Nonlinear Pricing. Alexei A. Gaivoronski. Adrian S. Werner. Adrian S. Werner. Planning and economics of Tele and Info services.

Transcription

1 Alexei A. Gaivoronski 1

2 Roadmap Introduction and examples Demand profile Design of a nonlinear tariff for one product Multipart tariffs Multidimensional pricing Capacity pricing 2

3 Price discrimination First degree: different price for each purchased unit (charged price = max. willingness to pay for that unit) Second degree: same price schedule for all users but different prices for different amounts nonlinear pricing Third degree: different users pay different prices, but each user pays same amount for all units 3

4 Nonlinear Prices and Telecommunications political decisions (deregulation) and technological advances monopoly became oligopoly competition often former monopolist still enjoys considerable market power 4

5 Nonlinear Prices and Telecommunications firms have high capital costs but low operation and administration costs big differences in customer demand for subscriptions and for connections, also due to emergence of new services price differentiation useful 5

6 Example: SMARTCALL's tariff for internet access Flat rate 3 different quality types: Basis, Pluss, Super each with different download and upload speed and accordingly different prices 6

7 Terms for some tariff types Nonlinear prices: any case where used tariff is not proportional to purchased quantity Block-declining tariff: marginal prices of successive units decline in steps Two-part tariff: initial fixed fee + constant price for each following purchased unit 7

8 Total Price Different tariff types Linear Block-declining Two-part Fixed fee Units 8

9 Example: three-part tariff in telecommunications monthly fee one hour of free calls additional usage: fixed price per minute Price Usage 9

10 Example: NetCom's tariffs for mobile telephony Several types of contracts for calls for different target groups: same installation fees different fixed monthly fees same initial fee for a call different charges per minute 10

11 Example: NetCom's tariffs for mobile telephony Abonnements- og samtalepriser for smalltalk, TALK, activetalk, youngtalk Abonnement Tilknytningsavgift Månedsavgift Samtale oppstartsavgift Kr pr. minutt SmallTALK 199,- 69,- 0,59 1,99 TALK 199,- 119,- 0,59 1,49 ActiveTALK 199,- 199,- 0,59 0,99 YoungTALK 199,- 0,- (1) 0,59 2,49 (1) Monthly fee of 200,- ; can make calls up to an upper limit of 200,- 11

12 Example: NetCom's tariffs for mobile telephony The first three are two-part tariffs with two dimensions: total call duration in a month number of calls in a month The last tariff is a fixed fee tariff 12

13 Quantity discounts Can be offered in two ways: Smaller prices for additional units (after reaching a threshold value) Smaller price for all units if the purchase size is large enough 13

14 Example: telephone tariff rates depend on - distance (local calls, distant calls) -time of day (day (peak), evenings, nights and weekends (off peak)) Further discounts in form of volume discounts, depending on monthly billings: $ 50 - $ 100: 2 %, over $ 100: 5 % price differentiation on several levels 14

15 Feature of a nonlinear tariff Average price per delivered unit depends on the total size of the purchase Tariff applies to all users: although the customer population is heterogeneous, do all users face the same conditions 15

16 Practical use Necessary for efficiency - costs may vary with size of an order Recovers administrative and capital costs in a regulated monopoly Useful strategy in competitive markets Enables firm with monopoly power to increase its profits 16

17 Advantages of nonlinear pricing Heterogeneity among customers different valuation of successive increments in quality or quantity market segmentation allows product differentiation Efficient way for regulated monopoly to recover costs But: benefits large customers via quantity discounts may disadvantage small customers compared to linear price 17

18 Increasing use in recent years Deregulation of major industries product differentiation for survival (high capital cost and competitive pressures) Standardization of products or service in mass markets advantages to differentiating tariffs Monitoring of demand behaviour easier (use of computers) 18

19 Problems Adverse distributional effects (different terms offered to different customers based on observable distinctions) Allocative and production inefficiencies Quality degradation only to enhance product differentiation Temporal price discrimination 19

20 Preconditions for implementation of nonlinear prices 1. Seller has monopoly power 2. Seller can prevent resale 3. Seller can sort customers 3.1 monitor their purchases 3.2 has disaggregated demand data 20

