Willingness to Pay

Many years ago when I was a poor undergrad, there was a band I wanted to see.  I didn’t have a ticket, but I showed up anyway.  The show was sold out but a scalper offered me a ticket for $40.  That was too much for me.  I stood there just waiting and waiting.  The whole time, the scalper kept egging me on to buy the ticket, “Man, the show is starting soon, if you want to see it, you have to buy this ticket soon!”  But I just waited.

Eventually, you could hear the band taking the stage inside. By then, the crowds on the sidewalk were all gone.  It was just me and scalper.

“Last chance kid, I’m leaving now.”

“Ok, I’ll buy it from you,” I said.


“I’ll pay you $10.”

“$10 dollars?” he replied.  “I paid $25 for this.  If I sell it to you for $10, I’ll be losing $15.”

“And if you don’t sell it to me, you’ll lose $25.”

He huffed and puffed, told me to go to hell and started walking down the street.  He got to the end of the block, looked down at the ticket in his hand, turned around, and walked back.

“Ok, give me $10,” he said.

I held out a $5 bill.  He gave me a dirty look, grabbed the $5 bill, handed me the ticket, and walked away cursing me under his breath.


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When Cutting Spending and Raising Taxes are actually the same thing

I think it’s fair to say that there are a number of Americans who would like to see tax cuts and reductions in government spending.   One odd thing about US tax law is that sometimes gov’t spending is a “tax cut,” meaning, if you cut spending, you’re actually raising taxes.  I’m talking about “tax expenditures.”

Here’s a simple example.  Imagine you owe $200 to the gov’t in taxes.  The simple accounting for this would be:

  • You: -$200 (taxes paid)
  • Gov’t: +$200 (taxes received)

Also imagine that there’s some machine that costs $100 the government would like you to own (for whatever reasons).  And, for whatever reasons, maybe because there’s a recession and money is tight, you’re not going out and buying it for yourself.  So what can the gov’t do to help you get the machine?  A simple answer is that the gov’t can just buy it for you and give it to you.  Let’s look at the final accounting of that:

  • You:  -$200 (taxes paid) +$100 (machine received) = -$100
  • Gov’t: +$200 (taxes received) -$100 (machine bought)= +$100

This is a straight-forward case of gov’t spending.  Let’s imagine a different arrangement.  In this case, the gov’t does not buy you the machine, but they tell you, if you buy it for yourself, you can deduct the cost from your taxes.  If you don’t buy it, you’re in the first case.  -$200 to you, +$200 to the gov’t.  If you do take this deal, you end up with:

  • You: -$100 (taxes paid after deduction) -$100 (cost of machine) +$100 (machine received) = $-100
  • Gov’t: +$100 (taxes received)

Notice, it’s exactly the same outcome.  You have $200 less in cash than when you started and you now own a machine worth $100, net change of -$100. Likewise, the gov’t has +$100. Even though the gov’t never bought anything and you bought the machine yourself, the affect of that tax deduction is exactly equivalent to the clear cut case of gov’t spending.  This is true no matter what people call it, a tax break, a tax deduction, a tax loophole, or even a tax cut (after all, the amount of taxes you paid went down).  But, it’s really the same exact thing as the gov’t just spending money to buy you “free” stuff.  Because it’s just like spending, only done through the tax code, it’s called a “tax expenditure.”  Politicians love “spending” money in this way because it allows them to call it a tax cut / tax break, and avoid the case where there’s no other name for it but government spending, even though that’s exactly what it is.

In terms of overall money, both on the individual level, and for the country in aggregate, one of the biggest real-life examples of a “tax expenditure” is the Mortgage Interest Deduction.  As the above example shows, this is exactly the same thing as if the gov’t just paid part of your mortgage payments for you.  That’s why some people refer to it as a mortgage subsidy.  But, my guess is, the average home-owner (especially someone in the Tea Party) would balk if you dare said that they lived in “government subsidized housing.”  Furthermore, imagine if you told that same person that you were going to “cut gov’t spending” by ending this “mortgage subsidy.”  They’d scream, “but then my taxes would go up!  That’s not a spending cut, it’s a tax hike!”  And, in a sense, they’d be right.  But so would you.  That’s because sometimes a tax increase and a spending cut are the same thing.

