LECTURE 1: (Monday 16 March, part 1)
Introduction to Monopoly
I. Discrete example to introduce General Equilibrium and Monopoly
Buyer
Value
Seller
Cost
Unit 1
6
Unit 2
5
Unit 3
4
Unit 4
3
Unit 5
2
Unit 6
1
Unit 7
0
0
1
2
3
4
5
6
A. General Equilibrium Version. Each buyer wants only one unit, each seller can
produce only one unit. If each seller acts as price-taker, then there will be a unique
equilibrium with price = 3 and quantity = 4.
If price is set greater than 3, then more than 4 sellers and fewer than 4 buyers will want to
trade. If price is set lower than 3, then fewer than 4 sellers and more than 4 buyers will
want to trade.
IMPORTANT MATHEMATICAL POINT: Note that market quantity and market price
are linked in a one-for-one relationship.
INVISIBLE HAND: Increase price if there is too much demand (“Surplus Demand”),
reduce price if there is too much supply. An invisible hand rule would lead to the unique
equilibrium price in this case.
SOCIAL SURPLUS: The social surplus for each unit produced is the difference between
buyer’s value and seller’s cost for that unit.
Here social surplus is positive for units 1 (surplus 6), unit 2 (surplus 4), unit 3 (surplus 2),
and either zero or negative for additional units. In this context, general equilibrium has
the important property of maximizing social surplus – whereby the equilibrium price
coordinates buyers and sellers to trade the “right number” of units.
We can subdivide social surplus into consumer surplus (CS) and producer surplus (PS) or
profits. Because of the symmetric nature of this example, CS and PS are equal for each
unit at the equilibrium price of 3 and in their sums.
Buyer and seller each achieve surplus 3 on unit 1.
Buyer and seller each achieve surplus 2 on unit 2.
Buyer and seller each achieve surplus 1 on unit 3.
Thus, the total CS = 3 + 2 + 1 = 6 and total PS = 3 + 2 + 1 = 6.
B. Monopoly Version: There are still 6 buyers, each wanting only one unit, but a single
seller with increasing “marginal costs” of production. It is free to produce the first unit,
costs 1 to produce the second unit, and so on.
The seller chooses a price and sells to anyone who wants to buy at that price. Another
way of saying this is that the seller can’t tell which possible buyers have the highest value
for the good. Let’s just try some possible prices:
1.
2.
3.
4.
5.
6.
7.
Price = 6 implies quantity = 1 with total cost 0. Profit = 1 x 6 – 0 = 6.
Price = 5 implies quantity = 2 with total cost 1. Profit = 2 x 5 – 1 = 9.
Price = 4 implies quantity = 3 with total cost 3. Profit = 3 x 4 – 3 = 9.
Price = 3 implies quantity = 4 with total cost 6. Profit = 4 x 3 – 6 = 3.
Price = 2 implies quantity = 5 with total cost 10. Profit = 5 x 2 – 10 = 0.
Price = 1 implies quantity = 6 with total cost 15. Profit = 6 x 1 – 15 = -9.
Price = 0 implies quantity = 7 with total cost 21. Profit = 7 x 0 – 21 = 21.
So there are two optimal choices of (quantity, price): (2, 5) and (3, 4).
Each of these optimal choices involves lower price and lower quantity than in the case of
general equilibrium.
DEADWEIGHT LOSS: This means that the monopoly outcome is economically
inefficient, because the equilibrium quantity is too low to maximize social surplus.
For example, with 2 units sold and price 5,
CS is equal to 1 for unit 1, 0 for unit 2, for a total of 1.
PS is equal to 5 for unit 1, 4 for unit 2, for a total of 9.
Social surplus is then equal to CS + PS = 1 + 9 = 10. The difference between social
surplus for general equilibrium and social surplus for monopoly = 12 – 10 = 2 is called
the deadweight loss. This loss matches the additional surplus that would have been
produced from trade of unit #3 (value = 4, cost = 2), but that did not occur as a result of
the monopolist’s profit maximization.
Why does the monopoly outcome yield lower quantity and higher price than the general
equilibrium outcome?
