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Notes On Second Degree Price Discrimination, Plus Some
Questions and Answers
An Example
- Assume two types of consumer with different demands for good; Type A
has low demand and Type B has high demand.
-
Producer wants to sell at a lower price to Type A consumers and a higher
price to Type B consumers. However, consumers are not readily
identifiable, except by their purchasing behaviour. The challenge for the
producer is to develop two price-quantity packages that will be designed
so that consumers self-select the package intended for them. If successful,
this will increase the profit of the producer beyond what could be earned
with a single package covering both consumer types.
-
Producer will offer two packages: each consists of a take-it-or-leave-it offer
of a particular quantity of the good at a particular price: high price and
high quantity for Type B consumers and low price and low quantity for
the Type A consumers.
-
Necessary condition for attracting Type B consumers to the more
expensive package is to give them at least as much consumer surplus as
they could get by purchasing the “Type A” package.
-
The profit-maximizing set of price-quantity packages will be one that
gives Type A consumers less than they would be willing to purchase at
the price charged, in order to charge more to Type B consumers.
-
This model can be interpreted as a price-quality model instead, with
demand curves for two quality levels by different consumer types. In this
case, the profit-maximizing set of price-quality packages will be one that
gives Type A consumers a lower quality than they would have been
willing to purchase, in order to create enough “distance” between the
Type A and Type B packages. Formally, this would be called quality
discrimination instead of price discrimination.
Example (note that these are individual demand curves, so if there are 100 of
each type of consumer, the market demand is 100 times larger than each of these
equations):
Type A demand: P = 10 - .02Q
Type B demand: P = 10 - .01Q
MC = AC = 0 (for simplicity)
There are 100 of each type of consumer.
Producer wants to design two take-it-or-leave-it packages of price and quantity.
First try: Sell the maximum quantity the consumer will purchase, at a price equal
to all the area under the individual consumer’s demand curve. For Type B
consumers this means a quantity of 1000 and a price of (10 x 1000)/2 = $5,000.
For Type A consumers, this means a quantity of 500 at a price of (10 x 500)/2 =
$2,500. Because there are 100 of each type of consumer, and because there are
zero costs, the producer expects the profit to be 100 x $5,000 + 100 x $2,500 =
$750,000. However, this design of the price-quantity packages is flawed. If Type
B consumers purchase the Type A package of 500 units at a price of $2,500, they
will get consumer surplus of [(10 – 5) x 500]/2 + (5 x 500) - $2,500 = $1,250. If
they purchase the Type B package, they will get consumer surplus of zero. So,
with this package design, they will prefer to purchase the Type A package.
Actual profit will end up being 200 x $2,500 = $500,000.
Second try: Sell the maximum quantity each consumer is willing to purchase, but
lower the Type B price so that the consumer surplus from purchasing the Type B
package is at least as good as what it would be if the Type B consumer purchased
the Type A package. For Type A consumers, this means a quantity of 500 at a
price of $2,500 (same as above). For Type B consumers, this means a quantity of
1000 at a price of $5,000 - $1,250 = $3,750 (i.e., leaving Type B consumers a
consumer surplus of $1,250). Consumers will now self-select into two groups,
with the Type A consumers paying a low price for a low quantity and Type B
consumers paying a high price for a high quantity. Profit for the producer will
be (100 x $2,500) + (100 x $3,750) = $625,000.
Third try: Reduce the quantity (and the corresponding price) for the Type A
consumers (making this package also less attractive for the Type B consumers).
Charge more for the Type B package than above but still leave the Type B
consumer with the same consumer surplus as he would have gotten from
consuming the Type A package (because the Type A package is now smaller, this
is a smaller amount of consumer surplus than before). As an example, make the
Type A package include a quantity of 333.33, rather than 500. We can calculate
the area under the Type A consumer’s demand curve up to this quantity as [(10
– 3.33) x 333.33]/2 + (3.33 x 333.33) = $2,222. The price-quantity package offered
to Type A consumers will be a price of $2,222 for a quantity of 333.33. Type B
consumers, if they chose this Type A package would get consumer surplus of
[(10 – 6.66) x 333.33]/2 + (6.66 x 333.33) - $2,222 = $555.50. The Type B package
would be designed to give this amount of consumer surplus. Therefore, the
Type B package would offer a quantity of 1000, at a price of $5,000 - $555.50 =
$4444.50. Each type of consumer would be willing to choose the package
intended for them. Total profit for the producer would be (100 x $2,222) + (100 x
$4444.50) = $666,650. As you can see, this design of the two packages delivers
increased profits for the producer.
Note that this example could readily be interpreted in terms of quality and price.
