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The lotteries increase inequality, but they also shake up the distribution, launching people from the lower ends up into higher stretches of the distribution. Income mobility may be as important to people as distribution.

This is not that much of a mystery. The first priority of the poor is not to be themselves poor. That strikes me as an impulse to equality, not inequality. The lottery is "notionally" one of the ways in which someone on the receiving end of inequality can become equal to their betters.

It's not surprising to me that people buy lottery tickets - people like to hope and to dream, and a lottery ticket is just the price of a week of dreaming of escape. And lots of people want to press the ESC key once in a while.

Though it's a perfect scheme, isn't it? Sell lottery tickets, making the poor poorer and the rich richer, thereby increasing inequality and the gap between rich and poor, thus making escaping a mundane existence even more attractive, therefore selling even more lottery tickets.

We've talked on this blog before about how hard it is to grasp large numbers, like the odds of not winning the lottery. Anyways the odds of winning the lottery are no smaller than the odds of some other dreams coming true, e.g. scoring a hat trick during my hockey game tomorrow night.

But why as a society we've chosen to use this urge, as Nick says, to increase inequality - as opposed to promoting saving or other good stuff - is absolutely beyond me. (The justification that these funds are going to public services is baloney - any other way of raising funds, even the fairly regressive ones like property taxes or GST, would hurt the poor far less than lottery tickets).

The thing, Nick, I'm surprised about is that you don't advocate a market solution - end the government monopoly on lottery purchases and advocate a more competitive market for gambling. Surely that's the libertarian solution.

That means, at an absolute minimum, one quarter of Canadians want (have a revealed preference for) more inequality.

Well, either that, or revealed preference and perfect information and rational agents are a load of hooey. One or the other.

Nick, "That means, at an absolute minimum, one quarter of Canadians want (have a revealed preference for) more inequality. "

No, what it means is that each person knows that the expected change in overall inequality as a result of his or her personal decision to buy a lottery ticket is, essentially, zero.

Think about it: someone is going to win even if I don't buy a ticket. My ticket purchase makes me $5 worse off, the winners $3 better off and generates government revenue of $2. From a lifetime perspective the winners might be in the top, say, 10% of lifetime earnings. Buying a lottery ticket probably does far less to increase overall inequality than, say, buying a microsoft product that increases Bill Gates' income, or filling up at an Irving gas station and increasing the Irving family's income, or even spending $100 at Canadian Tire.

Not everybody on the left is a fan of lotteries. The phrase "A tax on the stupid" springs to mind.
That the NDP is not of this mind is an indication of the centrist tendencies of parties that focus on winning elections.

Dr Rowe, I believe that you are thinking much, much, much harder about this than most Canadians that buy lottery tickets. In fact, I doubt that most could be induced to give careful consideration to your discussion. As far as libertarians are concerned, I think a wide variety of opinion is possible. I, for example, would rather that government did not involve itself in this activity at all for a variety of reasons, one of them being its insistence on using it as a form of tax. Unfortunately, in a country that lacks provision for a real franchise for its citizens it seems to be inevitable that all political parties must agree to the current arrangements.

Hello, lefty here. Well, people around here call me a lefty. I don't like lotteries. The only time I played was at the office pool and it was a dollar a week and you got some laughs and occasionally a toonie back.

Most of the time I prefer chocolate or cashews for my escape. The pleasure value of the purchase is clear and right in front of my face, until I put it in my face. :) What the purchase promises, the purchase delivers immediately.

On the other hand, there is a clear case to be made that since people gamble anyway, the state may as well take the profits. Many US States like New York, Pennsylvania and New Jersey operate Numbers Games where three digits are drawn randomly to match player tickets. The game started under Mob control in poor neighbourhoods like Harlem as early as the 1870's and flourished in the 1920's. The New York Mafia made it a mainstay of their rackets until New York started to offer the same game legally. New York broadcasts the draw weekly and I've often seen in on Buffalo stations.

This is one case where the government clearly legalized and usurped an organized criminal activity simply to appropriate the profits for itself.

A side effect of lotteries is definitely a redistribution of wealth - great point. But it has been demonstrated (Morgan 2000) that you can get perfectly rational risk neutral agents who don't care about income distribution to buy lottery tickets if the lottery is attached to the provision of a public good. The remarkable thing about this result is that it increases public welfare by increasing provision of the public good.

While I think you've captured an important side effect, I don't think your conclusion about preferences is entirely correct.

Determinant is completely correct when he states.

"This is one case where the government clearly legalized and usurped an organized criminal activity simply to appropriate the profits for itself."

Now should governments do the same for marijuana? How about prostitution? Or heroin?

I think someone smarter than I am could make a reasonable case for all 3.

I am at best indifferent to legalizing marijuana, I've never smoked it and never wanted to. I'm a diabetic and that just doubles the usual "don't smoke" reasons. The same reasons apply to tobacco too.

I am against legalizing prostitution because I am unwilling to place anyone in the position of feeling that they have to resort to whoring in order to earn a living. Until we as a society are prepared to offer a true and unconditional alternative to prostitution as an economic activity, until we can guarantee that those participating are truly "volunteers" then I cannot and will not support legalization. Economics be damned. However this amounts to advocating for a Guaranteed Minimum Income so politically it's a non-starter.

Though I do have a sneaking suspicion that many of our problems would be lessened if we legalized most drugs and simply offered treatment programs.

"I am against legalizing prostitution because I am unwilling to place anyone in the position of feeling that they have to resort to whoring in order to earn a living."

