Optimal pricing for voicemail services.

Spiegel, Uriel ; Tavor, Tchai

I. Introduction

The uniqueness of phone calls, mailing, chatting, fax messages and some (but not all) network instruments is precisely that communicating is achieved by the two parties only if an actual connection occurs (Kim et al (2002) call these "pingpong goods" whose values are generated only through joint consumption). The sender is willing to pay an appropriate price for his phone call only if the receiver actually picks up the phone and establishes communication Still the key question is who should pay for successful contact. In a recent paper by Hermalin and Katz (2004) this very question is raised. They conclude that in order to maximize welfare and the firm's profits the receiving party should subsidize the sender (since he is generating a benefit to the receiver). A similar conclusion was first established by Kim and Lira ((2001) and (2002)). The sender-receiver market issue has also been discussed by Rochet and Tirole (2004). They focused on the question of how to deal with a market where buyers and sellers need to be brought together for the market to exist, as well as on the nature of the pricing policy that may lead to an efficient solution. The question as to what happens in cases where the benefits to the two sides are not positive and/or not symmetric, as well as the resulting implications as to pricing policy were not addressed in their paper.

The asymmetry issue between parties, i.e. senders and receivers, who may generate either positive or negative externalities, was recently discussed by Loder, Van Alstyne and Wash (2006). Nevertheless, in their paper they deal with homogeneous senders and homogeneous receivers but don't relate to the issue of asymmetric attitudes towards contact between parties as well as the asymmetry of being a sender vs. that of being a receiver Our paper adds to the above literature another dimension that is part and parcel of the current communications industry, that of asymmetric behavior of different senders and receivers. In the last decades this industry has witnessed some significant technological improvements including features that did not previously exist and in particular that of voicemail service that is used today so frequently that people (especially the young) cannot imagine a world without it. This capability certainly increases the degree of communication efficiency. Statistically, a certain percentage of these messages cannot be directly received either because the receiver is unavailable due to the absence of an open phone in his vicinity, or because he simply does not desire to receive calls at that time.

Voicemail services increase the percentage of successful contacts, either by leaving recorded messages by senders to receivers, that do not require a further call back by the original receiver to the former sender, or by encouraging the original receiver to call back the original sender to discuss the issues through an additional direct call.

This service that benefits both parties (receivers as well as senders) should be charged for by the telecommunication companies, however, the question is who should pay and how much.

Assuming a monopoly telecommunication company who charges for each phone call and for each message left on the answering machine, the question still remains: who should pay for those services. Even if we assume that the usual policy is that the sender pays for regular phone calls that he successfully accomplishes, still the question that remains open is who pays and how much for leaving messages on voicemail.

Today the usual practice is that the sender who leaves a message pays for it, usually at the regular price of a phone call, while the receiver pays a fixed monthly tariff for the right to use the answering service.

We can argue that a different possibility should be considered by the communication industry. If the receiver would not make a return call to the sender, it is possible that the sender should pay for the use of the message left on the voicemail. But what if the receiver does call the original sender back? The telecommunication company gains as a result of the message left on the voicemail by the receiver's returned phone call. If the original sender knows that he must pay for leaving a message he might consider disconnecting the call and avoid leaving the message, but if the sender realizes that most likely the receiver will respond to voicemail messages (i.e., if these messages are not considered by the receiver as "junk voicemails") and in such cases the sender himself will be subsidized, this would encourage him to initiate the call. This paper discusses these kinds of intertemporal effects and demonstrates the possibilities for optimal profit maximization that should be considered. We depart from Kim and Lim's (2001, 2002) receiver pays principle by considering optimal positive as well as optimal negative pricing for voicemail for both parties. The simple theoretical model is followed by a numerical example.

II. The Model

Assume two individuals, i,(i = 1,2) where individual 1 is a sender of [X.sub.1] phone calls, while individual 2 is a receiver of the same phone calls. As a receiver of [X.sub.1] calls from the sender, he can either accept, [X.sub.11] the call, but [X.sub.12] calls either he could not accept at the time it was sent or he did not want to accept, but may respond to them when convenient, if a voicemail message is left by the sender.

Therefore [X.sub.1] calls of the sender distributed as follows:

[X.sub.1] = [X.sub.11] + [X.sub.12] (1)

[X.sub.11]--are direct successful contacts between sender and receiver

[X.sub.12]--are calls done by the sender that were not accepted directly by the receiver, but by voicemail messages.

We assume further that those two values [X.sub.11] and [X.sub.12] depend linearly and negatively on their prices, although the sensitivities to price changes that are measured by the values of [beta] and [gamma] can differ.