There are two components to profit:
A. Revenue = Quantity times price.
B. Cost
Quantity
Price
Total Revenue
Total Costs
Profit
1
6
6
0
6
2
5
10
1
9
3
4
12
3
9
4
3
12
6
6
5
2
10
10
0
6
1
6
15
-9
7
0
0
21
-21
Looking at Total Revenue, note that the gain in total revenue decreases with the number
of units sold. Also, increasing from 3 units sold to 4 units sold does not increase total
revenue, while increasing from 4 to 5 or more units sold will reduce total revenue!
The change in revenue as a result of selling an additional unit is called the Marginal
Revenue associated with that change in quantity.
Marginal revenue for the first unit is equal to 6, the actual price for that unit.
Marginal revenue for the second unit is equal to the change in total revenues:
MR(unit 2) = Total Revenue(2 units) – Total Revenue (1 unit) = 10 – 6 = 10.
The price for the second unit is 5, but its marginal revenue is 4, a smaller number.
The reason is that there are two effects of selling the second unit:
EFFECT 1: Gain in revenue of 5 from selling a unit that was not previously sold.
EFFECT 2: Reduction in price on unit 1, previously sold at price 6, but now sold
at price 5.
Marginal Revenue for unit 2 is the sum of effects 1 and 2
MR(unit 2) =
+6 – 1 = 5.
In analyzing the change from selling 1 unit to selling 2 units, we refer to unit 2 as a
“Marginal Unit” meaning that it is on the margin between beyond sold and not being
sold. We call unit 1 an “Inframarginal Unit”, because it is sold whether or not the price
is reduced (from p = 6) and the quantity is increased (from q = 1)
The computation for marginal revenue depends on a combination of price and quantity
sold at that price, because (1) price determines the magnitude of Effect 1, (2) quantity
determines the magnitude of Effect 2.
Since price and quantity are linked in a one-to-one relationship, lower price is associated
with higher quantity. Therefore:
Higher quantity means EFFECT 1 is smaller in magnitude since price is smaller.
Higher quantity means EFFECT 2 is larger in magnitude since there are more
inframarginal units sold.
Since EFFECT 1 is positive and EFFECT 2 is negative in the computation of Marginal
Revenue, higher quantity is associated with lower marginal revenue.
This property of marginal revenue illustrates the principle of Decreasing Returns.
Economic theory typically assumes decreasing returns for consumers (decreasing
marginal rate of substitution) and for producers (increasing marginal costs). However,
this analysis for monopoly pricing demonstrates how decreasing returns arises naturally
from an assumption of ordering of consumers / sales from highest value to lowest value.
LECTURE 2: (Monday 16 March, part 1)
Continuous Version of Monopoly Pricing
The concepts from the numerical example carry over to the more general case where
price and quantity are continuous variables rather than integers. The virtue of studying
price and quantity as continuous variables is that it allows us to use a Calculus based
approach and first-order conditions to characterize the marginal revenue and optimal
monopoly price and quantity.
Typically, we model the relationship between price and quantity as a function P(Q),
corresponding to a graph with quantity Q on the horizontal axis and price P on the
vertical axis. It may seem more natural to think of quantity as a function of price, but by
convention, economists write P(Q) rather than Q(P), where P’(Q), Q’(P) < 0.
Assume that the monopolist has a cost function C(Q) for producing Q units. Assuming
that production is costly, C’(Q) > 0.
Then the monopolist’s maximization problem is
OR
Max(Q)
Revenue – Cost
Max(Q)
Q P(Q) - C(Q)
Here, it is clearly helpful to be able to represent the price-quantity relationship as P(Q)
because this makes it possible to represent the monopolist’s maximization problem in
terms of a single choice variable Q.
We denote the monopolist’s profit as a function Π(Q).
First-Order Conditions for Monopolist Profit Maximization
Taking a derivative with respect to Q yields the first-order condition for monopoly
maximization.
dΠ / dQ = P(Q) + Q P’(Q) – C’(Q) = 0.
which can be rewritten as
P(Q) + Q P’(Q)
=
C’(Q)
(*)
The first term, P(Q), represents the price for the marginal unit of the good We referred to
this value before as “EFFECT 1”.
The second term, Q P’(Q), represents the instantaneous cost from reducing the price
(P’(Q)) on inframarginal units (Q). We referred to this value before as “EFFECT 2”.
The term on the right-hand side, C’(Q) represents the cost for producing the marginal
unit of the good.
