Can Risk Aversion Explain Three-Part Tariffs? Jose Miguel Abito University of Toulouse 1 [email protected] June 2007

Mémoire under the supervision of Prof. Patrick Rey, University of Toulouse 1.

Abstract A justification for the use of three-part tariffs is proposed using basic ideas borrowed from the competitive two-part tariffs literature. We show that risk aversion leads to optimal tariffs that closely resemble three-part tariffs when firms compete in a perfectly competitive market with homogeneous agents. Specifically, we formulate a highly stylized model with two “pseudo-goods,” where the first good reflects “normal” consumption while the second good reflects “excess” consumption. The first good is consumed in all states of nature while the second good is only consumed in one of the states. We calibrate the model to the market for local mobile calls and show that our model can give rise to a tariff that is similar to what is empirically observed in the U.K. market.

KEYWORDS: Three-part tariffs, Risk aversion, Mobile phone plans, Pseudo-goods, Consumption uncertainty

1

Introduction

Three-part tariffs are prevalent in different product markets. For example, mobile calling plans are usually designed as three-part tariffs, i.e. for a fixed monthly fee one gets a certain allowance of free minutes and is charged an overage rate for consumption beyond this allowance.1 Other notable examples include cable TV subscription (fixed number of premium channels with the option to purchase some more), internet services and credit cards (see Eliaz and Spiegler (2006) for a specific example). Despite its prevalence, there seems to be a dearth of papers offering theoretical justifications for the use of three-part tariffs (we will review this limited literature in the next section). The objective of this paper is to provide a theoretical justification for the optimality of three-part tariffs. We propose risk aversion as a possible justification and show this in two steps. In the first step, we formulate a highly stylized model (i.e. no income effects, independent goods2 ) of multiproduct competition with homogeneous agents and demand uncertainty. Specifically, the model consists of firms selling two products to consumers by offering price schedules. Both firms and consumers are homogeneous hence the model is innately a model of perfect competition, albeit with uncertainty. There are two states of nature in the model. In the first state, consumers would like to consume both goods while in the second state, only one of the goods is consumed. Both firms and consumers do not know the state of nature at the time the former is designing and offering the tariff, and at the time the latter is choosing from which firm to buy and at what prices. Consumption decisions are made by the consumer once the true state is realized. Using this model, we show that the optimal tariff involves the “certain good” being priced at marginal cost while the “uncertain good” priced above marginal cost. The firm also provides a fixed subsidy to consumers, the amount of which is equal to the expected profits of the firm (before the subsidy). The model also shows that consumption of the certain good is constant across the two states which will be an important fact for the second step. In the second step, we reinterpret the two goods as the decomposition of a single good into two “pseudo-goods”. The certain good reflects consumption in “normal” times while the uncertain good reflects consumption in “extraordinary” times. Because consumption of the certain good is constant across the two states, the consumer is indifferent between paying an additional fixed fee corresponding 1

In this paper, when we talk about three-part tariffs (or standard three-part tariffs) we refer to tariffs with a flat fee

over a given allowance and then some overage rate after this level. Generally speaking, a three-part tariff consists of a fixed fee and different prices corresponding to two separate quantity blocks. We call this tariff a general three-part tariff. Notice that if the two prices are equal, then we have a two-part tariff and when combined with a zero fixed fee, we have uniform pricing. 2 The assumption of no income effects requires that expenditures are small relative to income. We are interested in this case since risk aversion has often been disregarded in empirical studies precisely on this basis. For example, Miravete (2003) ignores risk aversion in his study of calling plans by arguing that ex post losses from choosing the “wrong” calling plan are very small relative to income. The independence assumption is used to facilitate our decomposition of the good into two pseudo-goods.

1

to the quantity of the certain good multiplied by the per-unit charge, and paying “by the unit”. Thus the consumption of the certain good can be seen as the optimal allocated “free” minutes in the mobile calls example while the uncertain good reflects excess call minutes. In interpreting the consumption of the certain good as the optimal free minutes, we are implicitly imposing that the subscriber is fully rational in the sense that she has already learnt and knew the level of consumption in normal times and also the probability that extraordinary events occur. Hence this implies that the consumer never consumes less than the amount of allotted free minutes.3 Taking the interpretation in the second step, it is easily observed that the optimal tariff resembles a general three-part tariff (resembles a standard three-part tariff if the marginal cost of producing the “normal” good is equal to zero). We then calibrate the model to derive optimal prices for excess minutes in the mobile call application. The optimal prices from the calibrated model are close to the prices of two particular U.K. mobile network operators thus risk aversion can account for such pricing behavior. We do not claim that risk aversion is the best or is the only explanation for three-part tariffs mainly because of the highly stylized nature of our model. However, this exercise shows that risk aversion should not be ignored in empirical studies. The model in this paper is closely related to the model formulated in Hayes (1987), which is one of the first few papers justifying the use of two-part tariffs in a (perfectly) competitive environment.4 As first pointed out by Hayes (1987), “the existence of two-part tariffs in many markets that are (or appear to be) competitive belies this standard assumption [that market power is a necessary condition for optimality of two-part tariffs].” In her paper, she argues that two-part tariffs can be optimal vis3

We drop this assumption in the appendix and show that our results still hold. In fact we derive a sufficient condition

such that the standard three-part tariff is the optimal tariff even when the marginal cost for producing the “normal” good is positive. If this condition is not met, the optimal price has a similar structure as in the case where consumers do not underconsume. 4 Two other papers on competition and two-part tariffs are Mandy (1991) and Locay and Rodriguez (1992). Mandy (1991) discusses the sustainability of two-part tariffs even with competitive firms. The main driving force of the model is heterogeneity in rates of time preference of individual consumers and firms. Two-part tariffs allow firms to redistribute the timing of consumption and profits in a way that maximizes welfare. For example, an up-front fixed fee in exchange for lower future marginal prices will be optimal if individuals discount future gains less heavily than firms. The opposite case where firms offer negative up-front fees for higher marginal prices is close to the results of our model, though Mandy (1991) abstracts from risk aversion and uncertainty. A previous version of our paper included a model with possible differential rates of time preference between consumers and firms combined with uncertainty. If consumers discount future utility more, then the subsidy that the firm can provide is larger, to the point that hypothetical monopoly prices can be supported. Locay and Rodriguez (1992) argue that two-part tariffs can be optimal in situations where firms compete in a perfectly competitive market for groups but groups constrain switching behavior among individuals. This allows firms to have some market power and price discriminate over individuals after the group has decided where to purchase. They show that above-marginal cost pricing with subsidized entrance fees is optimal for the case where the median of the group consumes less compared to mean consumption (i.e. more people in the group benefits from the reduced movie entrance fees in exchange for higher popcorn prices).

