Dear reader,

no, the title of this post is not meant to describe the quality and frequency of my posts! It wants to be, instead, a reflection of two commonly used heuristics: “it has a good quality/price ratio” and “something is better than nothing”.

For a long time I had a strong negative opinion about those sentences, one that I shared liberally, but after some thought and study, I’ve come to regard my views as robust and tenable, although not always expressed with the best etiquette, as you’ll discover at the end of this tirade.

The Hypothesis – How You Spend Your Money

Let’s start by considering a hypothetical object/service you are willing to spend money for, let’s say a laptop. You know there are innumerable models for sale, and you can fine tune or build from scratch any model you want, so there is a massive amount of options to choose from. Since you are not one of the wealthiest men in the world, you set a budget for yourself, intending to squeeze every bit of value from your hard-earned money. What then? You start looking at all models and combinations of parts within your budget, and you soon realise that it’s all a matter of tradeoffs: some you might consider, some you would not endorse.

Some examples of tradeoffs you might accept:

  1. A faster video card instead of more memory, since you like playing video games
  2. More GBs on your hard drive instead of a better quality webcam, since you want to carry around many photographs
  3. A wider screen instead of better quality speakers, for better watching the pictures mentioned above
  4. Less stylish design, but more money left in your wallet
  5. Slower hard drive, again for saving bucks.

Some examples of tradeoffs you would never consider:

  1. More GBs on your hard drive than point 2 above, but no webcam at all
  2. An even bigger screen than point 3 above, with crappy speakers
  3. A hard drive so slow that it is almost free, but makes the system unusable
  4. Saving money by buying unreliable components.

In the next section, we will go a little more technical and describe a way that could measure our preferences.

Utility Functions – How You Get Value for Money

Let’s reduce, for simplicity’s sake, the elements of our analysis to just “price” and a generic variable named “quality”, acting as a general index for the remaining considerations. Obviously, more quality at the same price or less money for the same quality is a win, but you might want to trade one for the other and still perceive that you had the same overall bargain. This concept is best investigated by what goes under the name of “utility function”: a function mapping the utility (i.e. the overall value) of a situation given the variables involved. You can check the ever-useful Wikipedia for more information.

Constructing a utility function can be useful to analyse and help decision-making in various instances; here are some examples:

  • accepting bugs in software development against respect of due dates for software deployment,
  • number and gravity of judicial errors against duration of the processes for civil justice,
  • calibrating the quantity of waste produced and safety for the environment compared to cost cuts and elapsed time in industrial processes.

The geometrical shape or mathematical form of a utility function is not so important per se. What matters is another concept, more useful in examining how our decisions can be shaped by the intrinsic characteristics of utility functions, that is called…

Indifference Curves – How to Tell the Difference

Indifference curves represent the points in your diagram which have the same perceived values, sketching out those changes you are indifferent to. Points 1 to 5 in the first list of items above are examples of movements along an indifference line; while moves like choosing another brand or getting a discount to have better quality at the same price, are putting you on a different indifference curve altogether, i.e. on a higher utility value.

In other words, indifference curves are the “same height lines” of utility functions: if you move along them, you neither go up or down.

I learned about utility function in the book “How to Measure Anything: Finding the Value of Intangibles in Business”, by Douglas W. Hubbard. It is an excellent read, one that I would encourage anyone to undertake. One of the cornerstones of the book is that “a measure is any process that helps reducing uncertainty about a given quantity”. As a physicist, at first I regarded the definition as too broad, but I came to appreciate its power and precision. Three days ago a friend of mine, a consultant, told me that he needed his working life to take a turn: I will point him toward Hubbard’s work for sure! The book is an exceptional choice for anyone who wishes to inject more care and consciousness in his business or activity, by making more informed business decisions.

A Toy Model – Back to School

Let’s get back to our business. I’ll try to construct an example that will be useful in showing my point. Consider the utility function U for our hypothetical laptop bargain. We will take into account two variables, price P and number/severity of defects (or deficiencies) D. Intuitively, in an ideal world, we’d like to have zero deficiencies in our laptop and get it for free. Vice versa, more defects lower the utility, as well as a higher price tag.

Let’s go to the chalkboard and consider the following mathematical model:

In this model, the utility is a generic “1 unit” (it could have been any other number, I just put 1 for simplicity) divided by the product of price and defects.

Here you can find a figure of three indifference curves for that model, those who sport a constant product of P times D. They are plotted in a graph that puts Defects on the vertical axis and Price in the horizontal one:

The curve closer to the axis, number 1, is the one with the highest utility (i.e. a better overall bargain), while the upper one, number 3, is the worst case, having a higher price and/or more deficiencies. The higher a curve is on the graph, the lower the associated utility.

Banging on the Threshold – Limits of the Model

This picture suggests that you could, indefinitely, cut defects and raise the price. Well, in some cases this may be true: some people are willing to spend humongous fortunes for unique and immaculate pieces. The human race expended an inordinate amount of money on some parts/services, like going to the Moon or the Large Hadron Collider (and they paid their dividends many times over).

However, I’m interested in the other branch of the curves. The question is: can you, as the figure implies, indefinitely increase defects by reducing costs? Can I sell you a half rotten tomato for half the price? Offer a laptop with an exhausted battery for a quarter of the price? Tempt you with a punctured spare tire for a tenth of its undamaged value?

I bet I can’t.

There is a threshold at which we are no longer willing to accept a tradeoff, as we have exemplified above. What does this mean, regarding our graph? We need to trace a line at the height of the maximum level of tolerable defects and cancel out every portion of the curves that lies above it, as in the following figure:

There is a whole region of the graph that is, in practice, inaccessible to our experience: no matter how hard we elaborate on our calculation of theoretical utility, no customer will ever buy our product. Often we get lured into thinking that going for lower and lower prices is a good strategy (and sometimes it is if you can sustain quality), but we have to realise that there is a threshold. As Seth Godin puts it in a brilliant post:

But the problem with the race to the bottom is that you might win.

Seth Godin

Even worse, you could arrive second in the race. Better to set the threshold and focus on never – NEVER – crossing it while engaging with your customers.

Conclusions

So, what can we conclude after this venture into mathematics? Let’s go back to one of the contested sentences: “it has a good quality/price ratio”. We have seen that has price goes down we can in principle reduce quality but maintain the same ratio, but we get to a barrier. So no, the quality-price ratio is not the be-all-end-all index for a bargain. For the same reason, not always “something is better than nothing” if that something does not cut it. Neither does add more and more quantity helps when we are beyond tolerable defects.

In other words, as I sometimes bluntly put it: “No thanks, I don’t like to eat free shit” and “No, not even if you add more to the plate, thank you, Sir”.

Oh, one last word to get to the title of this post: what does happen if you take into account the time factor?

Well… nobody likes faster and faster shit.

Until next time, get your shit together.