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date: 23 September 2021

Nonlinear Pricing with Reference Dependencefree

Nonlinear Pricing with Reference Dependencefree

  • Catarina Roseta-Palma, Catarina Roseta-PalmaISCTE-IUL
  • Miguel CarvalhoMiguel CarvalhoISCTE-Instituto Universitário de Lisboa
  •  and Ricardo CorreiaRicardo CorreiaUniversidade da Beira Interior


Many utilities, including water, electricity, and gas, use nonlinear pricing schedules which replace a single uniform unit price, with multiple elements such as access charges and consumption blocks with different prices. Whereas consumers are typically assumed to be utility maximizers with nonlinear budget constraints, it is more likely that consumer behavior shows limited-rationality features such as reference dependence. Recent studies of water demand have explored consumer reactions to social comparison nudges, which can moderate consumption and might be a useful tool given low demand-price elasticities. Other authors have noted the difficulties of correct price perception when tariff schedules are complex, and attributed those low elasticities to a lack of information. Nonetheless, it is also possible that consumers form reference prices, relative to which the actual price paid is compared, in a way that affects consumption choices. Faced with a nonlinear price schedule, such as increasing block tariffs, consumers could evaluate their actual marginal price as a loss or a gain relative to a particular reference price that is derived from the schedule. Introducing gain/loss terms into the utility function, in the discrete/continuous model of consumer choice that has been widely used for water demand analysis, leads to consumption decisions that vary when a higher-than-reference price is seen as a loss and a lower-than-reference price as a gain. Utilities might wish to explore these reference-price effects according to their strategic goals. For example, if there are capacity constraints or water scarcity problems, potential water savings can be achieved from highlighting the first-block price as a reference and framing higher-block prices as losses, inducing conservation even without raising overall prices. Furthermore, if higher-block prices are subsequently raised the demand response could be stronger.


  • Global Health
  • Theory and Methods


For most goods, consumers pay a single unit price which is easy to know and understand. Residential utility bills, on the contrary, are commonly based on nonlinear price schedules with many confounding elements, such as access charges and unit prices that vary along several blocks. In these circumstances price becomes endogenous, because it depends on the consumption block, and studying consumer reactions to price is challenging. In particular, average prices are different from marginal prices and demand analysis requires appropriate techniques to isolate price elasticities and ensure that these are properly interpreted (see Olmstead, Hanemann, & Stavins, 2007). This article reviews recent behavioral literature on water demand under increasing block tariffs (IBT), mostly empirical papers that allow for the possibility of deviations from consumer rationality in water demand. Although one of the essential elements of demand management is understanding consumer reactions to price, not much has been written on such reactions when consumers have reference-dependent preferences and price schedules are nonlinear. A novel contribution is then presented, discussing how consumers with gain-loss asymmetry—arising from reference-price anchoring—might determine their water consumption level under IBT.

Recent Behavioral Literature

The growing literature on behavioral economics applications to water demand was surveyed in Correia and Roseta-Palma (2014). This section describes more recent developments, focusing on three aspects: water demand estimations, social norm approaches, and welfare effects.

Water Demand

If consumers act rationally, that is, equalizing marginal utility of consumption and marginal price, there should be many consumers clustered around the kinks of the price schedule. The so-called bunching should happen because it is at the kinks that the marginal price has an increase. Several papers have tested this prediction and found weak or no evidence of such behavior. Moreover, when price schedules are complex, households are not necessarily aware of prices or even consumption levels. There is a good deal of literature on the matter—for instance, Clarke, Colby, and Thompson (2017), who find evidence of “schmeduling,” that is, consumers failing to fully grasp complex price schedules. To be more precise, people seem to choose their consumption depending on their current (or lagged) average price, which Liebman and Zeckhauser (2004) term the “ironing” heuristic of schmeduling.

Wang, Lee, Yan, and Thompson (2018) compared the distribution of water consumption after a price schedule change, where both blocks and marginal prices were new. After a period of adaptation to the new schedule, the authors found no bunching.

A strand of literature, starting with Shin (1985), tried to identify the perceived price to which consumers respond. For instance, Binet, Carlevaro, and Paul (2014) studied the household demand of water across municipalities with different IBT schedules using a nested model combining marginal price and average price demand. The estimated perceived price is even lower than average price, so further away from the marginal price.

Perceived price studies may be tricky when marginal and average price co-move under IBT. Wichman (2014) looks at the switch from a flat rate to an IBT, so that the effect of both can be disentangled, concluding that the average price is the price consumers perceive. Moreover, looking at the subsample of consumers who had average and marginal prices moving in different directions, consumers react to average price.