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a-vis a uniform price even within a perfectly competitive environment if there is some uncertainty in consumption. She argues that two-part tariffs in this setting act as insurance in the sense that consumers pay a fixed fee as premium and are entitled to prices below marginal cost across all states. She focuses her discussion on the “canonical form” of two-part tariffs which is composed of a positive fixed fee and some per unit charge. In the case of perfect competition, the zero (ex ante) profit condition implies that with a positive fixed fee that is paid in all states (hence decreasing income in all states) per unit charges should be below marginal cost (hence raising consumption on this good relative to other goods especially in high consumption states). The model then shows that two-part tariffs decrease utility in low consumption states, due mainly to the fixed fee, but increase utility in high consumption states (the benefit of lower charges outweighs the negative effect of the fixed fee) with the net effect being positive. Hayes (1987) briefly mentions that two-part tariffs with per-unit prices above marginal cost and negative fixed fees are optimal if the covariance between the marginal utility of income and the demand for the good is negative, e.g. with a positive income shock, marginal utility of income decreases and the good is a normal good.5 Our model exploits this result, albeit in a different manner since we abstract from income effects, and extends her model to a multiproduct setting. In our model, utility is higher in the state where both goods are consumed.6 A risk averse consumer does not want her utility across states to differ by a large amount hence the multi-part tariff allows her to transfer utility from the higher state to the lower. Specifically, since higher utility stems from the desire to consume a second good in one of the states, the optimal tariff levies a higher per-unit price for this good and transfers the profits earned to both states as a fixed subsidy. This makes the originally higher utility state less desirable while making the other more, hence bringing utility from both states closer to each other. The paper’s contribution is threefold. Firstly, we contribute to the growing literature on threepart tariffs. Contrary to most papers that try to justify the use of three-part tariffs, we assume that consumers are fully rational and do not exhibit biases. Moreover we concentrate on the case of homogeneous agents hence three-part tariffs are not justified based on the implementation of a convex optimal tariff in a screening context. Secondly, our model can also be used to analyze situations where a base good is initially bought and “add-ons” are later consumed, subject to some consumption 5

In Proposition 1 of Hayes (1987), she actually focuses on the standard two-part tariffs case (positive fixed fee but

with below-marginal cost per-unit prices) and briefly discusses the other case (negative fixed fee with above-marginal cost per-unit prices) at the end of the proof. 6 Implicitly we are assuming that changes in calling behavior are motivated by events that tend to increase utility, thus this seems weak given it is easy to imagine events that have a negative impact on utility and at the same time, change calling behavior (for example, having an accident would likely increase calls). However, we can use Hayes’ (1987) proposition directly to support our results since in times of emergencies for example, making a call is more valuable than say $50 hence the shock leads to a decrease in the marginal benefit from money, in relative terms. Therefore in this case, the marginal utility of income is negatively correlated with demand for (excess) calls and thus we expect pricing above marginal cost (steeper overage charges).

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uncertainty (see Ellison, 2005 for an analysis of add-on pricing). Finally, the paper shows how risk aversion can be an empirically relevant variable in the analysis of pricing. More importantly, we illustrate a situation where above-marginal cost pricing can be compatible with efficiency and measure how much mark-up in the market for local calls can be explained by risk aversion. We provide a brief review of the limited literature on three-part tariffs in Section 2 and present our basic model and results in Section 3. We first discuss how we model consumer and firm behavior and then derive the properties of the optimal price. Section 4 then considers an application of the model to the market of local mobile calls. We calibrate the model, derive the optimal price and compare to actual prices in the U.K. market. The final section concludes and suggests possible extensions (one of which is pursued in the appendix).

2

Three-part tariff literature

The literature on explicit justification of three-part tariffs is fairly new, possibly due to the complexity of situations where such tariffs can be optimal. An obvious justification for the use of three-part tariffs is for implementation of a convex optimal nonlinear tariff in the context of a screening model. Threepart tariffs seem to be a practical way for firms to implement a nonlinear tariff that exhibits some convexities, hence we expect three-part tariffs in these situations. Recently, Jensen (2006) documents a case where such convex tariffs can arise. She studies duopoly competition by taking Stole’s (1995) model on price discrimination under spatial differentiation, reinterpreting it with quantity as the sorting variable combined with the assumption that consumers can only buy from one firm (and only choose a single tariff) and finally showing that a binding participation constraint for low quantities can lead to convexities in the optimal tariff. Still under the rubric of a screening model, Eliaz and Spiegler (2006, Proposition 9) show that three-part tariffs can implement the optimal nonlinear tariff in a model where consumers differ in their cognitive abilities to forecast changes in taste (projection bias). They formulate a two-period model with a monopolist firm selling to dynamically inconsistent consumers and assume that the two have non-common priors with respect to the propensity of the consumer to change her tastes, i.e. the firm knows that consumers are dynamically inefficient but consumers only believe that they are with some probability (this probability is private information). A fully naive consumer believes that her preferences will not change while a fully sophisticated consumer believes otherwise. For example, a naive mobile phone subscriber thinks she will only consume q number of minutes per month at the time of subscription but consumes more eventually. The optimal nonlinear tariff then exploits this by having a low fee up to q but higher rates in excess of this level. On the other hand, a sophisticated subscriber knows that eventually her preferences will change, i.e. she will eventually demand more minutes, hence she accepts a tariff with a slightly higher fee up to q and a much higher overage rate compared to

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the naive consumer. She consumes q ex post, with the optimal nonlinear tariff acting as a self-control mechanism. The authors then show that this type of tariff cannot be implemented by a two-part tariff but is implementable by a three-part tariff. The two abovementioned papers justify three-part tariffs as a way to implement an optimal tariff that might display some convexities. In contrast, Grubb (2007) provides a justification that does not rely on screening and implementation. Similar to Eliaz and Spiegler (2006), the model is driven by the non-common priors assumption but unlike them, consumer bias in his paper is not limited to mistakes in estimating the average demand but also incorporates consumers’ overconfidence that the likelihood of large deviations from the average demand is small. He argues that in certain situations (e.g. mobile phone subscription), consumers are overconfident in their ability to estimate future demand and firms design tariffs that profit from consumers committing (large) mistakes. The optimal tariff thus is such that average price is increasing for quantities that are farther from the consumers’ expected level of consumption, which is a characteristic of a standard three-part tariff. For our paper, we provide a justification for three-part tariffs outside the realm of screening (and implementation) and consumer biases. Future work will incorporate these aspects, possibly leading to richer analysis and greater empirical relevance.

3

Model and analysis

There are two goods which we label as N and X. With probability equal to one, a representative consumer would like to buy good N ; however, she only buys good X with probability equal to λ ∈ (0, 1).