Putting these terms together, we interpret the left hand side of equation (*), which
represents the monopolist’s first-order condition, as saying
Marginal Revenue
=
Marginal Cost.
Note that each of these terms is computed at quantity Q, corresponding to price P(Q). In
words, we say that the monopolist chooses Q (and thus P(Q)) so that the marginal
revenue is equal to the marginal cost, with both MR and MC computed for the
combination (Q, P(Q)) for quantity and price.
Comparing FOC for Monopolist and Competitive Firms
In a world with many firms, each firm is assumed to be a price-taker, meaning that the
firm does not account for its effect of its quantity sold on the market price. That is, each
firm assumes P’(Q) = 0 in its profit maximization decision. The first-order conditions for
a competitive firm are similar to the first-order conditions for a monopolist, with the
single exception that P’(Q) = 0:
(FOC Monopolist)
P(Qm) + Qm P’(Qm) =
C’(Qm)
(*)
(FOC Competitive Firm)
P(Qc)
C’(Qc)
(**)
=
We can identify the qualitative difference between Qm and Qc by working backward from
the comparison of FOC’s. We add one additional assumption to help with analysis –
specifically that the monopolist has decreasing returns to production as indicated by
increasing marginal costs: MC(Q) = C’(Q) increasing in Q (i.e. C’’(Q) > 0).
In competitive equilibrium, (**) holds and therefore P(Qc) = C’(Qc).
But since P’(Q) < 0, we know that Q P’(Q).< 0, which implies MR(monopolist, Q) <
MR(competitive firm, Q) at any quantity Q.
At Qc, (FOC) holds for a competitive firm, so that
P(Qc)
=
C’(Qc)
That is, MR(competitive firm) = MC at Qc , so we know that MR(monopolist) < MC for
the monopolist at Qc. That is, the monopolist loses money on marginal units of
production at Qc.
What happens if the monopolist increases Q from Qc? First, MR will fall because of the
increase in Q. Second, MC will increase by our assumption of decreasing returns to
production. So increasing Q from Qc only causes the monopolist to lose even more than
before for producing the marginal unit.
Instead, now think about what happens if the monopolist reduces Q from Qc? This
causes MR to increase and MC to decrease, and will eventually restore MR = MC.
In sum, the comparison of FOC for monopolist and competitive firm confirms that
Qm < Qc and similarly that Pm > Pc. This type of analysis is used in many subfields of
microeconomic theory, and is sometimes formalized as an application of the “Implicit
Function Theorem”. The key point is that it may be possible to use FOCs to make
valuable comparisons between two equilibrium values without actually computing those
equilibrium values.
Graphical Version
The comparison between Qm and Qc is probably even more obvious from a graph of the
monopolist’s marginal revenue and price. Since MR(monopolist) lies below price at
each quantity, MR(monopolist) < MC at the efficient (competitive equilibrium) quantity
Qc where price P = MC.
And then, since MR is declining, MC is increasing in Q, it must be that MR = MC at a
quantity Qm < Qc.
Computations for Linear Demand Example
One common numerical example for monopoly pricing assumes constant marginal cost:
C(Q) = C * Q and a linear demand system: P(Q) = A – BQ, where A, B, and C are
positive and known constant values.
For example, with A = 20, B = 2 and C = 4, the price function is given by
P(Q)
=
20 – 2Q.
Then monopolist’s maximization is
Max(Q)
QP(Q) – C(Q) = Q (20 – 2Q) – 4Q
dΠ / dq
=
20 – 4Q – 4
= 16 – 4Q.
We have ignored the second-order conditions for monopolist’s maximization problem,
but can easily check them in this case.
d2Π / dq2
=
–4
<
0.
So the second-order conditions for a maximum hold and the solution to FOC for the
monopolist will be a global optimum. That is, the monopolist’s optimal quantity satisfies
16 – 4Q. = 0, so Qm = 4. Substituting into the price function, P(Qm) = 20 – 2Qm = 12.
In this example, it is straightforward to solve for profits, consumer surplus and DWL.
Profit:
The monopolist sells 4 units at price 12, cost 4 per unit. So Profit = 4 * (12 – 4) = 32.
Consumer Surplus:
We want to compute the area between the demand function P(Q) and the price p = 12.