[X.sub.11] = [A.sub.11] - [beta] x [P.sub.11] (2)

and

[X.sub.12] = [A.sub.12] - [delta] x [P.sub.12] (3)

The demand of individual 2, who is representing the original receiver for his own phone calls is affected negatively by the call prices and probably by calls received on the voicemail as follows:

[X.sub.2] = [A.sub.2] + [alpha] x [X.sub.12] - [gamma] x [P.sub.2] (4)

Equation (4) indicates that the received calls of individual 2 can be decomposed into two elements: responding to the sender (individual one), and initiating his own calls to individual 1, where [alpha] represents the ratio of the original sender's calls that are responded to by the receiver.

We can assume the following relationship between the parameters of (1)-(4) above: [A.sub.11] > [A.sub.12] and [A.sub.11] > [A.sub.2]. The relationship between these last three terms indicates asymmetry between individuals where we assume that individual 1 has a greater desire to make a phone call to individual 2 than vice versa. Still individual 1 (the original sender) has a greater desire to make direct calls than to leave messages on voicemail. In addition we assume that 0 < [alpha] < 1 where [alpha] is a coefficient representing the ratio of returned calls the receiver initiates for phone calls the sender sent and were not received on the spot by the receiver (individuals 2) and were left as messages on voicemail.

Based on these two demand functions for phone calls by both parties, the monopoly (the telecommunication company) maximizes its own profit function with respect to three decision variables: [P.sub.11], [P.sub.12], and [P.sub.2], where [P.sub.11] is the price of a direct call from sender to receiver, [P.sub.12] is the charge per message left on the voicemail, and [P.sub.2] the price for a return call from receiver to sender respectively. (i)

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where we assume that the marginal production cost function of any kind of calls is constant and equal to c, i.e.,

TC = c x ([X.sub.11] + [X.sub.12] + [X.sub.2]) (6)

The first and the third terms of the profit function are supposed to be positive. However, the second term might be negative, since we allow a negative, value of [P.sub.12]. This means that rewards are distributed to a sender who leaves messages for a receiver, since it may encourage more return calls by the latter.

The F.O.C. derived with respect to the three decision variables are

[partial derivative][pi]/[partial derivative][P.sub.11] = [A.sub.11] - 2[beta] x [P.sub.11] + [beta] x c = 0 (7)

[partial derivative][pi]/[partial derivative][P.sub.12] = [A.sub.11] - 2[delta] x [P.sub.12] + [delta] x c - [alpha] x [delta]([P.sub.2] - c) = 0 (8)

[partial derivative][pi]/[partial derivative][P.sub.2] = [A.sub.2] + [alpha][A.sub.12] - [alpha] x [delta][P.sub.12] - 2[gamma][P.sub.2] + [gamma]c = 0 (9)

The determinant A3 of the S.O.C. partial derivatives derived with respect to the three decision variables is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7')

At equilibrium the conditions for local maximum are: (a) -2[beta] < 0, (b) 4[beta][delta] > 0, and (c) -2[beta] (4[delta][gamma] - [[alpha].sup.2][[delta].sup.2]) < 0 or (c') 4[gamma] > [[alpha].sup.2][delta].

The actual prices at equilibrium are solved by (7)-(9) are:

[P.sub.11] = [A.sub.11] + [beta] x c/2[beta] (10)

which is always positive,

[P.sub.12] = [alpha] x [delta][A.sub.2] + ([[alpha].sup.2][delta] - 2[gamma])[A.sub.12] - 2[gamma] x [delta]c(1 + [alpha]) + [alpha][gamma][delta]c/[delta]([[alpha].sup.2][delta] - 4[gamma]) (11)

That only for a high value of 7 and small values of [alpha] and [delta] guarantees a positive price [P.sub.12].

However, in most cases it is more likely to determine an optimal negative value of [P.sub.12].

[P.sub.2] = 2[A.sub.2] + [alpha][A.sub.12] - [alpha] x [delta]c(1 + [alpha]) + 2[gamma]c/ 4[gamma] - [[alpha].sup.2][delta] (12)

which is definitely positive.

An extreme case that is based on the assumption that the marginal cost is negligible, thus, c = 0. (ii) In this case the actual price values are:

[P.sub.11] = [A.sub.11]/2[beta] (13)

[P.sub.12] = [alpha] x [delta][A.sub.2] + ([[alpha].sup.2][delta] - 2[gamma])[A.sub.12]/ [delta]([[alpha].sup.2][delta] - 4[gamma]) (14)

[P.sub.2] = 2[A.sub.2] + [alpha][A.sub.12]/4[gamma] - [[alpha].sup.2][delta] (15)