The timeline of the game is as follows. In the first stage of the game, firms offer pricing plans of the form (T, pN , pX ) where T is a compulsory fixed “subscription” fee (which can be a subsidy, i.e. T < 0), and pN and pX are the respective per-unit prices of goods N and X.7 A representative consumer then chooses a pricing plan and pays the relevant fixed fee (or receives a subsidy). Similar to Hayes (1987) the consumer commits to the chosen pricing plan before the state of the world is known but is free to choose the quantities of goods to consume after the state is revealed. We assume that the consumer can only buy from the firm with which she has subscribed to and that she is not allowed to subscribe to a different firm after committing.8 In the next stage, the state of the world becomes common knowledge. There are two states, ω 1 and ω 2 . In ω 1 , the consumer buys both goods (qN,1 and qX,1 ) while in ω 2 , she 7

Our goal is to show that (T, pN , pX ) does not degenerate to a two-part tariff, i.e. (T, p, p) or uniform prices, i.e.

(0, p, p) at the optimum. 8 Two factors that may prevent a consumer from collecting subsidies by having multiple subscriptions are transaction costs and penalties (e.g. one has to pay a large penalty for pre-terminating a mobile contract especially if the contract involves having a discount on the price of a new mobile phone). We assume that these factors are strong enough to give the consumer an incentive to stick to her contract. Mandy (1991) adopts a similar restriction in order for negative tariffs to be feasible.

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only buys good N (qN,2 with qX,2 = 0). As previously noted, state ω 1 occurs with probability λ and state ω 2 with probability 1 − λ.

3.1

Consumers

Consider the last stage of the game where the state is already known to the consumer. The consumer chooses quantities of goods N and X in order to maximize Ui + y subject to the budget constraint pN qN,i + pX qX,i + y ≤ W − T where W is income and y is the consumption of all other goods. Assume that Ui is monotonically increasing and concave (with respect to quantities), and that demand reflects

independent goods. In other words, optimal quantities are only a function of own price. Formally let ( uN (qN,1 ) + uX (qX,1 ) if i = 1 Ui = uN (qN,2 ) if i = 2 where uj (qj ) is the gross utility from consuming good j (j = N, X). Note that in state ω2 , qX,2 = 0, i.e. the consumer only chooses the level of good N to maximize her state-dependent utility. The use of a quasilinear form of utility has some useful interpretation. Utility derived from consumption of the two goods can be seen as being “converted” into units of income and thus, expected utility maximization will be in terms of net income across different states. Adopting a quasilinear form of utility may have some drawbacks, in particular, exogenous changes in W and/or T do not affect demand for either good and thus implying that expenditures on the two goods are small relative to gross wealth. This restricts the applicability of the model to other product markets. Nonetheless, it remains applicable to the market we are interested in (i.e. mobile calls). ³ ´ ∗ , q∗ ∗ Denote optimal quantities as follows: qN,i X,i for i = 1, 2 and with qX,2 = 0. These optimal quantities satisfy

∂Ui ∂qN,i ∂U1 ∂qX,1

= =

duN = pN dqN,i duX = pX . dqX,1

Also, define state-dependent indirect utility as ∗ ∗ Vi ≡ Ui∗ − pN qN,i − pX qX,i −T +W

where

⎧ ³ ´ ³ ´ ∗ ∗ ⎨ u∗ q ∗ + u q N X X,1 ³ N,1´ Ui∗ = ∗ ∗ ⎩ u q N N,2

if i = 1 if i = 2,

∗ ∗ and qX,i are functions of their own price only. In the first stage, we set-up the consumer’s and qN,i

expected utility by taking a concave transformation of state-dependent (indirect) utilities and then weighting them by their respective probabilities. Specifically, we introduce risk aversion by taking a 6

CRRA-transformation so that we can adjust the constant coefficient of relative risk aversion between state-dependent indirect utilities, i.e. EV (pN , pX , T ) ≡ λ

V 1−ε V11−ε + (1 − λ) 2 1−ε 1−ε

where ε > 0 reflects risk aversion (i.e. we say that the consumer is not risk averse when ε = 0; this definition of risk aversion implies that the consumer is averse to variations in utilities experienced in different states and not just on consumption or income alone). The necessity of taking a concave transformation to introduce risk aversion comes from the fact that state-dependent utilities are not linked since the decision variables of the consumer (i.e. qN,i and qX,i ) are state-dependent. In fact as we shall show later on, if ε = 0, multi-part tariffs do not play a role and the tariff structure will collapse to uniform pricing at marginal cost.

3.2

Firms

Firms are assumed to compete for consumers in a highly competitive market by offering a tariff with prices that are not state-contingent. Assume that firms have marginal costs of producing goods N and X equal to cN and cX respectively, with cN ≤ cX .9 For the sake of discussion, consider three possible

types of pricing plans that can arise wherein optimal prices are above or equal to marginal cost. In the first plan, firms set both pN and pX equal to their respective marginal costs cN and cX , and offer

zero fixed fees, T = 0. Observe that this pricing plan leaves the consumer with the highest utility in each state, subject to firms earning nonnegative ex post profits. Moreover, this tariff resembles uniform pricing. In the second plan, pN and pX are both equal and above marginal cost, assuming cN = cX . The fixed fee will then be negative hence representing a subsidy. This type of tariff has the interpretation of an ordinary two-part tariff when the two goods are pseudo-goods, i.e. a single good decomposed into two goods. Allocative inefficiency (deadweight-loss) is present in both states due to above-marginal pricing of the two goods. Furthermore, the subsidy will not be enough to compensate for the loss in efficiency because firstly, allocative inefficiency implies that profits are less than the loss in consumer surplus and secondly, the firm can only provide a subsidy up to the level of expected profits. Therefore we expect this type of pricing to be always dominated. Finally, the third pricing plan is as follows: pN is set to marginal cost, pX is some price above marginal cost and T is a subsidy equal to the expected profit of the firm. In contrast to the first plan, utility in state ω1 is subject to allocative inefficiency due to pricing above marginal cost thus consumers are worse off in this state. This is similar to the second plan though inefficiency is less severe because 9

The assumption that cN ≤ cX is made in anticipation of our pseudo-good interpretation. We can observe cN < cX

if production cost is increasing in output, i.e. output that satisfies excess consumption is more expensive to produce. In our application to the market for local mobile calls, we assume cN = cX .