Most of the time, this would require Calculus and evaluation of an integral. But with
linear demand, the computation can be done with simple geometry.
Consumer surplus is a triangular area with base 4 (the quantity sold) and height 8
(difference between value and price for the first unit sold). Thus CS = ½ * 4 * 8 = 16.
Intuitively, the consumer buys 4 units and has average value for each unit somewhere
between 20 (initial unit) and 12 (last unit). Since the demand curve is linear, the average
value for units sold is exactly 16, the average value for first and last unit. So consumers
are purchasing 4 units with average value 16 and price 12 per unit. Combining these
facts, CS = quantity * (Average Value – Price) = 4 * (16 – 12) = 16.
Deadweight Loss / Comparison to Perfect Competition
With perfect competition, the firms would produce until price = MC. Then, the
equilibrium quantity Q* would satisfy P(Q*) = MC(Q*) or 20 – 2Q* = 4, or Q* = 8.
There would be no profits to producers because they are selling at price = MC and MC is
the same for all units. (NOTE: If producers had increasing MC, they could still achieve
profits.) So social surplus all goes to consumers.
With Q* = 8, p* = 4, CS is the area of a triangle with base 8 and height 16, so
CS = ½ * 8 * 16 = 64.
Here, consumers purchase 8 units of the good and have average value = (20 + 4) / 2 = 12
per unit, so an alternate way of computing CS is
CS = Q * (Avg. Value – Price) = 8 * (12 – 4) = 64.
Deadweight loss is simply the difference between social surplus for perfect competition
and social surplus for monopolist.
DWL = 64 – 48 = 16.
Intuitively, DWL occurs because the monopolist sells only 4 units when it would be
economically efficient to sell 8 units.
Units 4 through 8 have average value (12 + 4) / 2 = 8 to consumers and constant marginal
cost 4 for the monopolist to produce. So the average surplus lost due to the quantity
reduction by the monopolist is 4 per unit, for a total DWL of 4 * 4 = 16.
Reversing the Relationship of Price and Quantity
We have focused on representing the monopoly maximization problem in the form
Π (Q) = QP(Q) – C(Q), where price is defined as a function of quantity. In general, we
assume that P(Q) is strictly decreasing in Q, so that in mathematical terms, there is a oneto-one relationship between price and quantity That is, each quantity is associated with a
unique price and so logically each price is associated with a unique quantity. This means
that we can invert the function P(Q) to identify the related function Q(P).
For example, if P(Q) is linear, P(Q) = A – BQ, then we can rearrange the terms in this
equation to produce the alternate equation Q(P) = (A – P) / B, where we have written Q
as a function of P rather than the other way around. Since A and B are positive constants,
this means that Q(P) is also linear – Q(P) = α – βp, where α = A/B, β = 1/B.
It is useful to be able to move back and forth between representations P(Q) and Q(P),
depending on whether we want to analyze optimal quantities or prices. For example, it is
generally appropriate to look at marginal revenue as a function of quantity sold, but it is
also generally appropriate to find the social optimum in terms of price, since the social
optimum is given by price = marginal cost.
Problems about price discrimination, which will be discussed in the next lecture,
generally involve different demand functions for different groups of consumers. The
monopolist typically thinks about total quantity sold among all consumers, meaning that
it will be necessary to combine the demand curves – which requires the addition of two
demand functions – e.g. Q(p) = Q1(p) + Q2(p). As suggested by this formula, this
addition is only possible when we write quantities as a function of prices rather than price
as a function of quantity.
So suppose once again that P(Q) = 20 – 2Q. If we want to work in price instead of
quantity, we would rearrange this equation to write Q(P) = 10 – P/2.
Then the monopoly profit function is given by
Π (P) =
P Q(P) – C(Q(P)),
which (fortunately) we can rewrite as
Π (P) =
Q(P) [P– C],
since the monopolist makes revenue P per unit sold at cost C per unit sold.
With Q(P) = 10 – P/2,
Π (P)
=
dΠ / dP
=
(10 - P/2) [P– C],
10 + C/2 – P, which has solution P = 10 + C/2.
With C = 4, the optimal solution has P* = 12, and thus Q(P) = 10 – 6 = 4, exactly as we
computed above.
