Kelly Bets and Education Policy

by Daniel on January 26, 2004

Non-UK readers might not be aware of this, but there is the most almightly kerfuffle going on in the UK at the moment on a subject which I strongly suspect Americans would regard it as bizarre to be having a debate about. We’re all throwing beer bottles and calling each other fascists over the question of … whether different universities should charge different fees. Why? Well, for one thing, Blair and his government promised us in their last manifesto that they weren’t going to do this, and apparently some of us still care about the government’s habit of allowing us to go into the ballot chamber believing things that aren’t true (by the way, where the hell are our oversized pint glasses and longer opening hours?). But there is another, more fundamental reason; a lot of people believe that this is a fundamentally inegalitarian measure. And on my analysis (though not that of most other economists) they are right. Read on …

The argument that differential tuition fees are inegalitarian is quite an easy one to grasp intuititively. It’s simply that poor kids will be more impacted than rich kids by the need to stump up the cash, so poor kids will tend to be drawn to the cheap ‘n’ cheerful universities while the rich kids go to Oxford or Cambridge in even greater numbers. Obviously, this is bad news for Durham, Bristol and St. Andrews universities, which have traditionally subsisted on a market made up of posh kids too thick to get into Oxbridge when they had to compete against the oiks. But it’s also regarded (most particularly, by people from modest backgrounds who were given a leg up by going to Oxbridge, a segment of the population in which Labour MPs are quite thick on the ground) as somewhat inegalitarian; it closes down one of the exit routes from the slums for those of us who can’t box and for whom Pop Idol is not a realistic option.

The counterargument which the neoliberal tendency among the Blairites have latched onto is that this is wrongheaded, and that differential tuition fees are a profoundly egalitarian measure. After all, the universities which would be charging a premium are the ones which provide the entree into the upper echelons of society. So the graduates of those universities are veritably the creme de la menthe of the UK class system; why should they have their education subsidised by dockworkers, painters, waiters etc?

When you put the argument like that, it’s really quite obvious that it’s based on a brutal insistence on ignoring the distinction between ex ante and ex post. The people whose education is being subsidised aren’t investment bankers, management consultants, and so on; they’re scruffy kids, with for the most part no income of their own. This is an argument for a graduate tax, (and quite a progressive one at that), not for an upfront user fee. But the problem is that the money from a graduate tax would take time to arrive unless it was imposed retrospectively, and Blair et al want the cash now, to spend on universities before the next election, rather than in ten years time when it might be contributing to some other bugger’s feelgood factor, so they are demanding the money up front. Seems pretty clear to me that this is a pathological outcome of the political process from which we should be protected, and that the setting of large general principles of taxation like a move toward user fees ought not to be driven by the exigencies of one year’s budget process (if my spies are to be belived, btw, the matter is complicated by the fact that Brown favours a graduate tax while Blair is keen on user fees for ideological reasons).

SIDEBAR: As a concession to the backbench Labour MPs who have been up in arms about this one, the current proposal gives what have been sold as very generous repayment terms to students. Specifically, they wouldn’t pay it back unless they were earning a certain percentage of the national average wage, and the debt would be wiped out after 25 years. I don’t buy this, for a number of reasons, chiefly because I happen to know a bit about how much both of these measures cost. The “percentage of average wage” criterion is the equivalent of payment protection insurance, which is available on normal commerically contracted debt for a fee of about 0.8% of the amount borrowed, while the present value in 25 years time of almost anything at any decent discount rate is fuck-all to a bag of chips. But more fundamentally, you ask yourself the question “Will the £30,000 of debt that we’re proposing to saddle new graduates with, make it more difficult for them to get a mortgage?” and the answer is “Yes, indubitably, there is no way in which it couldn’t”. The CML have suggested that the repayment schedule means that students would still get a mortgage, but the debt service involved probably reduces the student’s borrowing capacity by … well, about £30K. So there.

But in any case, that’s the argument against having upfront tuition fees at all; what I want to consider now is the argument against differential tuitition fees. In order to do so, we’re going to have to descend into the murk, and consider the most plausible form of the argument outlined above; one which I believe is still based on the same ugly confusion about ex ante and ex post, but less transparently so.

This would be a version of the “graduate salary premium” argument based on net present values. The idea here is that you take the salary premium earned by a graduate over a non-graduate, capitalise it at some risk-adjsuted discount rate, and treat the go to uni/don’t go to uni decision as the decision to exchange a sum of cash now for a stream of future benefits. If you put it like this, then the decision to go to university (or to go to Oxford rather than Thames Valley Uni) can be seen as an investment like any other. And on the basis of plausible numbers about graduate salary premia, it can be made to look like an investment with a very high return indeed. Hence, why do we need to subsidise this investment; it’s obviously rational for people to go to university so all this stuff and what-have-you about poor kids being scared off by debt levels is just scaremongering.

