Two-Part Tariff with One Customer Type

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This Week’s TopicsReview Class Concepts Simple Pricing Price Customization by Customer Type Two-Part Tariff Simple With Two Demand Curves Review Homework Practice Problems

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Class Concepts – Simple PricingSteps for Solving a Simple Pricing Problem 1) Identify your total cost function (TC) and inverse demand function Find the inverse demand function by finding the demand function and solving for P Revenue = P*Q = (inverse demand function)*Q 2) Solve for marginal cost (MC) and marginal revenue (MR) by taking derivatives of Revenue and Total Cost functions 3) Set MC = MR, and solve for Q* 4) Insert Q* into your inverse demand function and solve for P* 5) Check that you are profitable: Profit = Total Revenue – Total Cost π = P* x Q* – TC ≥ 0 ** Be sure to do the last step! Optimal Simple Pricing helps you figure out the most profit you can make from selling a good, given available demand. Is that optimal profit high enough to cover your fixed costs?

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Class Concepts – Advanced PricingAdvanced Pricing Use price customization to sell the same good at different prices per unit Ideally, we would like to sell each unit at exactly the maximum willingness to pay (WTP) of each customer. Since we don’t have enough information to do this, we use the following types of customization: Types of Price Customization Price Customization by Group Two-Part Tariff Price Customization by Quantity Versioning Bundling Intertemporal Price Customization

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Class Concepts – Price Customization by Customer TypePrice Customization by Customer Type Sell your product to different customer types at different prices How to use Price Customization by Customer Type If, you have two identifiable customer types with two different demand curves then: Use Simple Pricing, Steps 1-3 for each customer type If you have two unknowns in two equations, use algebra to solve for Q*1 and Q*2 Follow Simple Pricing, Steps 4-5 for each customer type Other Considerations Arbitrage?: Since you are charging different prices to different customers, think about the potential problems with arbitrage between customers in different groups Better than simple pricing?: Always

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Class Concepts – Price Customization by Customer TypeKnowledge Check You are selling Cal season football tickets to students and alumni. Student demand for tickets is QS = 15,000 – 100PS, and alumni demand is QA = 30,000 – 50PA. The total cost to the university is: TC = 0.001Q2 + 10Q + 2,000,000. At what price should the university sell tickets? Are there any other considerations? Solution Simple Pricing Steps 1 to 3 Step 1: Find Inverse Demand and Cost Functions PS = 150 – 0.01QS PA = 600 – 0.02QA RS = 150QS – 0.01QS2 RA = 600QA – 0.02QA2 Step 2: Solve for MR and MC by taking derivatives of Revenue and Cost Functions MRS = 150 – 0.02QS MRA = 600 – 0.04QA MC = 0.002Q + 10  MC = 0.002QA +0.002QS + 10 Step 3: Set MR = MC 0.002QA +0.002QS + 10 = 150 – 0.02QS 0.002QA +0.002QS + 10 = 600 – 0.04QA

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Class Concepts – Price Customization by Customer TypeSolution Solve for Unknowns 0.002QA +0.002QS + 10 = 150 – 0.02QS  Simplify 0.002QA +0.022QS = 140  Equation 1 0.002QA +0.002QS + 10 = 600 – 0.04QA  Simplify 0.042QA +0.002QS = 590  Multiply by 11 0.462QA +0.022QS = 6490  Subtract Equation 1 from this (0.462 – 0.002) QA +(0.022-0.022)QS = 6490-140  Simplify 0.46QA = 5350  Simplify QA* = 13,804.35  Substitute into Equation 1 0.002(13,804.35) + 0.022QS = 140 27.61 + 0.022QS = 140 0.022QS = 112.39 QS* = 5,108.70

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Class Concepts – Price Customization by Customer TypeSolution Step 4: Solve for P* QS* = 5,108.70 QA* = 13,804.35 PS* = 150 – 0.01QS PA* = 600 – 0.02QA PS* = 150 – 0.01(5,108.70) PA* = 600 – 0.02(13,804.35) PS* = 150 – 51.09 PA* = 600 – 276.09 PS* = 98.91 PA* = 323.91 Step 5: Check for Profitability π = (PS* x QS*) + (PA* x QA*) – TC π = (PS* x QS*) + (PA* x QA*) – [0.001(QS* + QA*)2 + 10(QS* + QA*) +2,000,000] π = (98.91 x 5,108.70) + (323.91 x 13,804.35) – [0.001(5,108.70 + 13,804.35)2 + 10(5,108.70 + 13,804.35) + 2,000,000] π = $505,316.64 + $4,471,408.32 – [$357,703.21 + $189,130.43 + $2,000,000] π = $4,976,724.95 – [$2,546,833.65] π = $2,429,891.30 ≥ 0

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Class Concepts – Two-Part Tariff with One Customer TypeTwo-Part Tariff with One Customer Type Charge a “Fixed Fee” for the right to buy units Then sell each unit at a set “Unit Price” How to use the Two-Part Tariff with One Customer Type 1) Set the “Unit Price” equal to the Marginal Cost (MC) of producing a unit 2) Set the “Fixed Fee” equal to the Consumer Surplus (CS) when selling at Marginal Cost (MC)  Find the area of the triangle If demand curve is linear: Set the Inverse Demand Curve = MC  Solve for QMC Area of CS = ½*(Constant from Inverse Demand Curve – MC)* QMC Requirements for using Two-Part Tariff This pricing method works best when all buyers have the same demand, and may become less valuable (and more complicated) with more demand types Arbitrage?: No, you are not charging different prices to different customers Better than simple pricing?: Always with one type of customer

