Constrained Monopoly Pricing with Endogenous...

Introduction Data Theory Estimation Evaluation

Constrained Monopoly Pricing with

Endogenous Participation

Gabriel A. Basaluzzo1 Eugenio J. Miravete2

1Universidad de San Andres

2University of Texas at Austin& Centre for Economic Policy Research

April 1, 2009

Basaluzzo-Miravete Constrained Monopoly Pricing

Introduction Data Theory Estimation Evaluation Motivation Intuition Agenda

Performance of linear vs. nonlinear pricing

Almost non-existing theoretical results.

This is due to the technical complexity of second-degree pricediscrimination.

This is an open empirical question in an area where there arevery few contributions available.

Spence (1977), Roberts (1979), Katz (1983).

The Robinson-Patman Act imposes some policy restrictions tosecond degree price discrimination:

Price discrimination of intermediate goods is forbidden(secondary line or alleged injury to rivals of the buyer).



What do we do in this paper?

Develop a “feasible” equilibrium model of second-degree pricediscrimination by a monopolist where:

Consumers’ intensity of preferences is private information.Consumers’ also differ in the subjective valuation of theiralternative to abstain from consumption.Due to scarce data availability, we use very limited information.

This is an empirical implementation of the exclusive agencymodel of Rochet and Stole (2002).

The solution to this model may accommodate commonfeatures of current tariffs such as the existence of an allowance(something that no other model can explain).



Single dimensional screening (1/3)

Key assumption 1 - Single crossing property: u(q, θ); uqθ(q, θ) > 0.

4

Single Dimensional Screening:Single crossing property: U(q,θ) s.t. Uqθ (q,θ) > 0.

p(q)

q




Key assumption 2 - F (θ) is IHR to avoid this:

5

Single Dimensional Screening:F(θ) is IHR to avoid this…

f(θ)

θ




Participation and consumption decisions are ordered by the samevariable, θ, leading to a fully separating equilibrium:

6

Single Dimensional Screening:Participation and consumption ordered by the same variable.

Fully separating equilibria.

T(q)

q



Two dimensional screening

Participation and consumption decisions are ordered by differentvariables, (t and x). Bunching at the bottom always occurs:

7

Two Dimensional Screening:Participation and consumption ordered by different variables.

Bunching at the bottom always occurs.

T(q)

q



Main task of the paper

Goal:

To estimate a structural equilibrium of model of nonlinear pricingwhere core parameters are conditioned on observablecharacteristics of a cross-section of local wireless markets as well ason the particular implementation of the nonlinear tariff by eachcarrier in each market.



Tariff comparison

Once we have obtained the structural parameters we compare theperformance of the following tariffs:

Fully nonlinear tariff.

Linear pricing (no quantity discounts).

Flat tariff (just a monthly fee).

Optimal two-part tariff.

Coasian tariff (fixed fee + marginal cost).

Performance evaluation includes:

Profits.Welfare.Market coverage.Quantity undersupply.



Applications

What do we use our model for?

Evaluation of the universal service policy:

Balanced: maximize market coverage while the monopolist stillbreaks even.Lump-sum subsidy to ensure efficient provision.

Address the importance of a second source of asymmetry ofinformation:

Rochet-Stole vs. Mussa-Rosen.


Introduction Data Theory Estimation Evaluation Description

Market Description

Data only include tariff information. Individual consumption is notavailable.

50 largest U.S. local cellular monopoly markets (1984-88).

Monopoly is temporary and entry of competitor is exogenous.No need to model entry or entry deterrence strategies.

Data include all tariff plans offered by the incumbent:

Focus on tariff options defining the lower envelope of peaktime.Include first and last quarter in the sample.

Complemented with Census, FBI, and U.S. HCN data.



Table 1: Tariff Features

Monthly Fee, Fi Rate per Minute, pi

Option No. Mean Std.Dev. Mean Std.Dev.

Markets with ONE option(56 observations)

1 28.78 (11.50) 0.39 (0.07)

Markets with TWO options(23 observations)

1 14.13 (4.96) 0.55 (0.13)2 40.68 (9.40) 0.38 (0.10)

Markets with THREE options(16 observations)

1 3.24 (1.29) 0.63 (0.14)2 15.65 (8.62) 0.46 (0.07)3 33.41 (18.29) 0.31 (0.10)

Mean and standard deviations (between parentheses) of monthly fixed fees Fi and rate per minute pi aremeasured in dollars.

