EconS 425 - First-Degree Price Discrimination · 2017-10-18 · First-degree price discrimination...

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EconS 425 - First-Degree Price Discrimination Eric Dunaway Washington State University [email protected] Industrial Organization Eric Dunaway (WSU) EconS 425 Industrial Organization 1 / 41

Transcript of EconS 425 - First-Degree Price Discrimination · 2017-10-18 · First-degree price discrimination...

Page 1: EconS 425 - First-Degree Price Discrimination · 2017-10-18 · First-degree price discrimination occurs when the monopolist has perfect information about the market. Second-degree

EconS 425 - First-Degree Price Discrimination

Eric Dunaway

Washington State University

[email protected]

Industrial Organization

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Introduction

We are moving on to our second unit of the semester; an examinationof pricing techniques available to monopolists.

We�ll begin with a discussion of price discrimination, which allows amonopolist to charge di¤erent prices to di¤erent groups based ontheir identifying characteristics.

Today, we�ll look at �rst-degree price discrimination, also known aspersonalized pricing, or perfect price discrimination.

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Price Discrimination

Up until now, we�ve been using uniform pricing.

Uniform meaning that we can only charge one price to everyone in themarket.

Price discrimination allows us to charge multiple prices to di¤erentgroups of people, based on observable characteristics.

Examples can range from students, to the elderly, to women, etc.

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Price Discrimination

We classify price discrimination into three types, based on how muchinformation is available to the monopolist.

First-degree price discrimination occurs when the monopolist hasperfect information about the market.Second-degree price discrimination occurs when the monopolist isaware that people are members of certain groups, but does not knowwhich groups people belong to.Third-degree price discrimination occurs when the monopolist canidentify which people are within each group.

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Price Discrimination

Why should a monopolist price discriminate?

It increases pro�ts!Price discrimination converts signi�cant portions of consumer surplusinto producer surplus. In some cases, it also eliminates some or all ofthe deadweight loss created by uniform pricing.

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Price Discrimination

A quick example:

A theater in town is holding a showing of a new movie this weekend.Students will see the movie if the price is 10 dollars or less. SeniorCitizens will see the movie if the price is 5 dollars or less.

Let�s say that there are 10 students and 5 seniors. Depending onwhat price the theater charges, the movie theater�s pro�ts look like

Price Student π Senior π Total π

$5 50 25 75$10 100 0 100

In this case, the �rm would do better by charging a higher price andlosing the senior market.

Could they do better?

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Price Discrimination

What if the theater charged $10 to students and $5 to seniors? Itspro�t would be

Price Student π Senior π Total π

$5 50 25 75$10 100 0 100

$10/$5 100 25 125

this way, the �rm could increase its pro�ts by charging less to theseniors and getting them to come see the movie, too.

The lower willingness to pay for the senior citizens is equivalent tohaving a lower price elasticity of demand (more elastic). Their �atterdemand curve requires a lower price from the monopolist.

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Price Discrimination

Real world example: Disneyland.

They charge $199 to out of state residents for a 3 day pass.But they only charge $154 to California residents.

Why? People who are travelling from out of state are likely onvacation and are planning on going to Disneyland regardless. Theyare less sensitive to the price than the local residents who havealready been several times.

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Price Discrimination

There are three requirements a �rm must meet in order to implementprice discrimination.

The �rm must have market power.The �rm must be able to di¤erentiate consumers (and they must havedi¤erent valuations of the good).The �rm must be able to prevent arbitrage (resale).

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Price Discrimination

The �rst two points may seem a bit obvious, but let�s talk about thethird.

If a �rm cannot prevent arbitrage, consumers who can obtain a lowerprice via their own valuation can resell the good to consumers with ahigher valuation.Potentially, they could compete with the �rm, o¤ering a price lowerthan the monopolist�s price, and cutting into their pro�ts.Most arbitrage prevention is done by identi�cation veri�cation orquantity controls.

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First-Degree Price Discrimination

In �rst-degree price discrimination, a �rm is able to observe everyconsumers�willingness to pay (reservation price) for their product.

Naturally, the �rm wants to charge every consumer exactly theirwillingness to pay.

Intuitively, the �rm will sell to anyone it can make a pro�t from,selling to those consumers whose valuations are at least the marginalcost of the �rm.

We�ll prove this in a little bit.

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First-Degree Price Discrimination

Before we get into the details of �rst-degree price discrimination, weneed to take a step back and talk about the demand function.

Remember that the demand function, in essence, is an ordering ofconsumers based on their willingness to pay for the good.At the top of the demand curve, we have the consumer who is willingto pay the most for the good, and the valuation decreases from there.Most importantly, we have assumed that consumers are distributeduniformly along the demand curve.