21 1 Seller has monopoly power Strongly competitive markets: prices close to direct costs exclude nonlinear pricing BUT: few markets are really perfectly competitive Monopolistically competitive markets: some product differentiation enough power for each firm for prices above direct costs Oligopolistically competitive markets: number of firms sufficiently small enables positive profits 21

22 2 Seller can prevent resale Otherwise: customers may profit from arbitrage Exclude purchases from multiple accounts by use of: - decreasing price schedule or - subadditive tariff: P ( q1 + q2) P( q1) + P( q2) 22

23 3.1 Seller can monitor customers' purchases Identifying customers: What is a customer? Measuring of single purchases Dimensions (''units'') of the tariff Quality dimensions Method of billing 23

24 3.2 Disaggregated demand data Essential: estimated / recorded data of purchases classified after - paid price - purchased quantity Measuring customers' heterogeneity Often identification of few segments sufficient; not too fine differentiation (complicates tariff and imposes costs for monitoring and billing) 24

25 How a nonlinear tariff works General insight: product differentiation and price discrimination important for efficiency in many industries Seller offers a ''menu'' of quantities/qualities with according prices Each customer chooses a quality he prefers and pays corresponding price: decides how many additional units of the good he still will buy Several quality levels: customer decides how ''high'' he will climb on scale of offered qualities 25

26 How a nonlinear tariff works Discounts stimulate demand In telecommunication also enhanced network effects: discounts encourage new users to enter increased utility also for existing users and increased demand (calls) even if new users would not generate calls: can be called by other users 26

27 Analysis of demand data Uniform prices: demand function predicts quantity D(p) that will be sold at price p Nonlinear prices: demand profile predicts demand for q-th unit at the price p(q) 27

28 Difficulties for a practical determination of demand profile Reliable data on individual demands rarely available Empirical estimation of demand behaviour depends on level of aggregation Completely aggregated demand data mask customer heterogeneity insufficient Adequate: data about distribution of purchase sizes at each price and possibility of demand prediction at a price change 28

29 Interpretation of demand profile Describes distribution of purchase sizes in response to each uniform price p (fix price p, let size q increase) measures (decreasing) number of customers purchasing at least q units N( p, q) = #{ t D( p, t) q} 29

30 Alternative interpretation of demand profile Describes distribution of customers' willingness to pay price p for q-th unit (fix size q, let price p increase) measures (decreasing) number of customers paying at least price p N( p, q) = #{ t v( q, t) p} 30

31 Displaying demand profile: as spreadsheet Purchase sizes qk Prices p j Demand profile N ( p, q j k ) 31

32 Estimating a demand profile Direct measurement: assumes that firm has accumulated sufficient data in the past (e.g. by using several uniform prices and recording the according data) Indirect measurement: deriving from estimations on customers' demand or benefit functions 32

33 Estimating a demand profile: direct measurement arrange different prices in increasing sequence arrange different increment sizes in increasing sequence q k demand data give number (fraction) of potential users who bought increment size p when price was q Demand profile: N ( p, q ) j k = l k j n jl n jk p j k 33

34 Estimating a demand profile: indirect measurement Derived from estimation of demand or benefit functions Assume: each customer is described by list t of parameters measure benefit from quantity q by function U(q,t) Derive marginal benefit function v(q,t) Set p = v(q,t) and solve for quantity q Convert this into predicted demand function D(p,t) of the user type t in response to the price p 34

35 Estimating a demand profile: indirect measurement Need distribution of customer types t in population Often estimate in form of density function f(t) Estimate type and parameters of f(t) using empirical analyses etc. Define set T(p,q) of types for which v( q, t) p or D( p, t) q N( p, q) = f ( t) dt T ( p, q) 35

36 Estimating a demand profile: indirect measurement, example List of parameters: Benefit function: U t = q, t) Marginal benefit function: ( t 1, t2 ) ( = t q t q p = v( q, t) = t1 t2q 1 D( p, t) = ( t1 t 2 p) Set T ( p, q) of types: T ( p, q) = { t t t2q 1 p } 36

37 Design of a nonlinear tariff for one product, key ideas Interpret nonlinear tariff as imposing different charge for each successive increment in purchase size Remember: customer who buys q-th increment must also buy all increments before Demand for q-th increment is demand for all purchase sizes at least as large as q 37