I genuinely don't understand your argument. As I see it, no one would become a prostitute unless the alternative were worse. Prostitution doesn't make the situation worse; it just turns a terrible situation into a slightly less terrible one (for many people). It seems to me that the only way to solve the problem that you're pointing out is to abolish poverty (which I would strongly support). The legality of prostitution has relatively little effect on the outcome. The only reasonable argument that I've ever heard against legalizing prostitution is that it would increase the number of individuals who are trafficked against their will. That would be a terrible outcome, but I don't think it's what you're describing.

Addressing the original topic, I would like to say that maybe the lottery has a real utility for those who buy tickets. Maybe irrational hope is a precious commodity. That would certainly explain a lot of things about the world.

"None of this makes sense to me"

Ah, but you can do math.

A family member who buys a lottery ticket every week, and has for years, when confronted with the statement "the expected returns on lottery tickets are negative" responded: "if I don't buy a ticket, I have zero chance of winning. But if I buy a ticket, then at least I might win, and win big". And I've heard many similar justifications.

Gambling isn't really a stupidity tax. It's a "failed/never took undergrad probability and statistics" tax.

How about retail investment?

It strikes me that it works in similar ways.

But it appeals more to people who like to think of themselves as smart or connected; so the middle and upper classes. But it's just chance, a few will win, most will lose. Inequality increases.

Arguably, it is a larger source of current day inequality then lotteries.

So why pick on lotteries? The easy opportunity for class condescension? Does the moral honesty in winning strike us as rather innocent, a bit jejune?

"Popular support for increased inequality?"

No. Simple answer to a simple question.

Why bother writing a post pretending there's a connection, is what I wonder?

"An ultra-quick Google says that one quarter of Canadians buy lottery tickets weekly. That means, at an absolute minimum, one quarter of Canadians want (have a revealed preference for) more inequality."

Actually, it means that one quarter of Canadians want to be rich and are either bad at math or value the small chance of f-you wealth higher than they value the money spent on the ticket. Will your next post earnestly wonder why so many people in the potato chip aisle have a revealed preference for having a heart attack and dying in their 50's?

"We do not begrudge the wealth of lottery winners. Even though we know they are not morally deserving of their wealth, and got it through sheer luck. Perhaps precisely because they got it through sheer luck, and so can't pretend they are more morally deserving than we are (which is what gets us really annoyed at the rich)."

Huh? The reason nobody begrudges the lottery winner their winnings is that it was a fair competition open to all without the playing field rigger in their favour. People get annoyed with the rich because they believe (with some justification, in my opinion) that those who are rich often got that way so by screwing over other people for most of their life (see Black, Conrad if you want a nice example) or by benefiting from a rigged game in the form of insider connections, inheritance, etc. Also, people get annoyed when rich people aren't content to simply be rich, but instead buy newspapers and politicians in an attempt to corrupt the government into increasing their riches further at the expense of the common man (see Black, Conrad if you want a nice example).

Here's an experiment that may help. Next time you are playing a game, try two methods for determining who goes first. In one case, highest roll goes first, in the second case, you go first because you said so.

At any rate, I'd say the view of the average Canadian is that your typical lottery winner has a stronger moral claim to their money than many if not most wealthy Canadians.

Declan: "Why bother writing a post pretending there's a connection, is what I wonder?"

You don't see the connection??? Let me spell it out for you.

The Rawlsian thought-experiment has been cited as one of the strongest arguments for equality. I have cited it many times myself. If you put people behind the Veil of Ignorance, they would vote for equality, unless the cost was too high. Well, guess what, at least one quarter of the population go behind the Rawlsian Veil of Ignorance every week. And they don't vote that way, even when the cost of equality would be negative.

Stop pretending there is no connection. (Unless you have never heard of Rawls, in which case you are forgiven for missing the point.)

Frances: "The thing, Nick, I'm surprised about is that you don't advocate a market solution - end the government monopoly on lottery purchases and advocate a more competitive market for gambling. Surely that's the libertarian solution."

The libertarian in me is arguing just that. And the paternalistic egalitarian in me, who believes in diminishing marginal utility, is arguing to ban the lot.

Again, I don't have a solution to this paradox. And while I am unconvinced by the resolutions offered in comments above, they are not obviously wrong, and I don't have anything better to offer.


As far as I understand Rawls' theory, people behind the Veil vote for equality out of self-interested risk-aversion, not out of altruism. Sure, there is a sense of justice involved, but that is in order to make sense of the Veil itself, not to explain people's decisions. So, it becomes a story about risk-aversion then. If risk-aversion is mostly a function of loss aversion, then even for generally risk-averse, insurance-buying people, the potential loss from buying a lottery ticket is probably small enough to justify taking a chance on earning a large amount, and getting as well a 100% sure participation in a society playing games, just for the friggin' hell of it. Life is short, right?

Assume Yvan is right about loss-aversion. It sounds plausible: people are willing to take a small loss when they by a lottery ticket, but insure against big losses on their house and car. But where does that leave the Rawlsian thought-experiment? Loss-aversion requires an existing level of income to make sense. What is that existing level of income in the Rawlsian Original Position?

mad: investing in risky stocks increases inequality too. But it also generally has, or may have, positive expected returns. We know for sure that investing in stocks has negative expected returns. We know for sure that people are not trading off risk for return.

Damn! Freudian slip. That should read: "We know for sure that investing in *lottery tickets* has negative expected returns.