If the gov’t had just written you a check to cover part of your mortgage, it’d be much easier to label it as gov’t spending, and thus, ending the program as cuts in spending.  Since politicians avoid the “spending” label by giving you this “subsidy” through the tax code, they can’t use that “cutting spending” term as easily, (not without giving up the con that you’re tax break is actually a subsidy).  Instead, politicians will use terms like “simplifying the tax code,” “tax reform,” or “closing loopholes.”  But, this is what they mean.

So why does it matter?  One reason is that Mitt Romney put forth a sketch of a budget/tax plan that include lowering the rates of various taxes (income, capital gains, etc.) for the highest income earners.  That means less tax revenue, which means, an increasing deficit.  To avoid that, Romney has promised that the tax revenue collected from those top earners won’t go down. He’ll make up the lost revenue by closing loopholes (ie, ending tax expenditures) for those top earners.  The problem is, even if you ended every single one of them, it doesn’t get you enough money to offset the losses from the rate cuts.  When the Tax Policy Center evaluated Romney’s plan, they took him at his word that it should be revenue neutral.  Since closing loopholes for the top earners wasn’t enough, they extended the closing of these loopholes into the middle class.  That’s the only way the math can work.  Romney says he won’t do this, so it’s unfair to say that he would, but if he doesn’t then his plan is simply mathematically impossible.

Regardless, I think it’s important for every American to understand the concept of a tax expenditure and how any talk of cutting spending, closing loopholes, or simplifying the tax code can often refer to such tax breaks.  And when a politicians talks about doing these sorts of things to close the deficit, the thing to realize is that this can only have a big fiscal impact if you go after the biggest “tax expenditures.”

Right now, the three biggest are (not in this order), the Mortgage Interest Deduction, the tax exclusions to retirement plants (eg. 401(k)), and the exclusion of taxes on health insurance benefits received from an employer (after all, those benefits are a type of “payment” but they don’t get taxed like our cash payments).  Ending that “spending” or simplifying the tax code to remove those will result in you having to write a bigger check to the IRS every year, or, as some may call it, a tax hike.

So just remember that the next time you hear a politician talking about “cutting spending” or “closing loopholes” he very well could mean he’s going to raise your taxes.

Also realize that if you pay a mortgage, or you get health insurance through your employer, or if you have a 401(k), you’re in an arrangement with the government that is identical to if they just wrote you checks. You are the benefactor of government spending, not matter what name the gov’t gives that tax policy.  Think about that when you start demanding that politicians cut spending.  They may just end up cutting the money they spend on you, regardless of whether or not you ever stopped long enough to think about it and realize that yes, you actually are the recipient of government spending, whether you personally think of it that way or not.  The gov’t does this because they want to encourage home ownership, they want to encourage saving for retirement, and want to encourage employers to give out good health insurance.  But have no doubts about it, they encourage these things by subsidizing them, and thus, subsidizing you.  And when they say they’ll pay for tax cuts by cutting spending, they very well could mean the money they spend subsidizing you.


As an aside, one example of where you regularly hear people talk about something that gets to this notion that a tax cut and an increase in spending can be the same thing is when you hear a conservative and a liberal argue over the stimulus.  The Conservative will likely bemoan the $780 billion in government spending.  The liberal will neatly always counter that by pointing out that conservatives love tax cuts and that around $282 billion of that (roughly a third) wasn’t spending, but rather tax cuts.  To truly settle that, you’d have to go through the stimulus provision by provision, but, in a general sense, both people are right. It was indeed $780 billion of spending, and a good chunk of that spending were indeed tax cuts.  Tax cuts and spending are sometimes the same thing.

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Not Getting Paid and the Price We Pay

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 ( \pi ) is total revenue ( TR) minus total cost (TC):

\pi = TR - 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 Q = 100-20P.  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 P*Q, the price they sell it at times the amount they sell, and total cost is \$1*Q , one dollar per cup they sell.

Since we can rewrite  Q = 100-20P as P=5-.05Q , and thus,  TR= P*Q= 5Q-.05Q^2.  That gives us a Marginal Revenue of MR = 5-.1*Q.  Set that equal to our MC of $1, and you end up with Q=40 and P= \$3 .

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 P for a Diet Coke, but 10% of the time, they get paid $0.  Our Total Revenue equation thus becomes:

TR = .9*P*Q + .1*0*Q

Do the math and you’ll find that Marginal Revenue is now given by the equation MR = 4.5-.09*Q.  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.


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