# 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

Overview Three sections Section 1 Section 2 Section 3
Discriminatory two-part tariff Section 2 Uniform two-part tariff Section 3 Applications of two-part tariff Volume Surcharge for IBM Marginal price discounts (like those used in public utilities)

So, what is a two-part tariff
A two-part tariff is one in which the consumer must pay a lump sum fee for the right to buy a product Examples: Disneyland: entrance fee + price per ride Bar: cover charge + price per drink Rate structures of utility companies Against: Standard pricing for the monopolist: single price where MC=MR

A discriminating two-part tariff
AKA first degree price discrimination To globally maximize profits by extracting ALL consumer surpluses Assumptions for Disneyland model (apparently Mickey is a greedy monopolist based on this article) Consumers are to derive no utility from going to the park itself Utility derived from consuming a flow of rides X per unit time period Problems: difficult and ILLEGAL

Mathematically… Consumer’s budget equation XP+Y=M-T [if X>O]
Y=M [if X=O] M=income Y= good; price set equal to 1

Utility = U=U(X,Y) If the consumer does not want to pay the tariff, T, they will have the following utility: U(X, Y,) = U(O, M,) **without entering the park, the consumer cannot consume the good, X ** Tax, T, allows Disneyland to extract the consumer’s income in order to turn it into Mickey Mouse’s profits (Mickey loves money)

Now, if the consumer has a demand for rides, they will have the following demand function: X=D(P, M-T) P is price per ride M is income T is admission This can be interpreted as follows: and increase in the lump sum tax will decrease consumption of the rides, x with respect to a consumer’s given level of income.

Not a fan of math… Total profits are given by:
п=XP+T-C(X) where C(X) is the total cost function And differentiating with respect to T we get: Interpretation: Assume that Y is a normal good. An increase in T will increase profits Forces consumer to move to lower indifference curve because they cannot consume more rides due to a decrease in their budget However, the monopolist extracts more of the consumer’s budget by employing this pricing strategy

There is a point T* where the consumer will not purchase this entrance fee and use their income to purchase other goods Therefore T* is the consumer’s surplus enjoyed by the consumer

To show the following: The larger the T* Disney can charge while keeping their consumers, the lower the price per ride will be This results in a larger consumer surplus The equation above represents the area under the constant utility demand curve And this step allows for us to reduce the following profit function to one variable: price per ride

Here, we differentiate profits with respect to price
Description of what has happened: Changes in the optimal tax, T* due to a change in P from this equation dT*/dP allows for P to satisfy the necessary condition Result: P=MC

After all that math There is a different lump sum tax, or entrance fee in the case of Disneyland, charged per consumer Price per ride, which is equal to marginal cost is the same across all consumers Conclusion: Customers who have larger surpluses for rides will be charged higher entrance fees or purchase privilege taxes Ie: admission fees transfer the consumer’s income thus putting them on a lower indifference curve Side note: yields pareto optimality In technical terms: marginal rate of substitution in consumption is equated to that in production Ie transfer of incomes to put consumers on a lower indifference curve

Section II: Uniform Two-Part Tariff
Will look at the process of developing a uniform two-part tariff Monopolist will want to maximize profit subject to the constraint of the number of consumers N Wants to keep as many consumers in the market as possible KEY:  Tax T will need to be adjusted whenever the price varies

X – market demand for rides T=T* -smallest of the N consumer surpluses
Total profits = X – market demand for rides T=T* -smallest of the N consumer surpluses C(X) – cost function And differentiating with respect to price and setting equal to zero Write equation of s on board Explain what E means

P will exceed c’ if (1- Ns1) >0
(1- Ns1) < 0 means that the price is less than MC And, if the monopolist chooses to raise the price, the smallest consumer would not purchase this product With a smaller amount of consumers, a new tariff needs to be found. Due to this, the tax will then increase and there will thus be a lower price charged per ride If p less than c’ than the smallest consumer demands less than 1/N of the total market If they choose to not, they are in the first equation

How to derive the new uniform tax
Have consumers n Profits from the lump sum tax: ΠA = nT Profits from the sale of ride: ΠS = (P – c)X An increase in T means that there will be less consumers buying the product ΠA thus changes therefore profits depends on T

If the market contracts, meaning there are less consumers because the monopolist increases T.
This gives the monopolist the ability to control the number of consumers in the park Increasing T reduces the number of consumers This then decreases the price per ride Why do they do this?? To capture the larger surpluses Result: a decrease in profits from sales Of the consumers who continue to buy the product (these people loooove disney world Because of the size of the market and the demand for X to decline

And the final answer is? The optimum and uniform two-part tariff that globally maximizes profits is reached: Profits is thus a sum of profits from the tariff and the profit from price per ride

Finally…Section III Applications
IBM as an example of volume surcharge or marginal price discounts Here’s the story for IBM’s case Someone who is renting a machine pays a lump sum monthly rental: T dollars The tax is for the right to buy/rent machine time The renter gets up to X* hours of machine time with no additional cost to the lump sum tax If they want to consume more, they are charged an additional fee of k per hour THUS Larger consumers have a volume surcharge This captures part of the surplus lost from smaller consumers

This is optimal when… MC of machine time is 0
Where the surcharge rate becomes effective, or the demand after X* is equal to the maximum demand by a smaller consumer No resale of computer time Rate k is determined by a process that is similar to a monopolist choosing a single price where MC = MR

Example: public utilities also employ this type of pricing structure (which is like marginal price discounts) We thus denote the following for prices with respect to varying demand P1 and demands up to X* P2 for a demand greater than X* Result of this: Kinked demand curve

Math of kinked demand curve
XP1 + Y = M [ if 0<X<X*] X*P1 + (X-X*) P2 + Y = M [ if X > X*] So, a marginal price discount can be like a two-part tariff for the larger consumers. We rewrite the second equation as follows: XP2 + Y = M – T [T = X*(P1 - P2 ) ] The tax is thus the right to buy X at P2

Results The monopolist can then receive larger surpluses from the larger consumers from this type of pricing discount. Monopolist has higher profits

Conclusion Charging a two-part tariff instead of a single price by the monopolist RAISES PRICES AND, it allows for less discrepancy between marginal rates of substitution in consumption and production

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