specific demand and cost information as well as an ownership indicator for each firm. Table 2 presents the

descriptive statistics of our data. All money valued magnitudes are expressed in dollars of July 1986. Data

definitions and sources are detailed in Appendix A.5

Perhaps the only variable that needs a detailed discussion is COVERAGE. This is an important

variable that critically determines the dispersion of the distribution of horizontal heterogeneity, x. Un-

fortunately we do not have any information available on individual subscribers, nor a precise defini-

tion of what constitutes the potential market in this early cellular telephone industry (the latter being

a common problem in applied industrial economics). We therefore proceed by defining COVERAGE =

1.3×TCELLS/(BUSINESS+250*POPULATION), where TCELLS denotes the number of antennae deployed in

each market, BUSINESS is the number of business (in thousands) that were considered likely users of cellular

telephony, and POPULATION is the population of each market (in millions). Thus, we identify the potential

market as that represented by a cellular telephone for each firm and families with an average number of four

members. At this early stage of development of the industry, the number of subscribers were effectively

constrained by the capacity of cellular carriers and a maximum of 1,300 could be simultaneously served

by each antenna.6 This definition of market penetration is certainly arbitrary, but we feel that it is not

unreasonable. We could have considered only the number of businesses as the potential market. However,

private use of cellular telephony also existed and cellular carriers targeted private users in their marketing

campaigns as well. Similarly, subscribing to a cellular service provider was far from common, and thus if

5 Many other variables were available but we only report here those that are used in the empirical analysis of Section 4.2.

6 Parker and Röller (1997, §4) report that one antenna could serve between 1,100 and 1,300 subscribers for the average use ofcellular telephony at that time. Using a small sample of markets they also report that the correlation between the number of antennasand the number of subscribers exceeds 90%.

– 7 –



Table 2: Descriptive Statistics

First Quarter Last Quarter

Variables Mean Std.Dev. Mean Std.Dev.

PLANS 1.4783 0.6909 1.6304 0.7989COVERAGE 0.0679 0.0681 0.0671 0.0679TIME 4.1957 2.4094 11.5652 4.0423MKT-AGE 2.1087 3.1849 24.6087 10.5756INCOME 28.2225 3.7465 27.5302 3.3053COMMUTING 26.1609 2.8513 26.2174 2.7778sdv(COMMUTING) 16.8640 2.9783 16.8627 2.9391RAIN 3.4107 1.9115 3.0609 2.0246POVERTY 11.0500 3.0281 11.2304 2.9443POP-AGE 32.7543 2.8836 32.6826 2.8606EDUCATION 12.5500 0.2420 12.5370 0.2507TCELLS 18.1739 18.5320 18.3478 18.4237HHSIZE 2.6223 0.2877 2.6195 0.2878sdv(HHSIZE) 1.4665 0.1602 1.4654 0.1604DENSITY 15.8190 14.0903 15.1110 13.1091OPERATE 6.5055 1.5614 6.4900 1.4977PRIME 10.8424 0.6310 9.1809 0.9526WAGE 6.9580 1.5533 7.3320 1.6649BELL 0.8261 0.3832 0.8261 0.3832REGULATED 0.4783 0.5050 0.4565 0.5036

Observations 48 47

All variables defined in Appendix A.

we consider the total population as the potential market, coverage becomes minimal, implying necessarily

that the distribution of the horizontal heterogeneity dimension, x, is almost degenerate. Evidently, our

results depend critically on the definition of COVERAGE. Were a better definition of market penetration

available we would have not hesitated in using it, but in its absence we feel that the mix of BUSINESS and a

fraction of POPULATION that approximates the number of families of each market is a reasonable definition

of the potential market, helps identifying the parameters of our model and highlights the methodological

contribution of this paper and the value of its potential applications.

3 A Flexible Model of Nonlinear Pricing with Random Participation

This section presents an equilibrium model of monopolistic nonlinear tariff with optimal endogenous

participation decisions of consumers. We believe that a static model of nonlinear pricing without network

externalities is an accurate representation of the early U.S. cellular telephone industry. Pricing does not

change much in the monopolistic markets of our sample even though the number of subscribers, while

still few, increased significantly. Furthermore, the numbering of cellular phones was identical to fixed

telephones and thus, cellular carriers were never able to offer within network discounts.

– 8 –


Introduction Data Theory Estimation Evaluation General Approximation

Fully Nonlinear Solution

Principal and agents’ objective functions:

Profit maximization:

π(q) = P (q)− cq .

Consumer surplus (linear demand):

v(t, q, x) = tq − γ

2q2 − P (q)− x .