Again, as far as my understanding of Rawls go, the participants in the OP are aware of all the general facts about the society for which they are designing principles of justice. What the Veil of ignorance blinds them to is their position in that society. So you would think that a decision about lotteries would itself be a function of the income distributions that the participants are considering, including transfers and other institutional features, etc. The relevant income here would be that of the lowest-income group in such a society, but this does not preclude that it would be high enough - as would be the maximin level in a wealthy society - so to consider the cost of lottery tickets to be relatively low.

Nick, it is totally consistent to say "in a Rawlsian original position I would not want lotteries to exist" and still buy lottery tickets.

One example: big pots, if unclaimed, carry over from week to week. If don't remember the exact dollar figure, but there is a point at which, if the carry over is large enough, the expected value of a lotto 6-49 ticket becomes positive.

Every action we take has both a manifest and latent function. You're arguing that the latent function of lottery tickets is to increase income inequality, therefore people who buy lottery tickets support increased income inequality. That's ignoring their manifest function, which is to, as Jim pointed out at the top, which is to increase social mobility, and give an escape.

Think about an average income person, without dental insurance, going to the dentist. That increases income inequality (if the dentist is richer than the patient). But people still go, because cavities hurt. Choosing to go to the dentist isn't a vote for increased income inequality, it's doing the best you can to relieve your pain given the options available to you.

Or think about a poor person who goes to a church and pays a wealthy corrupt priest for a prayer. That's actually a very close analogy - you pray because it's the only hope available to you, and yet this action perpetuates the inequality that gets you there in the first place.

I'd argue that lottery tickets are mostly bought for their entertainment value, not for risk-seeking.

The funny thing is that you can replicate a lottery's risk profile in the asset markets and come out with (potentially) a positive expected return. Just buy an out-of-the-money call on a broad market index (this is very cheap). If the call option expires valueless, then you're done. Otherwise, take the proceeds and stake them again in out-of-the-money options. Lather, rinse and repeat until you reach the desired payoff amount, then you may need an option collar in order to cap your "winnings".

That's not very enticing, is it. But it works, since the probability of winning the lottery grand prize is incredibly low.

"Or think about a poor person who goes to a church and pays a wealthy corrupt priest for a prayer."

I agree that this is a good analogy. The poor person who goes to the priest exchanges money for peace of mind, and the poor person who buys lottery tickets exchanges money for entertainment. The fact that one lucky person gets to win the lottery grand prize is--to some extent--a free lunch, which is incidental to the real transaction.

Lotteries provide hope to the poor (not the rich.) They make utility more equal, and measured wealth more unequal. People may want greater equality of utility, not greater equality of wealth.

In the end, though, does it really matter if *one* tax (or policy) is horribly regressive? Isn't it the bundle that matters in the end?

I'd argue that it is. Otherwise there's no way we could ever justify cigarette taxes. If you tried to design a regressive tax it probably wouldn't be as regressive as cigarette taxes.

Somewhat off-topic: Wired magazine had a really interesting piece on scratch lotteries in the latest issue.


anon: "The funny thing is that you can replicate a lottery's risk profile in the asset markets and come out with (potentially) a positive expected return. Just buy an out-of-the-money call on a broad market index (this is very cheap). If the call option expires valueless, then you're done. Otherwise, take the proceeds and stake them again in out-of-the-money options. Lather, rinse and repeat until you reach the desired payoff amount, then you may need an option collar in order to cap your "winnings"."

anon (or anyone): That makes sense. But (this is more of a BTW). who is taking the other side of that trade? Could the person taking the other side of that trade be someone who shares the same beliefs but is risk-averse?

Here's the reason I ask: it has always puzzled me that some people pay to take on risk (by buying lottery tickets) and other people have to be paid to take on risk (by owning risky assets). And it has always seemed to me there must be some sort of unexploited gains from trade there, if only some finance whiz-kid could figure out a way to convert insurance contracts into lottery tickets. You see people pay (in expected value terms) to avoid the small chance of a big loss, but will pay to have the big chance of a small loss and the small chance of a big win. Somehow there ought to be a way to slice and dice insurance policies so people who like lotteries would pay to write insurance contracts.

That's not as clear as I wanted it to be.

In other words, risk is one person's gold, and another person's garbage. One wants to pay for it, and the other wants to pay him to take it away.

We call lotteries a stupidity tax because they have, on average, a negative rate of return, yet we encourage a large number of people to "follow their dreams" into occupations (like the arts or sports) where all but a vanishingly small proportion also have a negative return and where the investment is much large (several years of one's life vs. a few bucks a week).

As mentioned above, I think we have to consider that there's positive utility in merely participating despite failure.

(Of course, I've never understood the utility of buying *more* than 1 lottery ticket. My idle daydreams don't become any more vivid.)

That makes sense. But (this is more of a BTW). who is taking the other side of that trade?

Well, that gets complicated. In the ideal CAPM model, call options are on the capital market line, so the trade would in fact earn compensation for market risk. However, empirically it seems that far OTM calls may earn low or even negative returns precisely because of their lottery-like quality (i.e. positive skewness), and arbitrage is hindered by high transaction costs. I imagine that things would be similar for other asset classes, such as "high yield" tranches from structured finance products.

it has always seemed to me there must be some sort of unexploited gains from trade there, if only some finance whiz-kid could figure out a way to convert insurance contracts into lottery tickets.

Meh. A lot of finance whiz kids tried to do this (more or less) by creating the aforementioned structured finance securities. The whole thing broke down in 2007, due to a number of factors. Transferable risk can often be securitized like any other asset class, but many kinds of risk are not really transferable due to hidden information.

