EconS 425 - Perfect Competition and Monopoly€¦ · Monopoly Now let™s look at monopoly, which...

EconS 425 - Perfect Competition and Monopoly

Eric Dunaway

Washington State University

[email protected]

Industrial Organization

Eric Dunaway (WSU) EconS 425 Industrial Organization 1 / 47

Introduction

Today we�ll review the structure of the perfectly competitive andmonopoly markets.

We�ll also dust o¤ how welfare calculations are done.


Supply and Demand

Consider a setting where there are n identical �rms. Each of these�rms faces an individual demand function where for any given price p,consumers will demand a corresponding quantity from �rm i , qi , ofthat good or service. As a function,

qDi = qD (p)

Under normal conditions (i.e., not a Gi¤en good), as the price of thegood or service increases, the quantity demanded decreases. Thus,

dqDiqp

< 0


Supply and Demand

We can add up all of these individual �rms to form the aggregatedemand curve

QD =n

∑i=1qDi

where the aggregate quantity also decreases with increases in themarket price.

Remember that we have to add demand curves together horizontally.By that, I mean that only quantities can be added together (Honestly,it would make no sense to add prices together).


Supply and Demand

A quick example. Consider two identical �rms with the followingdemand functions:

q1 = 10� 2pq2 = 10� 2p

We can aggregate these two demand functions together to obtain

QD = q1 + q2 = 20� 4p

Note: It�s not this easy when the �rms are not identical; you have totake �rm participation into question.


Supply and Demand

A lot of the time, it�s useful to refer to the inverse demand function,which is obtained by solving the demand function for price, p.

This is primarily a mathematical convenience. In reality, �rms chooseprices, rather than quantities.

In our above aggregate demand example,

QD = 20� 4p

we solve for price to obtain the inverse aggregate demand,

p = 5� 14QD


Supply and Demand

On the other end of the market, each of the n �rms has an individualsupply curve, where for every value that the market price can take,there is a quantity that �rm i will supply to the market, qi

qSi = qS (p)

Contrary to the demand curve, under all conditions, an increase inprice causes �rm i to supply more of their good or service to themarket.

dqSidp

> 0


Supply and Demand

Likewise, we can add all of the individual �rm supply curves togetherto obtain the aggregate, or market supply curve,

QS =n

∑i=1qSi

Remember that the supply curve is identical to the marginal costcurve, as long as the price is above the �rm�s average variable cost.

If price is below the average variable cost, the �rm would prefer to notproduce at all.


Supply and Demand

p

Q


Supply and Demand

p

Q

QD


Supply and Demand

p

Q

QD

QS


Perfectly Competitive Market

Lets move on to the perfectly competitive market.

Recall the four main assumptions that de�ne the perfectlycompetitive market:

Large number of buyers and sellers.Firms produce identical products.Everyone has perfect information.Firms can easily enter and exit the market.



With regard to the large number of buyers and sellers, this impliesthat the number of �rms, n, has to be quite large.

In fact, n has to be large enough to assume that any individual �rm�soutput decision, qi , does not have a noticeable on the total marketoutput, Q. Mathematically,

dQdqi

� 0

In other words, in a perfectly competitive market, one �rm�s decisionsdon�t in�uence what any other �rm chooses to do in that market.Every �rm is able to act individually.



In our perfectly competitive market, each �rm seeks to maximize theirpro�ts, which is de�ned as total revenue minus total cost

max TR � TC

Mathematically, it doesn�t whether price or quantity is chosen as thedependent variable, it will work out the same.

It�s almost always simpler to use quantity.In a practical setting, price is used.

If price is chosen, use the aggregate demand function. If quantity ischosen, use the inverse aggregate demand function.

Why the aggregate? All �rms face the market price.



Subtituting for total revenue and total cost,

maxqi

p(Q)qi| {z }TR

� c(qi )| {z }TC

where p(Q) is the inverse aggregate demand function, and c(qi ) isthe total cost faced by �rm i .