Taken in its own terms, it’s not a bad argument. What’s wrong with it?

Well, basically, that it’s a misapplication of the net present value rule. The use of a risk-adjusted discount rate on expected cashflows is a very,very wrong a way to take account of risk in this particular case. Net present value is a way of discounting mathematical expectations of future values. It’s a sensible way to operate if you’re looking at a large portfolio of investments; at base, it’s an actuarial approach. As a result, it relies upon the large-sample properties of the underlying returns generating process to a very great extent indeed.

Unfortunately, this portfolio approach is not at all available to a teenager making the university decision. Because they only have one life to live, they can’t invest their human capital in a portfolio of educational experiences; it’s all or nothing. And average statistics for the graduate salary premium aren’t as helpful as one might think; on average, British women say “yes” to a marriage proposal 78% of the time [1], but that’s not much use to you as you woo the fair Roxanne, is it?

So to recap, the decision to go to university is one which is a one-time decision, where one is making an all-or-nothing decision on the basis of subjective assessments of the probability and magnitude of a favourable outcome. The net-present value version of the “graduates reap the benefits” argument is based on a decision rule which is appropriate for making a portfolio allocation decision on the basis of actuarially expected probabilities.

For the time being, I’m going to set aside my fundamental objection to this whole approach (that this kind of way of thinking about investment decisions ignores the effect of animal spirits and is a fundamentally wrong way of thinking about decisions made under time and uncertainty, also that it massively overuses asymptotic properties to generate rules over discrete decisions) in order to suggest a different mathematical decision rule which I think captures a few important aspects of the decision in question.

Deciding to spend a chunk of money on going to university to me feels less like making an investment and more like betting on a horse. You’re not in a situation trading off risk and return from a well-defined menu; you’re trying to suit horses to courses (or indeed, yourself to courses) on a one-shot success-or-failure bet. You think you’ve got a bit of special information about yourself which makes it worth gambling at all, but the overall atmosphere is one of uncertainty. What do you do in these conditions?

It’s a surprisingly poorly developed body of economic theory, but one decision rule which I’d regard as at least defensible is the Kelly Criterion. It’s a piece of information theory (an article on a possible interpretation of the “information rate” in Shannon & Weiner’s theory) retooled by Ed Thorp for the black jack crowd. It’s a science of what you do when making a bet in which you have an edge. To start your intuition running, imagine that you happen to know that a particular roulette wheel is biased and comes up red 60% of the time. How much money would you bet on this wheel? If you bet all your cash on the first spin, you’ve got a 60% chance of winning big, but if you lose, you won’t be able to bet any more. If, on the other hand, you bet the table minimum you won’t go bust, but you’re not really making the most of your opportunity. What’s the happy medium?

Basically, the Kelly criterion says that, facing a bet where you have a probability of winning p and an expected return r, you bet the following fraction of your bankroll:

Fraction = {[p(1+r)]-1}/r

Note that for an even money bet, r=100%, and the fraction simplifies to 2p-1; this is the simple form of the rule that blackjack players will quote that you “bet your edge”. In the case of the roulette wheel, we assume an even money payout, so you should bet 20% of your bankroll per spin.

Betting in this manner has a number of attractive long-run characteristics; your wealth grows at a faster rate than any other system of betting, the time to reach any given wealth level is shorter, and you have zero chance of going bankrupt. It’s this last propert of the Kelly criterion (that it doesn’t go “all-in”) that makes the real difference with the NPV rule; the NPV rule will always tell you to go all-in on a positive NPV investment.

The Kelly criterion was largely disposed of by Paul Samuelson in an important paper called “Why We Should Not Make Log of Wealth Big Though Years to Act Are Long” (the paper is entirely written in words of one syllable!). It’s a towering achievement of Samuelson’s, and like so many of his other towering achievements, I do wish he hadn’t bothered. Basically, the point is that log-wealth criteria (of which species the Kelly criterion is one; it’s the solution to an equation which maximises the log of wealth over time) tend to give excessive weight to small probabilities of very bad outcomes (cf Tyler Cowen’s discussion of the Paradox of Immortal Drivers), and this is inconsistent with any plausible utility function.

