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

By Walter Y. Oi Presented by Sarah Noll

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**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?

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**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.

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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

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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

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**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

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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

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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.

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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

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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

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**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

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**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

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**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

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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

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**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.

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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

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**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.

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**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

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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

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**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

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**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

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