Efficient Pricing using Non-linear Prices

• Assume– Strong natural monopoly• => MC=P => deficit

– Non-linear prices are at their disposal

Example of a non-linear price

• Uniform two-part tariff– Constant price for each unit– Access fee for privilege– Disneyland, rafting permits, car rentals– Public utilities• Flat monthly charge,• Price per kwh• Price cubic feet of gas• Price per minute of telephone usage

Two-part Tariff Model

• p*q +t

• where• t Ξ access fee• p Ξ unit price• q Ξ quantity purchased• If t=0, the model is the special case of linear

prices

Declining Block Tariff

• Marginal price paid decreases in steps as the quantity purchased increases

• If the consumer purchases q• He pays– p1 * q +t, if 0<q≤q1

– p2 *(q- q1) + p1 * q1 +t, if q1< q ≤q2

– p3 *(q- q2) + p2 *(q2- q1) + p1 * q1 +t, if q2< q ≤q3

– If p1>p2>p3 => declining block tariff

Non-uniform tariff• t varies across consumers• For example, – industrial customers face a lower t b/c they use a constant q

level of electricity– Ladies night, where girls get in free

• Discriminatory, challenge in court• Often used to meet some social objective rather than

increase efficiency.• Initially used to distinguish between fixed costs and

variable costs– View demand (Mwh) and peak demand separately (MW)– The two are connected and that must be accounted for

Two-part Tariff Discussion

• Lewis (1941) – decreases distortions caused by taxes

• Coase (1946) – P=MC and t*n=deficit• Gabor(1955) – any pricing structure can be

restructured to a 2-part tariff without loss of consumer surplus

Rationale

• MC = P creates deficit, particularly if you don’t want to subsidize

• Ramsey is difficult, especially if it creates entry

One Option

• MC = P and fee= portion of tariff• Fee acts as a lump-sum tax• Non-linear because consumer pays more than

marginal cost for inframarginal units.• Perfectly discriminating monopolist okay with first

best because the firm extracts all C.S.– Charges lower price for each unit– The last unit P=MC

• Similarly, welfare max regulator uses access fee to extract C.S.

Tariff Size – bcef or aefP

QMC

D

Q

P

c

AC

b

f

a

e

Tariff not a Lump Sum

• Not levied on everyone• Output level changes, if demand is sensitive to

income change• Previous figure shows zero income effect

Additional Problems

• Marginal customer forced out because can’t afford access fee (fee > remaining C.S.)

• Trade-off between access fee and price– Depend on• Price elasticity• Sensitivity of market participation

Example of fixed costs

• Wiring, transformers, meters• Pipes, meters• Access to phone lines, and switching units

• Per consumer charge = access fee to cover deficit

• Book presents single-product– Identical to next model if MC of access =0

• Discusses papers with a model of two different output, but one requires the other.– Complicated by entry

Two-part Tariff Definitions• Θ = consumer index• Example– ΘA = describes type A– ΘB = describes type B

• f(Θ)=density function of consumers– The firm knows the distribution of consumers but

not a particular consumer– s* is the number of Θ* type of consumer

– s is the number of consumers

dfs

*

*)(*

dfs 1

0)(~

Θ* Type Consumer• Demand• q(p,t,y(Θ*), Θ*)

• Income• y(Θ*)

• Indirect Utility Function– v(p,t,y(Θ*), Θ*)– ∂v/ ∂ Θ ≤0

• => Θ near 1 =consumer has small demand• => Θ near 0 =consumer has large demand

• Assume Demand curves do not cross– => increase p or decrease t that do not cause marginal

consumers to leave, then inframarginal consumers do not leave

More Defintions• Let be a cutoff where some individuals exit

the market at a given p, t • If , no one exits– – Number of consumer under cutoff,

• Total Output

• Profit

dfstp

),(ˆ

0)(

),(ˆ tp

1ˆ

),(ˆ tp

dfytpqQtp

),(ˆ

0)()),(,,(

)(QCstQp

Welfare

– w(θ) weight by marginal social value

dfytpvwVtp

),(ˆ

0)()),(,,()(

Constrained Maximization

• max L=V+λπ• by choosing p, t, λ

• FOCs0

p

QMC

p

st

p

QpQVL pp

0

t

QMC

t

st

t

QpsVL tt

0 CstQpL

where

• Where is the change # of consumers caused by a change in p

• and

pp Qqp

Q

ˆ̂

tt Qqt

Q

ˆ̂

tt

s ̂

pp

s ̂

p̂

)ˆ()ˆ),ˆ(,,(ˆ fytpqq

where

• Simplifies to

• From the individual’s utility max

• Where vy(θ) MU income for type θ.

dfytpvwfytpvwV ppp )()),(,,()()ˆ()ˆ(,,()ˆ(ˆˆ

0

dfytpvwV pp )()),(,,()(ˆ

0

)),(,,()()),(,,( ytpqvytpv yp

Income

• Let vy=-vt because the access fee is equivalent to a reduction in income

• Ignore income distribution and let– w(θ)=1/vy(θ)– Each consumer’s utility is weighted by the

reciprocal of his MU of income

• Substituting into Vp revealsQdfytpq

)()),(,,(ˆ

0

Similarly

ˆ

0

)( dfVt

sVt

Substitution Reveals

• Where• s=Qp+Q/s Qy

• D= deficit

0ˆˆˆˆˆ

tptp s

Qt

s

QqSMCp

DQMCCtsQMCp

Solving Gives

• where

QZqZSs

ZDMCp

QZZqSs

DZqSt

ˆ

ˆ

tp S

QZ ˆˆ

yp QS

QQS

Interpretation• Let

• Marginal consumer’s demand (Roy’s Identity)

• To keep utility unchanged, the dt/dp=-qˆ• Differentiate to get

• Combining get

tp S

QZ ˆˆ

t

p

v

vq ˆ

),(ˆ tp

t

p

dp

dt

ˆ

ˆ

0ˆˆ tp S

Q

Result 1

• If the marginal consumers are insensitive to changes in the access fee or price, that is,

• then the welfare maximization is– P=MC– t=D/s

• Applies when no consumers are driven away– i.e electricity– Not telephone, cable

0ˆˆ tp

Result 2

• Suppose the marginal consumers are sensitive to price and access-fee changes

• Then, the sign of p-MC is the same sign as Q/s-qˆ

• And – p-MC≤0, then t=D/s>0– if p>MC, then t≥0

Deviations from MC pricing – Result 2

• Increase in price or fee will cause individuals to leave

• Optimality may require raising p above MC in order to lower the fee, so more people stay

• p>MC when Q/s>qˆ, because only then will there be enough revenue by the higher price to cover lowering the access fee.

Deviations from MC pricing – Result 2

• p<MC and t>0– Very few consumers enticed to market by lowering t– Consumers who do enter have flat demand with large

quantities– A slightly lower price means more C.S.– Revenues lost to inframarginal consumers is not too

great because Q/s<qˆ,– Lost revenues are recovered by increasing t without

driving out too many consumers– Q/s-qˆ is a sufficient statistic for policy making