The most compelling evidence against rational consumption choices under IBT, however, comes from a paper on electricity demand: Ito (2014) performed several tests on the best determinant of consumption choice. Lagged average price outperforms all other possibilities. Because it is difficult for consumers to keep track of which price bracket they are in at every moment of the billing cycle, it could be the case that they rely on some weighted average of marginal prices in the vicinity of the real one. The authors also tested for that but, once again, average price is a better determinant.

Several authors have noted that complex price schedules dampen demand reactions. Monteiro, Martins, Ramalho, and Ramalho (2018) find that consumer awareness of water bills increases demand-price elasticity, although water demand remains price-inelastic. Lott (2017) also shows that price saliency in the monthly bill is decisive for IBT to reduce consumption. Nonetheless, increasing the saliency of the price schedule does not necessarily lead consumers to save water. Wichman (2017) argues that increasing the billing frequency, thereby making the price more transparent, actually leads to an increase of 3.5–5% in water consumption. Likewise, Brent and Ward (2019) document an experiment where households were provided with detailed information on their consumption and prices, which led to higher consumption.

Strong and Goemans (2014, 2015) study how water demand changes when consumers are provided with real-time quantity information. As consumers become more aware of the price schedule, they react more rationally to it. In their case, this meant lower consumption with marginal pricing, and higher consumption under IBT on average, as most consumers approached the upper kink. Interestingly, there were also consumers that reduce consumption toward the lower kink.

Social Norms

Various experimental applications with energy and water utilities, beginning with Allcott (2011); Ferraro, Miranda, and Price (2011) have focused on social comparison through “reference consumption,” overlooking the role of price information. Although such social-norm interventions have shown some potential in reducing consumption, as reviewed in Nauges and Whittington (2019), these authors also point out that it is necessary to consider their welfare impacts. Nonetheless, there is a strand of the literature that studies the interaction between price and nudging incentives. Specifically on social comparisons and IBT, Brent et al. (2017) compare the effectiveness of different nudges in reducing water demand. Social comparison nudges (comparison of water consumption level to a relevant group) seem to have a stronger result, especially if the household’s consumption is way above the average, than other approaches such as pointing out possible monetary savings due to the increasing marginal price structure. The latter does, however, have longer-lasting effects.

Brent and Wichman (2020) look for complementarity or substitutability in social norms and IBT price incentives, finding that behavioral treatments have no effect on the price sensitivity of consumers, nor do different prices influence the reaction to norm-based nudges.

Lu, Deller, and Hviid (2019) review several case studies and discuss IBT and behavioral treatments as tools on water demand management.


Meran and von Hirschhausen (2017) wonder if the popularity of IBT price could be a result of distributional concerns. Using the Fehr Schmidt Inequality Aversion framework, they find a positive answer. Where household income and size are negatively correlated, an income-independent IBT schedule is, however, not “fair” because poorer households tend to pay higher marginal prices. This result is consistent with the findings of Griffin and Mjelde (2011), where the authors point out that IBT distribute welfare inequitably in scarce-water settings.

Ma, Zhang, and Mu (2014) explored the introduction of an IBT in Beijing. While observing that average price seems to be the main determinant of demand, they cautioned that low-income households appear to react to marginal prices, bunching at the kinks, especially if the marginal price increase is high.

Finally, Nauges and Whittington (2019) developed benefit-cost analyses for social norm and price interventions, noting that the former appear to be highly dependent on location and very uncertain, while the latter tend to achieve better welfare results in lower-income countries, albeit increasing households’s cost burden.

Nauges and Whittington (2017) discussed the rationale for an IBT price schedule. They simulated hypothetical scenarios with a heterogeneous population, focusing especially on two issues: water demand as a function of average versus marginal price, and none versus high demand-income elasticity. For several possible utility goals, the authors show that IBT is a rather poor choice in terms of redistribution and economic efficiency.

Demand under IBT with Reference Price

As noted by Olmstead et al. (2007, p. 194), with varying block rates “how price affects demand is, itself, somewhat elusive.” They found that the price elasticity of water demand is higher when the tariff structure is IBT and suggested that this could be due to an as-yet-unidentified behavioral response to price structure, as opposed to price itself. Such findings match a common perception among water managers that IBT send the consumer a stronger message about resource scarcity, which explains at least partly their growing popularity (Organisation for Economic Co-operation and Development, 2010). Here it is proposed that the multiple unit prices associated with a nonlinear schedule can induce such a behavioral response, if consumers anchor their price reference and then treat deviations from this reference as losses or gains.

