Many readers will already know that, in
addition to editing The Customer Service Blog, I also do lecturing and training
work in the areas of Business Studies, Management and Marketing. One area of
business that has always fascinated me is the subject of pricing. It’s a
subject that I lecture on at both undergraduate and postgraduate levels.

So just how do companies decide what
to charge for their products? How can they make a rational decision on the
level of prices to charge that will enable them to make enough sales, while
still being profitable? If the price is too high then customers will buy from elsewhere.
But if the price is too low, then the company might sell a lot of products, but
make no profit from these sales.

While flicking through an academic
journal this week, I came across a fascinating article by two mathematicians at
the University of Portsmouth, who have taken a more mathematical approach to
pricing.

As a lecturer in Business Studies, I
would argue that there are many other practical considerations when setting the
price of a product. But nevertheless, I think their article offers a really
interesting theoretical view of the ‘perceived value’ of products, and the
whole concept of attaching a ‘price’ to any product.

I have reproduced the article below, with permission under Creative Commons Licence.

Darren Bugg

Editor, The Customer Service Blog

**Are prices real? How ghosts of
calculus and physics influenced what we pay for things today**

With inflation in the UK and around
the world threatening to spiral out of control, prices of everything from milk
to oil, energy and Christmas presents are a concern for most of us. Most people
understand prices as simply the result of supply and demand - an agreement
between sellers and buyers about how much something should cost.

But there’s more to these numbers,
starting with the mathematically philosophical question: do prices even exist?

Suppose you are selling a bicycle to
clear up space in the shed. You are a supplier seeking demand. You advertise
for £100. You get an offer for £90, and ultimately strike a deal at £95. It is
tempting to declare that the price of the bicycle was £95.

It is more accurate to view the
situation as follows. Your initial £100 offer is a signal to buyers. The
response of £90 from a buyer is a disagreement with your initial offer. The
'true' price lies somewhere in between. Paradoxically, as soon as the deal is struck,
the price evaporates since there are no future deals to even consider.

During the disagreement phase, we
could say that the price was between £90 and £100. Once an agreement was
reached, the deal was executed and immediately the price no longer had any meaning.
Was the price ever £95?

The common view suggests that it is
agreement between buyers and sellers about future deals that sets the price,
representing the inherent value of the asset. But each party has their own
perceived utility from the execution of the deal, representing what they see as
the inherent 'value' of the asset.

Suppose the buyer’s perceived value of
owning this bike (including transportation benefit, status, enjoyment and so
on) is £95. That means she will certainly not pay more than £95. Similarly, if
the seller’s perceived value of owning the bike (including negative aspects
such as space taken up in the shed) is also £95, then she will not sell for
less than £95. It would seem that a deal is reached at £95. However, this
disregards the inevitable extra costs and effort associated with executing the
deal itself (time used and energy spent).

It is precisely because of a disagreement
on the item’s inherent value that they executed a deal. All we can deduce from
the £95 'price' is that the inherent value of the bike for the seller was less
than £95, and for the buyer it was more than £95. In this way, both parties
gained value from the exchange.

The £95 final price of the bike is
borne by the tension of disagreement between buyer and seller. But any attempt
to pinpoint it as a single number is illusory since as soon as the deal is
struck, the price disappears. All we can really say, based on the interaction
above, is that the 'true' price of the bike is between £90 and £100.

**Ghosts in the system**

How did prices, these ghosts of
departed trading deals, rise to prominence? The answer lies with other ghosts.
The fathers of modern physics, Isaac Newton and Gottfried Wilhelm Leibniz,
developed calculus in the 17th century as a mathematical tool for physics. They
manipulated infinitesimal numbers - elusive quantities that managed to be
positive and zero at the same time. In 1734, the Irish philosopher George
Berkeley in his book The Analyst delivered a famous critique of these ‘numbers’:
He said: “They are neither finite Quantities nor Quantities infinitely small,
nor yet nothing. May we not call them the Ghosts of departed Quantities?”

Towards the end of the 19th century,
mathematicians laid calculus on solid foundations by constructing the 'real
numbers'. Today, these are the default numbers we use every day, including 0,
1, 1.5, -1.323, and π.

In 1900 the French mathematician Louis
Bachelier took a bold step. He saw the tremendous power of calculus and
wondered if it could be utilised not just in physics, but also in finance.

Calculus works well in physics,
engineering and technology because the real numbers aptly describe relevant
quantities. For instance, your geographical position in Google Maps is
pinpointed by real numbers (latitude and longitude), and tools from calculus
are used directly on these numbers for navigation.

Bachelier needed a financial concept
that can be measured by real numbers. The price concept, as elusive as it may
be, fit the bill. It allowed Bachelier to model financial assets as if they
were moving particles, and thus exploit the mathematics of physics.

**Finance revolution needed**

Physics thrived on the ghosts of
departed quantities and modern finance thrives on the ghosts of departed
trades. Just like in physics, this theory worked well for a while, but
eventually reached its limits.

The realisation that the real numbers
were too restrictive for the purposes of modern physics led scientists to
invent new mathematics. Classical physics, which reigned supreme since Newton
and Leibniz, is now seen as the predecessor of quantum mechanics - a revolution
that is reshaping our lives.

In their book Trades, Quotes and
Prices, leading experts in econophysics poignantly describe the current
situation in finance:

*“Prices are fleeting quantities,
hypersensitive to fluctuations in order flow and prone to endogenous crashes.
So why do we put such blind faith in them? The belief that prices are faithful
estimates of values of assets, portfolios and firms has dominated the study of
financial markets for decades. Despite widespread adoption this approach has many
unfortunate consequences. Perhaps it is time for a better paradigm.”*

This call for a new paradigm is
leading researchers to draw inspiration from the frontiers of modern
mathematics. From the very recent advances of quantum mathematics to the depths
of category theory, nobody yet knows what the answer is, but when they find it,
it will surely shape the economy of the future.

Ittay Weiss

Mathematician, University of Portsmouth

Samuel Lloyd-Lindholm

PhD student, Mathematics, University of Portsmouth

*This article is republished from The
Conversation under a Creative Commons license.***To see hundreds more articles click here to visit our archive**