adopt TOU tariffs. Unfortunately, the flattest consumers are also the least expensive consumers to serve, and moreover, allowing them to opt‐in to voluntary TOU tariffs gives them the opportunity to free‐ride by reducing their electricity bills without making a significant reduction in peak‐time electricity consumption. Therefore, the firm faces a trade‐off between inducing more demand response from a small number of flat consumers and increasing coverage of the TOU tariff to include more consumers with peakier consumption preferences. Our model captures this trade‐off by the heterogeneity within the set of adopting consumers; we find that this measure must be sufficiently small relative to the wholesale price gap to warrant expanding the coverage of TOU tariffs to include peakier consumer types.

We also show that the optimal deployment of TOU tariffs will always lag the socially desirable level of TOU deployment and penetration. This is because the socially optimal degree of penetration considers both the reduction in the electricity bills of residential consumers as well as cost savings for the utility firm. Expanding coverage to peakier consumers is less palatable to the firm when the heterogeneity in consumption preferences is large, as this forces the firm to leave a large amount of savings to flatter consumer types which consequently affects the total revenue it can generate. This explains the tug‐of‐war between public utilities commissions and investor owned utility firms, wherein the former require the utilities to advertise and promote TOU tariffs aggressively while the latter may only do so reluctantly with little success by way of TOU penetration. States such as Indiana, Pennsylvania, and Wisconsin have not seen significant adoption of demand response programs in the residential sector despite public interest to that end.

TOU tariffs have also been touted as a mechanism to improve the equity of electricity pricing by imposing the real costs of electricity provision on all consumers. However, we find that when the deployment of voluntary TOU tariffs results in low adoption rates, the flatter adopting consumers end up cross‐subsidizing the peakier non‐adopting consumers more than in the status quo. Therefore, voluntary TOU tariffs can in fact exacerbate the inequitable distribution of benefits inherent in flat rate electricity pricing. This is an unavoidable consequence in those markets where voluntary TOU tariffs cannot induce adoption from consumers with greater than average preference for peak electricity consumption. Only a high degree of TOU adoption can produce superior outcomes from an equity perspective relative to FFR tariffs. This provides support to the growing practice of installing default TOU tariffs in states such as California that effectively results in very high adoption rates approaching full adoption. That is, TOU tariffs can reduce the degree of cross‐subsidization and thus consequently improve the equitable distribution of benefits only if the degree of TOU adoption is sufficiently high—approaching full adoption in the residential market. This occurs as a result of high TOU adoption ensuring that most consumers’ face the true costs of electricity provision through time‐dependent prices.

In closing, we acknowledge some limitations of our study and suggest potential extensions. First, we focus on the short‐run implementation of demand response programs wherein utility firms procuring electricity from the wholesale markets face exogenously fixed prices. In the long‐run, the supply‐side of the market can make significant adjustments to generation portfolios in reaction to the outcomes of these demand response programs. Hence, a natural extension to our model can consider the long‐run adjustment of the supply‐side of the electricity market by incorporating supply functions or by explicitly incorporating generation portfolios into the decision‐making framework of the utility firms. Second, we note that our analysis does not account for any barriers to demand response adoption arising from behavioral characteristics or biases of consumers. As states such as California exploit behavioral phenomena such as the status‐quo bias to increase TOU adoption by designating them as default tariffs, it is worth considering the impact of these deviations from rational consumer behavior on the potential for achieving demand response in the residential sector. Despite these limitations, our model can readily be applied to evaluate the potential for, and the optimal design of TOU tariffs in a given residential market. Recent studies have shown how consumers can be segmented into groups in the residential electricity market based on consumption characteristics (Haben et al. 2016, Kwac et al. 2014, Rhodes et al. 2014). With the widespread rollout of smart meters in recent years, utility companies can use the consumption data collected from these meters to segment the consumer market and evaluate the salient characteristics within each of these segments, thus preparing the groundwork for effectively utilizing our model to predict TOU potential and design appropriate tariffs.

Footnotes

Model Extensions

Proofs of Main Results

Acknowledgments

Dong Gu Choi thanks the support by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF‐2016S1A5A2A03925732). Michael K. Lim acknowledges the support by Creative‐Pioneering Researchers Program through Seoul National University and the Institute of Management Research at Seoul National University.

1

Products that use electricity for their main power source and have the capability to receive, interpret, and act on a signal received from a utility, third‐party energy service provider or home energy management device and automatically adjust their operation, depending on both the signal’s content and settings from the consumer (for example, clothes washers, clothes dryers, air conditioners, dishwashers, rooftop solar PVs, and electric vehicles) (Sastry et al. ).

2

Existing research on residential demand response programs has mainly focused on differences in consumers’ intrinsic consumption preferences (e.g., Haben et al. 2016, Kwac et al. 2014). In Appendix , we consider heterogeneity in demand‐shifting flexibility that can arise from differences in work schedules, demographics, and housing conditions.

3

Dong et al. (2017) find that there is essentially no significant change in the total electricity consumption when consumers switch from FFR to TOU tariffs barring some extreme cases. Nonetheless, we relax this stipulation and extend the base model by endogenizing electricity consumption in Appendix .

4

Ontario, Canada has been considering mandatory TOU tariffs since 2005 (Ontario Energy Board 2018). In addition, Italy has rolled out mandatory TOU tariffs in its residential sector since 2010 (Maggiore et al. 2013), and other countries in Europe such as Ireland and Spain are considering mandatory or default TOU tariffs (Hledik et al. ).

5

While generally true because a larger number of adopting consumer types will exhibit a wider deviation in peakiness from the kth consumer type, the sizes of the consumer segments could be specified in a manner such that

H ( k ′ ) > H ( k )

holds for

k ′ < k

.

6

One key difference relative to the base model is that all TOU adoption equilibria can be sustained, that is, there is no necessary condition such as the one described in Lemma (i) for a particular adoption equilibrium to be feasible.

7

If

( p p ) 2 − ( p f ) 2 < ( p f ) 2 − ( p b ) 2

, then adoption will always occur in incremental order of consumer types.

ORCID

Dong Gu Choi

Michael K. Lim

Karthik Murali

Valerie M. Thomas

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