Figure 1: f (θ) — Burr type XII density function

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

t/tmax

f(t)

0.2 0.5 1.0 λ 2.0 5.0

The Burr type XII distribution of θ in equation (3) is a restricted beta distribution with parameters

1 and λ−1. The economic interpretation of parameter λ is appealing as it is directly related to the average

markups charged by the monopolist to consumers with diverse valuations. 4 Different values of λ identify

whether high-valuation consumers are more or less numerous than low-valuation consumers. This is

clearly shown in Figure 1. If λ = 1 then θ is uniformly distributed. If λ > 1, there are more numerous

high-valuation than low-valuation customers and vice versa when λ < 1. If λ = 0, distribution (3) becomes

degenerate at θ = θ, asymmetric information turns out to be irrelevant and homogeneous consumers will

be efficiently priced by means of a single two-part tariff.

Finally from maximizing (2), the demand of any consumer type θ is linear in the marginal tariff:

x(p, θ) =θ − p

γ. (4)

4 The Burr type XII distribution of θ in equation (3) fulfills the IHR property as long as λ ≥ 0. See Johnson, Kotz, andBalakrishnan (1994, §12.4.5) for further details. Miravete (2002, Appendix) and Miravete (2005) analyze in detail the relationshipbetween λ and the markup charged to different single-dimensional consumers.

– 8 –



Figure 2: G(x) — Exponential distribution

0 1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

1/φ

exp(−

x/φ)

φ = 1φ = 2φ = 5

asymmetric information turns out to be irrelevant and homogeneous consumers can be efficiently priced

by means of a single two-part tariff.14

Similarly, the exponential distribution (3a) is a good approximation to model the horizontal het-

erogeneity dimension, x. This exponential density is monotonically decreasing with a maximum at x = 0

and has an expected value of φ. This parameter increases when the average consumer has better outside

options (perhaps offered by a competing firm in a horizontally differentiated industry). Thus, as it is shown

in Figure 2, a small value of φ leads to a fast rate of decay of the probability density function of x. In terms of

our model, it means that a small φ is associated with situations where consumer horizontal heterogeneity is

also small. Thus, the probability associated to the event that consumers attach a large value to the outside

option is very low. The contrary is true for large values of φ. The horizontal heterogeneity summarized

by x distinguishes the RS model from the standard single-dimensional MR screening model. If φ → 0 the

horizontal heterogeneity disappears and we are left with a standard, single-dimensional, nonlinear pricing

model. The role of x in the model is to capture the idea that consumers may have different valuations of their

outside option of not purchasing q independently of how their preferences are ranked with respect to this

14 For further results and properties of the Burr type XII distribution see Johnson, Kotz, and Balakrishnan (1994, §12.4.5).

– 12 –



Endogenous participation

Now participation decision is different for each consumer type t.Thus, the total market penetration is given by:

M(u, t) = Prob[t, x ≤ u] = G(u)f(t)

=[1− exp

(−u

φ

)]1

λ(t− t)

(1− t− t

t− t

) 1λ−1

,



Monopolist’s problem

The monopolist’s optimal control problem is:

maxq(t),u(t)

ZT

M(u(t), t) [P (q(t))− cq(t)] dt ,

u(t) = q(t) ≥ 0,

q(t) ≥ 0 ,

u(t) = tq(t)− γ

2q2(t)− P (q(t)) ≥ 0 .



Solution with optimal exclusion

Bunching at the bottom is always optimal for a monopolist:

0=φ exp(−φu(t))(t− t)

»u(t)− 1

2u2

–+ [1− exp(−φu(t))] (t− t) [2− u(t)]

−„

1

λ− 1

«[1− exp(−φu(t))] [t− u(t)] ,

u(t)=q(t) ,

q(t)≥qfb(t) ,

q(t)=qfb(t).



Approximately Optimal Screening

Firms offer a few tariff options to their customers.

Possible existence of marketing and commercialization costs,which are in general unobservable.

Foregone profits decrease rapidly with the number of tariffoptions offered.

Firms choose the number of tariff options provided theirexpected profits, which depend on distribution of consumertastes and the realization of marginal costs.

Simultaneously, our computed value for these core parametersdepend on the actual number of tariff options.



The monopolist solves:

π(ν|ω) = maxy∈[0,1)2ν

π“P (A(y), b(y), q)|ω

”,

y1 < y2 < . . . < yν < yν+1 < . . . < y2ν ,

bν+1−i(y) = c + (t− c)yi i = 1, 2, . . . , ν

ti(y) = c + (t− c)yν+i i = 1, 2, . . . , ν

A1(y) =[t1(x)− b1(x)]2

2γ,

Ai(y) = Ai−1(y) +[bi−1(y)− bi(y)]

hti(y)− bi−1(y)+bi(y)

2

i2

γ,

P (A(y), b(y), q) = mini

Ai(y) + bi(y)q .



