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 – p 1 * q +t, if 0<q≤q 1 – p 2 *(q- q 1 ) + p 1 * q 1 +t, if q 1 < q ≤q 2 – p 3 *(q- q 2 ) + p 2 *(q 2 - q 1 ) + p 1 * q 1 +t, if q 2 < q ≤q 3 – If p 1 >p 2 >p 3 => 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 Q MC 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 – Θ A = describes type A – Θ B = 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 v y (θ) MU income for type θ.

Income Let v y =-v t because the access fee is equivalent to a reduction in income Ignore income distribution and let – w(θ)=1/v y (θ) – Each consumer’s utility is weighted by the reciprocal of his MU of income Substituting into V p reveals

Similarly

Substitution Reveals Where s=Q p +Q/s Q y 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 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