On the other hand, this very property (loss-aversion) makes the Kelly criterion more attractive to the modern behavioral finance crowd, who are attempting to resurrect various versions of it because loss-aversion seems to describe human behaviour a bit better than expected utility theory (refs refs refs). Since doing something similar won John Nash a Nobel Prize, I’m going to assume for the meantime that it’s OK for me to elevate this highly arguable mathematical criterion with a few favourable characteristics into the very definition of rationality. I hope that Prof. Q will forgive me for this dreadful mangling.

Anyway, for my purposes, the important thing about the Kelly criterion is that it captures an important intuition which appears relevant to the question at hand; that no matter how favourable the tradeoff, how much you are willing to stake on something depends on how much you own now. The NPV rule misses this because at heart it’s a two-period model being made to do the job of a dynamic one, so it doesn’t really have much of a concept of “keeping in reserve”. So, using some random numbers skimmed off the web, I’ll try and calculate how much a Kelly-betting student would be prepared to spend on education.

The consensus figure is that graduates earn £400, 000 more than non-graduates “over a lifetime”. I can’t tell whether that is a net present value and strongly suspect it isn’t, in which case the government is doing its little bit to further the cause of financial illiteracy. I’m therefore going to scale it down to £300,000 because then the oddsd are a nice 10 to 1 when compared to £30,000 of debt (I’m treating the debt as the initial investment). Based on this site’s numbers, I’m going to assume that the chance of landing in a “non-graduate” occupation is 10%, so your probability of winning is 90%. Great betting odds, looks like. Plugging the figures into the formula reveals that it would be rational (by this criterion) to bet 89% of your wealth on a university education.

The trouble is that, for anyone with less than £34k of net worth, spending £30,000 on a university education is an “overbet”. Overbetting is particularly bad in the Kelly system, because you’re not only taking on too much risk, you’re actually reducing your expected return. And my guess is that surprisingly few teenagers have £30,000 in assets, even if you include that proportion of their parents’ assets they could call their own in the form of an expected inheritance. How does this affect the desirability of a university education?

I’m going to assume a teenager with wealth of £10k, assuming that there’s a family with three children and housing equity of £30k; the average house equity in the UK is £60k, but that’s skewed toward old people. This youth would be betting 300% of his wealth on a university education rather than the Kelly bet of 89%. What’s the effect on his returns?

If you’ve been paying attention, you’ll realise that you don’t need to do the math on this one; an overbet devastates returns. If you bet a fraction equal to or greater than 100% of your wealth, then you will go bankrupt with certainty in finite time (the gambler’s ruin theorem). So it looks like Samuelson had a point; a Kelly-bettor would not go to university unless he had wealth of more than £34k, which seems far too conservative.

The actual numbers here aren’t that significant. The important thing to bear in mind is that the question of inveting in a university education is qualitatively different for the poor than for the rich. If you’ve got a bankroll of £34k per child (equates to housing plus financial wealth of about £65k for your typical kid), then your decision to go to university is clear cut; it’s strictly rational in the sense that if you were faced with a succession of such choices over time, you would maximise the log of your wealth by accepting them all. If you’ve got less than £30k, then it’s much more of a leap in the dark. There is potentially a genuine rationing effect here which arises as the result of requiring the up-front payment rather than structuring the loan as a graduate tax.

I’ve not addressed except in passing a number of other issues that I regard as important; the precedent of introducing user fees, the fact that removing a middle-class “perk” is just the sort of thing that undermines the solidarity that’s the bedrock of a welfare state, and the fact that overhanging debt is a fine tool of political control (a theme that will probably be addressed this week by a number of CT posters). I also haven’t touched on the wider question of UK university funding, other than to say that I have no confidence at all that this proposal will really make more funds available to them rather than being offset. But I’d recap my point here as follows:

1. There is a very real difference between a graduate tax (a tax on graduates making lots of money) and a user fee (a tax on teenagers with no money who might in future become graduates), particularly in terms of the incentive effects. Note that a graduate tax is a tax that you incur by virtue of something that happened in the past, and thus doesn’t have further incentive effects.

2. This problem is tied up with fundamental questions of time and uncertainty which economic theory does a very bad job indeed at modelling.

3. Anything which means that you have to make an investment up front in your future is likely to have a disproportionate incentive effect on people with less wealth.

4. Once more, a graduate tax is not the equivalent of a debt to be repaid in line with income, because you cannot put the expected value of a stream of future earnings and tax payments on the same footing as a certain debt today, because key assumptions of the NPV rule (diversification and marginality) are not satisified.

That’s it. I’m not wholly satisified with this piece. But the vote is tomorrow, so it has to go up today.