Reference-price models are based on the general theory of reference-dependent preferences (Kahneman & Tversky, 1979), which emphasizes that people attribute value to changes in relevant variables and not their absolute magnitudes. Moreover, loss aversion, defined as a larger sensitivity to losses than to gains of similar size, appears to be a widespread trait of consumer preferences. Ho, Lim, & Camerer (2006) provided an overview on the topic in a range of economic domains. Kőszegi and Rabin (2006) introduced a utility function with two separable components, consumption utility and gain-loss utility, where the former is akin to classical utility and the latter includes the reference point. The most common assumption is that people compare outcomes to their status quo, but Kőszegi and Rabin (2006) proposed, more generally, that reference points come from expectations, relying on a personal equilibrium that takes into account anticipated behavior. In contrast, Wenner (2015) proposed, and provided experimental evidence for, a simpler model of expected prices as reference points, where the realized price is compared to a measure of the distribution all possible prices.

Reference prices have been discussed in the marketing literature. Putler (1992) developed a model of consumer choice and estimated asymmetric price elasticities for the wholesale egg market. This type of asymmetry also appears in Greenleaf (1995) for price promotions, and in Fibich, Gavious, and Lowengart (2005) which discuss the difference between short-term and long-term effects. Krishna (2009) reviewed the evidence on behavioral aspects of pricing, distinguishing price-presentation effects from internal reference prices.

The following section models the possibility that viewing a set of block prices, instead of a single price, affects consumption decisions.

Modeling Reference-Price Effects with Block Rates

Assume that consumers make rational choices and know their rate structure yet have subjective reference prices which are affected by available block rates. Demand analysis under block-rate prices requires a model that distinguishes the discrete choice of block from the continuous choice of consumption within the block (Hewitt & Hanemann, 1995; Moffitt, 1986). Deriving consumer demand requires (a) definition of the budget subsets for the discrete choice; (b) maximization of utility in each budget subset to find demand conditional on location in that subset; and (c) given conditional demands from the previous step, build the unconditional demand. The novelty here is the incorporation of a gain-loss component based on a reference price in the utility function within step (b).

Suppose there are N goods, x1 to xN, one of which, x1, is characterized by a nonlinear price schedule with a fixed access charge, FC, and K rate blocks, which imply K1 switching points (kinks). Let x11 to x1K1 and p11 to p1K denote the block-switching points and the block prices, respectively, with x10=0 Assume that once the consumer changes to a higher block, only the additional units are charged at the new price, allowing several marginal prices.

The budget set B will be kinked but continuous and it will have 2K1 subsets, denoted by Bb, where B=b=12K1Bb. All budget subsets are convex even if the full budget set is not, as will be the case for decreasing block tariffs (DBT). There are two type of subsets: kink points, which occur when x1=x1k; and line segments defined as (x1,,xN)|x1kx1x1k+1,p(x1)+j=2NpjxjM (Hewitt & Hanemann, 1995).

The budget set can be written using a difference term, dk, which is the difference between a payment and what it would have cost to purchase the same quantity at the kth block price:


Thus defined, the difference term will be negative for DBT and it may be positive or negative for IBT, depending on the size of the fixed access charge.

For the consumption choice within each budget subset, either the consumer is at a kink point (x1*=x1k) or he is on a line segment k where the marginal price is p1k. Either way, the decision will be the outcome of a utility-maximization problem where the utility conditional on being located on a specific budget subset is:


Gk and Lk are gain and loss (GL) terms, respectively, and θ represents other parameters of the utility function. For simplicity consider that good 1 (x1) is the only good where these terms exist. Having a price schedule (external reference prices) provides the consumer with exogenous information that affects the (internal) reference price, influencing her perception of the price she is paying. The unit loss is lk=I(p1kp1r), where p1r is the reference price, and the unit gain is gk=(1I)(p1rp1k), where:


To allow for the possibility of nonlinear gains or losses experienced by the consumer, define effective per unit gain or loss as:


where El(lk)>0,Eg(gk)>0, and each approaches zero when lk and gk, respectively, approach zero. Now one can define the total loss or gain affecting the consumer, Lk=El(lk)x1 or Gk=Eg(gk)x1. Assume that consumption marginal utilities are positive and decreasing, while ULk<0, 2UL2k>0, UGk>0 and 2UGk2<0. Thus the utility function is strictly concave in consumption and gains and convex in losses. Loss aversion would be reflected in a presumably stronger reaction of El(lk) to price ”losses” than that embodied in Eg(gk) for price “gains.”

The third step (Hewitt & Hanemann, 1995) is to take the conditional demands for all budget subsets, verify which achieve indirect-utility maximization taking all blocks into account and build the unconditional demand functions (if the whole budget set B is convex and there is a unique optimum) or correspondences (if the budget set is not convex, in which case there may be more than one solution). However, demand functions will now depend not only on prices and income but also on the GL effects.