Since 4[gamma] > [[alpha].sup.2][delta] then [P.sub.2] > 0, but if [alpha] x [delta][A.sub.2] + ([[alpha].sup.2][delta] - 2[gamma])[A.sub.12] > 0 then a negative price on voicemail messages is possible, i.e., [P.sub.12] < 0. Furthermore, from equations (13)-(15) we can find how changes in parameter [alpha] can affect the pricing policy of a monopoly profit maximizer. Since 0 < [gamma], [delta] < 1 and [A.sub.1] and [A.sub.2] are larger than the marginal cost, c, it is definitely clear that an increase in [alpha], the propensity/inclination to return voicemail messages with a phone call from the receiver, will lead to an increase in the actual price charged for each call of the receiver. This result is expected, since any increase in demand for calls encourages the monopoly to charge more for each call. However, the question is whether this attitude towards responding to voicemail calls will encourage the monopoly to charge a lower price or a higher price for those messages, i.e., a lower or higher [P.sub.12]. Furthermore, in the case of a negative charge it is possible that the subsidy on left messages will increase (negative [P.sub.12] becomes more negative) or vice versa.

We can see it may be either negative or positive (iii):

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This indicates in general that if the receiver initiates on his own more phone calls, i.e., [A.sub.2] is large and at the same time the receiver's calls are not sensitive to price, i.e., [gamma] is small, while the marginal cost of each call, c, is small, then it is more likely that [partial derivative][P.sub.12]/[partial derivative][alpha]. However in other cases the opposite conclusion holds.

In the section below, we demonstrate our general results by a numerical example named "the story of the father and son telephone communication".

1. Numerical Example

We introduce certain values for the parameters which may represent the real relationship between a father who needs and likes to communicate with his son but the reverse holds true for the son towards his father. Therefore we consider such parameter values where: [A.sub.11] > [A.sub.12] and [A.sub.11] > [A.sub.2] .

Setting the values of the parameters as:

[alpha] = 0.4, [beta] = 0.25, [gamma] = 0.15, [delta] = 0.5, [A.sub.11] = 100, [A.sub.12] = 50, [A.sub.2] = 60

we demonstrate the actual pricing values of the monopoly for two cases. Case I -discriminatory monopoly with the possibility of negative pricing.

We insert the parameter values in equations (10)-(12) and find the three prices:

[P.sub.11] = 100/2 x 0.25 = 200

[P.sub.12] = 0.4 x 0.5 x 60+([0.4.sup.2] x 0.5 - 2 x 0.15) x 50/ 0.5 x ([0.4.sup.2] x 0.5 - 4 x 0.15)

[P.sub.2] = 2 * 60 + 0.4 x 50/ 4 x 0.15 - [0.4.sup.2] x 0.5 = 269.23

And from equations (2)-(4) the quantities are:

[X.sub.11] = 100 - 0.25 x 200 = 50

[X.sub.12] = 50- 0.5 x (-3.85) = 51 x 92

[X.sub.2 ] = 60 + 0.4 x 51.92 - 0.15 x 269.23 = 40.38

The receiver (son) in our example initiates 20.77 calls as a result of voicemail messages sent by the sender (father). Thus most of the profits from the receivers' calls depends on these voicemail messages. The company encourages leaving these kinds of messages by paying the sender for those messages, rather than charging the sender a positive price.

The profit of the monopoly in this case is:

[pi] = 200 x 50 + (-3.85) x 51.92 + 269.23 x 40.38 = 20673.08

Case II -discriminating monopoly with a zero price on voicemail calls.

In this case [P.sub.12] = 0. i.e., it cannot be negative. Thus, the father does not pay for the messages that are not followed by the son's responses.

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The F.O.C. in this case are:

[partial derivative][pi]/[partial derivative][P.sub.11] = 100 - 0.5 x [P.sub.11] = 0

[partial derivative][pi]/[partial derivative][P.sub.2] = 80 - 0.3[P.sub.2] = 0

The prices at equilibrium are: [P.sub.11] = 200, [P.sub.2] = 266 2/3,

The quantities are: [X.sub.11] = 100 - 0.25 x 200 = 50, [X.sub.2] = 60 + 0.4 - 50 - 0.15 x 266 2/3 = 40

The profit of the monopoly is: [pi] = 200 x 50 + 266 2/3 x 40 = 20,666.67 < 20,673.08

By allowing a negative price for invoice messages that are followed by a later response by the receiver, the monopoly can generate more profit.

III. Implications

The possibility of negative pricing for messages left on voicemail is considered. This policy can be used for one purpose: to encourage the sender to leave these kinds of messages, and is very applicable under an asymmetric relationship between sender and receiver. An example of this kind of asymmetric behavior is that of a parent who is a sender and a child who is the receiver.