MSM

The vector of structural parameters is ω = (λ, γ, φ, t, c)′ ∈ R5:

We condition these parameters on observable marketcharacteristics:

ωi = Mzi + εi , ε ∼ N(0,Σ) .

Conditional on zi, we can obtain:

ωi = E(ω|zi) = Mzi .



Given ωi we need to solve for the optimal tariff conditional onthe number of options offered in each market (which is takenas given).

For each ωi = Mzi the model predicts for each market:

Monthly fee A.Price per minute b.Market penetration M(u, t).

In addition, using the information across markets we alsoobtain a prediction of:

Average household consumption (active subscriber) (Sample:160 minutes/month).Average monthly bill (Sample: 100 dollars/month).



Difficulties

“Few” moments available, in particular for two-part tariffs. Ifwe want to increase the number of regressors in M , we needto consider higher moments of the distribution of fixedmonthly fees, price per minute, or share of market penetration.

Every new option offered by a carrier doubles the number ofmoments related to A and b.

Simulation requires solving the optimal nonlinear tariff and itsparticular implementation (with a given number of tariff plans)countless number of times. This is a very expensive process.



Table 5: Welfare, Usage, and Market Penetration

First Best Nonlinear Flat Linear Two-Part Coasian

ONE Option

MONTHLY FEE 0.0000 150.6434 0.0000 29.1809 108.7219RATE PER MINUTE 0.1022 0.0000 0.4907 0.3784 0.1022PROFITS 0.0000 1.0000 0.8579 0.9687 0.9915 0.9147PROFITS (?) 0.0000 2.1609 1.9360 2.1047 2.1434 1.9680WELFARE 1.0000 0.7383 0.6869 0.7376 0.7365 0.7171WELFARE (?) 5.2581 3.8793 3.7176 3.8737 3.8716 3.7650COVERAGE 0.1356 0.0558 0.0444 0.0633 0.0542 0.0458AIRTIME USAGE 269.2043 266.2942 388.4190 226.8420 275.9493 356.1682UNDERSUPPLY 1.0000 0.4169 0.4580 0.4038 0.4134 0.4595

TWO Options


THREE Options


Median of the distributions across markets. All variables are defined in the text. PROFITS(?) and WELFARE(?) indicate the money valueof these variables in millions of dollars for a market with mean potential customer base of about 400,000 customers.

case of a flat tariff. A distribution of types heavily concentrated around low valuation explains that the

significant reduction in airtime usage when we compare markets where more tariff options were offered.

Because of the efficiency at the top result, only the highest consumer type pays the marginal cost

for the latest unit consumed. All others consume less than under the FB solution due to the existence of

asymmetric information. Since we do not observe individual subscribers we need to define some aggregate

measure of the undersupply of airtime in each market when using different tariffs. In Table 5, undersupply

indicates the median of the ratio of total minutes sold in a market (average airtime usage × coverage)

under each tariff relative to the total market usage with the FB solution. Thus, as the number of tariff

options increases from uniform to two-part tariff and then to the fully nonlinear tariff, the distortion gets

reduced as these mechanisms target heterogeneous consumers more accurately. Overall, usage provision

– 24 –


Introduction Data Theory Estimation Evaluation Distribution Effects Robustness

Universal Service

The constrained monopolist’s problem is:

maxP (q)

ZT

M(u(t), t)dt ,

s.t.

ZT

M(u(t), t) [P (q(t))− cq(t)] dt ≥ 0 ,

M(u(t), t) =

»1− exp(−u(t)

φ)

–1

λ(t− t)

„t− t

t− t

« 1λ−1

,

u(t) = tq(t)− γ

2q2(t)− P (q(t)) ≥ 0 ,

q(t) ∈ argmaxq

ntq − γ

2q2 − P (q)

o,

q1 ≤ q2 ⇒ P (q1) ≤ P (q2), ∀q1, q2 .



Table 6: Average Effects of Universal Service Policy

First Best Break Even Free

ONE Option

WELFARE 1.0000 0.8262 0.9670COVERAGE 0.1356 0.1727 0.1616AIRTIME USAGE 269.2043 148.3324 279.8256UNDERSUPPLY 1.0000 0.5510 1.0395

TWO Options

WELFARE 1.0000 0.8425 0.6317MARKET PENETRATION 0.0620 0.0764 0.0723AIRTIME USAGE 151.4041 89.5111 172.9245UNDERSUPPLY 1.0000 0.5912 1.1421

THREE Options

WELFARE 1.0000 0.8709 0.7902MARKET PENETRATION 0.0398 0.0468 0.0577AIRTIME USAGE 121.0355 79.5081 135.5475UNDERSUPPLY 1.0000 0.6569 1.1199

Results reported for the median of the parameters values. Variables are defined as in Table 5.