To assess the impact of the gain-loss formulation on consumer choices for different reference-price possibilities, start by looking at interior solutions, using the FOC from problem (2):


whence arise the block-conditional demand functions, which include the marginal GL terms. In particular, for good 1:


The consumption choice could also be at the kink point between two segments. Therefore, the unconditional demand (x1) for K blocks, taking into account the unit GL of each conditional demand function, is:


Notice that the utility maximization only occurs at the kink point if utility maxima along each segment are found in the unfeasible range (Moffitt, 1986). Under IBT, the budget set will be strictly convex; therefore any interior solution will be unique as long as indifference curves are continuous, even if they are no longer expected to be differentiable at the (block-switching) kink point: although the following figures only show the relevant section of each indifference curve, it is clear that the marginal rate of substitution will be higher where the slope of the budget constraint is lower, invalidating the existence of an alternative solution on the other block.

Whenever the price for good 1 is the same as the reference price, the GL terms disappear. Otherwise, the impact of the new components can be gathered from an analysis of the marginal rate of substitution between good 1 and any other good j at a given consumption point (x1*,xj*):


Inequality (5) indicates that indifference curves will be steeper for losses (p1k>p1r) than when there is no gain-loss effect (p1k=p1r), while less steep indifference curves will occur in the domain of gains (p1k<p1r).

A graphical analysis in two-good space can show how the resulting change in the slope of indifference curves alters consumption choices. Consider only two blocks and no fixed charge. To apply the gain-loss framework requires us to make assumptions about the reference price. The impact on consumer choices is checked in two alternative cases: (i) p1r=p11, (ii) p1r=p11+p122; the case p1r=p12 is a simple extension. Case (i) would be relevant if the consumer normally remains in the first block, or if this price is highlighted by the firm. Case (ii) would apply whenever the typical consumption level is close to the block limit, falling on either the first block or the second.

If the reference price is p11, then consumers on the second block will be affected by a sense of loss, since their price is higher than the reference price, as shown in Figure 1. On the other hand, consumers on the first block or the kink will react no differently than in traditional models. Using equation (5), it is clear that if the consumer’s choice without gain-loss components was on the second block, at that point, x1trad*, the new indifference curve will have a lower slope. Thus, the new choice will be a point to the left of x1trad*, such as x1new*.

Figure 1. Decrease in consumption due to the loss effect (with p1r=p11)

Furthermore, if there is an increase of the second-block price (p12), the loss term increases so the consumption response will be stronger than one would expect in a traditional utility-maximization setting. This is a potential theoretical justification for the finding that consumers with IBT show higher price elasticities, as noted in the opening of the section “Demand under IBT with Reference Price.”

If the reference price were to be the average of the two block prices, p1r=p11+p122, indifference curves will shift on both segments, yielding a rise in consumption for those on the first block and a decrease for those on the second. This case is shown in Figure 2, which highlights that one would expect to see more “bunching” around the kink point, a regularity that has not been found in the empirical studies.

Figure 2. Increase (decrease) in consumption in the first (second) block (with average price as the reference)

Final Remarks

Residential water consumers often have to deal with nonlinear prices. New developments in our understanding of consumer reactions to water prices based on IBT were reviewed and an extension was proposed. It is possible that facing a range of prices influences consumer decisions by introducing gain-loss components, as the actual price one pays is compared to a reference price. Previous authors have noted that reference prices could be based on earlier prices (the status quo) or expected prices; instead, here the prices of different blocks are used, analyzing the consequences of expanding the traditional utility-maximization model to include these behavioral reactions. In particular, a model is developed where the nonlinear price schedule can affect consumption, depending on which price is seen as the reference by consumers. Since there is no reason to suppose a certain unit price will catch the consumer’s attention more than another, firms could advertise the price that best suits their strategic goals. For example, utilities that are dealing with capacity constraints or natural-resource scarcity could focus on the first block price, or even the second, if there are more than two and the first is a subsidized block meant to ensure affordability for poorer households. Higher-block prices would then be framed as losses, to induce conservation behavior even without raising prices. Furthermore, if those prices were eventually raised, for example, to reflect marginal cost as so often recommended on efficiency grounds, such framing would also encourage a stronger consumption response.

The theoretical analysis presented was developed under the assumption that consumers know their tariff structure. Nonetheless, the available empirical evidence questions whether, in fact, consumers are aware of prices embedded in complicated schedules, and even of their own consumption levels. Therefore, a useful (and more policy-relevant) extension to the current work would be to develop a model where consumers have a sense of their reference expense in water bills but do not adequately distinguish price and quantity effects.


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