The parent is usually the party who initiates a call to his child who may not be available or is too busy to respond right away. On the other hand, since the younger child respects the parent, he will probably eventually respond to a message left by the parent. Rewarding the parent for messages left on voicemail to the child encourages the parent to leave a message and thereby increases the calls returned by the child to the parent, and as a result increases the monopoly profit on call services supplied to the market.

A policy by the monopoly of profit sharing with the parent sender for calls returned by the child can generate more profit as well as result in a more efficient communications industry. The optimal use of this new feature of voicemail generates more phone calls and revenues that otherwise would not exist if the service was not available or not marketed in an optimal manner, i.e., rewards to customers who use and leave messages on the voicemail service.

Furthermore, we assume that only one side uses the voicemail. We can introduce a more realistic situation when both parties own the voicemail services. Then the communication procedure is even more efficient since both parties may leave messages and both may respond (again more likely asymmetrically). Furthermore the efficiency can be even greater because a lot of communication activity can be accomplished by leaving messages on voicemail to voicemail. Or parties may leave voicemail messages as a response to previous voicemail messages which may increase by a "geometric series" that can be thought of as a multiplier effect.

Larger asymmetry, i.e.., a bigger gap between the desire to initiate phonecalls encourages the monopoly to subsidize leaving of messages on voicemail/answering machine the larger the desire of individual 2 to respond to voicemail messages, i.e., the larger the value of [alpha]. Individual 2 calls either as a response to voicemail message, or upon initiating calls independently for his own purposes.

Another extension to our model is that of adopting different values of [alpha] for a whole distribution of the senders' and receivers' interactions. For example, we can expect a different value of from a receiver who receives a call from his or her parents, attorney, physician, his or her loving partner, police, versus a call from a cantankerous old aunt or uncle. In such a case a negative or positive discriminatory pricing policy should be considered by the communication company but we leave this for future research.

Appendix

The derivative of [P.sub.12] with respect to [alpha] is:

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The sign of the numerator depends on the relationship below:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the same way we find the derivative of P2 with respect to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since

4[alpha][delta] x ([A.sub.2] - [gamma]c) + (4[gamma] - [[alpha].sup.2][delta] x

([A.sub.12] - [delta]c) > 0 then [partial derivative][P.sub.2]/[partial derivative][alpha] > 0

The last case that is discussed by us is the derivative of [P.sub.12] with respect to [alpha]:

[partial derivative][P.sub.12]/[partial derivative][alpha] = 2[gamma] - [[alpha].sup.2][delta]/[delta] x (4[gamma] - [[alpha].sup.2][delta]) >/< 0

From the analysis above we no that 4[gamma] - [[alpha].sup.2][delta] > 0, therefore the sign of the derivative depends on 2[gamma] - [[alpha].sup.2][delta]

if 2[gamma] > [[alpha].sup.2][delta] then [partial derivative][P.sub.12]/[partial derivative][alpha] > 0

If 4[gamma] > [[alpha].sup.2][delta] > 2[gamma] then [partial derivative][P.sub.12]/[partial derivative][alpha] < 0

References

Hermalin, B. and Katz, M. (2004). "Sender or Receiver: Who Should Pay to Exchange an Electronic Message?", RAND Journal of Economics, 35: 423-448.

Kim, Jeong-Yoo, Hyung Bae, and Dongchul Won, (2002). "Dutch Treat Versus Oriental Treat," Journal of Economic Behavior & Organization, 48: 413-422.

Kim, Jeong-Yoo and Lim, Yoonsung (2001). "An Economic Analysis of the Receiver Pays Principle", Information Economics and Policy, 13: 231-260.

Kim, Jeong-Yoo and Lim, Yoonsung, (2002). "Welfare Effects of the Receiver Pays Principle", Telecommunications Review. 12: 92-99.

Loder, T., M. Van Alstyne, and R. Wash, (2006). "An Economic Response to Unsolicited Communication", Advances in Economic Analysis & Policy, 6 (1): 1-37.

Rochet, J.C., and J. Tirole (2006). "Two-Sided Markets: A Progress Report", The Rand Journal of Economics, 37 (3) 645-667.

Notes

(i.) We assume that the communication company can identify and generate segmentation between senders and receivers' calls, and thus may set different prices.

(ii.) In the telecommunication industry most of the production costs can be treated as fixed costs, e.g., on infrastructure, while the marginal cost per call can be negligible.

(iii.) By taking the partial derivative of Equation (14) with respect to [alpha] see Appendix).

by Uriel Spiegel * and Tchai Tavor **

* Department of Management, Bar-Ilan University, and Visiting Professor, University of Pennsylvania, email: spiegeu@mail.biu.ac.il

** Department of Economics, Yisrael Valley College, email: tchai2000@yahoo.com We would like to thank an anonymous referee for his very helpful suggestions.