Simply assume that each consumer chooses only one of the products and that Q
reflects a varying level of quality. Type B consumers are willing to consume a
higher quality, while Type A consumers are willing only to consume a lower
quality. The producer designs price-quality packages to try to maximize profit
through consumer self-selection.
Sample Question, with a Solution:
1. There are two types of consumers in the market for digital books: Type A
consumers are wary and uncertain, Type B consumers are enthusiastic. The
demand per week by each Type A consumer is given on the graph on the next
page (labelled “Type A Demand”). The demand per week by each Type B
consumer is given on the graph below (labelled “Type B Demand”). You can
assume that the cost of producing digital books is zero. The monopoly
producer of digital books is trying to figure out how to price digital books so
that each type of consumer will pay a different price, so that the monopolist
can price discriminate. Since the consumers are not distinguished by any
obvious characteristic, the monopolist will have to design different pricequantity packages that encourage consumers to self-select into the two types.
For the purposes of this question, you can assume that there is one Type A
consumer and one Type B consumer in the market. The questions on the next
page refer to this diagram.
$ per book
80
60
Type B Demand
Type A
Demand
0
40
60
90
120
140
160
Number of digital
books
1 (a). If the monopolist were able to completely separate the Type A
consumer from the Type B consumer (so that no resale was possible, and so
that the Type A consumer could only buy the Type A package and the Type B
consumer could only buy the Type B package), what would be the price and
quantity of the “take-it-or-leave-it” package offered to the Type A consumer?
What would be the price and quantity of the package offered to the Type B
consumer?
(b) Now assume that it is not possible to completely separate the market into
two sections. Instead, both the Type A consumer and the Type B consumer
can purchase either package. Now, the monopolist must try to design pricequantity packages that will encourage the two types of consumers to selfselect the package designed for each. Assuming now that the Type A
package will contain 120 digital books and the Type B package will contain
160 digital books, what would be the price charged for the Type A package
and the price charged for the Type B package to encourage self-selection (but
provide as much revenue for the monopolist as is possible from these
packages)?
(c) Keeping the same assumptions as in part (b), how much consumer surplus
would the Type B consumer get, if he chose the Type B package?
(d) Now assume that the monopolist can change the size of the Type A
package. It no longer has to contain 120 digital books. Instead it could
contain 40 or 60 or 90 books. Of these four possibilities (i.e., 40 books, 60
books, 90 books, or 120 books), what is the profit-maximizing amount of
books for the monopolist to include in the Type A package (given that this
might affect the sale price of the Type B package)? What price should the
monopolist charge to the Type A consumer for this package?
(e) If the monopolist offers the Type A consumer the profit-maximizing
package from amongst the choices described in part (d) of this question, what
price will be offered to the Type B consumer for 160 digital books? How
much consumer surplus will the Type B consumer get?
Answer to Question 1:
1(a). The demand curve for Type A consumers is P = 60 – .5Q. The demand
curve for Type B consumers is P = 80 - .5Q. The maximum amount that Type
A consumers will consume is 120 units; for Type B consumers, this maximum
is 160 units. If the monopolist was able to separate these consumers and
charge them a maximum price, it would charge (60 x 120)/2 = $3600 to Type
A consumers and (80 x 160)/2 = $6400 to Type B consumers. Therefore the
price and quantity for Type A consumers would be 120 units at $3600 and for
Type B consumers it would be 160 units at $6400.
(b) To encourage Type A consumers to just be willing to consume 120 units,
the monopolist can charge them their maximum willingness-to-pay which is
the entire area under the demand curve (60 x 120)/2 = $3600. However, Type
B consumers could decide to consume this “Type A” package once it is
offered on the market. If they did they would gain consumer surplus of [(80 x
160]/2 – (3600) - (20 x 40)/2 = $6400 - $3600 - $400 = $2400 (this is calculated
by measuring the area under the entire Type B demand curve up to 160 units
of output and then subtracting areas A and C from it). Because of this, any
other package offered to Type B consumers has to offer them at least this
same amount of consumer surplus, or else they will not select it. Therefore, if
the monopolist wishes to get Type B consumers to purchase 160 units of
output, the maximum it can charge is $4000 (i.e., the total willingness to pay
under the Type B demand curve, minus the consumer surplus they need to be
given). The two packages are, therefore, 120 units at $3600 and 160 units at
$4000.
(c) The consumer surplus the Type B consumer would get, as calculated
above, would be $2400.
(d) By offering the Type A customers a somewhat lower quantity in the Type
A take-it-or-leave-it package, the monopolist will make less profit off each
Type A consumer. However, this will also mean that each Type B
consumer would get less consumer surplus by choosing to consume the
Type A package (it becomes less attractive to Type B consumers). As a
result, it is possible to charge more to Type B consumers for the “Type B”
package. What is gained on Type B consumers will, up to a point, more
than compensate for the losses on Type A consumers.
