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

Â purchased.

Â Non-linear pricing as this practice is known,

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

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

Â as the following examples illustrate.

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

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

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

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

Â or if we buy ten sessions instead of only five.

Â There are plenty of examples as well in telecommunications industry.

Â Telco companies usually have a pricing scheme consisting

Â of a fixed monthly charge and a charge per minute.

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

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

Â more specifically in the airline frequent flyer programs.

Â They offer discounts for people who obviously fly a lot.

Â Starbucks charges as well less per ounce if you

Â order a venti instead of a grande or a tall.

Â The fundamental idea behind non-linear pricing is the Law

Â of Diminishing Marginal Utility.

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

Â while keeping consumption of all other products constant.

Â There is a decline in the margin of utility that

Â person derives from consuming each additional unit of that product.

Â Let me illustrate this.

Â The more burgers you have already eaten today,

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

Â Let's look at the following example.

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

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

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

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

Â fifth night at the hotel.

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

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

Â As you can imagine,

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

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

Â determine the revenue at that price point.

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

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

Â In this particular case,

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

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

Â So we can easily determine what is the overall revenue that

Â we are able to charge.

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

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

Â This now brings us to the next relevant price point.

Â In this particular case it would be $100.

Â So we now have as well segment 2 joining in.

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

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

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

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

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

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

Â Letâ€™s do one more example.

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

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

Â as well.

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

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

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

Â So yet again another increase.

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

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

Â a price point of $55.

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

Â resulting in a overall revenue of $495,000.

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

Â same result.

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

Â Obviously this module is about nonlinear pricing.

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

Â the different nights staying at the hotel?

Â So essentially having one price for

Â the first night at the hotel, another price for

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

Â So let's approach this in exactly the same way

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

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

Â that particular night at the hotel.

Â So let's look at the very first night.

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

Â $90, 100 and 120.

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

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

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

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

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

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

Â the first night, dropping it to $90.

Â Now being able to sell not only to the second and

Â third segment but including as well the first segment.

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

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

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

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

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

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

Â Again we start at the highest price at $100.

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

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

Â Now we can drop the price further.

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

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

Â Again in this particular case we see dropping the price to

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

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

Â fifth night.

Â The resulting price or the price that optimizes our revenues for

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

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

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

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

Â overall avenue at these different price points.

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

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

Â 11,000 nights.

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

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

Â That's a whopping increase of 36%.

Â So as we have seen in this very simple example,

Â nonlinear pricing is very powerful.

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

Â However it is not for everyone.

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

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

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

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

Â As you can imagine what actually happened is

Â that some of the travelers were sharing this card.

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

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

Â A similar problem occurs with frequent flyer miles where

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

Â This was definitely not intended by the airlines.

Â