In order to induce participation a policy of universal service requires important markup reductions.

In the Free universal service policy reduces the value of the outside option uniformly to all consumer types

t, and thus, a larger proportion of them, whether high or low, will participate in the market. This is seen

in Figure 1 as an approximately parallel upward shift of the participation line relative to the FB solution.

Consumers in this market are simply receiving a rent transfer from the government and thus all consumers

types may potentially benefit from it as even those willing to pay a positive amount less than c end up

subscribing to the service. Since under this Free service marginal rates are also lower for all consumer types

we can conclude then that the oversupply result reported in Table 6 is widespread and common to all types

t, as it is captured by the upward shift of consumption relative to the FB solution in Figure 2.33

The case of a Break Even universal service policy is quite different and perhaps more interesting

because it involves a substantial redistribution of rents among participants in the market. In this case the

goal of maximizing market penetration needs to be achieved without funding from outside the industry.

Thus, the monopolist reduces the tariff discount for large customers and increases it for small ones and

such re-balancing of the tariff needs to be done optimally so that the number of new low valuation sub-

scribers offsets those high valuation one that decide to leave the market. In environments characterized

by our relatively large fraction of low valuation customers, the overall effect of this policy is to increase

participation at the cost of reducing consumption of all but the highest consumer type relative to the FB

33 Figures 1 and 2 refer only to those markets where one single option was made available to consumers (more than half ofour sample). Qualitative results are identical for the other markets in which more options were offered.

– 28 –



Figure 1: F(t) — Who participates under universal service?

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t [$/min]

1 - e

-u(t)

/φ

N = 1

Break evenFree serviceFirst best

reduces the value of the outside option uniformly to all consumer types t, and thus, a larger proportion

of them, whether high or low, will participate in the market. This is seen in Figure 1 as an approximately

parallel upward shift of the participation line relative to the FB solution. Consumers in this market are

simply receiving a rent transfer from the government and thus all consumers types may potentially benefit

from it as even those willing to pay a positive amount less than c end up subscribing to the service. Since

under this Free service marginal rates are also lower for all consumer types we can conclude then that the

oversupply result reported in Table 7 is widespread and common to all types t as depicted in Figure 2.

The case of a Break Even universal service policy is quite different because it involves a substantial

redistribution of rents among participants in the market. In this case the goal of maximizing market

penetration needs to be achieved without funding from outside the industry. Thus, the monopolist reduces

the tariff discount for large customers and increases it for small ones. The effect of an increase in discounts

– 34 –



Figure 2: F(t) — Consumption profile under universal service?

0 0.5 1 1.5 2 2.50

100

200

300

400

500

600

t [$/min]

q(t)

[min

]

N = 1

Break evenFirst bestFree service

for low valuation customers needs to offset that some high valuation consumers leave the market as they

enjoy smaller discounts than before.

Provided our relatively large fraction of low valuation customers, the overall effect of this policy

is to reduce the price distortion as documented in Table 7 by the important increase of the median ratio of

undersupply in the case of the Break Even policy relative to the fully nonlinear tariff solution. As Figure 2

shows, some of these new low valuation customers do not even consume while all infra-marginal types

t < t consume less than in the FB solution. 44

The success of universal service policies is normally measured only by the actual increase in partic-

ipation but little attention is paid to the efficiency loss and redistributive effects of such policy. Our model

takes into account participation and consumption decision and thus can be used to asses the impact of these

44 The efficiency at the top of the Break Even policy depicted in Figure 2 is not the result of a constrain imposed while solvingproblem (21a)–(21f) but rather a feature of the solution.

– 35 –



Robustness Check

Since due to lack of individual usage data our result depend onparticular functional form assumptions, we need to test whetherthey will change to different specifications:

Alternative maximum consumption.

Beta Distributed types:

F (t|a, λ) =Γ(a + λ−1)

Γ(a)Γ(λ−1)

(t− t)a−1(t− t)λ−1−1

(t− t)a+λ−1−2.

Nonlinear demand:

v(t, q, x) =tρ+1

γ(ρ + 1)

»1−

“1− γ

tq”ρ+1

–− P (q)− x .

Non-CRS technology:

TC(q) =c

1 + wq1+w .


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