# A Disneyland Dilemma: Two-Part Tariffs for a mickey mouse monopoly

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A Disneyland Dilemma: Two-Part Tariffs for a mickey mouse monopoly
By Walter Y. Oi Presented by Sarah Noll

How Should Disney price?
Charge high lump sum admission fees and give the rides away? OR Let people into the amusement park for free and stick them with high monopolistic prices for the rides?

How should Disney price?
A discriminating two-part tariff globally maximizes monopoly profits by extracting all consumer surpluses. A truly discriminatory two-part tariff is difficult to implement and would most likely be illegal.

Option 1 Disneyland establishes a two-part tariff where the consumer must pay a lump sum admission fee of T dollars for the right to buy rides at a price of P per ride. Budget Equation: XP+Y=M-T [if X>0] Y=M [if X=0] M -is income Good Y’s price is set equal to one Maximizes Utility by U=U(X,Y) subject to this budget constrain

Option 1 Consumers demand for rides depends on the price per ride P, income M, and the lump sum admission tax T X=D(P, M-T) If there is only one consumer, or all consumers have identical utility functions and incomes, the optimal two-part tariff can easily be determined. Total profits: Π= XP+T-C(X) C(X) is the total cost function

Option 1 Π= XP + T – C(X) Differentiation with respect to T yields:
𝑑𝜋 𝑑𝑇 =𝑃 𝑑𝑋 𝑑𝑇 +1− 𝑐 ′ 𝑑𝑋 𝑑𝑇 =1− 𝑃− 𝑐 ′ 𝑑𝑋 𝑑𝑀 c’ is the marginal cost of producing an additional ride If Y is a normal good, a rise in T will increase profits There is a limit to the size of the lump sum tax An increase in T forces the consumer to move to lower indifference curves as the monopolist is extracting more of his consumer surplus

Option 1 At some critical tax T* the consumer would be better off to withdraw from the monopolist’s market and specialize his purchases to good Y T* is the consumer surplus enjoyed by the consumer Determined from a constant utility demand curve of : X=ψ(P) where utility is held constant at U0=U(0,M) The lower the price per ride P, the larger is the consumer surplus. The maximum lump sum tax T* that Disneyland can charge while keeping the consumer is larger when price P is lower: T*= 𝑃 ∞ ψ 𝑃 𝑑𝑃 𝑑𝑇∗ 𝑑𝑃 =−ψ 𝑃 =−𝑋

Option 1 In the case of identical consumers it benefits Disney to set T at its maximum value T* Profits can then be reduced to a function of only one variable, price per ride P Differentiating Profit with respect to P: 𝑑𝜋 𝑑𝑃 = 𝑋+𝑃 𝑑𝑋 𝑑𝑃 + 𝑑𝑇∗ 𝑑𝑃 −𝑐′( 𝑑𝑋 𝑑𝑃 ) 𝑃− 𝑐 ′ 𝑑𝑋 𝑑𝑃 =0 or 𝑃= 𝑐 ′ In equilibrium the price per ride P= MC T* is determined by taking the area under the constant utility demand curve ψ(P) above price P.

Option 1 In a market with many consumers with varying incomes and tastes a discriminating monopoly could establish an ideal tariff where: P=MC and is the same for all consumers Each consumer would be charged different lump sum admission tax that exhausts his entire consumer surplus This two-part tariff is discriminatory, but it yields Pareto optimality

Option 2 Option 1 was the best option for Disneyland, sadly (for Disney) it would be found to be illegal, the antitrust division would insist on uniform treatment of all consumers. Option 2 presents the legal, optimal, uniform two-part tariff where Disney has to charge the same lump sum admission tax T and price per ride P

Option 2 There are two consumers, their demand curves are ψ1 and ψ2
When P=MC, CS1=ABC and CS2=A’B’C Lump sum admission tax T cannot exceed the smaller of the CS No profits are realized by the sale of rides because P=MC

Option 2 Profits can be increased by raising P above MC
For a rise in P, there must be a fall in T, in order to retain consumers At price P, Consumer 1 is willing to pay an admission tax of no more than ADP The reduction in lump sum tax from ABC to ADP results in a net loss for Disney from the smaller consumer of DBE The larger consumer still provides Disney with a profit of DD’E’B As long as DD’E’B is larger than DBE Disney will receive a profit

Option 2.1 Setting Price below MC Income effects=0
Consumer 1 is willing to pay a tax of ADP for the right to buy X1*=PD rides This results in a loss of CEDP Part of the loss is offset by the higher tax, resulting in a loss of only BED Consumer 2 is willing to pay a tax of A’D’P’ The net profit from consumer 2 is E’BDD’ As long as E’BDD’> BED Disney will receive a profit

Option 2.1 Pricing below MC causes a loss in the sale of rides, but the loss is more than off set by the higher lump sum admissions tax

Option 2.2 A market of many consumers
Arriving at an optimum tariff in this situation is divided into two steps: Step 1: the monopolist tries to arrive at a constrained optimum tariff that maximizes profits subject to the constraint that all N consumers remain in the market Step 2: total profits is decomposed into profits from lump sum admission taxes and profits from the sale of rides, where marginal cost is assumed to be constant.

Step 1 For any price P, the monopolist could raise the lump sum tax to equal the smallest of N consumer surpluses Increasing profits Insuring that all N consumers remain in the market Total profit: 𝜋 𝑁 =𝑋𝑃+𝑁𝑇−𝐶 𝑋 X is the market demand for rides, T=T1* is the smallest of the N consumer surpluses, C(X) total cost function

Step 1 Optimum price for a market of N consumers is shown by:
𝑐 ′ =𝑃[1+( 1−𝑁𝑠1 𝐸 ) S1= x1/X, the market share demanded by the smallest consumer E is the “total” elasticity of demand for rides If the lump sum tax is raised, the smallest consumer would elect to do without the product.

Step 2 Profits from lump sum admission taxes, πA=nT
Profits from the sale of rides, πS=(P-c)X MC is assumed to be constant The elasticity of the number of consumers with respect to the lump sum tax is determined by the distribution of consumer surpluses

Step 2 The optimum and uniform two-part tariff that maximizes profits is attained when: 𝑑𝜋(𝑛) 𝑑𝑛 = 𝑑𝜋𝐴 𝑑𝑛 + 𝑑𝜋𝑠 𝑑𝑛 =0 This is attained by restricting the market to n’ consumers Downward sloping portion of the πA curve where a rise in T would raise profits from admissions

Applications of Two-Part Tariffs
The pricing policy used by IBM is a two-part tariff The lessee must pay a lump sum monthly rental of T dollars for the right to buy machine time IBM price structure includes a twist to the traditional two-part tariff Each lessee is entitled to demand up to X* hours at no additional charge If more than X* hours are demanded there is a price k per additional hour

IBM Profits from Consumer 1= (0AB)- (0CDB)
Profit from Consumer 2= (0AB)- (0CD’X*)+(D’E’F’G’) The first X* cause for a loss, but the last X2-X* hours contribute to IBMs profits

Questions?

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