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{ 54 comments }

1

Chris Bertram 01.26.04 at 1:50 pm

Two observations:

(1) Your argument supposes — indeed asserts — that students are being asked for an upfront user fee. But this is false. They are asked for such fees at the moment (of just over £1000) but the proposals abolish this. And as I understand it the repayment proposals have some features of a graduate tax (how much you pay varies depending on how much your earn). Those who earn little will only repay a proportion of the debt and will have it written off eventually.

(2) Your ex ante/ex post argument supposes that egalitarians should be concerned with the pattern of distribution among people at particular stages of their lives — “scruffy kids” — rather than over their lives taken as a whole. Perhaps, but it would be nice to know why you think this.

2

Matthew 01.26.04 at 1:55 pm

Dsquared,

I thought that was excellent, and it is certainly true that the welfare state needs middle class support.

However I think you misunderstand the nature of the proposals. According to you,

“But the problem is that the money from a graduate tax would take time to arrive unless it was imposed retrospectively, and Blair et al want the cash now, to spend on universities before the next election, rather than in ten years time when it might be contributing to some other bugger’s feelgood factor, so they are demanding the money up front”

but according to the BBC,

“unlike now nothing would be paid “up front” – so students would no longer have to pay while at university. Instead fees would be recovered from graduates, starting once their annual income passed £15,000 “

Now unless you are referring to some clever accounting tricks this seems to that very little money will come in from these measures unless there are loads of people who opt to pay up front, but if this is possible it seems unlikely. Furthermore the current (up front) fee will be phased out, and maintenance grants will return.

This would seem to have the opposite budgetary effect in the short term than you say.

Matt

3

Chris Bertram 01.26.04 at 1:56 pm

And isn’t there a little bit of tension between your assertion that

Blair et al want the cash now, to spend on universities before the next election…

toward the beginning of your post and

_I have no confidence at all that this proposal will really make more funds available to them rather than being offset._

towards the end.

Not strictly a contradiction, I grant you.

(And, actually, it is hard to see how there will be much more available before the next election on these proposals – they’ll take much longer to kick in.)

4

dsquared 01.26.04 at 2:12 pm

Right, I see what you mean. I have looked into this and I think I’m right, but it’s not at all clear in the piece.

The user fees show up as current revenue to to the government because of the accruals principle; it’s in the current year that the kids make a commitment to pay them. The cash arrives over time, but it makes an impact on the govt. borrowing statistics right now.

Similarly, the kids are being offered a debt-like contract right now with a face value, and the option to pay it now or stick it on the tab; rather like the options available at a car showroom. The repayment scheme really doesn’t have that many features of a graduate tax; if your income reaches £15k and stays there forever, you pay the whole lot of it. As I say, it looks a lot more like payment protection insurance than anything else, and that currently sells for about eighty basis points. And it’s a sunk cost investment; if you get a degree, then chuck it in and become a used car salesman on £16K you still have to pay.

Your ex ante/ex post argument supposes that egalitarians should be concerned with the pattern of distribution among people at particular stages of their lives — “scruffy kids” — rather than over their lives taken as a whole. Perhaps, but it would be nice to know why you think this.

Because there’s a difference between scruffy kids with rich dads and scruffy kids without. If you put into place a system which has the effect of making it more risky for poor kids than rich kids to invest in education, then the forseeable effect of that policy will be that fewer poor kids than rich kids make that investment, which will result in greater inequality over lives as a whole. The argument about scruffy kids is meant to be a counter to the NPV argument suggesting that this isn’t a price-rationing system.

5

dsquared 01.26.04 at 2:15 pm

I’d add that although the flat rate £1000 is being abolished, the total investment being asked of consumers in their own education is clearly rising. There are a whole knot of issues here as regards the extent to which cash flows at different times can be brought onto the same footing, but there’s decent empirical evidence (and decent theoretical reason) for future liabilities having more presence in people’s decisions than future benefits.

6

harry 01.26.04 at 2:50 pm

Daniel,
just to be clear about part of your response to Chris, you are denying his imputation to you of the view that

‘egalitarians should be concerned with the pattern of distribution among people at particular stages of their lives — “scruffy kids” — rather than over their lives taken as a whole’.

Right? There is no view about the need for people to be more equal within stages of life; you are just saying that the inequalities will affect the magnitude of risk that a fixed loan represents to that person. Just asking for clarity.

BTW, will the concessions being made (hour by hour it seems) improve the proposal? Either way, great post.

