Matt Yglesias wrote a post about hospitals and the fact that some people can’t pay their bill. His post is about whether or not these losses translate to higher prices for the insured who do pay for care. What struck me about his post though was his example:

Think about the CVS downstairs from my office. It charges prices that it believes are profit-maximizing. Now suppose some indigent person comes in and burns all the magazines on their magazine rack for fun. The guy’s got no money, so CVS can’t recoup its losses. Does this force CVS to raise prices on Diet Coke to make up for the cost? No—CVS was already charging profit-maximizing prices, and past losses are irrelevant to determining optimal forward-looking pricing strategy.

I think he’s getting at the idea that one should ignore sunk costs. Past losses shouldn’t influence future pricing decisions.

It’s a good starting point for thinking about simple profit maximization. Profit ( ) is total revenue ( TR) minus total cost (TC):

To maximize (or minimize) a function, you take the derivative and set it equal to zero. The derivative of Total Revenue is Marginal Revenue (MR), and likewise, for Total Cost, it’s Marginal Cost (TC), leaving you with:

Or, as any econ undergrad knows, the familiar profit maximization condition MR=MC. That is, to maxmize profits, you produce at the quantity where the amount of revenue you get from selling one more good is equal to the amount it costs to produce one more good. Matt’s right that past costs and losses don’t figure into it. Furthermore, if you assume Perfect Competition where the price is set by the market as a whole and no single firm has enough market power to change that price, then your marginal revenue is just the price. Moreover, that’s simply the market price. It doesn’t matter if someone burns down part of your store, it won’t effect the market price of Diet Coke, and thus not the price you charge either.

For all sorts of reasons, I think that’s a silly way to think about hospitals. Not only that, it’s not always the best way to think about Diet Coke either. Let’s look at a slightly different market for Diet Coke. Instead of a CVS, imagine a movie theater down the street. Once people buy a ticket, they’re somewhat trapped in terms of soda options. Here, the demand curve won’t be a perfectly elastic straight demand line as in perfect competition.

Just for fun, let’s throw in some (unrealistic, but simple) numbers and work out the profit maximizing price.

Let’s say demand for Diet Coke at the movie theater works out to . Thus, if the price of Diet Coke is $1, they’ll sell 80 cups. If it’s $2, they’ll sell 60, etc. In other words, demand is downward sloping, the higher the price, the less they sell, the lower the price, the more they sell. Further more, to keep things simple, let’s assume that the cost of selling one more cup of Diet Coke is $1 (eg, the combined cost of the syrup, cup, straw, lid, etc.)

The total revenue from Diet Coke sales will be , the price they sell it at times the amount they sell, and total cost is , one dollar per cup they sell.

Since we can rewrite as , and thus, . That gives us a Marginal Revenue of . Set that equal to our MC of $1, and you end up with and .

Let’s go back the hospital for a second. The question there is, does the fact that some people don’t pay for their care affect the profit maximizing price? We can add that easily to our Diet Coke example in the movie theater. Like the hospital, the theater is uncompensated for some of their product. People spill their drinks or the theater makes a mistake and gives Dr. Pepper instead of Diet Coke and the patron asks for it to be fixed (without having to pay for this second drink.) It happens often enough that most drink machines actually have a waste button on them so that the theater can keep track of how much soda they’re not getting compensated for.

Let’s say that this happens 10% of the time. Thus, 90% of the time, the movie theater gets paid a price for a Diet Coke, but 10% of the time, they get paid $0. Our Total Revenue equation thus becomes:

Do the math and you’ll find that Marginal Revenue is now given by the equation . Set that equal to our Marginal Cost of $1, solve for P and Q, and you’ll get roughly Q=38.888 and P= $3.05 (OK, so my made up numbers aren’t quite as clean here, but that’s not the point). The point is, once you factor in that some of your product is being given away for free it does indeed change the profit maximizing price. Here it’s gone up by 5 cents a drink.

I’m not saying my model of Diet Coke is a good example of how hospital care works. It’s not. Health care is very unlike nearly every other market for a host of reasons. My point is simply that profit maximizing prices can indeed be different between one scenario when every product is actually sold and anothere where the product is sometimes given away for free. Matt’s point that one should ignore sunk costs isn’t the right way to think about it. It’s not a sunk cost. First of all, it’s probably better to not think of it from the cost side, but rather the revenue side (it’s the fact that they’re not paying that’s important.) Moreover, it’s an ongoing situation that affects all future sales (and thus future marginal revenues). This should (and will) affect the profit maximizing price.

Whether or not it holds true for hospitals is beyond this post.

I gotta say, I think you’re doing it wrong. The issue is not that the cost is sunk (irrecoverable) but that it’s fixed (does not vary with the number of paying customers).

Your example was not a fixed cost; the more sodas are sold, the more are spilled or stolen.

Imagine instead that the theater had to buy a soda machine that cost $1000 (or $10,000 or a million dollars). Rerun the numbers: you’ll notice it doesn’t affect the profit-maximizing price at all. Constants disappear when you take the derivative.

This all assumes a fixed demand curve. Your sodas are competing with the option of… no sodas at all.

Imagine now you’re in a food court, and every restaurant can buy or not buy that soda machine. The difference between the soda machine costing a thousand dollars and it costing a million affects the number of competitors you have, which certainly affects the demand curve.

The issue is not that the cost is sunk (irrecoverable) but that it’s fixed (does not vary with the number of paying customers).In my particular example they do vary with the number of paying customers since all “free” sodas are given as replacements to sodas that have already been paid for. You’re entirely right it wasn’t one of fixed costs. It wasn’t intended to be for exactly the reasony you point out. Fixed costs don’t change anything. My point is, I’m not sure fixed costs is the right way to think about it, hence my example where things vary with quantity.

As I said, this isn’t necessarily a good analogy for hospitals (and neither is one involving how a fire at CVS affects the price of Diet Coke).

I mainly wanted to highlight the fact that there isn’t always one unchanging profit maximizing price for a given product. In some situations, when you’re not getting compensated for all your “sales” it can lead to a different price than when you are. It’s not clear from Matt’s example if he doesn’t understand this, or if he just doesn’t think it applies to hospitals.

You’re entirely right about the “vary” aspect. It essentially comse down to, “Can a firm make a decision that changes how much “free” product they’re giving away?” and “Does this decision change the calculus of the optimal price?”

In general, the answer to both is, “It’s entirely possible in certain situations.” As for hospitals, I’m not so sure. Matt’s saying, “Look! The answer is ‘no’ for CVS when a bum burns down magazines, therefore it’s ‘no’ for hospitals.” I’m saying, “Look, it could be ‘yes’ for a movie theater, therefore it’s obviously not ‘no’ for all circumstances.” For hospitals, maybe it is, maybe it isn’t.