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pN is set to marginal cost. Consumers are better off in state ω 2 due to the subsidy provided by the firm (compared to the first plan) and the lack of allocative inefficiency (compared to the second plan). Thus this plan can somehow be considered as the best way among the three for firms to bear some consumption risk, albeit in an imperfect way. We show later that risk aversion is both a necessary and sufficient condition for this type of pricing to be optimal. In our model, pN does not play any role in the derivation of the optimal subsidy or fixed fee hence changes in this price only affects utility through its effect on the chosen quantity of good N . Thus, we can guess that at the optimum, pN = cN in order to attain efficiency with respect to consumption of good N . On the other hand, pX affects utility both directly (through the subsidy or fixed fee) and indirectly (through the chosen quantity of good X). Thus the optimal pX should balance these effects. The case wherein either price is below its respective marginal cost follows a similar argument as the preceding ones. We can expect that having pN = cN would be optimal since this achieves efficiency with respect to good N. Moreover, having pX < cX decreases utility in both states. Pricing below marginal cost implies a positive fixed fee hence utility in ω 2 is decreased. In ω1 , the increase in consumer surplus from a lower pX will not be sufficient to cover the firm’s losses from pricing below marginal cost that will be passed on as a fixed fee and thus on net, utility will decrease. We now proceed with formally deriving the optimal tariff.

3.3

Optimal tariffs

Because firms operate in a highly competitive market, they design their pricing plans so as to maximize the expected utility of consumers under the constraint that they earn nonnegative ex ante profits. Formally they solve

max EV (pN , pX , T ) £ ¤ ∗ ∗ ∗ + (pX − cX ) qX,1 + T ≥ 0. st λ (pN − cN ) qN,1 + (1 − λ) (pN − cN ) qN,2

pN ,pX ≥0,T

∗ ∗ ∗ (p ) and = qN,2 = qN Since there are no income effects and goods are independent, we have that qN,1 N ∗ ∗ (p ) hence the nonnegative profit constraint simplifies to qX,1 = qX X ∗ ∗ (pN − cN ) qN (pN ) + λ (pX − cX ) qX (pX ) + T ≥ 0 ∗ (p ) + Since EV is decreasing in T , the constraint must be binding in equilibrium thus (pN − cN ) qN N

∗ (p )+T = 0 or stated differently, −T = (p − c ) q ∗ (p )+λ (p − c ) q ∗ (p ). Here λ (pX − cX ) qX X N N N X X X X N

we can see that if both prices are above marginal cost, then the fixed fee is necessarily a subsidy. If both prices are set at marginal cost, then the fixed fee is zero and thus we are back to the uniform pricing case. The fixed fee is only positive if at least one of the prices is below marginal cost. Hayes (1987) shows that if the covariance between the marginal utility of income and the demand for the 8

good is positive, then below-marginal cost pricing is optimal while if this covariance is negative, then above-marginal cost pricing is optimal. With the way we construct our model, it is easy to see that this covariance is negative and therefore it is more likely that the price of good X is above marginal cost. To see this, note that utility in state ω1 is always higher than in ω 2 given that the consumer buys positive quantities of both goods in ω 1 (otherwise she can achieve the same level of utility in state ω 2 by not consuming the now available good), and thus (expected) marginal utility of income will be lower in ω 1 than in ω 2 . Finally since demand for good X is positive in ω 1 and zero in ω 2 , the covariance between marginal utility of income and the demand for good X is indeed negative. Plugging the value of T into Vi we have ∗ ∗ ∗ − cN qN (pN ) + λ (pX − cX ) qX (pX ) + W Vi = Ui∗ − pX qX,i ∗ = q ∗ (p ) if i = 1 and q ∗ = 0 if i = 2. The problem then reduces to with qX,i X X X,i

λ ∗ ∗ ∗ (U1∗ − (1 − λ) pX qX (pX ) − cN qN (pN ) − λcX qX (pX ) + W )1−ε pN ,pX ≥0 1 − ε 1−λ ∗ ∗ ∗ (U − cN qN + (pN ) + λ (pX − cX ) qX (pX ) + W )1−ε . 1−ε 2 max

(1)

In the following proposition, we derive the optimal tariff for this problem. The main sufficient condition is risk aversion on the part of consumers. Proposition 1 Assume that there exists a pair of maximum prices (PN , PX ) such that for pN ≥ PN

(pX ≥ PX ), qN = 0 (qX = 0).10 If consumers are risk averse, then the optimal tariff is characterized by p∗N = cN , p∗X > cX and solves

pX − cX ∗ (p ) qX X ∗ (p∗ ) < 0. and T = −λ (p∗X − cX ) qX X

¢ ¡ −ε ¯ ∗ ¯ −ε ¯ dqX ¯ (1 − λ) V − V 2 1 ¯ ¯ ¯ dpX ¯ = £λV −ε + (1 − λ) V −ε ¤ 1 2

Proof. Differentiating the objective function with respect to pN and pX respectively yields, µ ∗ µ ∗ ∗ ¶ ∗ ¶ dqN dqN −ε duN −ε duN λV1 + (1 − λ) V2 (FODpN ) − cN − cN dpN dpN dpN dpN and λV1−ε

10

∙

∙ ¸ ¸ ∗ ∗ ∗ du∗X dqX dqX dqX −ε ∗ ∗ ∗ λ (pX − cX ) − pX − qX (pX ) + λ (pX − cX ) + λqX (pX ) +(1 − λ) V2 + λqX (pX ) . dpX dpX dpX dpX (FODpX )

This condition is not necessary and is only used to simplify the proof. A weaker sufficient condition is that demand

goes to zero as price goes to infinity.

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Recall that at the optimal demands, du∗N dq ∗ = pN N ≤ 0 dpN dpN and

dq ∗ du∗X = pX X ≤ 0. dpX dpX

From these we see that FODpN

and FODpX

¡

λV1−ε

− λ) V2−ε

= − + (1 ⎧ ⎪ ⎪ ⎨ > 0 if pN < cN = 0 if pN = cN ⎪ ⎪ ⎩ < 0 if p > c N N

¢

¯ ∗ ¯ ¯ dq ¯ (pN − cN ) ¯¯ N ¯¯ dpN

¯ ∗ ¯¸ ∙ ¯ dq ¯ £ −ε ¤ ∗ −ε ∗ = λ λV1 + (1 − λ) V2 (pX ) . qX (pX ) − (pX − cX ) ¯¯ X ¯¯ − λV1−ε qX dpX

Since the firm never chooses prices strictly above (PN , PX ) (and is restricted to setting nonnegative prices), the feasible set is closed and its upper-contour set is bounded. Moreover, the objective function is continuous in the (restricted) domain of (pN , pX ) and therefore a solution exists. There are several possible solutions, depending on whether some of the constraints on (pN , pX ) are binding or not. Note that we can easily rule out solutions involving either pN = PN or pX = PX since the firm can decrease the price a little bit to induce consumption and increase utility (notice that FODpN < 0 for pN > cN ∗ (p ) = 0, the subsidy is also zero). Furthermore, solutions with p = 0 (binding) are and that if qX X N

not optimal given that FODpN > 0 for pN < cN (cN > 0). Thus we are left with two possible solutions. One of the candidate solutions is for pN to be an interior solution and pX = 0. This implies that ∗ (p ) > 0, FODp ≤ 0 if and only if FODpX ≤ 0 when evaluated at pX = 0. Observe that given qX X X £ ¤ ¯ ∗ ¯ ¯ (1 − λ) V2−ε − V1−ε (pX − cX ) ¯¯ dqX ¯ £ −ε ¤ −ε ≤ q ∗ (p ) ¯ dp ¯ . λV1 + (1 − λ) V2 X X X