7

Andrew Boucher 01.26.04 at 2:51 pm

“if you get a degree, then chuck it in and become a used car salesman on £16K you still have to pay.”

This strikes me as obviously a good. Why is it “still” ?

On the other hand the bill seems to create an incentive to emigrate – get the free university education, emigrate and refuse to pay afterwards.

8

harry 01.26.04 at 2:51 pm

Sorry, ‘clarity’ should have read ‘clarification’.

9

harry 01.26.04 at 2:55 pm

Andrew’s point actually suggests one way in which it might be superior to a graduate tax. It is a graduate tax which would create an incentive to emigrate (because the UK, unlike the US and the Roman Empire, doesn’t tax its citizens, only its residents). But the government can demand repayment of a loan wherever you are, just like a crdit card company, unless you move to sufficiently obscure places that its impossible to trace you.

10

harry 01.26.04 at 2:55 pm

Andrew’s point actually suggests one way in which it might be superior to a graduate tax. It is a graduate tax which would create an incentive to emigrate (because the UK, unlike the US and the Roman Empire, doesn’t tax its citizens, only its residents). But the government can demand repayment of a loan wherever you are, just like a credit card company, unless you move to sufficiently obscure places that its impossible to trace you.

11

Seth Gordon 01.26.04 at 2:59 pm

The obvious free-market solution is to bring back the old-fashioned custom of indentured servitude. Employment agencies should be allowed to pay kids’ college tuition in exchange for a 25-year indenture. They can offer each prospective servant a tuition package appropriate for his or her future earning power, and universities can charge the agencies whatever tuition they like.

Everybody wins! After all, the whole point of going to school is being able to make money when you’re grown up, right?

12

dsquared 01.26.04 at 2:59 pm

Right? There is no view about the need for people to be more equal within stages of life; you are just saying that the inequalities will affect the magnitude of risk that a fixed loan represents to that person. Just asking for clarity.

To be honest I hadn’t thought about it that way. There wasn’t any intention to suggest that egalitarianism has to mean egalitarianism over life-slices rather than lives. Which doesn’t mean that I might not end up being committed to that view.

13

Mark 01.26.04 at 3:00 pm

Daniel, two questions:

1) How are your calculations affected by the fact that the maximum figure being discussed at the moment is 9-12 thousand pounds, rather than 30?

2) I’m not particularly familiar with the extent of UK financial and merit aid; how much of that 3-4 grand a year would be offset by grants, scholarships, etc?

14

dsquared 01.26.04 at 3:09 pm

The £30k is the figure I got from trawling the web of the total forecast average debt on a newly minted graduate under the current proposals. Everything, including grants, scholarships, and the level of fees, seems to be a hell of a moving target at the moment. If you want to use a £12k figure, then just substitute it throughout into the calcuations; the Kelly edge would in that case be 0.896, (it’s a risk-averse log wealth criterion, so it’s not sensitive to quite substantial changes in the odds at high edge) and the minimum wealth needed to make university strictly Kelly-rational would be £10,752. Though a) I think this is a low estimate and not really appropriate and b) since we’re for the most part worrying about the effect on teenagers from households with no wealth at all, the question is moot.

15

dsquared 01.26.04 at 3:10 pm

The obvious free-market solution is to bring back the old-fashioned custom of indentured servitude. Employment agencies should be allowed to pay kids’ college tuition in exchange for a 25-year indenture. They can offer each prospective servant a tuition package appropriate for his or her future earning power, and universities can charge the agencies whatever tuition they like.

Hahahahaha very funny (bitter laugh). This is how I financed £18k of business school debt.

16

Jesse R 01.26.04 at 3:45 pm

Isn’t the right measure of the teenager’s wealth to use in your calculations the present value of her earnings in the job she would end up with if she didn’t choose to make the investment? This would imply a bankroll of rather more than 10k pounds.

Of course, there are issues with the risk associated with the non-university job. And if the tuition would be required up-front, then liquidity constraints might make students unwilling to commit more than their current assets on the basis of their future earnings. But when the government is offering to finance the tuition, it is hard to see the argument for considering only current assets in your equation.

Now, I’m no great fan of the permanent income hypothesis, but this does seem like a case where it is relevant. Which isn’t to say that it changes your basic point, that increased tuition is likely to have an effect on poor students’ decisions. I think this holds up in any case, whether or not we can prove that it is rational for them to react to it.

