Because most philosophies that frown on reproduction don't survive.

Tuesday, July 03, 2007

Applied Math

Quite some time ago I put out a request for recommendations on books with which to refresh my pretty much entirely forgotten calculus knowledge. Several readers provided some very helpful suggestions, and as a result, I ended up with a few new books on the shelf, a couple of which I've been slowly working through ever since (given my habit of having several books of different sorts going at any given time, my pace is at times glacial).

However, I was excited the other day to finally (ten years after finishing the Saxon Calculus book in high school) manage to use calculus at work. I believe the following is sufficiently scrubbed of any company specific or confidential information to be sharable, and for those others out there with an interest in how things work (and/or mathematics in particular -- though I fear any true math afficianados will find my efforts rather simplistic) I thought I'd share the result.

Much of what I do at the moment consists of setting the list price for products, and trying to determine what is the best price at which we will sell a large number of units (usually done by cutting price) while still booking enough profit on each unit to make money for the company.

If you track the results of this experimentation over time, you can plot the data points out on a graph of price (Y axis) versus unit volume (X axis) such as this:

This line conforms to the equation: [quantity] = [slope]*[price]+[intercept]

You can then use the equation, and the values of slope and intercept for the line, to calculate estimated quantities at additional price points.

Now by using these now known values to solve for revenue and profit (assuming that you know cost), you can calculate the likely amounts of revenue and profit at various price points, as well as looking at their historical values.

You can plot these values on a graph, and see that there is a curve for revenue and one for profit.

Now say that you want to find the price at which the most profit would be realized. You take the equation for the profit curve and take its derivative. When the derivative of the equation equals zero, you have the point of the curve where it is flat, as it turns over from ascending to descending. Solve for the price that will produce that point equal to zero, and you have the price that will result in the maximum profit -- or at least as close to it as the correlation of your data to reality will allow.

Fun, eh?


Anonymous said...

Indeed, fun it is. Math has always been "fun" for me. I came home from kindergarten one day and told my mom that math was my favorite subject. Been hooked ever since.

One of the hats I wear is power quality focal. One thing that I have to study is current and voltage harmonics. The math can get rather hairy here: Fourier analysis.

Glad to see you being able to use advanced math at work.

Pro Ecclesia said...


An Economics major who nevertheless hated this stuff

John Farrell said... geek! :)

I myself am actually doing the same thing...going through some remedial calculus in the quixotic attempt to be able to at least appreciate better classic papers in physics.

Anonymous said...

Hey Darwin,

What are your standard errors on your OLS estimates? Seems like those might play an important role if you're going to use your regression equation for prediction of something else (e.g., profit in this case).

Darwin said...

j. christian,

As you surmise, deviation is the bit problem in this whole situation. The data we deal with is weekly, and because of product lifecycles it's a bad idea to work with more than about 15 weeks of data (and even that can be enough to introduce issues in some cases) so we really don't have enough data points to be statistically rigorous.

We also have issues in that there are other factors (advertising, seasonality, etc.) that affect sell rates at least as much as price.

So at best the information is aproximate, and we use it for dictional guideance, but certainly not for financial forecasts.

1990bluejay said...

Ufff.... bad memories of 300 and 400 level microeconomics came rushing back as did the calculus. Yeah, using regression for forecasting is tricky, especially since what's taught in econometrics almost always is for regression until you hit the advanced levels. Don't envy you a bit and I have a degree with an economics concentration.

Anonymous said...

I hear you. I get nervous when thinking about forecasting with fat confidence intervals and "out of range" regressors, but as you say, management ain't betting the future on this kind of stuff!

Still, it warms the heart of this (somewhat disillusioned) economist to see this stuff put into practice!

Darwin said...

Yeah, it's approximate, but it does help to convince some of the less analytical marketeers, who have a tendency to argue that by increasing price you can always increase total revenue.

On the rev/profit graph I show there, I had to extrapolate in order to show the revenue downturn, although my real data points charted a very nice margin curve. Still, the point is important to get across.