∗ (p ) > 0 implies V > V , the term at the left of the inequality is strictly positive given that Since qX 1 2 X

good X is consumed and the consumer is risk averse, i.e. ε > 0. Thus whenever pX ≤ cX , FODpX > 0

hence a contradiction. Therefore the only remaining solution is the interior solution characterized by

p∗N = cN , p∗X > cX and solves pX − cX ∗ (p ) qX X ∗ (p∗ ) < 0. and T = −λ (p∗X − cX ) qX X

¡ −ε ¢ ¯ ∗ ¯ −ε ¯ dqX ¯ ¯ = £(1 − λ) V2 − V1 ¤ ¯ ¯ dpX ¯ λV1−ε + (1 − λ) V2−ε

Since the price of good N does not affect state-dependent utilities in a direct way (i.e. not through

the subsidy; see equation (1)), pN is set in order to maximize the joint surplus of the consumer and the 10

firm with respect to good N in each state. Therefore it is optimal to set pN = cN . On the other hand, pX is set to balance the positive and negative effects on utility. When pX increases, two contrasting effects occur. The first effect decreases utility in state ω 1 by raising allocative inefficiency due to above-marginal cost pricing. The second effect increases utility in both states by increasing the subsidy received.11 The optimal level of pX is achieved when these two effects are balanced, the condition for which this occurs can be seen as a condition on the price elasticity of demand for X evaluated at the candidate price, i.e. ¡ −ε ¢ ¯ ∗ ¯ ¯ ∗ ¯ −ε ¯ ¯ ¯ pX ¯¯ dqX ¯ = £(1 − λ) V2 − V1 ¤ + cX ¯ dqX ¯ . −ε −ε ∗ ∗ ¯ ¯ ¯ qX (pX ) dpX qX (pX ) dpX ¯ λV1 + (1 − λ) V2

If the elasticity is below this level, the second effect dominates the first hence it is optimal to increase pX further. In contrast, if the elasticity is above this level, allocative inefficiency is too high and thus it is optimal to reduce pX . Observe that if we interpret N and X as pseudo-goods, then the combined tariff is a general threepart tariff, as can be seen in Figure 1. For consumption levels below the normal level, a lower price is charged while beyond this, the consumer faces a steeper price.

Figure 1: Optimal tariff with pseudo-good interpretation Since optimal consumption of good N is the same in both states, the consumer will be indifferent ∗ and paying a fixed fee equivalent to this, i.e. between paying a per-unit price equal to cN for q ≤ qN

∗ . Thus the general three-part tariff can be approximated by a standard three-part tariff. Figure cN qN 11

The subsidy increases with pX if and only if

Observe that

 ∗   (pX − cX )  dqX  < 1. ∗  qX (pX ) dpX 

  (1 − λ) V2−ε − V1−ε  −ε  λV1 + (1 − λ) V2−ε

= <

V1−ε  1 −  −ε λV1 + (1 − λ) V2−ε 1

and thus in the neighborhood around the optimum p∗X , the subsidy is increasing.

11

2 illustrates this.

Figure 2: Approximation of optimal tariff with pseudo-good interpretation Finally, when cN = 0, the optimal tariff is equal to a standard three-part tariff as seen in Figure 3.

Figure 3: Optimal tariff with pseudo-good interpretation and cN = 0 Therefore we have shown that risk aversion can lead to optimal tariffs that have the same structure as general and standard three-part tariffs. The next corollary proves the necessity of risk aversion in our model for the optimality of three-part tariffs. Corollary 1 Risk aversion is a necessary condition for the optimality of three-part tariffs in our model. Proof. Notice that when ε = 0, FODpX T 0 for pX S cX and therefore the unique solution to the firm’s problem is p∗N = cN , p∗X = cX and T = 0. Thus pricing collapses to uniform pricing at marginal cost when the consumer is not risk averse. The intuition behind the role of risk aversion can be understood as follows. A risk averse consumer would like to smoothen utilities across different states. In other words, when the consumer is risk averse, she is willing to give up some utility in state ω 1 to increase utility in state ω2 given that V1 > V2 . In our 12

model, the increase in utility in ω 1 comes from good X, which she is now consuming. Thus the optimal way to transfer utility from one state to another is to make consumption of good X less attractive by increasing its price and shift utility to state ω 2 by way of a higher subsidy. It can be shown using implicit differentiation of FODpX that p∗X is increasing in the degree of risk aversion which means a more risk averse consumer commits to having utility in ω 1 a lot closer to utility in ω 2 by accepting a much higher pX in exchange for a subsidy. We can also show that p∗X is decreasing in income which can be easily understood by examining the concavity of the CRRA-transformed statedependent utilities. Holding the difference between V1 and V2 constant, a higher level of W means that these two are located in a flatter region and therefore there is less incentive to smoothen utilities across states. Simply said, consumers do not care so much about the differences in utilities across states when they have greater income. We conjecture that the preceding logic is often used by researchers when they disregard risk aversion as a relevant explanation for the particular pricing behavior we are examining (see Miravete (2003) for example).

3.4

Criticism of risk aversion

Risk aversion has earlier been proposed and rejected as a possible explanation for the fixed-fee bias in the telecommunications literature (Train, 1991; Mitchell and Vogelsang 1991, Miravete 2002, 2003; Clay, Sibley and Srinagesh 1992; Nunes 2000; as cited in Lambrecht and Skiera, 2004). Theoretically, risk aversion will induce subscribers to choose fixed-fee plans in order to avoid the risk of having to pay a large phone bill during times of higher than average demand. However, given that expenditure on phone bills is small relative to income, unrealistically high degrees of risk aversion seem to be necessary. Our model takes a look at this issue again, albeit from a different angle. We assume that risk aversion is not only with respect to payments but also over state-dependent utilities hence including utility gained from consumption of calls. Tariffs then have a slightly different insurance interpretation in our case. Instead of being risk averse solely about having to pay an expensive phone bill, we consider subscribers who desire smoothing their utilities across different states. Though a larger phone bill is expected in high demand states, we take into account the additional utility from making calls during these states, hence on net, utility actually increases. Here we utilize a “revealed preference intuition” in the sense that subscribers call more when the utility they get from making calls is higher. It is worth noting that Grubb (2007) rules out risk aversion as an explanation for three-part tariffs by arguing: “[i]f consumers are risk averse to such variability in their monthly bill, the use of threepart tariffs with steep overage charges is more surprising rather than less surprising.” The logic of this argument is that risk averse subscribers will prefer paying a higher fixed fee in exchange for lower overage charges, but this exactly is the interpretation of risk aversion used to explain the fixed-fee bias. In contrast, our interpretation of risk aversion would rationalize steep overage charges since high demand states mean higher utility from making calls, hence smoothing utilities across states involves 13

a reduction in consumption utility (through higher per-minute charges) combined with a monetary transfer in all states (through a subsidy).