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First-Degree Price Discrimination

qD

p

q

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First-Degree Price Discrimination

qD

Number ofConsumers

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First-Degree Price Discrimination

qDUniform Distribution

Number ofConsumers

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First-Degree Price Discrimination

qD

Number ofConsumers

f (x )

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First-Degree Price Discrimination

Another facet of �rst-degree price discrimination is that we couldhave consumer willingness to pay arbitrarily distributed along thedemand curve.

We can even have consumers with the same valuations.This can pose some challenges for calculating pro�ts, as we�ll see.

Let v(x) represent the willingness to pay for consumer x . We say thatthe consumers are distributed along some function f (x).

Let v(0) = v̄ , i.e., the �rst consumer along our spectrum has thehighest valuation of v̄ .As we move along our valuation (demand) curve, v(x) decreases, i.e.,dv (x )dx < 0.

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First-Degree Price Discrimination

To start, we will assume that every consumer will either buy one unitof the good, or none at all.

This is known as unit demand.

The �rm will sell to consumers whose valuations are high enough. Atsome point, consumer s will have a valuation where they areindi¤erent between buying the good or not.

We are interested in �nding consumer s. Everyone with valuationsequal to or above s will purchase the good, while everyone withvaluations lower than s will not.

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First-Degree Price Discrimination

We can obtain the total number of units sold in this market, q, byintegrating the density function up until our indi¤erent consumer s,

q =Z s

0f (x)dx

This is equivalent to saying that we just add up everyone who buysthe good.

In the uniform case, f (x) = 1, and we �nd that

q =Z s

01dx = x js0 = s � 0 = s

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First-Degree Price Discrimination

qD

Number ofConsumers

f (x )

s

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First-Degree Price Discrimination

With this, we can set up an extremely general version of the �rm�spro�t maximization problem. With non-uniform pricing and anon-uniform distribution of consumers, we have

maxs

Z s

0v(x)|{z}p

f (x)|{z}q

dx � c�Z s

0f (x)dx

And from here, we can make some simpli�cations. If we imposeuniform pricing, v(x) = v(s) for all x , and we can pull it out of theintegral,

maxs

v(s)Z s

0f (x)dx � c

�Z s

0f (x)dx

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First-Degree Price Discrimination

If we wanted to have non-uniform pricing, but a uniform distributionof our consumers, we would have q = s,

maxq

Z q

0v(x)dx � c

�Z q

0dx�= max

q

Z q

0v(x)dx � c (q)

Or if we wanted to have both uniform pricing and a uniformdistribution of our consumers, again, we would have q = s,

maxq

v(q)Z q

0dx � c

�Z q

0dx�= max

qv(q)q � c(q)

which is essentially the pro�t function we have seen since thebeginning.

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First-Degree Price Discrimination

Let�s go back to our original, general de�nition.

maxs

Z s

0v(x)f (x)dx � c

�Z s

0f (x)dx

�We can di¤erentiate this function with respect to s to obtain

v(s)f (s)� c 0(q)f (s) = 0v(s) = c 0(q)

Here is that result that I spoke of before. The indi¤erent consumer swill have a valuation that is equal to the �rm�s marginal cost at theequilibrium quantity.

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First-Degree Price Discrimination

E¢ ciency wise, under �rst-degree price discrimination, we have asituation where every individual whose valuation is at least as high asthe marginal cost of the �rm purchases the good.

That sounds pretty e¢ cient to me.

In fact, under �rst-degree price discrimination, there is no deadweightloss.

There isn�t any consumer surplus either. Since every consumer paystheir valuation, they receive no surplus.All surplus in this case is producer surplus.

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First-Degree Price Discrimination

qD

p

q

qS

MR

CS

PS DWL

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First-Degree Price Discrimination

v(x )

p

x

MC

PS

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First-Degree Price Discrimination

To obtain the welfare level, we simply integrate the di¤erencebetween the price paid (the consumer�s valuation) and the marginalcost of each consumer up until consumer s,

PS =Z s

0

�v(x)� c 0(x)

�f (x)dx

or we could simply use a triangle formula when it is convenient.

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First-Degree Price Discrimination

Consider a banana market where a single �rm acts as a monopolist.There are four consumers, Stuart, Kevin Bob, and Dave.

Stuart values a banana at 5, Kevin and Dave value a banana at 3, andBob values a banana at 1.It costs 2 to make a banana.

If the banana monopolist knew all the reservation prices and couldprevent the consumers from reselling bananas, what price would itcharge to each of them?

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First-Degree Price Discrimination

The banana monopolist would charge Stuart 5, Kevin and Dave 3,and wouldn�t sell a banana to Bob.