LECTURE 3: (Tuesday 17 March, part 1)
Price Discrimination
The existence of DWL in standard monopoly pricing suggests two things. First, there is
scope for the government to impose regulations on the monopolist in order to increase
quantity sold and reduce DWL. Second, there is an incentive for the monopolist to try to
capture this DWL as additional profit.
Returning to the marginal revenue calculation, the reason that the monopolist restricts
quantity is to maintain high prices on inframarginal units sold. In other words, the
monopolist limits quantity because of the restriction that all units have to be sold at the
same price.
If the monopolist could charge different prices to different people – and especially if the
monopolist could charge high prices to customers with high values and low prices for
customers with low values, then the monopolist would do so. This type of pricing is
called price discrimination. The exact nature of price discrimination used by a particular
monopolist depends on two factors: (1) information about the specific values of different
consumers; (2) legal restrictions on prices offered to different consumers.
As discussed in Nolan Miller’s lecture notes, it is also possible to practice price
discrimination by charging different prices for different numbers of units of a good. The
Miller notes discuss both quantity discounts (“Non Simple Pricing”) and two-part tariffs
(a fixed fee paid in advance, plus a price per unit consumed). The idea of each pricing
rule is generally that customers are willing to pay more for the first units that they
consumer – in other words, that first-unit buyers are the high-value customers and laterunit buyers are the low-value customers.
These types of non-linear pricing are common for athletic facilities, amusement parks,
and utilities such as electricity and fuel.
The most common forms of price discrimination involve different prices for different
customers, with each customer only wanting one unit. First degree or Perfect price
discrimination is defined as a case where each customer pays her true value for the
good. This is only possible if each person’s value is somehow observable to the seller.
For example, very clever merchants may be able to figure out a customer’s value from
brief conversation, or careful observation (how the customer is dressed, for example).
Third degree price discrimination is the case where price is formally determined by an
observable characteristic of each buyer. The most common example of this is “student
discounts”, where anyone with a “student identification card” is given a discount.
The Miller notes give an example of ticket prices for a sports event. It is useful to work
through this example in detail to get a better sense of the importance of marginal revenue
and for the tradeoffs involved with selling to one group rather than the other. Suppose
that demand from non-students is given by
pA(QA) = 100 – QA
(or QA = 100 – pA)
while demand from students is given by
pS(Qs) = 20 – QS / 10. (or QS = 200 – 10 pS).
Assume that there are no costs to selling additional seats: MC = 0.
Uniform Price Optimum
If the monopolist is restricted to offering the same price to both types of consumers, then
the demand curve is piecewise linear – there are two separate parts to the demand curve
with two different slopes.
PART 1: Price > 20. Only group A buys and Q(p) = 100 – p, so that the demand curve
just follows the demand curve for type A for prices in this range.
PART 2: Price < 20. Then demand is the sum of QA and QS, which are each positive.
That is, Q(p) = 100 – p + 200 – 10 p = 300 – 11p, corresponding to p(Q) = (300 – Q) / 11.
Note that these two separate formulas, Q = 100 – p and Q = 300 – 11p each give the same
value Q = 80 at price p = 20.
The optimal price along part 1 on the demand curve is given by Max Q (100 – Q), which
occurs at Q* = p* = 50, yielding revenues 2500.
The optimal price along part 2 of the demand curve is given by Max Q(300 – Q) / 11,
which occurs at Q = 150, which corresponds to price 150 / 11, yielding revenues of
approximately 2050. That is, the optimal choice is for the monopolist to sell only to type
1 customers, setting p = 50 and q = 50.
Note that if the monopolist has to sell out the stadium, an expansion of seats in the
stadium actually may reduce revenues.
Price Discriminating Outcome
:
If the monopolist can set different prices for each group, then the optimum price and
quantity for each group is given by the condition
MRA(QA) = MRS(QS) = 0. The optimal price and quantity for group A is the same as
above, pA* = QA* = 50.
The monopolist’s optimum for group S is given by MR(QS) = 0, or 20 – QS / 5 = 0, or
QS = 100, which corresponds to pS = 10. This yields quantity 50 and revenue 2500 from
group A and quantity 100, revenue 1000 from group S.