17

Conrad barwa 01.26.04 at 3:53 pm

Just some brief thoughts:

1) one of the main problems with a graduate tax for Blarites, is partially one its virtues that you extoll – namely that it does come close to being a sort of progressive tax on earnings. I think you sort of allude to this when you mention that Blair favours user fees for ideological reasons; this is very much in line with his general proposals for other public services such as health, social security etc. the principle of those who earn more paying more, does not really present itself as an attractive one compared to a more broad and in some cases flat user-fee. Part of this is down, I would hazard, to his ‘Daily Mail’ obsession that the well-off middle classes shouldn’t explicitly feel that they are having to stump up more than what they get out of the system (I could be wrong, but seems to be my general impression).

2) the argument on differential fees is an excellent one. I don’t fully understand the English education system but at least some of the distortions seem to come from the fact that a disproportionate share of entrants into Oxbridge, tend to be those educated in the Public school system as opposed to those in the state system and this leads from what I can observe to an odd form, of an almost soft-caste system – albeit one that is being steadily eroded. In thise sense some of the inequalities presented here bestow an acquired advantage to Public school students in achieving the results necessary to enter elite universities; and given the shambles that the state school sector seems to be in, this is quite a hefty favour in their factor that imposes another additional obstacle for students from poorer backgrounds. On the other hand, as I hear ad infinitum from the said quarter, the education at Public schools is quite expensive and often paid for by struggling middle class parents who cut back on other luxuries to give their children a better chance in life (so out goes the annual holiday to Tuscany and the good wine, then). This has led to the feeling that having paid for this expensive education, any acquired advantages are well deserved and that differential fees simply extend the same sort of streamlining to the university sector; if anything some of these students feel aggrieved that having paid all this money for their school education, they don’t benefit from the historic subsidy to university education that their forebears enjoyed and that they have to cough up for that too. I don’t know exactly how this impacts in a net fashion on the fees argument, but I feel things are in some manner going to go down the wrong direction if simple user-fees are going to be introduced; I guess in line with your final point (3) this is an additional penalty that poorer students will have to face if such a proposal is introduced – not that under current conditions the costs of relatively poorer schooling is compensated by a subsidised university education by much.

3) Also my understanding is that the bulk of the evidence from countries where user-fees are in place; is that this has not really impacted on the number of less-well-off students going to them. This is, in turn a reflection of the labour market where it is becoming more and more imperative to have university-type qualifications to sucessfuly obtain a foot on the career ladder (the collapse of many manufacturing industries that relied on manaul apprenticeships has something to do with this as well). So a lot of the people who will end up paying these user-fees will be the less well-off who have to overcome substantial social and attitudinal obstacles to enter university in the first place. From a labour perspective, I would have thought that through this period at least, govt policy should be geared towards making things easier as opposed to harder for them to do so.

18

dsquared 01.26.04 at 3:56 pm

Isn’t the right measure of the teenager’s wealth to use in your calculations the present value of her earnings in the job she would end up with if she didn’t choose to make the investment? This would imply a bankroll of rather more than 10k pounds.

An attractive idea on the face of it, but I don’t like using the net present value rule in this case. Also those earnings aren’t available to invest now, and I don’t see any way of making them so.

19

Jesse R. 01.26.04 at 4:09 pm

“An attractive idea on the face of it, but I don’t like using the net present value rule in this case.”

Fair enough. NPV may assign too much weight to uncertain income down the road. But there has to be some weight attached to future income: It really isn’t the case that a destitute 16-year-old has nothing he can risk at the blackjack table, as long as someone is willing to finance the bet for ten years or more. Which brings us to:

“Also those earnings aren’t available to invest now, and I don’t see any way of making them so.”

No, they aren’t available to invest now. But the government is offering a loan, on which payments needn’t be made until those future earnings start to become available. Doesn’t that, in essence, make the earnings available to invest now?

If I win a lottery in the U.S., payments are generally structured as equal nominal amounts, annually for 20 years. The 20th year’s payments aren’t available to invest now, except that my neighborhood multinational bank will happily offer me a loan, on good terms, against those future earnings. So if I won the lottery last week, and I see a blackjack bet with a high expected value, I can safely place a large bet on it without risking ever going hungry. (Correct me if I’m wrong, but it seems that the risk of going hungry is more or less what the Kelly criterion is all about–running flat out of cash is a very, very bad outcome, one not to be risked under any circumstances.)

20

Pete Stevens 01.26.04 at 4:12 pm

Graduates earning 300k more than non graduates over a lifetime.

I suspect this number is true today for the present graduates, many of whom went to university in the 60s & 70s when going to university was unusual and a degree was a much bigger differentiator than it is now.

