Efficient Pricing using Non-linear Prices

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 • pΞ unit price • qΞ 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 b c P f e AC D MC Q 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

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

Welfare • w(θ) weight by marginal social value

Constrained Maximization • max L=V+λπ • by choosing p, t, λ • FOCs

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

where • Simplifies to • From the individual’s utility max • 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

Similarly

Substitution Reveals • Where • s=Qp+Q/s Qy • D= deficit

Solving Gives • where

Interpretation • Let • Marginal consumer’s demand (Roy’s Identity) • To keep utility unchanged, the dt/dp=-qˆ • Differentiate to get • Combining get

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

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

Efficient Pricing using Non-linear Prices