Poor Bob.

In this case, we don�t have an indi¤erent consumer s since ourconsumers are discrete. If we did have one, their valuation would haveto be 2, the marginal cost of the banana monopolist.

Furthermore, the banana monopolist captures all of the surplus in thismarket,

PS = (5� 2) + (3� 2) + (3� 2) = 5

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First-Degree Price Discrimination

First-degree price discrimination is very costly to implement.

Typically, the cost to acquire all of the information necessary for eachconsumers�valuation would o¤set anything gained by non-uniformpricing.Furthermore, consumers will have strong incentives to hide their truevaluations from the �rm.

Only a few �rms, like personalized accounting and some legalpractices, are able to implement this sort of thing.

Be careful not to confuse �rst-degree price discrimination withbargaining.

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Two-Part Pricing

Thus far, we have limited our discussion of �rst-degree pricediscrimination to consumers with unit demand, i.e., they only buy atmost one unit of the good.

The method above becomes more challenging when we haveconsumers who would want to purchase multiple units of the samegood.

Willingness to pay for the second unit is typically less than the �rst, etc.

We can allow for �rst-degree price discrimination while consumers areable to purchase multiple units of the good by employing two-partpricing.

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Two-Part Pricing

Two-part pricing breaks down the price a consumer pays into a �xedand variable part.

T (q) = A+ pq

where consumers must pay the �xed cost A for the right to purchasethe good, and then a (usually small) price p for each unit purchased.

Two-part pricing is fairly common in the real world.

Taxi services, phone plans.Costco, bars.

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Two-Part Pricing

T (q) = A+ pq

The most challenging aspect of two-part pricing is designing the �xedcost, A, such that the consumer is willing to pay it.

We require the �xed cost to be incentive compatible, i.e., the betterchoice for the consumer.

We can �gure out the value of A by looking at the surplus received byan individual consumer x ,

Sx (q) = v(x , q)� p(x)q

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Two-Part Pricing

Sx (q) = v(x , q)� p(x)q

If consumer x doesn�t pay the �xed cost, their surplus is 0 (nochange). Thus, in order to get them to buy the product, their surplusmust be positive. Thus, A � Sx (q).

For simplicity. We are going to assume that if the consumer isindi¤erent between buying and not buying, they choose to buy. Thislets us set A = Sx (q).Intuitively, the �rm can charge consumer x their entire surplus for theright to purchase the good.

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Two-Part Pricing

Looking at just consumer x , the monopolist then needs to �gure outhow much to charge per unit, p. Setting up the pro�t maximizationproblem,

maxq

A+ p(x)q � c(q)

and we can substitute A = Sx (q) = v(x , q)� pq into this expressionto obtain

maxq

v(x , q)� p(x)q + p(x)q � c(q)

maxq

v(x , q)� c(q)

Di¤erentiating with respect to q,

∂v(x , q)∂q

� c 0(q) = 0

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Two-Part Pricing

∂v(x , q)∂q

� c 0(q) = 0

Let�s talk about that �rst term, ∂v (x ,q)∂q . This is how consumer x�s

valuation of their quantity purchased changes as the quantityincreases.

That sounds a lot like an inverse demand function for consumer x . Infact, that�s exactly what it is.We can replace ∂v (x ,q)

∂q with p(x), the price for consumer x , and obtain

p(x)� c 0(q) = 0p(x) = c 0(q)

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Two-Part Pricing

Thus, a �rm simply wants to charge each individual the marginal costper unit that they purchase.

Again, this leads to an e¢ cient solution (no deadweight loss).

To make up for charging such a low unit price, the �rm sets the �xedcost to exactly the consumer�s surplus, thus claiming all of the surplusfor themself.

It�s basically the same outcome as we saw in �rst-degree pricediscrimination.

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Two-Part Pricing

This is very much how Costco prices its products.

Yearly membership fee, then marginal cost for everything else.

The only di¤erence between this method and our model is that the�xed price is uniform; Costco doesn�t di¤erentiate between itsconsumers.

This could lead to some small distortions in the market, but such is thecase in reality.

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Summary

First-degree price discrimination, while hard to implement, allows a�rm to charge an individual price to each consumer based on theirwillingness to pay.

It�s also e¢ cient!

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Next Time

Block Pricing and Third Degree Price Discrimination

Organizing consumers into groups is much easier than looking ateveryone individually.

Reading: 5.3

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Homework 2-3

Consider a market with a single �rm, that faces the following inversedemand function,

p = 100� 5qand a constant marginal costs of MC = 5.

1. If the �rm decided to implement a two-part pricing strategy, whatwould the monopolist charge for the entry �xed cost and the price perunit?

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