Sunday, July 31, 2011

The Curse of Basic Numeracy (or Why I Keep Gaining Weight)

Like all O.R. people (I hope!), and perhaps 2% of political officeholders in the U.S. (fewer if you restrict the count to Congress), I am functionally numerate. In particular, when ordering food at restaurants or high-calorie beverages (like the one I'm slurping as I type this) at coffee shops, I recognize when multiple sizes of the same order are priced so as to give the consumer an economy of scale.  For instance, the price of the frozen mocha in front of me comes from the following table:


Marginal Cost
Average Cost
10 3.75 0.375 0.3750
16 4.35 0.100 0.2719
20 4.65 0.075 0.2325
24 5.05 0.100 0.2104

There is a modest diseconomy of scale in marginal cost going from the 20 oz. size to the 24 oz. size, the reason for which eludes me. That in turn bothers me at one level (I teach in a business school, so I feel that I should understand pricing models) and not at another (I occasionally fly on commercial airlines, whose ticket pricing algorithms appear to be the first "practical" application of chaos theory).

Obviously, being an O.R. person (and a business prof at that), I was compelled to order either the 20 oz. size (best marginal cost) or the 24 oz. size (best average cost). Thus do I, being perfectly rational, give the locker room scale one more reason to shudder when I walk in the door. I did order a reduced-calorie version, which is somewhat like playing stickball on a busy one-way street as opposed to a busy two-way street.

Update: I'm now sitting in another coffee shop, where I buy beans. They have a pretty sweet deal: buy a pound of beans (slightly more expensive bargain brands at the local supermarket, but not much more) and get a free drink, any type, any size.  I'm not sure exactly how many ounces in this drink, but the fact that it has its own lifeguard on duty is not a good sign. I truly do not want to think about how many calories it contains.


  1. Fantastic post! I think about these things, too. Coffee often has non-monotonically decreasing marginal costs that are (sorry for the double negative). I also find that irritating.

    My other observations:
    - coffee places usually charge a flat rate of about $0.60 to make an espresso drink with soy milk. That just encourages me to order the biggest possible size (then I usually switch to decaf to avoid a sizable jolt).
    - I was in a major taco chain (Qdoba?) where buying three single tacos was cheaper than buying a three pack.
    - some Dunkin' Donuts charge $0.99 per donut and $5.99 per half a dozen donuts. This made no sense to me until I learned that a single donut is taxed at the restaurant rate (11% in VA) whereas half a dozen donuts are taxed at the grocery rate (2%), which means that buying half a dozen donuts comes out cheaper than buying six single donuts. They are taxed at different rates, since people presumably cannot eat six donuts in a single meal. I won't comment on that. I go to DD for the coffee so their donut prices are irrelevant.


  2. @Laura: Thanks for the comment. I not infrequently see products in the local supermarket (Kroger) where there is a diseconomy of scale for going to a larger package relative to the equivalent number of smaller packages. In some cases, maybe they're trying to get rid of excess inventory of the smaller size; but I think I see it too often for that, and particularly in nonperishable items with fairly high turnover and low holding cost (like toilet paper).

    I suspect some of this is further proof that our national angst over K-12 mathematics education is warranted.

    Re Dunkin' Donuts, when I was a kid (and prices were considerably lower), we had a sales tax (5% I think) that was not collected if it came out to less than one cent. At the local ice cream stand, people would come in with a carload of kids and order one cone, then one cone, then one cone ... to avoid paying any sales tax. Management tried to put a stop to the practice, and at least a few customers asserted their Constitutional right to place as many orders as they wanted. (This was in NY, where people tend to assert their rights rather loudly.)

  3. I routinely see diseconomies of scale at grocery stores here in South Florida (Publix and Whole Foods). My wife and I have learned from it and we always double check the price per ounce. Fortunately, most stores these days pre-calculate that for you and put the $/oz on the price tag.

    @Laura: I did not know that this tax differentiation existed. I wonder if this is true everywhere or if it depends on state laws. It started me thinking about creating an optimization model: what kinds of bundles/promotions to create to take advantage of tax laws? If there's enough complexity it might be worth doing.


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