"Sure, some part of the revenues are used to fund good things. But a simple lump sum tax on everyone could do the same funding, without increasing inequality at the same time."

But is a lump sum tax a credible alternative? Would a party proposing such a tax ever get elected? Perhaps the real choice is to have public goods funded through a lottery or not have the public goods at all.

Yvan reminds me of the bee-stings and car-dents theory of poverty:

TGGP: Charles Karelis's theory of poverty is very interesting. If the diminishing marginal disutility of "bads" (contra the usual theoretical assumptions) is what causes poverty, then marginal utility of money must be low, and increase with wealth until the "bads" are mostly exhausted. But this means that poor people should be risk lovers, and buy risky assets such as lottery tickets.

TGGP - I was just making a point about Nick's appeal to Rawls, really. I don't think I was arguing at all, implicitly or otherwise, for a theory of poverty. Was I?

I think Patrick is right. Most people just dont do the math. But beyond that, they don't even believe that the laws of probability apply to them. Among other things, many people (especially those who are indigent) believe that they might be able to influence the outcome of events (even totally random ones) through prayer.

And they always bet on the home team - that's how bookies make a living. It's totally irrational but I think it's true that most people think they are special. The default assumption ought to be that one is a random representative agent, but it doesn't seem to be a natural perspective.

Nick: you can't compare lottery tickets and risky assets. Risky assets have beta. For zero beta assets we are supposed to be indifferent to zero-expected value trades no matter what the distribution (so long as there are enough of them to diversify). But nobody should ever take a zero-beta negative expected return trade. It's not rational (in the CAPM sense).

I did once observe a very rational friend buy a lottery ticket. Others in the office were buying and he decided that he had *extremely* negative utility for the state of the world in which they won and he didn't. It struck me as a perfectly rational decision and made me wonder if office pools are a game with an inefficient equilibrium. Something like a stag hunt, maybe, where buying tickets is like everyone hunting rabbit.

Can someone explain this part to me? "Lotteries have an administrative cost, and so reduce average wealth."

Brett: you and I toss a coin for $1. The winner is $1 richer and the looser $1 poorer. Average wealth stays the same. But if there's a person employed to toss the coin, sell tickets, distribute the winnings, etc., who is paid out of the winnings, and who could have been employed producing other goods, average wealth falls.

But why is producing the coin flipping service not wealth generating?

Brett: both lotteries and insurance redistribute wealth, at an administrative cost. That administrative cost decreases wealth. But you can argue that the benefits of the service provided outweigh that cost, so that overall wealth increases once those benefits are included in the calculation. The point is simply that the redistribution is not free. It has a cost. And the paradox is that insurance reduces inequality, while lotteries increase inequality.

But I would prefer to say that both insurance and lotteries decrease wealth, but (depending on whether people are risk-averse or risk-loving) may increase the expected utility of wealth.

I can see that there is a cost to the people who buy the lottery and the people who win, but I don't understand why there is a cost to society at large. (I'm not trying to make a point. I just don't understand how to think about this.)

Brett: there's no cost to the rest of society. (Well, some people might argue there is, but that wasn't my point). And from a libertarian P.O.V., there's nothing at all wrong with lotteries. The average wealth of people playing the lotteries is reduced, but if they are willing to play, they presumably value the lottery more than the drop in average wealth.

Nick - this issue strikes home for me. When I was at McGill I lived in the St Henri district of Montreal and I saw a mom and (young)daughter in the depanneur spend $20 on lottery tickets instead of the milk or whatever they were there to buy. The mom sent the daughter home to Grandmother to get more more money with the instruction "Don't tell Grandma why."

I am libertarian enough to not object to lotteries or gambling on football or playing poker for money. It does bother the Hell out of me that government use them to raise money. It is a terrible way to tax. Government should most decidedly not use peoples weaknesses against themselves. At some point we have to deal with ourselves as *we really are* not as we would like to be. People have weaknesses. That is why you need money down to buy a mortgage. That is why a corporate director can go to jail for lying. Because humans do strange and wrong things.

Governments exploiting human weakness us just plain immoral.

"They ... vote for more inequality [by purchasing lottery tickets]."

I would argue that most people do not believe that there is such a thing as voting with one's wallet, and that their purchases should not necessarily be taken as any sort of moral or political endorsement.

It's expectation of return, isn't it? The masses vote for redistribution in the "soak the risk/handouts for the poor" format, because they expect to receive a handout or to need it as a form of income insurance. The rich expect to have a negative returns on social insurance, i.e. they expect to always be rich. That's the way social insurance has always been pitched to the middle classes who realize negative returns (when using comparable private investments as an opportunity cost).

Lottery tickets are not advertised as a form of redistribution, a tax on certain demographics, or a tool of [in]equality smoothing... even if the winnings create those effects. They are sold to the people who have an expectation of receiving something in return. This is almost by definition; the "you can't win if you don't play" analysis, combined with the voluntary participation, means only people who expect to win some day will play.

I would think a more interesting focus would be on whether support for redistribution programs increases or decreases when you tinker with the ability to predict payoffs. Both social insurance and lottery tickets have known odds of paying off; both have visible costs of participating; but only one is voluntary. What would happen if lottery odds were unpublished? If social insurance retirement benefits were undefined? What if a $1 lottery ticket was mandatory and taken out of your paycheck each week, or if social insurance was opt-in?

You may not have been arguing for a theory of poverty, but your point that people basically discount the small loss of the ticket because a win would represent real change fits with Karelis.

