Posts Tagged ‘statistics’

Some Thoughts On Pricing, Part II

October 16th, 2008 1 comment

Let's say instead of a uniform distribution, as described in "Some Thoughts On Pricing, Part I" and its simulation, comparable products' (say apple juice) prices hover around 10.00 pesos, with standard deviation 3.00 pesos, and the distribution is normal (by omniscience).  If there is just one other competing company (F1) against me I could have ascertained the mean and variance (or standard deviation) by sampling its pricing across 1000 shops, and obtained them by maximum likelihood.  If there are more than one companies (several tens) in the market competing against me perhaps I could have derived the distribution parameters by maximum likelihood across a shelf of a single shop (by ergodicity), always assuming prices across companies are realizations of the same distribution.

An advantage of the Gaussian distribution is that the domain is the entire real line, and so probabilities are assigned for every conceivable price (even negative prices, meaning I as a supplier choose to pay you the consumer to take the product!).  Expected profit is likewise therefore defined everywhere of practical interest.

Following as before, let's say again that the cost of producing a single unit of product is 1 for me (*).  The probability of my selling by  being the lowest price at a particular shop is one minus the cumulative density of a Gaussian:

1 - \frac{1}{2}\big(1+erf(\frac{x - \mu}{\sigma \sqrt{2}})\big) ,

or in this case:

1 - \frac{1}{2}\big(1+erf(\frac{x - 10}{3 \sqrt{2}})\big) ,

and the probability of failing at a sale is the cumulative density:

\frac{1}{2}\big(1+erf(\frac{x - 10}{3 \sqrt{2}})\big)

As before, for one competing company, my expected profit is:

 E_p(x) = \Big(1 - \frac{1}{2}\big(1+erf(\frac{x - 10}{3 \sqrt{2}})\big)\Big) \cdot (x-1) + \frac{1}{2}\big(1+erf(\frac{x - 10}{3 \sqrt{2}})\big) \cdot (-1)

and if there are n competing companies,

 E_p(x) =\Big(1 - \frac{1}{2}\big(1+erf(\frac{x - 10}{3 \sqrt{2}})\big)\Big)^n (x-1) +

 + \Big(1 - \Big(1 - \frac{1}{2}\big(1+erf(\frac{x - 10}{3 \sqrt{2}})\big)\Big)^n\Big)(-1)

Here I have graphed price to expected profit for 1...5, 10, 20, and 40 competing companies.  I also used a computer to obtain approximations of zeroes and extrema.

One of the remarkable things about this family of functions is that maximum expected profit can be achieved for any competition in the market, no matter if there are few companies competing against me or thousands, although if there are several hundreds or thousands at some point my maximum expected profit is negative (but never less than -1, of course, since I cannot lose more than what I cannot sell).  Here's a graph of this phenomenon:

Another remarkable thing is that maxima seem to lie on a line defined for  E_p(x) > -1 , although I am not entirely sure of the behavior at near  E_p(x) = -1 (maxima may in fact be asymptotic to such).  Break-even prices become lower and lower the more competition there is, until finally there is a point at which, even though I may sell some product of apple juice, I can never make up for my expense (graph above).  Lastly, here again maximal expected profit is achieved at lower pricing the more competition there is in the market.

At any rate, the lesson would seem to be: the more competition there is, the more the piece of the pie is divided up, the lesser the maximal profit for me.

In my next post, I will describe what happens to expected profit when the Gaussian distribution of prices across competing firms is more certain: by this I mean the standard deviation or variance is smaller, which translates to there being a general agreement or consensus across suppliers about what the price of a product should be.

Some Thoughts On Pricing, Part I - Simulation

October 15th, 2008 No comments

Here are the results of a simulation of competition against 1 other company and against 5 other companies, at maximum expected profit price (and 1000 trials/shops).  Everything is as in "Some Thoughts On Pricing, Part I" (axioms, distributions, etc.).  Just for kicks, I also simulated the cost functions of the companies as uniformly distributed between 1.00 and 4.00 pesos, and calculated their profits (severely negative, which suggests pricing differently at each shop, even within a range, is not a good strategy at all!).  The results are in good agreement with theory.



Some Thoughts On Pricing, Part I

October 7th, 2008 3 comments

Meyer, a close friend of mine, and I live in a country of unique and colorful beauty, Mexico, and our town Guadalajara additionally inherits a traditional Spanish-colonial and picturesque quaintness. Meyer and I are good pals and you can often find us shopping together: we live close by enough to make it convenient and worthwhile (loading and unloading and placing in the cupboard is easier and faster with a helping hand). From another point of view, it gives us the opportunity to chit-chat and share ideas, and we definitely appreciate each other’s company.