4

Application

We now apply the model to the market for local mobile calls. Specifically, we calibrate the model using a given set of parameters and derive the optimal tariff. Goods N and X are interpreted as “normal” and “excess” call minutes respectively. In our model, excess minutes should be interpreted as excess call consumption due to extraordinary events and not simply due to small fluctuations in monthly call consumption. Given the state the subscriber is in, both goods are neither substitutes nor complements (in the sense of cross-price elasticities) hence, the rationale for our independence assumption.12 The functional form we choose for the state-dependent utility in fact implies that when state ω 2 is realized, total consumption (normal plus excess) is almost doubled. This comes from the symmetry (i.e. given equal prices, the total benefit from calls doubles when the extraordinary event shock is realized) of normal and excess calls with respect to state-dependent utility. Observe that in our model, consumption of normal calls (qN,i ) is constant across the two states. ∗ can be considered as the monthly fixed fee with corresponding free minutes amounting to Thus pN qN ∗ . Implicit in the model is the assumption that the subscriber consumes all of her free minutes in the qN

normal state. This assumption is consistent with Miravete’s (2003) observation that people eventually learn to choose the “right” calling plan since having excess free minutes consistently would motivate a subscriber to choose a different plan (say, after a contract period of one year) with less free minutes and lower monthly fees. Consider the last stage of the game where the state is already known to the subscriber. Assume that the utility derived from the two consumption goods is of the quadratic form, i.e.13 ⎧ ³ ´ ⎨ α (qN,i + ηqX,i ) − β q 2 + q 2 if i = 1 N,i X,i 2 Ui = ⎩ αqN,i − β q 2 if i = 2. 2 N,i

The consumer maximizes Ui + y subject to the budget constraint pN qN,i + pX qX,i + y ≤ W − T where

W is monthly income and y is the consumption of all other goods. Again note that qX,2 = 0. Utility 12

In reality, more complicated plans allow for some substitution between free and “excess” minutes. For example, a

subscriber can opt to use her phone in off-peak hours in order to consume her free minutes instead of using it during peak hours (which might not be included in the free minutes in some plans). We abstract from this issue for now. Moreover in state ω1 , the two goods are somehow discrete complements in the sense that consumption of X only happens after a certain level of good N is consumed. Our model captures this effect even if we assume that cross-price elasticities are nil since consumption of good X is driven by an exogenous shock and not by the price of good N. 13 The general form for this utility is β 2 2  q + 2γqN,i qX,i + qX,i . U (qN,i , qX,i ) = α (qN,i + ηqX,i ) − 2 N,i We consider the case with γ = 0.

14

maximization leads to the following simple state-dependent linear demand equations: ∗ ∗ qN,1 = qN,2 =

α − pN β

αη − pX β = 0.

∗ qX,1 = ∗ qX,2

State-dependent utilities now become V1 = V2 =

(α − pN )2 (αη − pX )2 + −T +W 2β 2β (α − pN )2 − T + W. 2β

The optimal price for good N is equal to marginal cost while the optimal price for good X is implicitly defined by

¢ ¡ (1 − λ) V2−ε − V1−ε pX − cX ¤ = £ −ε αη − pX λV1 + (1 − λ) V2−ε

which is a highly nonlinear equation that is difficult to solve analytically but quite easy to solve numerically. This equation has a unique real root for the range of parameters we consider. In calibrating the model, we set our parameters in accordance to observed values in the U.K. market for local mobile calls. We compute the equivalent per minute cost of free calls for a given plan and set marginal cost equal to this value, i.e. cN = cX = monthly fee / total free monthly minutes. This parameterization may be reasonable if we look at the highest volume plans since these have the lowest monthly fee to free minutes ratio, which somehow can be used as a proxy for marginal cost.14 In fact, for one of the operators we are considering, the equivalent per minute cost is close to 5.1 pence per minute, which is the (new) cost of terminating15 a call . On the demand side, we normalize α = η = 1. To find β which serves as a scale parameter for demand, we utilize data from the highest volume plan of two U.K. mobile operators. Derivation of β ∗ given p∗ = c . This allows entails equating the total number of free minutes with the expression for qN N N ∗ and the monthly fixed fee to p∗ q ∗ . We assume that us to match the total allocated free minutes to qN N N

monthly income amounts to £15,000 which is not unusual given that annual salaries commanded by non-entry level positions in investment banks can range from £100,000 to £200,000 (and even more). With this level of income, the monthly fixed fee is roughly half a percentage point of monthly income 14

Another justification for focusing on the highest volume price plans is that subscribers who opt for these plans are

relatively few hence competition is stronger for these customers and therefore, more consistent to our model. 15 In this calibration exercise, the optimal price that we derive corresponds to the charge for excess minutes consumed to make intra—network or mobile—to—fixed line calls. If we assume that the cost of originating and terminating a call are roughly the same (see for example, Hausman and Wright (2006)), then termination charges can approximate the marginal cost for intra-network or mobile-to-fixed line calls.

15

which we think is small enough to be consistent with our no income effects assumption.16 Finally we compute the optimal price under different values for the probability of consuming excess calls (λ = 0.01, 0.05, 0.10, 0.20). Using the calibrated model, we derive the optimal price for excess minutes as a function of the degree of risk aversion. The calibration exercise considers the highest volume plan offered by Vodafone and Orange U.K., the details of which are publicly available in their websites. We summarize the plans we consider in Table 1.17 Table 1: Summary of pricing plans considered fixed fee (£)

free minutes

excess charge (£/min)

β (×10−4 )

fixed fee / income (%)

Vodafone

63.83

1200

0.085

7.91

0.43

Orange

75

1000

0.120

9.25

0.50

We take the realistic range of risk aversion as [1.2, 1.8] (Szpiro, 1986; Layard, Mayraz and Nickell, 2007) and derive the optimal price for this region. Figures 4 and 5 capture the optimal price as a function of the degree of risk aversion for different values of λ. Our model implies higher prices compared to the actual Vodafone price, especially for λ < 0.20. For λ = 0.20, the true price is achieved when the degree of risk aversion is roughly equal to 1.3. Suppose we are interested in the case for which when ε = 1.5, the optimal price is approximately equal to the true price. This can be achieved by increasing λ further. Alternatively, we can set λ = 0.01 and find the level of monthly income that would give us the result. We are interested in λ = 0.01 to back-up our assumption that excess minutes in our model correspond to rare or extraordinary events, hence implying a low probability of occurring. We find that this can be achieved when W = £21, 100 which means that the monthly fixed fee is 0.30% of monthly income, hence more consistent with our assumption of no income effects.18

16

The expenditure on good X is close to the expenditure on good N, hence on average, total expenditures on calls

(including excess charges) are roughly 1% of monthly income. 17 As noted, we consider the charge for intra-network and mobile—to—fixed line excess minutes. 18 For both Vodafone and Orange, the excess payment is roughly equal to the monthly fixed fee since with the way we specified our model, total consumption (normal plus excess minutes) almost doubles in state ω1 relative to state ω2 . This means that total payment (without the subsidy, which in our case is very small, i.e. less than £1) is approximately 0.60% of monthly income for Vodaone and 1.32% for Orange.

