minutes of service used. The firm has been changing a single price

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1 John Riley Background material for UCLA Case Study 17 April 2016 Introduction to indirect price discrimination 1 A firm with constant marginal cost c has two classes of customers with demand price functions p( 1, q) 1 q and p( 2, q) 2 q There are n 1 type 1 customers and n 2 type 2 customers The quantity q is the weekly minutes of service used The firm has been changing a single price p to maximize its profit You have been invited to give a presentation on the advantages of selling plans with different usage ceilings Plan 1 offers q 1 units per week for a weekly cost of r 1 Plan 2 offer a larger quantity q 2 for a weekly cost of r 2 Preferences over outcomes Regardless of the method that a firm uses to extract revenue from a customer, the two things that the customer ultimately cares about are the quantity consumed and how much this costs We call this the outcome for the customer ( qr, ) Given such an outcome the consumer s utility is u(, q, r) B(, q) r High demanders have a higher marginal benefit p(, q) p(, q) 2 1 Participation constraints A customer will only purchase a plan if the total benefit of the plan exceeds the cost Suppose that the firm offers a single plan ( qr, ) If the firm wants both types of customer to participate then it must be the case that B(, q) r 0 1 and 2 B(, q) r 0 The high demander s benefit from q minutes per week is greater, ie 1 This is often called Second Degree Price Discrimination However the early literature is not at all clear on the meaning of second degree This name was popularized and possibly coined my McAfee

2 B(, q) B(, q) 2 1 Therefore if the goal is to sell to both types of customer, the only binding participation constraint is B(, q) r 0 1 If the firm wants to sell only to the high demanders then the participation constraint is B(, q) r 0 2 To begin, we consider the latter case For any choice of minutes q the firm can raise the weekly fee until the participation constraint is (almost) binding Consider any plan below ( qr ˆ, ˆ) The benefit to the high demander is the area under his demand curve This is shown The firm extracts all the surplus by charging r B( ˆ 2, q) The total cost is cq ˆ Thus the profit is the diagonally shaded region Figure 11: Demand price and total benefit From Figure 11 it is clear that the firm can do even better by increasing the minutes to * q where the demand price (marginal benefit) is equal to marginal cost 2

3 Two plans If the number of low demand customers is sufficiently large, it will not be profit maximizing to sell only to high demanders Instead the firm offers two plans ( q1, r1) and ( q, r ) 2 2 where q q 2 1 Participation constraint: As argued above we need only consider the participation constraint of the low demand customer B(, q ) r Who purchase the high minute plan? The low demanders will purchase the plan if B(, q ) r B(, q ) r, That is, the difference in benefit exceeds the difference in plan cost B(, q ) B(, q ) r r The increased benefit is depicted below The high demanders have higher demand price functions and thus value a bigger plan more Therefore B(, q ) B(, q ) r r implies that B( 2, q2) B( 2, q1 ) r2 r We therefore have our first important conclusion Conclusion 1: If the two types of customers purchase different plans, the low types must purchase the low unit plan and the high demanders must purchase the high unit plan 3

4 Incentive constraints The low demanders must prefer the small plan Equivalently, B(, q ) r B(, q ) r, B(, q ) B(, q ) r r For the high demanders the extra benefit from the high plan must be bigger than the extra cost Thus the two incentive constraints and the participation constraint are as follows: B(, q ) B(, q ) r r B(, q ) B(, q ) incentive constraints B(, q ) r 0 participation constraint Conclusion 2: To maximize profit the participation constraint must be binding To see why this is true, note that we can increase both r 1 and r 2 without affecting the incentive constraints As we do so revenue rises Therefore we continue to do so until the participation constraint is binding Conclusion 3: To maximize profit the high demander must be indifferent between the two plans To understand this, suppose that we increase r 2 until the incentive constraint for the high demander is binding This increases revenue It also relaxes the other incentive constraint We can now use these results to solve for the profit-maximizing plan From conclusion 1 we know that q2 q1 From conclusion 2 we know that all the surplus is extracted from the low demander So the profit is the dotted area in the left figure If the high demander purchases the low plan his benefit is greater His net gain is the diagonally shaded area 4

5 Figure 13: Profit on each plan Now look at the right figure The total surplus is the sum of the diagonally shaded and dotted areas The diagonally shaded region cannot be extracted because that would reduce the high demanders benefit below his payoff from purchasing the low minute plan Therefore the maximized total profit in each plan are the dotted regions Figure 14: Total profit on each plan 5

6 Using the same argument as when there was only one plan, more profit can be extracted from high demanders by offering a bigger pan and then extracting additional surplus This is shown in Figure 14 The remaining step is to consider the impact of increasing the number of minutes in the low plan The left figure below shows the extra profit made on low demanders (the dotted area) and the extra surplus high demanders receive if they switch to the low plan with its increased minutes (the diagonally shaded area) Since the high demanders like the new low plan more, the firm cannot extract as much from them in the high plan Let the increased profit be on each low type be 1 and let the increased surplus to high types if they switch be S 2 With n 1 low types and n 2 high types, the net change in profit is n n s Figure 1-5: Marginal benefit and marginal cost of increasing minutes in plan 1 p(, q ) p(, q ) 1 p( 1, q1 ) c

7 Note that as ˆq 1 gets larger, the extra profit from further increases in minutes, 1 declines towards zero So it is never profit-maximizing to increase demand price (marginal benefit) is equal to marginal cost q 1 to the output where the For efficiency, marginal benefit and marginal cost must be equated This is the case for high demanders However low demanders are undersupplied The net change in profit can be rewritten as follows: n n n n s n ( s ) n [ ( p(, q ) c) ( p(, q ) p(, q ))] q) n2 n2 Therefore, as the proportion of low demanders falls, the profits made from customers with low demands become relatively less important so the profit-maximizing q 1 falls If the ratio n / n gets sufficiently small, it is best to squeeze them out of the market completely Then the firm can extract all the surplus from the high demanders 1 2 7