Tuesday, June 12, 2012

Fatties Are Optimising

Let me repeat a frequent refrain that I hear from smug skinny people from time to time:
Hur hur hur. Look at that fat guy at McDonalds, getting a super-sized Big Mac meal with a diet coke. As if the diet coke makes up for the huge number of calories he's consuming! What a moron!
Reader, I am here to tell you that at least along one metric, the fattie is behaving in an entirely rational way, and the skinny guy is in fact the moron for not understanding optimisation.

When I say 'rational', I don't mean that they are doing the optimal thing in some cosmic sense. Rather, I mean it in the classical applied microeconomics sense that they are maximising something.

So what exactly might they be maximising that would be consistent with their behaviour?

Consider that they get utility from eating equal to

U = Taste*Quantity

It's better to eat tastier stuff, and it's better to eat more of it. Not too controversial, right?

In addition, suppose they can't eat an unlimited amount - let's grant them a binding calorie budget for each meal. The exact budget doesn't actually matter for the analysis.

So the fatties want to eat as much tasty food as possible given a maximum total calorie allowance. How do they choose the amounts of foodstuffs to get to this point?

Well, if you work through the fairly simple constrained optimisation, the relevant metric for comparing across food items is the taste per unit calorie. This is a measure of how good the 'value' of each food is, if you think of calories like money. In other words:

"Value(Food X)" = Taste (Food X) /  Calories (Food X)

In equilibrium, you will want to allocate more consumption towards foods that deliver higher value, and reduce consumption in low value foods.* When faced between two foods, that's how you'll decide between them.

Let's add further the assumption that the person must have at least one food item and one drink item.

So how do you choose between the items?

Let's start by comparing Coke versus Diet Coke.

A 12 ounce can of Coke has 140 calories. Let's call it's tastiness = y.

A 12 ounce can of diet coke has, say, 1 calorie (it's closer to zero, but never mind). Let's say you find diet coke much worse than coke - it's only 30% as tasty, say.

So the value of coke is V(coke) = y/140
The value of diet coke is V(diet coke) = 0.3y/1

In other words, Diet Coke is 42 times better value than Coke.

Now let's compare a serve of fries relative to our equivalent of 'Diet Fries', say a Premium Southwest Salad with Chicken.

A large McDonalds French Fries has 500 calories. The salad has 290 calories.

But everyone knows that the salad is not 60% as tasty as the French Fries. At best, it's about a quarter as tasty.

In other words, the Salad is worse value than the large fries.

So the fatty that is rationally optimising the problem we've set out will choose the large fries and the diet coke, and ignore the southwest salad and the coke. This will give him more tasty food for the same amount of calories.

And this conclusion holds no matter what the calorie budget. It doesn't matter if you let the guy eat a huge meal - he's still better off ordering more fries and a diet coke. Coke has a much (calorie)-cheaper substitute than fries do for the same level of taste.

I think it's a mistake to assume that fatties don't care about being fat. My guess is that they care deeply about it. They just really like food.

And these are exactly the people whom I'd expect to figure out the optimal way to eat the most amount of tasty food for a given level of calories.

Frankly, if I only optimised over the things above, I'd eat McDonalds a lot more. It's tasty as hell, and doesn't even have that many calories. As we've seen, you can eat it every day and not necessarily get fat.

The only thing that stops me getting to this point is adding in a health cost to each item. If you care about your health (and on this front, I think it's safe to assume that fatties may not care as much), then you're more likely to pick the salad. But most people are unlikely to pick the salad based on the taste/calorie tradeoff alone. Unless they're idiots. In addition, Kevin Murphy's back-of-the-envelope calculations about how large the health costs of a hamburger are suggest that they're only in the range of $2-$3 per hamburger. What the costs are in terms of attractiveness, however, is another story.

But the bottom line is that it's the fattie at McDonalds who isn't ordering the diet coke who is more likely to be making the mistake. You're always better off ordering the diet coke and getting a larger fries instead.

*Technical aside: if you don't specify a declining function of taste with greater consumption (i.e. each fry tastes as good as the last), the equilibrium will be a corner solution - e.g. you only eat french fries. The fact that people tend to want a burger and fries suggests that taste declines with consumption, and thus the optimisation is at an interior solution. Fact.

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