Let’s try these different alternatives (i.e., 40, 60, 90 and 120) in order and
calculate the results. If the monopolist were to sell a quantity of 40 to Type A
consumers, Type A consumers would be willing to pay [(3600 – (40 x 80)/2] =
$2000. However, Type B consumers could decide to consume this “Type A”
package once it is offered on the market. If they did they would gain
consumer surplus of [(6400 – 2000 – (60 x 120)/2] = $800. Because of this, any
other package offered to Type B consumers has to offer them at least this
same amount of consumer surplus, or else they will not select it. Therefore, if
the monopolist wishes to get Type B consumers to purchase 160 units of
output, the maximum it can charge is 6400 – 800 = $5600 (i.e., the total
willingness to pay under the Type B demand curve, minus the consumer
surplus they need to be given). The monopolist would make profit of: $2000
+ $5600 = $7600.
If the monopolist were to sell a quantity of 60 to Type A consumers, Type A
consumers would be willing to pay [(3600 – (30 x 60)/2] = $2700. However,
Type B consumers could decide to consume this “Type A” package once it is
offered on the market. If they did they would gain consumer surplus of
[(6400 – 2700 – (50 x 100)/2] = $1200. Because of this, any other package
offered to Type B consumers has to offer them at least this same amount of
consumer surplus, or else they will not select it. Therefore, if the monopolist
wishes to get Type B consumers to purchase 160 units of output, the
maximum it can charge is 6400 – 1200 = $5200 (i.e., the total willingness to
pay under the Type B demand curve, minus the consumer surplus they need
to be given). The monopolist would make profit of: $2700 + $5200 = $7900.
If the monopolist were to sell a quantity of 90 to Type A consumers, Type A
consumers would be willing to pay [(3600 – (15 x 30)/2] = $3375. However,
Type B consumers could decide to consume this “Type A” package once it is
offered on the market. If they did they would gain consumer surplus of
[(6400 – 3375 – (35 x 70)/2] = $1800. Because of this, any other package
offered to Type B consumers has to offer them at least this same amount of
consumer surplus, or else they will not select it. Therefore, if the monopolist
wishes to get Type B consumers to purchase 160 units of output, the
maximum it can charge is 6400 – 1800 = $4600 (i.e., the total willingness to
pay under the Type B demand curve, minus the consumer surplus they need
to be given). The monopolist would make profit of: $3375 + $4600 = $7975.
As we saw earlier, the profit if the Type A package contains 120 units would
be $3600 + $4000 = $7600. Clearly, the best package to offer is a Type A
package of 90 units for $3375 and a Type B package of 160 units for $4600.
The price charged to Type A consumers would be $3375 for 90 books.
(e) As described above, the Type B package would contain 160 books for
$4600. The Type B consumer will get $1800 worth of consumer surplus.
Question without a solution provided:
2 . There are two types of consumers in the market for digital books: Type A
consumers are wary and uncertain, Type B consumers are enthusiastic. The
demand per week by each Type A consumer is given by P = 20 - Q. The demand
per week by each Type B consumer is given by P = 20 – 0.5Q. You can assume
that the cost of producing digital books is zero. The monopoly producer of
digital books is trying to figure out how to price digital books so that each type
of consumer will pay a different price, so that the monopolist can price
discriminate. Since the consumers are not distinguished by any obvious
characteristic, the monopolist will have to design different price-quantity
packages that encourage consumers to self-select into the two types. For the
purposes of this question, you can assume that there is one Type A consumer
and one Type B consumer.
(a) Label the graph below and show the demand curves of the two types of
consumers. Assume that the monopolist designs price-quantity packages
with Q = 20 per week for Type A consumers and Q = 40 per week for Type B
consumers. (i) What prices will the monopolist charge for the Type A
package and the Type B package if she wants consumers to self-select into
purchasing the appropriate packages? (ii) on the graph, show the amount of
consumer surplus that each Type B consumer will get when he purchases the
appropriate package. Below the graph, calculate the amount of this consumer
surplus.
a. Now assume that the monopolist is trying to work out an even better
arrangement for price discrimination. The monopolist decides to offer a
different set of price-quantity packages to the consumers. She offers packages
with Q=12, intended to attract Type A consumers, and Q=40, intended to
attract Type B consumers.
(i)
how much will the monopolist now be able to charge Type A
consumers for the first package?
(ii)
how much will the monopolist be able to charge Type B
consumers for the second package?
(iii)
does this new price discrimination arrangement increase the
profits of the monopolist? By how much?