If your one chance to get out of hell was to buy a lottery ticket, would you take it? (Now that makes a moral issue out of it: You should not buy a lottery ticket not because the odds are against you, but because it would make almost everyone else slightly worse off. The same with any form of gambling.)

Anyway, many (some, most?) of the poor don’t perceive any other way out. Many work hard, to small gain. But… perhaps they don’t work hard enough. Perhaps all the poor should all work harder, yes, longer hours and, and study more. Yes. They should all struggle to improve themselves. Make themselves better. Then what would they all accomplish, except to work even harder at being poor? For one thing, the harder they work, the more labor they supply, the lower the price of labor. If the demand curve for labor is inelastic, they will be the poorer for their increased efforts. Do the wealthy begrudge them their beer and football?

The law of averages says there will always be the poor. This puts the lie to the claim that the poor are poor because they want to be. It’s statistics. Granted, sometimes the exceptional, or the lucky, will prevail, but most will not.

It is not up to the poor, as a class, to further make themselves better. They are not the ones with the capital. Each one optimizes his situation, according to his perceptions. The conservative bases his theory on rational agents, and then claims the poor are behaving irrationally. No. The poor are responding rationally to the web of circumstance and incentive, or lack thereof, they find themselves in.

Hmm... Sort of off the track of the lottery and Rawlsian inference. But I think a real case could be made for bounded rationality. And the poor would not be the only ones who rationally, ie out of narrow self interest, choose to hide behind a veil of ignorance.

"Well, guess what, at least one quarter of the population go behind the Rawlsian Veil of Ignorance every week. And they don't vote that way, even when the cost of equality would be negative."

Sorry, I guess I still don't see your point Nick. As Frances explained above, the impact of lotteries on inequality is just a side effect that none of the parties involved are really concerned with. The government isn't promoting inequality, just collecting revenue. People aren't supporting inequality, just giving themselves a bit of hope to escape their soul crushing job treadmill, 'imagining the freedom' as it were.

Behind a Rawlsian Veil of Ignorance, people would vote to have a society where nobody dies of a heart attack because they got high blood pressure from eating too many potato chips. And yet in reality, governments freely permit the selling of potato chips, companies go to great lengths to produce and sell them, and millions of people buy them, cutting years off their life in the process. Does that mean that governments support people dying of heart attacks, or that people themselves do? Or that it would be wrong to assume that if people were designing how society should look, they wouldn't include unnecessary heart attacks in their utopian vision?

The existence of lotteries is a failure of our educational system.

Everyone should be required to have covered basic financial literacy - concepts like expected value and compound interest - before the end of grade 10. Even those who drop out of high school need this knowledge - if anything, they need it the most. A few simple math exercises will convince most kids that buying lottery tickets is a bad, bad idea.

There is nothing you study in high school math more important than this. Throw out algebra, geometry, statistics, almost anything else if you have to for the weaker students, but don't leave vulnerable kids at the mercy of predatory lenders, credit cards, casinos and government lotteries.

There is a social function to lotteries, they are a way for some people to save, who have wrong time preferences. For example if you would like to save for a home, but when presented with a temptation you can't resist, but after the fact you regret a lottery is a way to commit yourself in a way that removes the option to be tempted.

Declan: the heart attacks are a contingent side-effect of chips. We can imagine chips that don't cause heart attacks. People don't eat chips because they cause heart attacks; they eat chips because they taste good, despite the heart attacks.

A lottery that didn't cause increased inequality wouldn't be a lottery.

Playing cards with friends can be fun. Playing for money can add to the excitement. But it's not all about the money. The money is just the salt on the chip.

You could explain lotteries the same way, but it's harder to.

The main economic case for government policy to increase equality goes like this: the marginal utility of wealth decreases with wealth. As evidence for this statement we point to people's risk aversion. People with diminishing marginal utility of wealth who maximise expected utility will buy insurance (if it's not too costly, in the expected payoff sense). People do buy insurance, therefore MU must diminish with wealth. Therefore redistribution from rich to poor (if it's not too costly) will increase total utility, and therefore the government should do it.

But when we observe a quarter of the population buying lottery tickets, even when it's costly (because the total payoffs are less than revenue from ticket sales), that very same argument gets stood on its head. They must have increasing MU of wealth. Therefore redistribution from rich to poor will reduce total utility, even if there's no cost of doing so.

Or, forget about expected utility maximisation altogether. Suppose we observe the government converting people's grape juice into wine, and losing some juice in the process. (That's Arthur Okun's "leaky bucket" metaphor, to represent the costs of redistribution). Then we observe a quarter of the population joining a club to convert their wine back into grape juice, and losing still more wine in the process. That might make us wonder if those people really wanted the government to convert their grape juice into wine.

OT: if instead of buying the lottery ticket and waiting for the draw to find out if it was a winner, you had to commit to paying a loser fee if the ticket you had did not win. Would people still play?

Patrick: my hunch is that "degenerate gamblers" (Mario Puzo's phrase) would buy still more.

TGGP - I didn't suggest that a win would necessarily "represent real change". I didn't even assume that there was something special about poverty that made the purchase of lottery tickets more rational for poor people than for anyone else. I just did not believe that buying lottery tickets was at all irrational or even incompatible with risk-aversion. I'm in the top half of the income distribution, I have insurance policies and I buy pretty much one lottery ticket a week. And BTW I work with statistics, have a masters degree in economics and am not at all impressed by the "stupidity" assumptions or the preference for inequality arguments.

