efficient pricing using non-linear prices assume – strong natural monopoly => mc=p =>...
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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 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
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
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