# selfselection.pricediscrimination_000

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NOTES ON PRICE DISCRIMINATION BY SELF-SELECTION – SECOND
DEGREE PRICE DISCRIMINATION

-   Assume two types of consumer with different demands for good; Type A has
high demand and Type B has low demand.

-   Producer wants to sell at higher price to Type A consumers and lower price
to Type B consumers. However, consumers are not readily identifiable,
except by their purchasing behaviour. Challenge for 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 A consumers and low price and low quantity for the Type
B consumers.

-   Necessary condition for attracting Type A consumers to the more expensive
package is to give them at least as much consumer surplus as they could get
by purchasing the “Type B” package.

-   The profit-maximizing set of price-quantity packages will be one that gives
Type B consumers less than they would be willing to purchase at the price
charged, in order to charge more to Type A 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 B
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.
This would be called quality discrimination instead of price discrimination.

Example:

Type A demand: P = 10 - .01Q
Type B demand: P = 10 - .02Q
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 demand curve. For Type A consumers this means a
quantity of 1000 and a price of (10 x 1000)/2 = \$5,000. For Type B consumers, this
means a quantity of 500 at a price of (10 x 500)/2 = \$2,500. Profit is expected, by
the producer, to be 100 x \$5,000 + 100 x \$2,500 = \$750,000. However, this design of
the price-quantity packages is flawed. If Type A consumers purchase the Type B
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 A package, they
will get consumer surplus of zero. So, with this package design, they will prefer to
purchase the Type B 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 A price so that the consumer surplus from purchasing the Type A
package is at least as good as what it would be if the Type A consumer purchased
the Type B package. For Type B consumers, this means a quantity of 500 at a price
of \$2,500 (same as above). For Type A consumers, this means a quantity of 1000 at
a price of \$5,000 - \$1,250 = \$3,750 (i.e., leaving Type A consumers a consumer
surplus of \$1,250). Consumers will now self-select into two groups, with the Type B
consumers paying a low price for a low quantity and Type A 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 B
consumers (making this package also less attractive for the Type A consumers).
Charge more for the Type A package than above but still leave the Type A
consumer with the same consumer surplus as he would have gotten from consuming
the Type B package (because the Type B package is now smaller, this is a smaller
amount of consumer surplus than before). As an example, make the Type B
package include a quantity of 333.33, rather than 500. We can calculate the area
under the Type B 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 will be a
price of \$2,222 for a quantity of 333.33. Type A consumers, if they chose this Type
B package would get consumer surplus of [(10 – 6.66) x 333.33]/2 + (6.66 x 333.33) -
\$2,222 = \$555.50. The Type A package would be designed to give this amount of
consumer surplus. Therefore, the Type A 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 product and that Q
reflects a varying level of quality. Type A consumers are willing to consume a
higher quality, while Type B consumers are willing only to consume a lower quality.
The producer designs price-quality packages to try to maximize profit through
consumer self-selection.

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