This being said, I do think that there are inconsistencies between reality and the simplified "rational idiot" type of framework that is used, correctly I think, for analyzing most economic choices. There are puzzles, and this is one when you assume away all non-income related advantages, but I think it still dissolves when you give a little more flexibility to your definition of rationality. In fact, what I suggested is that if you allow for loss aversion to explain some of the risk-aversion (as opposed to leaving it all to diminishing MU of wealth, which is just too ad hoc for my taste), and if that aversion only kicks in beyond a certain threshold, then there is no reason that you would need to treat this as some weakness of the poor schmo's will, and thus you would not see it a priori as a problem, i.e. as a socially imposed loss of income instead as just another available consumption good, when you put yourself "behind the Veil of ignorance" for ethical purposes.

Also, I have no quarrel at all with Karelis' theory - it does look quite interesting.

I'm sympathetic to Yvan's view above. It's how I want to think. But I still think we are cheating a bit.

We have this preconceived view that: MU diminishes; and equality is good. And when people buy insurance we take this as unproblematic evidence confirming our view. But we refuse to treat people buying lottery tickets as unproblematic evidence of the opposite view. We try to explain away that disconfirming evidence. We add epicycles. We cheat.

Maybe people buy insurance for fun. Maybe they are irrational, in some narrow expected utility sense.

Yep. Maybe people drastically overestimate the risks of an accident (the odds of winning a lottery). Maybe people lay awake nights dreaming of all the bad things that could happen without insurance (dreaming of all the good things they could do if they win the lottery). That's why they buy insurance. It's got nothing to do with E(U) maximisation with diminishing MU.

Nick, I think it's true there is probably something like the opposite of the precaution principle going on here (i.e. giving more weight to extreme events in calculating the expectation, which is not the same as overestimating probabilities) - call it the dream principle, I guess. But what rattles me in your notion of cheating is that you seem to hold to some monotonically decreasing MU as a real heliocentric hypothesis. And I don't see why the "good" of equality is at all at stakes here anyway - maybe one very specific notion of equality is threatened, but then it's the one that is closely knit with decreasing MU anyway.

So why should it be cheating to entertain the idea of a globally decreasing MU for most people, although you would have lots of local peaks and valleys, depending on your anticipated projects and/or specific fears on that day? Also, what do we do with real-world loss aversion, i.e. this "bias" we have that makes use lose more utility from losing our 101st dollar than we gained from earning it in the first place? You can't even have an MU function with this, right? But is it irrational? Is decreasing MU an index of rationality per se?

Jim Sentence got it.

It has to do with increasing equality of opportunity.

But Lotteries don't sell real improvement (most studies suggest winning the Lottery is a mixed blessing) they sell dreams. Looking at the fact that some people win, is missing the point.

There is another way of looking at it. People are balancing their investment portfolios. There is an unsupply of very high reward, very high risk investments for small investors.

@Tom West - It's kind of a math game. Us occasional lottery buyers justify buying a bunch at once because given the choice between spending $3 every week vs spending $18 every six weeks, the latter has better odds.

Still not significant, but better.

I buy lottery tickets much less than every six weeks, but usually buy $5 or $10 worth when I do. Usually I'll walk away with one of the smaller prized ($2-$5, or at least a free play). I also have a policy of not checking my ticket until long after the draw, keeping the dreams alive for free for several weeks.
Lottery pools are the worst. You effectively have to buy in, because being the one person in the office who didn't participate in a win is so much worse than being out few dollars every week. Even the fantastically low odds make it a good investment.

I have no major insight to offer, but I shall offer some Orwell:

They were talking about the Lottery. Winston looked back when he had gone thirty metres. They were still arguing, with vivid, passionate faces. The Lottery, with its weekly pay-out of enormous prizes, was the one public event to which the proles paid serious attention. It was probable that there were some millions of proles for whom the Lottery was the principal if not the only reason for remaining alive. It was their delight, their folly, their anodyne, their intellectual stimulant. Where the Lottery was concerned, even people who could barely read and write seemed capable of intricate calculations and staggering feats of memory.

"The existence of lotteries is a failure of our educational system.

Everyone should be required to have covered basic financial literacy - concepts like expected value and compound interest - before the end of grade 10. Even those who drop out of high school need this knowledge - if anything, they need it the most. A few simple math exercises will convince most kids that buying lottery tickets is a bad, bad idea."

Well, I agree that our education system does a lousy job of teaching basic financial literacy (although I'm curious when that became the responsibility of the school system instead of being something imparted by parents).

But I disagree that the existence of lotteries has to do with ignorance. I think for most lottery players, lotteries sell hopes and dreams. At $5 for a week's worth of dreams of being a millionaire, that's not an irrational or poorly thought-out spending choice, that's a bargain.

Moreover, for someone who wants to be a millionaire (understood as someone who will never have to work again in their life - which nowadays probably means you actually have to be a mutli-millionaire), you're probably no less likely to achieve your goal by playing the lottery every week than by saving up an extra $5 a week in your investment account (and you're probably more likely to achieve it sometime soon. Moreover, lottery winnings, unlike investment income, is tax-free). Granted, in either case, the incremental effect of your $5 investment decision on your likelihood of being a millionaire is almost, but not quite, zero, but it's the "no quite" that keeps people hoping.

> I also have a policy of not checking my ticket until long after the draw, keeping the dreams alive for free for several weeks.

Now *that's* playing the lottery rationally!

Personally, about 5% of my tickets (once every year or two) get checked. I've used up their value by the time I might check them, and the opportunity cost of the time used to check for a win is greater than the expected return.

Actually, I find the office pool works well, since the idea of winning $10 million and having my life turned upside down isn't nearly as enticing as the dream of paying off the house and topping up the RRSP's with something for a family vacation.