I strongly suspect that the 50% of present teenagers will not on average earn 300k more than their peers because now being a graduate is no longer rare or unusual.

This also doesn’t include the effect of interest – non graduates start earning earlier than graduates which makes a huge difference to compounded interest investments like pensions.

21

Barry 01.26.04 at 4:14 pm

Chris – about Daniel’s comment:

“I have no confidence at all that this proposal will really make more funds available to them rather than being offset.”

If I’m interpreting Daniel’s comment correctly, he anticipatess that the Blair government might do something sneaky, like cut university funding as they impose fees. From what I heard, this is what my state (US, Michigan) did after it instituted a lottery, with the profits earmarked towards education – they reduced general fund payments.

Frankly, I’m shocked at Daniel’s lack of patriotism. After all, this is the same Blair who told us that Saddam had WMD’s ready to use in 45 minutes. Just because Blair was speaking through an excretory orifice on trivial matters of war, Daniel doesn’t trust him with money. Tsk, tsk :)

22

bill carone 01.26.04 at 4:31 pm

Dsquared,

I agree that using NPV *with a risk-adjusted discount rate* gives problems.

Why not just use decision-theoretic methods to model risk and use NPV *at a risk-free rate* to model time preference?

That way, you would say “I have a 90% chance at a $300K increase in NPV and a 10% chance at a $30K decrease in NPV” or something. Given a particular person’s risk aversion, you could find the best decision.

Is there a problem with this that I don’t see?

23

Maynard Handley 01.26.04 at 4:52 pm

Daniel,
As is frequently the case in these things, I think you fail to consider the full sociology of the issue.
On the one hand, yes, there is the case you are presenting: that going to college is a risk, but that there is some generalized social utility to having people go to college (both from the point of view that more equality is better, and from the point of view that an educated workforce is better); therefore it makes sense for the government to encourage people to go to college, not make it more difficult.
BUT there is the other hand, which is the belief that people are going to college to have a good time for three or four years, with no realistic hopes of ever turning into more productive citizens, and with ever expectation of partying and screwing for as long as they can get away with it. You have not mentioned this, but my guess, is that this view (helped out by plenty of movies like Animal House along with Girls Gone Wild DVDs) is secretly animating part of this proposal.
I think it behooves you to deal with this issue, either with numbers that show that this is, in spite of Hollywood, not an important issue (at least for British Education), that most of those going to higher education do a good job of using their time well and being converted into more productive citizens; or, if the numbers sadly show that this is not the case, that their is a core of 20% or whatever parasites, proposing some alternative mechanism that will prevent their leeching off the system.

I’ve said it before on CT; the left, bless its heart, pays far too little attention to the issue of social parasites. It would far rather claim that they don’t exist, a claim that most people regard as ludicrous, than deal with the problem honestly; for example either saying “yeah, there’s a percentage x of parasites, and that sucks, but we consider it to be acceptable given the payoff”; or proposing their own de-leeching proposals rather than simply complaining whenever someone else proposes policies aimed at attacking leeches.

24

Chris Bertram 01.26.04 at 5:03 pm

Much as I hate the phrase “the bottom line”, in this context it seems appropriate. Here’s the “Guardian’s summary of the proposals”:http://education.guardian.co.uk/students/tuitionfees/story/0,12757,1128089,00.html

bq. Upfront fees will be abolished (as in Scotland). Universities will be allowed to charge up to £3,000 a year for undergraduate courses from 2006. The amount is up to the university and could vary by subject. Students will repay the money after they graduate and are earning £15,000 or more. They will be charged zero rate real interest on the loan. They will still be able to take out student loans to cover living costs, to be repaid with the fees loan. Repayments are based on income, not the size of the loan – for a graduate earning £18,000 weekly payments will be £5.30.

So, to be clear, they’re are being asked to stump up the price of 2 pints of lager or one packet of cigarettes on a weekly basis. Most students of my acquaintaince — even “scruffy kids” — do not seem deterred from such expenditure.

25

bill carone 01.26.04 at 5:09 pm

Dsquared,

“imagine that you happen to know that a particular roulette wheel is biased and comes up red 60% of the time. How much money would you bet on this wheel?”

Aren’t there standard risk-scaling results on this?

Assuming I get one bet, then the game is over, I would increase my bet until adding one more dollar to the bet gives me less than a one-dollar increase in certain equivalent.

For me, with my $150,000 risk tolerance, this means that I would bet around $31,000 on the 60% roulette wheel, since

– if I pay $31,000, I get a deal with a 60% chance of ending up with $62,000 and a 40% chance at ending up with nothing. I value this at about $34,060, and

– if I pay $32,000, I get a deal with a 60% chance of ending up with $63,000 and a 40% chance at ending up with nothing. I value this at about $35,053.

