An important tool of price discrimination is to adjust prices according to quantity

purchased.

Non-linear pricing as this practice is known,

involves a discount in the price with increase in the number of units.

Non-linear pricing abounds and can take many different forms,

as the following examples illustrate.

In a software market or looking at software companies such as SAP,

we see that they typically set non-unit prices for the licenses.

So the more licenses you purchase, the lower the price per license.

Going to the gym typically gets cheaper as we buy it five sessions instead or one,

or if we buy ten sessions instead of only five.

There are plenty of examples as well in telecommunications industry.

Telco companies usually have a pricing scheme consisting

of a fixed monthly charge and a charge per minute.

As a consequence, the price per minute decreases with increased consumption.

Another example very similar to this, is as well used by airline companies,

more specifically in the airline frequent flyer programs.

They offer discounts for people who obviously fly a lot.

Starbucks charges as well less per ounce if you

order a venti instead of a grande or a tall.

The fundamental idea behind non-linear pricing is the Law

of Diminishing Marginal Utility.

A law of economics stating that as a person increases consumption of a product,

while keeping consumption of all other products constant.

There is a decline in the margin of utility that

person derives from consuming each additional unit of that product.

Let me illustrate this.

The more burgers you have already eaten today,

the less you would be willing to pay for yet another burger.

Let's look at the following example.

In this example we have information about the willingness to pay for

three different customer segments for different night staying in a hotel.

Most specifically we have three segments each consisting a 1000 customers.

And we have the willingness to pay for the first night, second, third, fourth, and

fifth night at the hotel.

So how do we find the optimal price considering that we only want to set

one price for all the different customers, for all the different nights.

As you can imagine,

we're going to approach this problem in a very similar way as we have done before.

We're going to start with the highest willingness to pay,

determine the revenue at that price point.

And then slowly go from this price point to all the other price points

in order to see how is this going to effect the overall revenue.

In this particular case,

we see that segment number 3 has the highest willingness to pay, which is $120.

They would be willing, at that price, to go for one night to the hotel.

So we can easily determine what is the overall revenue that

we are able to charge.

In this particular case 1,000 customers of segment 3 will be willing to pay $120,

so the overall revenue that we're able to charge is $120,000.

This now brings us to the next relevant price point.

In this particular case it would be $100.

So we now have as well segment 2 joining in.

Segment 2 now will be willing as well to go for one night to the hotel,

while actually now segment 3 will be willing to go two nights to the hotel.

So we'll have a total of 3,000 nights at the price of $100,

which is a total of $300,000 in revenues.

So we've already seen a significant increase in the revenue we're able to

generate just by dropping our price from 120 to $100.

Let’s do one more example.

We see that the next relevant price point is $90.

In this particular case we now see as well that segment 1 would be joining in

as well.

For segment 2, segment 3, nothing is really changing.

So in total we're now able to sell a total of 4,000 nights at a price of $90.

So that would maximize or would increase our revenues to $360,000.

So yet again another increase.

If you would continue this in going down from one price to the other.

What we would actually see is that the overall revenue is being maximized at

a price point of $55.

At that price we're going to be able to sell a total of 9,000 nights,

resulting in a overall revenue of $495,000.

You can obviously try this now in order to see whether you come up with a very

same result.

Now we are going to approach the problem from a slightly different perspective.

Obviously this module is about nonlinear pricing.

So that idea would be, could we maybe charge different prices for

the different nights staying at the hotel?

So essentially having one price for

the first night at the hotel, another price for

the second night at the hotel, the third night, fourth night, and fifth night.

So let's approach this in exactly the same way

only that we look now in this chart at each line individually.

And we're going to figure out what is the optimal price for

that particular night at the hotel.

So let's look at the very first night.

We have still this three different segments with the willingness to pay of

$90, 100 and 120.

Obviously if you're charged a price of $120, we would only sell one night to

segment number 3 or a total of 1,000 nights to segment number 3.

If we drop our price to $100, we now would be able to sell a total of 2,000 nights at

$100 which would already increase our revenues we're able to charge for

the first night from $120,000 to $200,000 in revenue.

But let's look what would happen if we drop our price even further for

the first night, dropping it to $90.

Now being able to sell not only to the second and

third segment but including as well the first segment.

So in total we would be able to sell 3,000 nights at a price of

$90 which would result in $270,000 in revenues.

So we clearly see that the maximizing price for the first night is $90.

Let's repeat the very same approach as well for our second night.

We now have the willingness to pay of 60 for the first segment, 75 for

the second segment, and $100 for the third segment.

Again we start at the highest price at $100.

We would only sell 1000 nights right, resulting in $100,000 in revenues.

We can drop that to 75 which would basically result in $150,000.

Now we can drop the price further.

We're now going to drop it to $60, which would mean we're going to be able to

sell a total of 3,000 nights resulting in $180,000.

Again in this particular case we see dropping the price to

the lowest price point at $60 clearly maximizes our revenues.

What we can now do is continue the very same approach for the third, fourth and

fifth night.

The resulting price or the price that optimizes our revenues for

the different nights at the hotel would be as if we have seen in the first case $90.

In the second case $60, in the third case 55,

in the fourth $40 and 35 in the last, in the fifth night.

What we actually would see is that now we can significantly increase our

overall avenue at these different price points.

Remember before, at a price of 55, we were able to sell a total of 9,000 nights.

We're now actually going to be able to increase the overall volume being sold to

11,000 nights.

The same is actually true as well for the revenue we have achieved.

The revenue has gone up from $499,000 to $675,000.

That's a whopping increase of 36%.

So as we have seen in this very simple example,

nonlinear pricing is very powerful.

And there are many arguments for the power of nonlinear pricing.

However it is not for everyone.

For nonlinear pricing to work resale by the buyer must be preventable.

In the same vein, buyers should be restrained from combining their demand.

One example might be the German railway company, Deutsche Bahn and their BahnCard

100 for frequent travelers which entitles them to unlimited travel for 12 months.

As you can imagine what actually happened is

that some of the travelers were sharing this card.

So as a consequence after certain misuse, Deutsche Bahn included as well a picture

of the card owner to prevent them from sharing this card across different users.

A similar problem occurs with frequent flyer miles where

brokers such as Flip My Miles are buying and selling these bonuses.

This was definitely not intended by the airlines.