Calculating a �rst-order condition,

p0(Q)dQdqi

+ p(Q)| {z }MR

� c 0(qi )| {z }MC

= 0



p0(Q)dQdqi

+ p(Q)� c 0(qi ) = 0

This is where our assumption that dQdqi � 0 become important. Wecan cancel out the �rst term above in order to obtain

p(Q)� c 0(qi ) = 0

Or, rearrangingp(Q) = c 0(qi )

In words, this is the classic result from the perfectly competitivemarket: price equals marginal cost.



p(Q) = c 0(qi )

Graphically, this is fairly easy to solve.

The left-hand side of this equation can be replaced with the inversemarket demand function.The right-hand side of this equation is simply the supply curve, asmentioned above.


Perfectly Competitive Market Example

Consider a �rm in a perfectly competitive market that faces thefollowing inverse market demand curve

p = 100� 2Q

and the following competitive supply curve

p = 10+Q

Find the equilibrium price and quantity for this �rm.



p = 100� 2Qp = 10+Q

First, we need to �nd the marginal cost function, which is simply thesupply curve,

MC = 10+Q

Then, just set the inverse demand function (price) equal to themarginal cost

p = MC

100� 2Q = 10+Q



100� 2Q = 10+Q

Solving this expression for Collecting terms,

3Q = 90

Q� = 30

We can �nd the market price by simply plugging this value back intothe inverse market demand curve

p� = 100� 2Q� = 100� 2(30) = 40



p

Q

QD QS100

10

50



p

Q

QD QS100

10

40

30 50


Monopoly

Now let�s look at monopoly, which �nds itself on the opposite end ofthe competition spectrum. Monopoly has the following assumptions:

One seller, and a large number of buyers.Firms produce identical products.Everyone has perfect information.No �rms can enter the market.


Monopoly

Since there is only one �rm in this case, n = 1 and Q = qi , i.e., themonopolist�s output level is the same as the market output level.Furthermore, we can di¤erentiate Q = qi to obtain

dQdqi

= 1

While the monopolist gets the bene�t of setting its own price (marketpower), it also has to take the consequence of its price intoconsideration, as a higher price will lead to a lower quantitydemanded by its consumers.


Monopoly

Returning to our �rst-order condition from the pro�t maximizationproblem,

p0(Q)dQdqi

+ p(Q)| {z }MR

� c 0(qi )| {z }MC

= 0

We can use Q = qi and dQdqi= 1 to rewrite this as

p0(Q) + p(Q)| {z }MR

� c 0(Q)| {z }MC

= 0

and rearranging, we obtain the classic equilibrium de�nition for amonopolist:

p0(Q) + p(Q)| {z }MR

= c 0(Q)| {z }MC

or more generally, marginal revenue equals marginal cost.


Monopoly

p0(Q) + p(Q) = c 0(Q)

There is also a neat math de�nition in this statement. Let�s rearrangeit a bit.

p(Q) = c 0(Q)� p0(Q)Remember that p0(Q) < 0. Since as price goes up, quantitydemanded goes down (we can also say that as quantity demandedgoes up, price goes down).

This means that for a monopolist, price has to be greater thatmarginal cost, because the right-hand side of this equation mustalways be greater than simply marginal cost.


Monopoly Example

Let�s look at the same example as before, with the following inversemarket supply and demand curves.

p = 100� 2Qp = 10+Q

Now, we�ll solve it from the perspective of a monopolist.


