It’s the Thought that Counts

For those of you who didn’t see the earlier posts, this is the third part of a series on the problems I see with state lotteries.  In parts one and two, I give what I hope is a convincing argument that playing the lottery is irrational for the vast majority of players.  Today I’m trying to go a step further and argue that we should actually get rid of these lotteries.

Let’s first be clear about one thing. I know state lotteries bring in money for useful things (education, for example). This is of course misleading, since the legislature then plans on this money and reduces other spending on it accordingly. However, it’s true that were one to simply remove state lotteries, many government services would suffer. There might be many situations where this tradeoff is still preferable to the status quo, but there’s another option which is very clearly better. The government could raise taxes (preferably income taxes) in order to compensate for the lost revenue. It’s not politically popular, and it’s not always the best of all possible policy decisions, but it is always better than having a state lottery.

Why would income taxes be better than a lottery? The first and probably biggest reason is that they’re progressive. The wealthy pay a larger portion of their income than the poor do. State lotteries are horribly regressive. According to Brooks, a household with income under $13,000 spends on average 9% (!!!) of their income on lottery tickets. That’s insane. We have come to a general conclusion as a society that a progressive income tax is better (because it seems more fair, because with a lower marginal utility from money the rich are hurt less by each dollar of taxes, and because large wealth disparities have negative social consequences). A few people on the right still advocate a flat tax, but no one believes that regressive taxes make any sense.

Now, this would be okay if it was reasonable to think of the lottery more as a government-run business than as a tax.  This only makes sense, though, if you believe that the lottery tickets had a value comparable to their cost and that people were choosing the play the lottery rationally.  As I explained in the prior posts, neither of these things is true.

That means that the government is essentially conning people. read the rest »

Tags: economics, state lotteries

Yesterday I started a multi-post series on why state lotteries are dumb. I tried to show that it was irrational to play the lottery. As I said yesterday, though, I left one potential counterargument to deal with today. This is Posner’s argument about a U-shaped marginal utility curve:

And finally and most interestingly, there are people whose marginal utility of income is U-shaped rather than everywhere declining. Usually we think of it as declining: my second million dollars confers less utility on me than my first million, and that is why I would not pay a million dollars for a lottery ticket that gave me a 50.1 percent or probably even an 80 percent probability of winning $2 million. But maybe I lead a rather drab life, and this might make such a gamble rational even if it were not actuarially fair. Suppose that for a $2 lottery ticket I obtain a one in a million chance of winning $1 million. It is not a fair gamble because the expected value of $1 million discounted by .000001 is $1, not $2. But if having $1 million would transform my life, the expected utility of the gamble may exceed $2, and then it is rationally attractive.

Now, Posner is right in that this is a more interesting argument than the ones I dealt with yesterday, but I also think it’s the least realistic. However, the reasons why will take some explaining.

Marginal utility, for those of you out there who aren’t up on your economics, is the “utility” (think “happiness”) that you get out of an additional dollar. So if I have $700, and you give me a dollar, that dollar will make me more happy, and the amount of additional happiness it gives me is its marginal utility. The marginal utility of each dollar is different, however. If I had $10,000 already, and you gave me a dollar, probably it would add less to my happiness than the dollar that put me up to $701 did.

That idea, that the later dollar, when you’re more wealthy, gives you less happiness, is an example of what economists call “decreasing marginal utility.” Economists assume that in general each dollar is worth less to you than the dollar before it. They have pretty good reasoning behind this. Let’s say I get $10. There are a huge number of ways I can spend that $10. Presumably, I pick whichever of those ways makes me happiest and spend it on that. Then I get another $10. If the thing I spent the first $10 is repeatable — say, buying dinner at a restaurant — then maybe I’ll do it again. Probably it’ll provide less happiness for me now, but at best it provides the same happiness as before. And if it isn’t repeatable — say, seeing a movie — then I pick one of the other options, which I had previously decided gave me less utility. Overall, each additional amount of money gives me less benefit than the one before, so marginal utility is decreasing.