16

Figure 4: Vodafone optimal price (true price = 0.085) We now turn our attention to Orange. In contrast to the results for Vodafone, our model implies lower prices compared to the actual Orange price. Either lower levels of λ or lower monthly income can bring optimal prices closer to the true price. We repeat our thought experiment and find the monthly income such that at ε = 1.5 and λ = 0.01, the optimal price is roughly equal to the true price. We find W = £11, 100 which implies that the monthly fixed fee is 0.66%, which is still a reasonable level for our assumption of no income effects.

Figure 5: Orange optimal price (true price = 0.12)

17

From our results, we can conclude that our model can reasonably explain actual prices, at least for the highest volume plan. Thus risk aversion should still be a relevant factor in analyses of mobile phone pricing.

5

Concluding remarks

We have shown that risk aversion can be a possible justification for three-part tariffs. The main intuition is that three-part tariffs allow risk averse consumers to transfer utilities across different states. Specifically, the optimal tariff raises the per-unit price for the uncertain good that provides additional utility in one of the states and transfers expected profits from the sale of this good to the lower utility state by way of a reduction in the fixed fee. We have also provided an application of the model to the market for local mobile calls and show that the optimal tariff from our model can be quite close to tariffs that are empirically observed. Possible extensions of the model basically involve relaxing some of the assumptions in the current model. For instance, one can still retain the quasilinear form of state-dependent utilities but derive the results for interdependent goods (substitutes and complements). Subtle issues such as cannibalization/monopolization of either good inherent in these types of models should be taken care of and hence complexity of deriving explicit results will increase. Another extension is to analyze the case with income effects. Other extensions involve changing the environment of our model. For instance, we can study how the results of the model will change in an environment with a monopolist and monopolistic competition. Results will not change substantially for the former case, the only difference being that the monopolist absorbs the full consumer surplus while retaining the same tariff structure.19 Though probably involving more complicated analysis, the latter case would certainly be fruitful—allowing us to see subtle issues involving the interaction between risk aversion and strategic behavior of firms. It will also be interesting 19

This can be easily seen by recalling the firms’ objective function in the competitive case. Since we already know that

the solution for (pN , pX ) is interior, the equivalent Lagrangian for this problem is L = EV + μEΠ where  ∗ ∗  ∗ + (1 − λ) (pN − cN ) qN,2 + (pX − cX ) qX,1 +T EΠ = λ (pN − cN ) qN,1

and μ is the multiplier for the binding constraint, i.e.

μ = λV1−ε + (1 − λ) V2−ε > 0. By inverting μ, this problem is equivalent to max

pN ,pX ≥0,T

EΠ

st EV = 0 which is the monopolist’s problem. The only difference between the two problems is that for the former, T < 0 and set such that expected profits are zero, while for the latter, T > 0 and set to extract consumers’ expected utility.

18

to see how risk aversion can be incorporated in sequential screening models and where consumers exhibit some bias. An important extension is that of allowing underconsumption in normal times. Recall that in our model we have assumed that the fully rational consumer knows her level of normal consumption. Relaxing this assumption brings our model closer to that of Grubb (2007) and can be used to model firsttime subscribers who have not yet learnt this information. We pursue this extension in the appendix and argue that even with underconsumption, risk aversion can still lead to an optimal tariff that resembles a general three-part tariff. We also provide a sufficient condition such that risk aversion can explain the use of standard three-part tariffs when there is underconsumption.

6

References

Clay, K., D. Sibley and P. Srinagesh (1992) “Ex Post vs. Ex Ante Pricing: Optional Calling Plans and Tapered Tariffs,” Journal of Regulatory Economics, 4(2): 115-138. Eliaz, K, and R. Spiegler (2006) “Contracting with Diversely Naive Agents,” Review of Economic Studies, 73(3): 689-714. Ellison, G. (2005) “A Model of Add-on Pricing,” Quarterly Journal of Economics, 120(2): 585-638. Grubb, M. (2007) “Selling to Overconfident Consumers,” mimeo, Stanford GSB. Hausman, J. and J. Wright (2006) “Two Sided Markets with Substitution: Mobile Termination Revisited,” mimeo, MIT and National University of Singapore. Hayes, B. (1987) “Competition and Two-Part Tariffs,” Journal of Business, 60: 41-54. Jensen, S. (2006) “Implementation of Competitive Nonlinear Pricing: Tariffs with Inclusive Consumption,” Review of Economic Design 10: 9-29. Lambrecht, A. and B. Skiera (2004) “Paying Too Much and Being Happy About It: Existence, Causes, and Consequences of Tariff-Choice Biases,” (older mimeo version). Layard, R., G. Mayraz and S. Nickell (2007) “The Marginal Utility of Income,” mimeo, London School of Economics. Locay, L. and A. Rodriguez (1992) “Price Discrimination in Competitive Markets,” Journal of Political Economy, 100(5): 954-965. Mandy, D. (1991) “Competitive Two-part Tariffs as a Response to Differential Rates of Time Preference,” Economica, 58: 377-389.

19

Miravete, E. (2002) “Estimating Demand for Local Telephone Service with Asymmetric Information and Optional Calling Plans,” Review of Economic Studies, 69: 943-71. Miravete, E. (2003) “Choosing the Wrong Calling Plan? Ignorance and Learning,” American Economic Review , 93(1): 297-310. Mitchell, B. M. and I. Vogelsang (1991) Telecommunication Pricing: Theory and Practice, 1st Edition, Cambridge, UK: Cambridge University Press. Nunes, J. (2000) “A Cognitive Model of People’s Usage Estimations,” Journal of Marketing Research, 37(4), 397-409. Stole, L. (1995) “Nonlinear Pricing and Oligopoly,” Journal of Economics and Management Strategy, 4(4): 529—562 Szpiro, G. (1986) “Measuring Risk Aversion: An Alternative Approach,” Review of Economics and Statistics, 68(1): 156-159. Train, K. (1991) Optimal Regulation: The Economic Theory of Natural Monopoly, Cambridge, Mass: MIT Press.