Monopoly Example

p = 100� 2Qp = 10+Q

First, we need to �nd the marginal revenue, which we obtain from thetotal revenue,

TR = pQ = (100� 2Q)Q= 100Q � 2Q2

Next, we di¤erentiate with respect to Q to obtain the marginalrevenue

MR = 100� 4QAnd set it equal to marginal cost (the supply curve)

MR = MC

100� 4Q = 10+Q


Monopoly Example

100� 4Q = 10+Q

Rearranging, and solving for Q gives the monopolist�s equilibriumoutput level

5Q = 90

Q� = 18

And to �nd the equilibrium price, we plug the equilibrium quantityinto the demand function (Be careful not to plug it into marginalrevenue!)

p� = 100� 2Q� = 100� 2(18) = 64


Monopoly Example

p

Q

QD QS100

10

50


Monopoly Example

p

Q

QD QS100

10

50MR


Monopoly Example

p

Q

QD QS100

10

50MR

18


Monopoly Example

p

Q

QD QS100

10

50MR

18

64


Comparing PC and Monopoly

Let�s check what we predicted before about the relationship betweenperfect competition and monopoly.

Perfect Competition Monopolyp� 40 64Q� 30 18

As predicted, the monopoly yields a higher price than perfectcompetition while producing a less.


Comparing PC and Monopoly

p

Q

QD QS100

10

50MR

18

64

40

30

Perfect Competition

Monopoly


Welfare

What about welfare levels?

Let�s brush up on those, too.

Recall that the di¤erence between what a consumer was willing topay for a good or service and the price they actually pay is known asconsumer surplus.

Likewise, the di¤erence between what a producer receives and whatthey were willing to sell a good for is known as producer surplus.

Adding them up, we get the total welfare.


Welfare

p

Q

QD

QS

ConsumerSurplus

ProducerSurplus


Welfare

We can obtain the consumer surplus by integrating the di¤erencebetween the inverse market demand function and the equilibrium pricefrom zero to the equilibrium quantity, i.e.,

CS =Z Q �

0

hpD (Q)� p�

idQ

Similarly, we can obtain the producer surplus by integrating thedi¤erence between the equilibrium price and the inverse market supplyfuncton from zero to the equilibrium quantity,

PS =Z Q �

0

hp� � pS (Q)

idQ

Or, if your instructor is feeling nice and gives you linear supply anddemand functions, just use triangle and trapezoid formulas.

Returning to our example,


Welfare

p

Q

QD QS100

10

40

30 50

CS

PS


Welfare

Starting with consumer surplus, we can use a triangle formula,

CS =12(100� 40)(30) = 900

And doing the same thing with producer surplus,

PS =12(40� 10)(30) = 450

Adding them together gives us the total welfare level,

W = CS + PS = 1350


Welfare

p

Q

QD QS100

10

50MR

18

64 CS

PS


Welfare

Again, calculating the consumer surplus,

CS =12(100� 64)(18) = 324

For producer surplus, we actually have a trapezoid, so we have tomake sure we use the correct formula,

PS =12(64� 10+ 64� 28)(18) = 810

Lastly, we calculate the welfare level of the monopoly,

W = CS + PS = 1134


Welfare

Comparing our results,

Perfect Competition MonopolyCS 900 324PS 450 810W 1350 1134

We can see that consumer surplus falls, producer surplus rises, andtotal welfare falls under monopoly.

In fact, the di¤erence between the perfectly competitive andmonopoly levels of welfare is known as the dead weight loss.

This is the amount of economic activity (measured in dollars) that isbeing lost due to a distortion in the market.


Welfare

p

Q

QD QS100

10

50MR

18

64 CS

PS

DWL


Summary

In order to understand imperfect competition, we need to have abaseline to compare our results to (perfect competition)

Monopoly represents the opposite end of the competitive spectrumfrom perfect competition, and is a signi�cant source of market power.

Monopolists produce less, charge more, and create market distortions.


Next Time

Intertemporal considerations.

How patient are �rms and consumers?What happens when we add time as a factor to our models?

Reading: Section 2.2


Assignment 1-2

Consider a market that faces the following inverse demand and totalcost curves,

P = 500�QTC = 50+ 50Q +Q2

1. Calculate the equilibrium price and quantity if this market is perfectlycompetitive.

2. Calculate the equilibrium price and quantity if this market ismonopolized.


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EconS 425 - Perfect Competition and Monopoly€¦ · Monopoly Now let™s look at monopoly, which...

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