38 Design of a nonlinear tariff for one product, notations Total price for q-th increment: P(q) Unit price for q-th increment: p(q) Increment size: δ (''q-th increment'' = last increment to reach purchase size of q units) 1 p( q) = ( P( q) P( q δ )) δ Cost per unit: c Cost for increment of size δ: cδ 38

39 Design of a nonlinear tariff for one product Price schedule: list of prices p(q) for all possible purchase sizes q Revenue expected from price schedule p(q): Rev = N( p( q), q)[ p( q) q c] δ Sufficient: find prices that maximise profit contribution from each market segment: R( p( q), q) = N( p( q), q)[ p( q) c] 39

40 Example, design of nonlinear tariff: demand profile p q (units) D(p) $ $ $ $

41 Example, design of nonlinear tariff: max. profit contribution p q (units) D(p) $ $ $ $

42 Example, design of nonlinear tariff: Purchase size p(q) (per unit) price schedule $4 $4 $3 $3 $2 P(q) $4 $8 $11 $14 $16 Profit contrib. $195 $150 $90 $40 $5 42

43 Example, design of nonlinear tariff: comparison to uniform price Optimal uniform price: $4 earned revenue: $450 sold units: 150 Optimal nonlinear tariff earned revenue: $480 sold units: 185 average price: $

44 Example, continued Illustrates advantage of nonlinear pricing: price breaks offered for large purchase sizes stimulates demand that would be lost otherwise firm can obtain higher profit without disadvantaging customers 44

45 Formulation as optimization problem Profit = N( p( q), q)[ p( q) c] δ q s. t. N( p( q), q) = n( p( x), x) x q N( p( q), q) 0 N( p( q), q) = 0, for p( q) M p or (bounded capital and demand) N( p( q ), q ) < N( p( q ), q ) for q p( q 1 ) 1 1 p( q 2 ) for q 1 2 > q q M > q 2 max p( q) q 45

46 Fixed costs and fees Minor importance in nonlinear pricing: often recovered by fixed fee appended to tariff, extra access fee or minimum purchase requirement When customers don't buy good when negative benefit: optimal nonlinear tariff imposes no fixed fee if firm has no fixed cost for serving users Nonlinear pricing emphasizes design of a schedule of marginal prices 46

47 Multipart tariffs Tariffs that are piecewise linear over wide ranges Simplest: two-part = fixed fee + uniform price n-part tariff: fixed fee + (n-1) different marginal prices in volume bands In case of decreasing marginal prices: same net effect as menu of (n-1) two-part tariffs Example: pricing of leased machines according to different usage rates 47

48 Multipart tariff as menu of (n-1) two-part tariffs Menu of 3 twopart tariffs 4-part tariff 48

49 Design of a multipart tariff Similarly to already described procedure In demand profile, take into account the wider range of increments for each marginal price volume bands q i, r ] [ i Assume that i-th marginal price applies to the i-th volume band [ q i, ri ], i.e. applies to each of the q-th units with q q i r i 49

50 Design of a multipart tariff Denote then demand in terms of volume bands: N ( p, i) = N( p, q) δ qi q r i Find marginal prices that maximize profit p contribution in single volume i bands i: N ( p, i)( p c) i Problem: must take account of users' demand behaviour at boundaries between volume bands, may cause suboptimal price schedules i 50

51 Multidimensional pricing Until now: single product, users chose only number of purchased units Extension: product with several quality attributes (e.g. speed, place of use, reliability) user chooses combination of these attributes for each unit, pays for selected combination (e.g. fast but not very reliable connection) 51

52 Multidimensional pricing Multidimensional vs. multiple products: Multidimensional: can freely assign different combination of quality attributes to each unit Multiple products: offering several products, each with fixed configuration of quality attributes 52

53 Multidimensional pricing Example: Mail service. Prices depend on delivery speed (service priority), size and destination of packages Assume quality levels as cardinal (at same price higher quality is preferred) All users rank quality levels in the same way 53

54 Multidimensional pricing Description of users' purchases Assignment form Profile form Load-duration curve other suited forms 54

55 Multidimensional pricing, assignment form Usual intuitive accounting for purchase of multiple products: how many items assigned to each of the possible combinations of quality magnitudes Example: how many connections of each pair (slow, reliable), (slow, unreliable), (fast, reliable), (fast, unreliable) 55