7

Appendix: Underconsumption in normal times

We extend the model to the case where underconsumption of free minutes is allowed. Define a random variable z which affects the demand for call minutes in state ω 2 in a positive manner (higher z implies ∗ (p , z)). Assume that z is distributed over the support [0, Z] with probability density function higher qN N

f (·). E (p ) as the expected level of consumption of good N given p . The fixed fee will now Define qN N N

be

£ ∗ ¤ E ∗ −T = (pN − cN ) λqN (pN , Z) + (1 − λ) qN (pN ) + λ (pX − cX ) qX,1

and the firm’s problem transforms to

λ 1−λ V 1−ε + max pN ,pX ≥0 1 − ε 1 1−ε

20

ZZ 0

V21−ε (z) f (z) dz

where ∗ ∗ V1 = U1 (pN , pX ) − pN qN (pN , Z) − pX qX (pX ) £ ∗ ¤ E (pN ) + (pN − cN ) λqN (pN , Z) + (1 − λ) qN ∗ +λ (pX − cX ) qX,1 +W

∗ (pN , z) V2 (z) = U2 (pN , z) − pN qN £ ∗ ¤ E (pN , Z) + (1 − λ) qN (pN ) + (pN − cN ) λqN ∗ +λ (pX − cX ) qX,1 +W

The behavior of FODpX is almost the same as in the model in Proposition 1. After some computations, FODpX can be rewritten as ⎧ ⎫ ⎡ Z ⎤ ¯ ∗ ¯⎨ ZZ Z ⎬ ¯ dqX ¯ ∗ ¯ λV −ε + (1 − λ) V −ε (z) f (z) dz +(1 − λ) λqX (pX ) ⎣ V2−ε (z) f (z) dz − V1−ε ⎦ −λ (pX − cX ) ¯¯ 2 ⎭ dpN ¯ ⎩ 1 0

Assuming

ZZ

0

V2−ε (z) f (z) dz − V1−ε > 0, when pX ≤ cX we have FODpX > 0. This is analogous to the

0

proof of Proposition 1, where instead of simply having V2−ε > V1−ε , we have taken into account the fact that consumption can be random even in normal times. Thus p∗X > cX . We now look at FODpN . After some computations we have

FODpN

⎧ ⎫ ¯ E ¯¶ ⎨ ¯ µ ¯ ZZ ⎬ ¯ dqN ¯ ¯ dqN (Z) ¯ 1−ε −ε ¯ + (1 − λ) ¯ ¯ λV = − (pN − cN ) λ ¯¯ + (1 − λ) V (z) f (z) dz 2 ¯ dpN ¯ ⎩ 1 ⎭ dpN ¯ 0 ¯ ¯ ∗ E ∗ −λV11−ε ¯λqN (pN , Z) + (1 − λ) qN (pN ) − qN (pN , Z)¯ + (1 − λ)

ZZ 0

£ ∗ ¤ E ∗ V2−ε (z) λqN (pN , Z) + (1 − λ) qN (pN ) − qN (pN , z) f (z) dz.

We are primarily concerned with very low values of λ since we expect extraordinary events to occur less often. Since expected utility is continuous in λ, we can take a look at its behavior when λ → 0 to have an approximation of what happens for sufficiently low λ. Taking limits, FODpN simplifies to lim FODpN

λ→0

¯ E ¯ ZZ ZZ ¯ dqN ¯ £ E ¤ −ε ∗ ¯ ¯ = − (pN − cN ) ¯ V2 (z) f (z) dz + V2−ε (z) qN (pN ) − qN (pN , z) f (z) dz. ¯ dpN 0

0

21

Notice that if V2−ε (z) is independent of z, i.e. V2−ε (z) = V2−ε , then ¯ E¯ ¯ dq ¯ lim FODpN = − (pN − cN ) ¯¯ N ¯¯ V2−ε λ→0 dpN ⎧ ⎪ ⎪ ⎨ > 0 if pN < cN = 0 if pN = cN ⎪ ⎪ ⎩ < 0 if p > c N

N

hence p∗N = cN using the same argument as in the proof of Proposition 1. Therefore in this case, we have a similar optimal tariff as before. On the other hand, if V2−ε (z) is increasing in z, then ZZ 0

£ E ¤ ∗ V2−ε (z) qN (pN ) − qN (pN , z) f (z) dz < 0.

Thus for all pN ≥ cN , lim FODpN

λ→0

¯ E ¯ ZZ ZZ ¯ dqN ¯ £ E ¤ −ε ∗ ¯ V (z) f (z) dz + V −ε (z) qN = − (pN − cN ) ¯¯ (pN ) − qN (pN , z) f (z) dz < 0 2 2 ¯ dpN 0

0

hence p∗N ∈ [0, cN ). More precisely, if for all pN ≤ cN ¯ Z ¯ ¯ E ¯ ZZ ¯Z ¯ ¯ dqN ¯ ¯ ¯ £ ¤ −ε −ε E ∗ ¯ V (z) f (z) dz < ¯ V (z) qN (pN ) − qN (pN , z) f (z) dz ¯ cN ¯¯ 2 2 ¯ ¯ ¯ dpN ¯ ¯ 0

0

then p∗N = 0. Thus this serves as a sufficient condition for optimality of a standard three-part tariff.20

This inequality can also be rewritten as ¯ Z ¯ ¯Z ¯ E ¯ ZZ ¯ ∙ ¸ ∗ ¯ ¯ qN (pN , z) cN ¯¯ dqN ¯¯ −ε −ε ¯ V2 (z) f (z) dz < ¯ V2 (z) 1 − E f (z) dz ¯¯ ¯ ¯ E qN (pN ) dpN qN (pN ) ¯ ¯ 0

0

and since if this holds for all pN ≤ cN then it must also hold for pN = cN : ¯ Z ¯ ¯Z ¯ h i ¯ ¯ ∗ ¯ V −ε (z) 1 − qNE(cN ,z) f (z) dz ¯ 2 ¯ qN (cN ) ¯ E ¯ ¯¯ ¯ ¯ cN ¯¯ dqN ¯< 0 . E (c ) ¯ dp ¯ ZZ qN N N V2−ε (z) f (z) dz 0

This means that in order for the optimal tariff to be a standard three-part tariff, the price elasticity of expected demand for good N evaluated at marginal cost should be sufficiently inelastic. 20

E Notice that a necessary condition is for dqN /dpN to be bounded. This is similar to the assumption of a satiation point

in Grubb (2007).

22

Observe that the condition V2−ε (z) is increasing in z is analogous to Hayes ’ (1987) result whereby if the marginal utility of income (in our case, V2−ε (z)) is positively correlated with demand, then the ∗ (p , z) is assumed to be monotonically increasing in z, optimal price is below marginal cost. Since qN N

the marginal utility of income and demand (for good N ) are indeed positively correlated (with respect to the random variable z).

23