### Why Do All-You-Can-Eat Buffets Exist?

When someone offers all-you-can-eat to any customers, those that show up should be ones for whom the amount that they can eat is worth more than the price they expect to pay. After all, if the buffet costs \$10 no matter how much you eat then those who eat the most will get the most value out of it. But the average amount consumed can’t exceed the price, otherwise the restaurant will lose money and go out of business. So if the average amount consumed is \$16 worth of food, then the restaurant will have to raise the price to above \$16. But this means those who more than \$10 but less than \$16 worth of food will no longer find it worthwhile to eat there, so they will stop going, and the average customer left will be those who eat more than \$16 worth.

This process continues, until there is only one guy left going to the buffet, and he eats \$300 worth of fish and is charged exactly \$300 for it. In effect, this theory says that all-you-can-eat buffets should not exist. And yet they do
I have three responses:

1) All exchanges take place when evaluations are unequal, not equal: the buffet can be worth \$16 to me, but the cost to the restaurant can be less than that.

2) People like to done with friends: the guy who eats \$20 dollars of food may come in with two people who eat only \$10.

3) My personal reason: I like the variety. I could not eat the same number of foods at an a la carte restaurant for the same price, because I cannot get just a spoonful of this food and a spoonful of that one.

1. Your #1 is my answer, as well. Another answer is that many buffets price discriminate. Hometown Buffet, here in San Diego, charges extra for a drink -- far more than the cost of the drink to them. I figure that they expect that a non-random sample of their customer base will pay for drinks. Many buffets charge different prices for the time of the day (more expensive food is usually served for dinner).

But, your #1 is probably the most important answer. If I go to Hometown Buffet, I usually get 2 plates of food and dessert. A bigger person might get 5 plates of food. On average, I'm willing to bet that the mean consumption is lower than the mean cost.

2. I don't see this. "Worth more" equivocates: it's true that people who come place a higher value on the food they consume at a marginal price of zero than the entrance fee: the food they eat is "worth more" in that sense. The consumer surplus exceeds the entrance fee. Why does this mean the place loses money: that would only be true if the *cost* of providing the amount consumed at a marginal price of zero exceeds the entrance fee. That's a different sense of "worth more" and that the food is worth more than the fee in the first sense doesn't entail that it is "worth more" in the second sense. I must be missing something!

3. Yes, Kevin: that is what I was trying to say with point 1 above.

By the way, when I say that number three is "my "answer, what I mean is that's the reason I go into all you can eat buffets. I think the other two answers are true as well.

4. Gene: D'oh!

Anyway, I was thinking about all-you-can eat places again. Let's take a bench mark case where people all have the same (inverse) demand schedules - the same willingness to pay as a function of quantity. Then the profit-maximizing two-part tariff is to sell the food at price per unit equal to cost (I'll be thinking constant unit cost in what follows), which maximizes the total surplus, and then charge a fee equal to this maximized surplus. Surely the marginal cost of the food at the buffet is not zero (although it comes pretty close in some of them, I grant you!) Why is this money left on the table (hah!)? Things just get worse when people have different WTP's. For example's sake, say you have two types of customer whose WTP schedules are parallel and linear, with the absolute value of the slope equal to b and intercepts of H for the high-valuation type and L for the low-valuation type, where H>L. There are two possibilities for the profit-maximizing tariff: you have to check to see which is better. First is to charge a fee equal to the high-valuation type's consumer surplus when they buy at cost and charge the unit cost as the per-unit price - thus excluding the low-value types completely. Or charge a unit price equal to C + (H-L)/b, where C is unit cost, along with a fee equal to the consumer surplus the low-value customer would get buying at this unit price. So if we end up with the second scheme, the optimal per unit price is above cost, and a fortiori above 0. This is the real mystery! Hic rosa, Hic salta!