Why Matching Formulas Don’t Make Sense

Politico reports:

As part of the economic stimulus, the DOT allocated $3.3 billion to California’s planned high-speed rail line, which has become bogged down in a high-stakes fight over its price tag and location. California risks losing those federal funds if the state Legislature doesn’t approve $2.7 billion in bonds by mid-June.

To understand the incentives of a state legislator, consider a hypothetical example. Imagine there is a new restaurant in town. It is called “matching formula.” The restaurant has fifty tables and offers a special deal: no matter what you order, you get to split half of your bill with the rest of the restaurant’s patrons.

Now think about the value you would derive from a nice steak meal. Let’s say it is worth $26.00 to you. Unfortunately, at this restaurant, the meal costs $50.00.  Normally you wouldn’t pay that kind of money. But given the matching formula, it is only going to cost you $25.50 (that’s half the price, $25.00, plus one-fiftieth of the other half, $0.50). Since you derive $26.00 in value from the meal and since it only costs $25.50, you go ahead and order it.

But there’s more. The other patrons at the other 49 tables face the exact same incentive. If they too value the meal at $26.00, they too will order it. That’s another 49 meals, half the cost of which will be split 50 ways. Your share of their meals works out to $24.50 (that’s $25, times 49, divided by 50). So your final bill is $50.00 (that’s $25.50 for your meal, plus another $24.50 for everyone else’s).

Remember, you only valued it at $26. So on net, you are down $24 ($26 value, minus $50 cost). You might be thinking that it would be better to just order nothing. But if you do that, you still end up paying $24.50 for everyone else’s meal, but in this case you get nothing at all. It is better to be down $24 than $24.50.

Given the incentives of the restaurant, it is completely rational for you to order your meal and it is completely rational for everyone else to order theirs. The system as a whole, however, is nuts.

You could try convincing everyone else in the restaurant to order nothing. In that case, you would all be out $0. This is a winning strategy, but it can be very difficult to convince everyone. If 49 tables agree not to order anything, the 50th can get a GREAT deal: one steak valued at $26.00 that will cost them only $25.50. Once others see that the hold-out is profiting, they will refuse to abstain and the agreement will fall apart.

Strange as this story seems, this is exactly the way the federal government structures a number of federal-state programs. According to the story cited above, for example, California legislators can foist 55 percent of the cost of their high-speed rail on to federal taxpayers. And presumably other states’ legislators can do the same.

Under normal circumstances, the average state can export 58 percent of the cost of its Medicaid program on to federal taxpayers and some states can export up to 74 percent of the cost. But the stimulus bill temporarily enhanced these matching formulas so that now the average state can export 71 percent and some states can export up to 81 percent.

Notice that you don’t have to dislike steaks, trains, or Medicaid to find fault in these formulas. You just have to understand that there are costs and benefits to everything and recognize that these formulas bias cost/benefit calculations in favor of spending more than these things are worth.

The great 19th Century French economist Frédérick Bastiat once defined the state as, “that great fiction by which everyone tries to live at the expense of everyone else.”

Matching formulas institutionalize this fiction.

My apologies to Russ Roberts who once told a very similar story far better than I.