Download Efficient Pricing using Non

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 aef
P
a
P
b
c
f
e
AC
D
Q
MC
Q
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
s*   f ( )d
 *
~s  1 f ( )d

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 ˆ( p, t ) be a cutoff where some individuals
exit the market at a given p, t
• If ˆ  1, no
one exits
ˆ
 ( p ,t )
f ( )d
– s
0
– Number of consumer under cutoff, ˆ( p, t )
• Total Output
ˆ ( p ,t )
Q
q( p, t , y( ),  ) f ( )d
0
• Profit
  p  Q  t  s  C (Q)
Welfare
ˆ ( p ,t )
V 
0
w( )v( p, t , y ( ),  ) f ( )d
– w(θ) weight by marginal social value
Constrained Maximization
• max L=V+λπ
• by choosing p, t, λ
• FOCs

Q
s
Q 
L p  V p   Q  p
t
 MC
0

p
p
p 

Q
s
Q 

Lt  Vt    s  p
 t  MC
0

t
t
t 

L  p  Q  t  s  C  0
where
Q ˆ
  p qˆ  Q p
p
Q ˆ
  t qˆ  Qt
t
s ˆ
p
p
s ˆ
 t
t
• Where ˆp is the change # of consumers caused
by a change in p
• and qˆ  q( p, t , y (ˆ), ˆ) f (ˆ)
where
ˆ
V p  ˆp w(ˆ)v( p, t , y (ˆ) f (ˆ)   w( )v p ( p, t , y ( ),  ) f ( )d
0
• Simplifies to
ˆ
V p   w( )v p ( p, t , y ( ),  ) f ( )d
0
• From the individual’s utility max
v p ( p, t , y ( ),  )  v y ( )q( p, t , y ( ),  )
• Where vy(θ) MU income for type θ.
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 reveals
ˆ
 q( p, t , y( ),  ) f ( )d  Q
0
Similarly
ˆ
Vt    f ( )d
0
Vt  s
Substitution Reveals

 ˆ Q ˆ   ˆ Q ˆ 
ˆ
 p  MC S  q  p  t    t  p  t   0
s  
s 


 p  MC Q  ts  C  MC  Q  D
• Where
• s=Qp+Q/s Qy
• D= deficit
Solving Gives
 ZD
p  MC 
sS  qZ   QZ

S  qˆZ D
t
sS  qˆZ   QZ
• where
Q ˆ
ˆ
Z   p  t
S
Q
S  Q p  Qy
S
Interpretation
• Let
Q ˆ
ˆ
Z   p  t
S
• Marginal consumer’s demand (Roy’s Identity)
vp
qˆ 
vt
• To keep utility unchanged, the dt/dp=-qˆ
• Differentiate ˆ( p, t ) to get dt ˆp

dp ˆt
• Combining get
Q ˆ
ˆ
 p  t  0
S
Result 1
• If the marginal consumers are insensitive to
changes in the access fee or price, that is,
ˆp  ˆt  0
• then the welfare maximization is
– P=MC
– t=D/s
• Applies when no consumers are driven away
– i.e electricity
– Not telephone, cable
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