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Class Concepts – Two-Part Tariff with One Customer TypeKnowledge Check You are running an amusement park, where all customer demand is Q = 30 – 0.4P and the marginal cost of each ride is 5. You would like to charge an entry fee to the amusement park, and charge for tickets for each ride. How much should your entry fee be? How much should a 1-ride ticket cost? Solution Each ride ticket should cost $5. Solve for the Inverse Demand Curve: P = 75 – 2.5Q Inverse Demand Curve at P = 5 5 = 75 – 2.5QMC 2.5QMC = 70 QMC = 28 Consumer Surplus = ½ * (75 – 5) * (28) Consumer Surplus = 980… hmmm, this seems unrealistically high

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Class Concepts – Two-Part Tariff with Two Customer TypesTwo-Part Tariff with Two Customer Types Charge a “Fixed Fee” for the right to buy units Then sell each unit at a set “Unit Price” How to use the Two-Part Tariff with Two Customer Types You Need: Number of High Type, Low Type Customers (NHT, NLT), these will be numbers Demand for High Type, Low Type Customers, these will be functions Total Cost (may be based on number of customers and number of units sold), this will be a function of the number of customers & number of units Follow the steps on the following slide:

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Class Concepts – Two-Part Tariff with Two Customer TypesHow to use the Two-Part Tariff with Two Customer Types (cont.) Find High and Low Type Quantity Purchased functions (in terms of price, P) High Type Quantity = NHT*(High Type Demand) Low Type Quantity = NLT*(High Type Demand) Revenue (from Fixed Fee & Unit Price) and Costs in terms of an optimal price, P Unit Revenue = Q x P = [High Type Quantity + Low Type Quantity] x P Fixed Fee Revenue is based on the consumer surplus of the Low Type Customer Fixed Fee = ½*(Constant from Inverse Low Type Demand Curve – P)*(Low Type Demand) Fixed Fee Revenue = Fixed Fee* (NLT + NHT) Costs Costs = Cost for NLT & NHT + Cost for Quantity Costs = Cost for NLT & NHT + MC*(High Type Quantity + Low Type Quantity) Find MR and MC by Taking Derivative with respect to P Set MR = MC Solve for P Requirements for using Two-Part Tariff with Two Customer Types One customer type must demand more than the other customer type at every price Always check to see if you could make more by just selling to the high type!

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Class Concepts – Two-Part Tariff with Two Customer TypesKnowledge Check Number of High Type customers = 50, Number of Low Type customers = 30 Low Type Demand: P = 12 – 4Q  Q = 4 – 0.25P High Type Demand: P = 16 – 2Q  Q = 8 – 0.5P TC = 4N + 3Q (Where N is the total number of customers) What is the optimal price? Solution High Type Quantity = NHT*(High Type Demand) = 50 * (8 – 0.5P) High Type Quantity = 400 – 25P Low Type Quantity = NLT*(High Type Demand) = 30 * (4 – 0.25P) Low Type Quantity =120 – 7.5P

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Class Concepts – Two-Part Tariff with Two Customer TypesSolution High Type Quantity = 400 – 25P Low Type Quantity =120 – 7.5P Unit Revenue Unit Revenue = [High Type Quantity + Low Type Quantity] x P Unit Revenue = [(400 – 25P) + (120 – 7.5P)] x P = [520 – 32.5P]P Unit Revenue = 520P – 32.5P2Fixed Fee Revenue Fixed Fee = ½ * (12 – P)*(4 – 0.25P) = (6 – 0.5P)*(4 – 0.25P) = 24 – 1.5P – 2P + .125P2 Fixed Fee = 0.125P2 – 3.5P + 24 Fixed Fee Revenue = Fixed Fee* (NLT + NHT) Fixed Fee Revenue = (0.125P2 – 3.5P + 24) * (30 + 50) = (0.125P2 – 3.5P + 24) * (80) Fixed Fee Revenue = 10P2 – 280P + 1920Costs Costs = Cost for NLT & NHT + Cost for Quantity Costs = 4N + 3Q = 4(NLT + NHT) + 3*(High Type Quantity + Low Type Quantity) Costs = 4(30 + 50) + 3*[(400 – 25P) + (120 – 7.5P)] = 4*80 + 3*[520 – 32.5P] Costs = 320 + 1560 – 97.5P = 1880 – 97.5P

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Class Concepts – Two-Part Tariff with Two Customer TypesSolution Revenue = Unit Revenue + Fixed Fee Revenue Revenue = (520P – 32.5P2) + (10P2 – 280P + 1920) Costs = 1880 – 97.5P Solve for MR and MC MR = 520 - 65P + 20P – 280 = 240 – 45P MC = -97.5 Set MR = MC 240 – 45P = -97.5 45P = 337.5 P = 7.5 Check Profitability π = Revenue – Cost = (520*7.5 – 32.5*7.52)+(10*7.52 – 280*7.5 + 1920) – (1880 – 97.5*7.5) π = (3,900 – 1828.125) + (562.5 – 2,100 + 1,920) – (1,880 – 731.25) π = 2,071.875 + 382.5 – 1,148.75 π = 1,305.625

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Class Concepts – Two-Part Tariff with Two Customer TypesSolution What happens if we sell only to the high type? 50 Customers High Type Demand: P = 16 – 2Q  Q = 8 – 0.5P TC = 4N + 3Q = 4*50 + 3Q = 200 + 3Q Unit Price = MC = 3 Find Quantity Purchased when Price = MC 3 = 16 – 2Q  2Q = 13 Q = 6.5 Consumer Surplus = ½ * (16 – 3) * (6.5) = ½ * 13 * 6.5 = $42.25 Profit = Revenue – Cost = 50*[$42.25 + ($3*6.5)] – (200 + 3*6.5) Profit = 2,112.50 – 200 = $1,912.50 Since this profit is higher, choose to sell only to the high type customers

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Last Updated: 8th March 2018

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