I guess that's why I'm not a "playa" :-).

And yes, it ticks me off that the government (through the lottery corp) spends a *lot* more on telling people that the way to get ahead is by being lucky than it spends on telling people that the way to get ahead is by studying hard and then working hard. Obviously the "short cut to riches" dream will always be with us, but should it be the government selling it?

Nick, I appreciate you taking the time to elaborate your position. I follow what you're saying, I just don't see there's that much of a connection there. If the winning condition for the lottery was that *everyone* would become wealthy enough to not work again (leaving the macro problems with that scenario aside for the moment :) people would still play. It's not the inequality that people want (in my opinion), it's the wealth, the ability to live their life without having to turn the other cheek to a bunch of jerks every day because they have to stay on the job treadmill or fall through the cracks in society. In fact, given a choice, many if not most people would probably prefer it if everyone else could join them in a life of ease. Of course, this is challenging from a macro perspective to pull off, but then so is making a healthy potato chip that still tastes good.

People aren't really equipped with a natural instinct for continuous variables (at least that's one of the things I took away from Steven Pinker's 'The Stuff of Thought' and it matches well with what I see day to day). I think the internal model most lottery ticket buyers have is one where their current state is 'unhappy' and buying a lottery ticket gives a very small chance of 'happy' with a near 1 probability of 'unhappy' Given this internal model, they have nothing to lose by buying a ticket. (Insurance is similar, but in reverse - people see a possibility of two outcomes, 'No Change' and 'Disaster', buying insurance converts the situation to one where the only option is 'No Change' so it makes sense.)

One of the reasons your juice-wine-juice analogy is flawed is because it assumes, by analogy, that all inequality is the same, whereas for most people, the inequality of a random lottery winner getting rich is quite different from the inequality of a system that gives them 0 chance of succeeding while a bunch of jerks get rich because they chose the right parents when they were born or because they were more willing/able to bend the rules and screw people over than other people were (you know, the type of person that the business press calls 'innovative'). It's much like the difference between a democracy vs. a dictatorship. Both systems end up creating an inequality of political power, but in one case it is considered legitimate and in the other it is not, because of the difference in the process that generated that inequality.

You can see this in the studies that show how people's support for redistribution is heavily contingent on whether they believe the current inequality is the result of a fair process or not.
Tyronen: "A few simple math exercises will convince most kids that buying lottery tickets is a bad, bad idea."

That's a nice dream. Meanwhile many of my co-workers (all of whom are pretty mathematically literate, some of whom have masters degrees in statistics) can often be found purchasing lottery tickets.

One thing I think I could add here as something to think about, is the point often made by Chris Dillow - for most people a pure meritocracy is just as bad a deal as oligarchy/plutocracy. Even if the system is not necessarily rigged against them, it gives them no chance of "winning". To go back to Rawls, a lottery (where not played by addictives) makes relatively little difference to the balance of society - but it does make the mind experiment in Rawls (that your position in society is unknown) real.

Declan: interesting. You are coming at this from a more "proceduralist" perspective (a perspective I tend to associate more with libertarians, though it doesn't have to be, I suppose).

Let me expand a little:

Most economists (e.g. in optimal tax theory models) see the distribution of income as being determined by: choices; and luck. Some luck can be insured against, and people will insure against it. But some can't (because of moral hazard, adverse selection, or because the luck happened before you could insure against it (you were born that way). An optimal tax system is designed to try to insure people against the uninsurable luck (to remove the effects of luck), while recognising that there's a moral hazard problem because this will distort people's choices, and tries to do the best it can on the trade-off between removing luck and creating moral hazard (disincentives).

So if you start from that perspective, and then see a lot of people deliberately re-introducing luck, and undoing all your costly attempts to try to remove some of the effects of luck, you start to wonder: why the hell did we bother!?

see my post and lots of posts by Chris Dillow. One element of luck people can't possibly insure against, is being born untalented. Lotteries of their way of being able to "win" in life anyway.
But surely you see the point here - popular lotteries (those that make sense to me anyway) are always those with very large rewards at very low probabilities - whereas what the tax system tries to do is put a floor under losses. They are two very different animals. If people were going in lotteries that were expensive and only every second player lost everything, then I could understand your concerns.

When we think about gambling addicts, I also think the type of games they will concentrate on are not the sort of lotteries we are talking about. Gamblers go in a game in order to enjoy the kick of winning. Going in a lottery where you in all probability will NEVER win, cannot be their thing.

reason: yep. It's uninsurable risks, like being born untalented, that the tax/transfer system is supposed to be trying to deal with. And yes, I see that the risk created by a lottery isn't exactly the same as the risk removed by the tax/transfer system. But it does suggest that people don't want the luck of super-high earners completely redistributed, because that's exactly what they try to re-create in the lottery.

Nick: I have to confess respectfully that I'm somewhat amused by your frustrated paternalism... But obviously, you're going to see anomalies all over the place when you take simplifying assumptions seriously, including when it comes to normative economics. Yet, you could still rescue your personal ideals by assuming that what you see as revealed preferences are conditional on the presence of inequality, so that people buy lotteries given they have to put up with inequality anyway (why should they be the victim of such a mean society!), but they would so much more want to live in the world that you want for them... Would that perspective be heart-warming enough?

Yvan: Yep, but that sense of "frustrated paternalism" is I think exactly how an optimal tax theorist would think.

