Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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 sS qZ QZ S qˆZ D t sS 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