Handout #3: Peak Load Pricing

Transcription

1 andout #3: Peak Load Pricing Consider a firm that exeriences two kinds of costs a caacity cost and a marginal cost ow should caacity be riced? This issue is alicable to a wide variety of industries, including ielines, airlines, telehone networks, construction, electricity, highways, and the internet The basic eak-load ricing roblem, ioneered by Marcel Boiteaux, considers two eriods The firm s rofits are given by π + β max {, } mc( + ) Prices eual to marginal costs are not sustainable, because a firm selling with rice eual to marginal cost would not earn a return on the caacity, and thus would lose money and go out of business A caacity charge is necessary The uestion of eak load ricing is where the caacity charge should be allocated Demands are ordinarily assumed indeendent, but this is neither a good assumtion nor a necessary one Our revious analysis suggests how the solution will change, however, and so I will stick with indeendent demands for simlicity Social welfare is W ( x) dx + ( x) dx β max {, } mc( + ) The Ramsey roblem is to maximize W subject to a rofit condition As always, write the Lagrangian Therefore, L W + l L ( ) β mc + λ + β ( ) ( ) mc Or, ( ) β λ mc λ + ε where is the characteristic function of the event Similarly,

2 ( ) β λ mc λ + ε Note as before that læ yields the monooly solution There are two otential tyes of solution Let the demand for good exceed the demand for good Either >, or the two are eual Case : > ( ) β mc λ + ε λ and ( ) mc λ + ε λ In case, with all of the caacity charge allocated to good, uantity for good still exceeds uantity for good Thus, the eak eriod for good is an extreme eak In contrast, case arises when assigning the caacity charge to good would reverse the eak assigning all of the caacity charge to good would make eriod the eak Case : The first order conditions become ineualities, of the form ( ( ) + ( ) mc) ( + λ) ( ) mc + λ β ( mc λ λ + ε β and ( mc λ λ + ε β These must solve at The rofit euation can be written ( - mc + ( mc b This euation shows that the caacity charge is shared across the two markets roortional to the inverse demand Priority Pricing The eak load roblem is essentially a cost allocation roblem It has an efficiency asect, in that ricing matters to relative demand, but that efficiency asect is incororated in a familiar way, using inverse elasticities The riority ricing roblem introduced by Robert Wilson has a suerficial similarity to the eak load roblem when caacity is reached, who should be rationed? Imlicitly, the eak load formulation imlies a sot market, so that each market is rationed efficiently In many circumstances, it is not ossible to use rices ex ost to ration the market For examle, absent smart aliances, it is difficult for homeowners to adjust electric demand in real time as rices vary homeowners aren t even informed about the abrut rice

3 changes Priority ricing is a means of contracting in advance when caacity, or demand, is stochastic At this time, the roblem of stochastic demand and riority ricing has not been adeuately addressed In articular, with stochastic demand, there is an issue of whether all customers are able to articiate in the ex ante riority market Consider a case of a continuum of consumers, each of whom desires one unit As will become clear, it doesn t matter if some consumers desire multile units each unit can be treated as demanded by a searate consumer Rank the consumers by their valuations for the good, so that the th consumer has a value ( for the good, and is downward sloing The uantity available is a random variable with distribution F Priority ricing is a charge schedule c which rovides a unit to a customer aying c( whenever realized suly is or greater It is a straightforward exercise to calculate the incentive comatible c schedule A customer of tye should choose to ay c( for the th sot in the riority list This leads to the incentive constraint: u( ( ( c( )( ) ( ( c( ˆ))( ˆ)) The enveloe theorem gives u ( ( ( ) It is a straightforward exercise to demonstrate that the first order condition is sufficient; see handout # Let F(), so that u() Then ( ( c( )( ) u( Thus, u ( ds ( ( ) ds ( ( ) ( f ( ds f ( c( ( ds E[ sot rice ( ( ] F( Revenues to the firm from the riority ricing are R c( ( ) d ( f ( ds d ( f ( d This is the revenue associated with a cometitive suly; a monoolist might have an incentive to withhold caacity to boost rices ow does a monoolist do so? Withholding of caacity has the roerty of changing the distribution of available suly, in a first order stochastic 3

4 dominant manner In articular, the monoolist can offer any distribution of caacity G, rovided G F What is the monoolist s solution? Rewrite R to obtain R ( g( d MR( ( G( ) d Provided marginal revenue MR is single-eaked, F if G if MR MR < That is, the monoolist cuts off the caacity at the monooly suly Matching Problems Priority ricing is a solution to a matching roblem, matching the high value buyers with caacity Many other roblems have this feature, that it is desirable to match high tyes with high tyes and low tyes with low tyes Such models have been used as models of marriage, emloyment, university admissions, incentive contracts, and other categories Wilson examines not just the continuum matching, in which each robability of service interrution is searately riced, but also finite grous Rather than offer a continuum of categories, consider offering just two high riority service and low riority service ow well does such a riority service do? The answer is remarkably well Consider first the linear demand case with a uniform distribution of outages Perfect matching gets a ayoff ( ( d ( d 3 No matching that is a random assignment roduces an exected value of ¼, a fact which is evident from ( d ( d ( ) d 4 Now consider two grous of eual size The high value grou has an average value of ¾, and is / served with robability d + d ¾ The low value grou has average value ¼ and is / served with robability /4 Thus, the exected value from two categories is 4

5 9 5 + Note that 5/6 is 75% of the way from ¼ to /3! That is, a single grou catures 75% of net value of a continuum of tyes! The linear/uniform distribution is secial; however, I show elsewhere that, rovided a common hazard rate assumtion is satisfied, two grous of distinguished by being above or below the mean generally catures 5% or more of the ossible gains over no riority ricing That is, even two classes is sufficient to cature a majority of the gains arising from riority ricing Wilson shows that the losses from finite classes are on the order of /n 5