Price Discriminating Outcome with Fixed Number of Seats
If there are a fixed number of seats, then this can affect the monopolist’s price
discriminating optimum. First, if the fixed number is greater than 150, then the
monopolist would continue as above, selling a total of 150 seats and leaving some empty.
Second if the fixed number is less than 150, then it is possible to set quantities QA < 50
and QS < 100 – so that MRA (QA) > 0 and MRS (QS) > 0, while also selling all seats in the
stadium. For this reason, the monopolist wants to sell all of the available seats in this
case. The monopolist maximizes revenue by setting MRA(QA) = MRS (QS).
This gives two equations in two unknowns:
QS + QS = F
MR(QA) = 100 – 2 QA = MR(QS) = 20 – QS / 5.
Solving these two equations simultaneously:
80
=
2 QA – QS / 5
40 + QS / 10 =
QA.
Then substitute in the first equation:
40 + QS / 10 + QS
=F
11 QS / 10
= F – 40
QS
= 10 F / 11 – 400 / 11
This result given a formula for the quantity of seats allocated to group S in the optimal
price-discriminating scheme when there are a total of F seats available.
Formally, we write
QS(F)
= 10 F / 11 – 400 / 11,
because this number of seats assigned to group S is a function of the value F. Note
further that the formula gives a negative number if F < 40. In fact, this means that if F <
40, the monopolist should allocate all seats to group A. This requirement that F > 40 for
the monopolist to sell to both groups is consistent with marginal revenue calculations
because MRS (0) = 20 (no student is willing to pay more than 20 for a ticker) and
MRA(QA) = 100 -2 QA > 20 if QA < 40. In words, marginal revenue for alumni is greater
than the marginal revenue for the first unit sold to a student if alumni are purchasing 40
tickets are fewer. So if there are fewer than 40 seats, the profit-maximizing rule allocates
them all to alumni since each unit sold to alumni increases the total profit by more than
20, the price for the first ticket sold to a student.
Beyond F = 40, the equation QS(F) = 10 F / 11 – 400 / 11 indicates that for every 11
additional seats available, the optimal price-discriminating rule allocates 10 to students
and 1 to alumni. (This relationship stems from the fact that MR decreases 10 times as
quickly in quantity for alumni than for students.)
For example, if F = 150, the number of seats that monopolist would choose if given any
choice of F, then QS = 10 * 150 / 11 – 400 / 11 = 100, exactly the same result that we
calculated above.
Logically, the formula QS = 10 F / 11 – 400 / 11 ensures that F < 150 seats are divided
among groups A and S so that MRA(QA) = MRS (QS) > 0. Substituting the rule
QS = 10 F / 11 – 400 / 11
into the formula MR(QS) = 20 – QS / 5,
we find MR(Q*S(F)) = 20 – [10 F / 11 – 400 / 11] / 5
= 20 + 80 / 11 – 2 F / 11
= (300 – 2F) / 11.
At F = 40, the smallest number so that the price-discriminating rule includes sales to
students, MR(Q*S(F)) = (300 – 2*40) / 11 = 20. Beyond F = 40, the marginal revenue for
the last seat allocated to students (and for the last seat allocated to alumni) decreases at
rate 2 / 11 per additional seat.
In this sense, the limitation of a fixed number of seats is analogous to the imposition of a
fixed marginal cost MC > 0. If the number of seats were unlimited and MC = 0, then we
already know from earlier analysis that the optimal quantities in a price-discriminating
scheme are QS = 100, QA = 50. As MC increases, both QS and QA would decrease in an
optimal price-discriminating scheme. Furthermore, the optimal price discriminating
scheme for a given MC is exactly equivalent to the optimal price discriminating scheme
for some fixed number of seats F and MC = 0.
For example, if F = 95, then QS = 10 F / 11 – 400 / 11 = 50 and thus QA = 45.
Substituting in our equation MR(Q*S(F)) yields MR(Q*S(F)) = (300 – 190) / 11 = 10.
That is, the optimal prices and quantities for a price discriminating monopolist are exactly
the same if (1) restricted to 95 seats with MC = 0 OR (2) unrestricted in number of seats
and MC = 10.
This correspondence of price discriminating rule subject to two different constraints is
another form of duality – at any optimum, the monopolist distributes seats among groups
A and S efficiently from the perspective of profit-maximization.