# All Dollars are Equal: A Common Economic Fallacy

Here’s an example of an all-too common fallacy in assessing the rationality of economic actors. It comes from Daniel Kahneman and Jonathan Renshon in Foreign Policy in an otherwise interesting article:

Imagine, for example, the choice between:

Option A: A sure loss of $890

Option B: A 90 percent chance to lose $1,000 and a 10 percent chance to lose nothing.

In this situation, a large majority of decision makers will prefer the gamble in Option B, even though the other choice is statistically superior. People prefer to avoid a certain loss in favor of a potential loss, even if they risk losing significantly more.

Kahneman and Renshon believe Option A to be the rational choice, but in fact Option B should be preferred by most people, most of the time. Do you see why?

Option A is the preferred choice only if you’re going to make this choice more than once: at least two times, perhaps more. However if you’re only presented with this as a one time decision, then you can either guaranteedly lose $890 or possibly lose $900 and possibly lose nothing. Given the probabilities, the expected loss from option B is $900. If all dollars are equal, then the rational actor will choose to lose $890, and certainly this is the rational choice if you have to make this choice again and again until you go bankrupt.

However, suppose you only make the choice once. Then if you lose in option B you’re only $110 behind option A. If you win in option B you’re $890 ahead of Option A. If all dollars are equal, then the relative chances of winning losing in option B make Option A the superior choice. However all dollars are not equal. To many of us, much of the time $1000 is worth more than ten times the value of 110 dollars. $110 is not quite pocket change, but it’s not rent or the mortgage payment or a new computer either. $1000 is. We can’t really do much with the extra $110 dollars, maybe have a nice meal. We can get much more value from the $1000.

Now of course this is all relative to wealth and needs of the person making the decision. For some people, especially wealthy ones, Option A may always be preferred. However the choice of Option B is not necessarily irrational and explains some markets.

For example, suppose somebody needs an operation to save their child’s life that costs $100,000. If they can’t pay, the child dies. They only have $50,000 after exhausting all credit cards, family and friend connections, and so forth. Is it rational for this person to walk into a casino and bet it all on black, even though they have less than a fifty percent chance of winning the needed cash and a slightly greater than 50% chance of losing everything? Yes, because in this situation $100,000 is worth more than double $50,000. Not all dollars are equal.

Not all situations are this extreme or obvious, but this calculus does come into play quite often. For instance, why do many of us purchase insurance of one kind or another? If the actuaries have done their jobs right, then insurance is an expected loss for every purchaser. The reason we buy insurance anyway is because the cost of the loss we’re insuring against times the probability of the loss occurring is greater than the actual value of the money we spend on the insurance, even though a dollar-per-dollar comparison shows the opposite. An uninsured catastrophic event can hurt us far more than the cost of the premiums, even after discounting for the likelihood of occurrence.

The simple truth is that not all dollars are created equal. Many times there are hard and soft inclination points where adding or subtracting a few dollars makes a lot more than a few dollars difference in people’s lives. The very wealthy may reach points where the incremental value of an additional dollar is essentially zero. But for most of us the value of a dollar does not follow a linear progression.

January 4th, 2007 at 9:28 AM

If the majoriity of Americans read the first few chapters of Samuelson, we’d never be in the fix we’re in.

January 4th, 2007 at 11:26 AM

For some people, of course (like me), medical insurance is a net win. My annual medical expenditures greatly exceed the cost of premiums plus out-of-pocket costs, probably even when you figure in whatever my employer is paying. This is because I’m relatively sicker than most of the people at my company, but the insurance company can’t single me out of the pool.

January 4th, 2007 at 1:06 PM

The difference between a single decision and repeated decisions is at the heart of game theory. There’s a big difference between Prisoner’s Dilemma played once and played over and over. The strategies one uses are completely different.

Another thing we often forget: after you’ve stopped playing the game, your descendants may still be playing it. So some games have a long-term reproductive success component that alters a strategy’s effectiveness.