Yep, there will always be anomolies. But one quarter of the population (and that's certainly an underestimate, given people who play the lottery less frequently, and who do negative sum gambles in other ways) is a big one. Do more people gamble than insure (not counting compulsory insurance)?

I can see how your "rescue" *might* work. But the preferences you would need to assume to make it work would be a long way from those assumed in optimal tax theory.

The lesson I'm groping towards is this: there might be a case of redistributive taxation. But that case, and the sort of tax system that would be justified, might be very different from the case made by optimal tax theory.

The exchange with Declan was interesting. At first I couldn't see how he could fail to see my point. Then I realised he genuinely didn't see it (though I'm still a bit miffed at him accusing *me* of "pretending"). Because he didn't see an egalitarian tax system as trying to partially remove the effects of luck. He saw it as doing something quite different. Something like trying to remove the effects of procedural unfairness?

If real people don't want the tax/transfer system to do what optimal tax theorists think they want it to do, maybe it shouldn't try to do it.

Nick: 1/4 of the population is not 1/4 of a relevant economic variable. You give a bit too much importance, I think, to something that is the adult equivalent of a kid throwing a penny in a fountain. But all I'm saying is that optimal tax theorists could simply accept that their utilitarian framework is at best an imperfect yet useful work hypothesis, and just go on doing their thing. Isn't that really what they end up doing anyway, at least once they've tamed their inner central planner?

Nick:  I'm not seeing the need to distinguish between "luck" and "procedural unfairness."  The world is rife with procedural unfairness: inheritance, monopoly rent, preexisting division of finite resources. Luck (other than winning the lottery) is just landing on a particularly juicy piece of rent.   
This passage from a favourite Kurt Vonnegut novel always comes to mind for me:

Sure[, a poor person can become rich] -- provided somebody tells him when he's young enough that there is a Money River, that there's nothing fair about it, that he had damn well better forget about hard work and the merit system and honesty and all that crap and get to where the river is.  "Go where the rich and powerful are," I'd tell him, "and learn their ways.  They can be flattered and they can be scared.  Please them enormously or scare them enormously, and one moonless night they will put their fingers to their lips, warning you not to make a sound.  And they will lead you through the dark to the widest, deepest river of wealth ever known to man.  You'll be shown your place on the riverbank, and given a bucket all your own.  Slurp as much as you want, but try to keep the racket of your slurping down.  A poor man might hear."

This kind of world view is, at least for me, the principal justification for redistributive taxation.  

I should have referenced where I found that version of the Vonnegut quote: http://blawgletter.typepad.com/bbarnett/2009/03/should-law-firms-have-to-disclose-partner-pay.html

I don't think that the existence of lotteries necessarily invalidates the assumptions of optimal tax theory. For instance, if Karelis' theory is correct, then poor people have S-shaped utility functions, but it is still optimal to redistribute income if this places them above the inflexion point.

K, interesting point, but I see no reason to conflate luck with rent seeking. Yes, if rent seeking by those with higher income (which is obviously a negative externality) outweighs their positive effects on the rest of society, then an income tax can increase efficiency, if only by discouraging everyone.

Clearly, it's better to avoid rent in the first place by improving policy, and to capture remaining rents for public use. Even then, this would not substantially decrease uncertainty, since 'honest' profit seeking is also subject to risk, much of which is uninsurable. And lottery winnings should be untaxed, since playing the lottery is a profit-seeking activity, although a rather trivial one.

anon:  I mostly agree.  Public policy (including taxes) ought to directly target the existence of rents (e.g. land tax).  My point was that it is the existence of rents that is the principle cause of extreme inequality.  Therefore luck (i.e. whether you happen to be a payer or receiver of rent and how much) accounts for vastly greater dispersion of wealth, than it would in an economy without private rents.  So I do think think that collection of rents for public purpose would have a significant impact on reducing uncertainty.

And I also agree that honest profit seeking (including lotteries) ought to be as untaxed as possible (ideally not taxed at all).

"An ultra-quick Google says that one quarter of Canadians buy lottery tickets weekly. That means, at an absolute minimum, one quarter of Canadians want (have a revealed preference for) more inequality."

No, it doesn't. Meaning is not preserved across logical inference. What it means is that people value their increase in hope and excitement over the cost of the lottery ticket. They don't do the probabilities, or, if they know that the odds are against them, not only in terms of money but in terms of the utility of the money payoff, they don't care. (And for many the utility of the money payoff is seen as more than the money itself. It means a lifestyle change. As one woman said, "The lottery is my pension plan." Sure, the odds are that she would have a more comfortable old age if she saved the money instead of playing the lottery, but her odds of having a comfortable old age at all are better with the lottery.)

As for the fact that having a lottery at all makes for inequality, I think that you are right that people want inequality. But you don't need the lottery to conclude that. Look at the popular fascination with celebrity and wealth, even with royalty. People want the fairy princess to live happily ever after (or to be in rehab ;)).

you think too linearly. Embrace non-linearity.

Min: you are right. But, if people don't understand lotteries, they don't have a hope of understanding the tax/transfer policies of the political parties they vote for!

reason: but the Utility function of a risk-averse agent, as assumed by optimal tax theory, is non-linear!

Nick Rowe: " if people don't understand lotteries, they don't have a hope of understanding the tax/transfer policies of the political parties they vote for!"

You said it! :)

but but is it S-shaped?

And Nick, I thought it was clear that EXPECTED utility maximisation is besides the point here. People are spending a small amount of their income in order to be able to dream. It is cheaper than going to the movies. Actually winning the lottery (a bit like winning the girl) is almost a let down compared the dream.

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