56 Multidimensional pricing, profile form Takes into account assumed quality cardinality Interpret higher quality level as upgrade: user must buy lowest level and all upgrades until desired quality level Keeps account of increments of the various qualities that the users selected Suited for design of a multidimensional tariff, basically similarly as before 56

57 Multidimensional pricing, load-duration curve Industries subject to peak loads (communications, electric power, transportation etc.) Demand changes on short time scale, but capacity and potential supply relatively stable Curve specifies for each load level x number of hours H(x) when user load exceeds this level x 57

58 Multidimensional pricing, load-duration curve Alternative for measuring a load-duration curve: Arrange the hourly intervals (e.g. over a year) so that demand declines along the sequence: first hours with most demand etc. Describe then user demand by function L(h): specifies demand in h-th ranked hour of this year 58

59 Generalization of NetCom example Tariff consists of: M monthly fee p fixed fee per call (establishment fee) C minute charge p Total charge per month (N calls, total duration D): P M = p M + N p E + D p C E p 59

60 Generalization of NetCom example Heterogeneous customer population with different number of calls N and duration D M E C offering different p, p, p This tariff is a two-part tariff with two dimensions: duration of connections frequency of connections (amount per month) 60

61 Generalization of NetCom example Total charge Call amount N M p Duration D 61

62 Generalization of NetCom example Further possible dimensions: time of day destination (discrete entities): other providers' customers mobile or fixed network abroad 62

63 Capacity pricing Telecommunications, power industry, service provision etc.: times with big capacity demand and with little demand users' demands vary over time but capacity relatively stable products / service difficult or impossible to store 63

64 Capacity pricing Cannot equate a customer's usage and its capacity requirement (offpeak: idle capacity) Tariff must account separately for costs of usage and of capacity Aim: design tariff to stimulate customers to a relatively regular usage (cut down in peak times and instead reduce idle capacity at offpeak times) 64

65 Capacity pricing Provider may serve Base loads with equipment with high acquisition and low operation costs Peak loads with equipment with low acquisition and high operation costs assigns lower idleness costs in offpeak periods marginal operation costs vary with demand 65

66 Capacity pricing Variety of pricing schemes: demand contingent spot pricing includes real-time or time-of-use-tariffs tariffs inciting users to mimic the firm's cost structure uncontingent uniform service price together with rationing of customers by a selection scheme (e.g. random, FIFO/queues, priorities ) 66

67 Capacity pricing: demand-contingent spot pricing Uniform price per service unit varies continually in response to changing demand Aim: keep demand within supply limits Adapted to firm's short-run costs Variants: peakload pricing real-time pricing 67

68 Capacity pricing: long-run tariffs Take longer view on firm's cost structure Different tariffs, for example: user pays a ''demand charge'' based on his annual peak load and a price for usage dependent on the time of use load factor tariff: discount for users with high load factor 68

69 Capacity pricing In analyses often assumed that firm must meet customers' load individually e.g. when demand is synchronized and peaks occur at same time In practice often asynchronous demands, idle capacity can be used to serve other customers need less total capacity than sum of peak loads 69

70 Capacity pricing, tariff design Make use of the load-duration curve for demand description Design of an optimal tariff comprises specification of: optimal capacity configuration (peak / offpeak equipment) to serve a user's load optimal price schedule 70

71 Capacity pricing, tariff design Optimal price schedule - different systems: time-of-use tariff: nonlinear tariff for demand charge plus additional nonlinear tariff for load charge Wright tariff: demand charge based on maximum load plus nonlinear tariff for load charge 71

72 Literature R. Wilson:. Oxford University Press, Y. Kanemoto: Price and Quantity Competition among Heterogeneous Suppliers with Two-Part Pricing: Applications to Clubs, Local Public Goods, Networks, and Growth Controls. Regional Science and Urban Economics 30 (2000),

73 Literature, continued H.R. Varian: Price Discrimination. In: R. Schmalensee and R.D. Willig (eds.), Handbook of Industrial Organization, Volume I. Elsevier, N. Curien, B. Jullien and P. Rey: Pricing regulation under bypass competition. RAND Journal of Economics 29 (1998), No.2,