This decreasing marginal utility is the reason why economists expect people to be risk averse. Say you have $100, and you have the option of betting $10 on a coin flip, so that you’ll end up with either $90 or $110 with equal probability. On average, you have $100 whether you take the bet or not, so a risk-neutral person would be indifferent towards taking the bet. However, if you have decreasing marginal utility, the money you could lose (dollars 91 through 100) would give you on average more happiness than those you could win (dollars 101 through 110). Therefore, you would choose not to take the bet, even though the expected value in each situation is the same. Now, for small amounts of money, the difference is not so big, so if you had a 55% chance of winning this bet, you’d probably take it. But for large amounts of money, the differences are huge, so if someone offered you double-or-nothing on your entire life’s savings, you probably wouldn’t take the bet even if you had a 70% chance of winning. The extent of risk-aversion varies between people, but it’s pretty consistently there, and it rests on very rational foundations.

Now, Posner suggests that people might have a U-shaped marginal utility curve, meaning that they have decreasing marginal utility up to some point, but that it then increases again. If that’s true to a large enough extent, it’s conceivable that the average marginal utility of the dollars in your lottery winnings is higher than the marginal utility of the dollar you gave up to buy the ticket, and if it’s enough higher (like, twice as high) it could mean that the bet is worth making even though on average you’ll lose money.

I just don’t find this believable. read the rest »

Tags: economics, state lotteries

David Brooks wrote a column last week in which he worries about the deterioration of cultural values related to intelligent use of money. This got responses on the Becker-Posner blog. All three articles make good points, but I got hung up on a sub-debate occurring between Brooks and Posner on the question of state lotteries. This is one of my pet peeve idiotic government policies, and I wanted to write something about it. As I started writing, it became clear to me that this was longer than one post, so I’m breaking it up into a several-post series.

My ultimate goal is to argue that states should not have state lotteries, giving fair consideration to arguments for the other side. Today’s goal, though, is to show a smaller point — a lemma, for the mathematicians out there — that from the point of view of the typical lottery customer, playing is irrational.

The most obvious value of a lottery ticket, the chance to win, doesn’t even come close to making it worth buying. Typically the expected value of the lottery ticket is something like half the cost of buying it. It is an idiotic way to try to get rich. Proponents of the lottery tend instead to point to more intangible things. Maybe the suspense of playing the lottery has entertainment value. Maybe it gives people hope and the chance to dream.

I find these arguments unconvincing. I’m sure playing the lottery is suspenseful and that people playing it dream about what they’d do if they had the money. I’m sure some people buy lottery tickets because they like the suspense or the daydreaming it brings even though they know the payoff is incredibly not worth the price. I just don’t believe these people account for an appreciable portion of lottery tickets sold. My main reason for believing this is just common sense, but it’s also backed up by they way lotteries are advertised. Those writing the ads presumably use focus groups and so forth to see what message it is that makes people play. The slogans they come up with aren’t things like “It’s fun and suspenseful” or “What a great way to donate to your state government.” What they do emphasize is the chance of winning. California uses “Dreams do come true.” New York uses “Hey, you never know” and “All it takes is a little bit of luck.” Pennsylvania uses “You have to play to win.” (In the interest of fairness and giving credit where credit is due, Illinois seems to be an exception here, emphasizing the fun of playing with “Have a ball”. I only checked a handful of states, so there are probably other exceptions, but the trend is clear.)

The “allowing people to dream” benefit is on particularly weak ground. If all you need in order to daydream about being rich is a non-zero probability of that happening, you don’t need to buy a lottery ticket. You could get rich by discovering oil in your backyard or a priceless antique in your attic. You could find and return a cat whose grateful billionaire owner makes you the sole beneficiary in their will. If however, you need a reasonable possibility of becoming rich in order to daydream about it, the lottery ticket is insufficient, as the chances of winning any of the big prizes are mind-numbingly small. In other words, buying a lottery ticket so that you can dream requires just as bad an understanding of the odds as buying one to make money does.

There’s one more potential reason for the lottery being rational that I want to deal with. It’s Posner’s hypothetical U-shaped marginal utility curve. That one, though, involves some complicated economics, so I’m going to push it to tomorrow’s post.

Update: Installments two and three have been posted.

Tags: economics, state lotteries