In examining a variety of comparable products the other day, say concentrated apple juice, I noticed that, even across generic brands, there exists a price fluctuation that I simply could not explain. One might cost 7.50 pesos, another might cost 10, and yet another might cost 8. This completely intrigued me. Because I had recently worked in the tequila sector on a consulting project, I moved on the the spirits section. For different, generic, and comparable-in-quality tequila products, again significant fluctuation. The branded products also displayed a not-so-tight price range.

I put myself in the shoes of the supplier of the product. If I am determining what the acquisition price of my product should be to maximize my expected profit, and I am competing against a plethora of other products of comparable quality, I am in effect bidding against the other products for the lowest possible price. I am bidding for the lowest possible price because a rational consumer will always choose the cheapest product across comparable-quality. The bid is a sealed one, in which it is impossible for me the supplier to determine how other products will price (since I cannot know with exactness other suppliers’ cost functions, inventory levels, etc.), although I can get a fair idea by sampling directly from the shelves, say.

These observations, then, can be elevated to the category of axioms: first (C1), that the rational consumer knows that the products that he is offered at the shelf are comparable in quality, and that he will always chose the cheaper of the products, much as I did when I chose the cheapest generic brand of apple juice available; second (S1), that the supplier is bidding for the lowest price in a sealed bid against other comparable products, and this gives rise to price fluctuation of the products; third (S2), that the supplier can inform himself by sampling the shelves of the particular price fluctuation, and can determine by careful methods a general distribution of the prices (or in fact knows a priori the distribution because he is omniscient); fourth (S3), that the supplier knows his cost function.

For example, say I am an apple juice company, *, and competing against me there is a single other company (or many other companies with very tight price center of mass), F1, and our products are comparable-in-quality (S1).  Now let's say there are 1000 different shops or so that can sell our product, but only one rational consumer (C1) per shop is interested in the product and comes in that day.  Let's say that at these shops either you sell the product that day at the time of purchase or it's returned to you, and you must inventory it or destroy it (perhaps the apple juice is no good after 1 day).  Furthermore let's say you can decide what price you'll set for the product, and F1's price can be modeled as the value of a random variable that is uniformly distributed within a range, maybe anywhere between 2 to 15 pesos (S2). I, *, only need to invest 1 peso per product produced (S3).

The probability of me selling at any shop, if I price at x, is \frac{15-x}{13} (one minus the cumulative density): if I price at the lower bound 2 I am guaranteed to sell, but if I sell at 15 I am guaranteed not to sell.  My probability of not-selling or failure is 1-\frac{15-x}{13}, or \frac{x-2}{13} (cumulative density).  Being a profit maximizing company, what should I price my products in order to sell as much as possible and earn as much as possible?  If I do sell, then I earn x-1 pesos.  If I don't sell, I have to destroy or inventory my product, and I have lost 1 peso.  I can easily calculate my expected profit as a function of x:

E_p(x) = \underbrace{ \frac{15 - x}{13} \cdot (x-1)}_{success/win/sell} + \underbrace{\Big(1-\frac{15-x}{13}\Big)\cdot (-1)}_{failure/lose/didn't-sell}


E_p(x) = \frac{15 - x}{13} \cdot (x-1) + \frac{x-2}{13}\cdot (-1)


E_p(x) = \frac{1}{13} \cdot (-x^2 + 15 x - 13)

which is, luckily in this particular case, an inverted parabola with maximum that can be readily calculated at:

-2x + 15 = 0

or x=7.5 pesos.  At this price my expected profit per unit is about 3.33 pesos. For a 1000 shops my expected profit is about 3330 pesos.

I must mention that I based this idea on a bidding problem from Ross's excelent book titled "A First Course in Probability," but the core concept has morphed significantly in this particular application.

Now suppose that, besides me, there are 2 companies (or two tight center-of-masses), F1 and F2, each pricing with the same uniform probability distribution as before (alternatively, one can choose to think the prices of F1 and F2 are two realizations of the known uniform distribution), and each are not influenced by the other (they speculate on their own), and are therefore independent.  To sell product at the shops, now I must be twice successful: I must win against both F1 and F2.  My probability of winning now is: (\frac{15-x}{13})^2 by independence (and of losing one minus that), for which the expected profit is:

E_p(x) = \Big(\frac{15 - x}{13}\Big)^2 \cdot (x-1) + \Big(1-(\frac{15-x}{13})^2\Big)\cdot (-1)

Further developing the idea, for n companies labeled Fn, my expected profit is:

E_p(x) = \Big(\frac{15 - x}{13}\Big)^n \cdot (x-1) + \Big(1-(\frac{15-x}{13})^n\Big)\cdot (-1)

Graphing these functions it is obvious that the break-even points for 2 competing companies, 3 competing companies, and so on (zeroes of expected profit functions) are closer and closer to the lower bound.  Also maximum expected profit can be achieved only at the lower bound for more competition in the market.  Indeed, prices do fall the more competition there exists.

I think using a Gaussian distribution instead of a uniform is much more interesting (probabilities are not constrained to an interval, e.g.) and potentially more realistic, and that is the subject of my next post.