So, paying $1000 more only gets me an increase of (35,053 – 34,060)=$993. So, if I can choose what amount to bet, I will only bet up to around $31,000.

Note that the $32,000 bet is still attractive; if I was given the choice of $32,000 or nothing, I’d bet the $32,000. The $31,000 is just better (it nets me $3060, as opposed to only $3053).

This still jibes with one of your points; smaller-income students will tend to have smaller risk tolerances, and thus not be willing to pay as much as higher-income students for risky deals.

If you can play over and over again, then the results probably change (e.g. if you could play infinitely quickly, you would probably bet as low as possible and let “gambler’s bliss” take over). However, you were discussing one-time decisions, and standard decision-theoretic results seem to hold. Or am I missing something?

26

Keith 01.26.04 at 5:18 pm

creme-de-la-menthe
No way, Pedro!

27

dsquared 01.26.04 at 5:25 pm

Is there a problem with this that I don’t see?

Not really, but my problem is that I don’t think that the existence of the expectation of the future cashflows is on solid probabilistic ground.

28

dsquared 01.26.04 at 5:29 pm

So, to be clear, they’re are being asked to stump up the price of 2 pints of lager or one packet of cigarettes on a weekly basis.

Mate, do you buy a lot of things from catalogues? Sorry to be so blunt, but this is “easy payments” sales patter. A three grand fee costs £3,000, whether it’s on the never-never or not.

i>Most students of my acquaintaince — even “scruffy kids” — do not seem deterred from such expenditure.

But there has to be an element of selection bias here?

29

Matt 01.26.04 at 5:45 pm

If you want an instance of a university loan program gone wrong, try googling the words “yale tuition postponement option”

30

Chris Bertram 01.26.04 at 5:48 pm

_Mate, do you buy a lot of things from catalogues?_

Nope, though if the terms were as good as these: zero rate real interest and repayments tied to my income and not the size of the loan, I might.

In any case, central to the case made so elegantly in your original post was a claim about the effect of the scheme on real-world behaviour. I’m being crude and anecdotal here I know, but since actual people do buy things from catalogues (as well as lager and cigarettes), the fact that they do so may reasonably be trotted out by those who are sceptical (as I am) about the alleged deterrent effect.

31

bill carone 01.26.04 at 6:01 pm

“I don’t think that the existence of the expectation of the future cashflows is on solid probabilistic ground.”

I’m not sure what you mean.

Two situations.

– What would I pay now to get a 50-50 chance at $1000, paid right now?
– What would I pay now to get a 50-50 chance at $1000, paid a year from now?

The first case is fine, no? For these low amounts of money, I’m risk neutral, so the answer is at most $500.

The second case, I would simply discount the $1000 at my risk-free rate of 3% per year (ignoring inflation), then do the same calcuation ($498.50 or so, no?). If I weren’t risk neutral, I would discount my risk-aversion by the same rate, then do a certain equivalent calculation.

What is the difficulty? (real question; I’d like to know your point of view). I can’t imagine saying that these two situations are so totally different that you can’t use similar methods for both.

32

dsquared 01.26.04 at 6:05 pm

Well, it’s like this; what would you pay right now to be awarded 10% of my post-tax income for the year 2030, in twenty-seven years’ time?

The only answer I can see to that question is “There is no sensible basis whatever for me to form an expectation of that variable”.

33

dsquared 01.26.04 at 6:07 pm

In other words, I’ve got no problem with discounting, NPV and risk-adjusted rates per se, it’s just that there are a number of contexts where I don’t think they can sensibly be applied (basically, anything that an actuary would refuse to give you an estimate for).

34

Jason McCullough 01.26.04 at 6:33 pm

“you put into place a system which has the effect of making it more risky for poor kids than rich kids to invest in education, then the forseeable effect of that policy will be that fewer poor kids than rich kids make that investment, which will result in greater inequality over lives as a whole.”

Did you mention something a while back about putting together a “increasing risk-exposure in society” series?

35

dsquared 01.26.04 at 6:40 pm

Yeh, but the size of it overwhelmed me to be honest.

36

bill carone 01.26.04 at 7:43 pm

Dsquared,

“Well, it’s like this; what would you pay right now to be awarded 10% of my post-tax income for the year 2030, in twenty-seven years’ time?”

This just comes down to assessing my own personal probability distribution on your income in 27 years time, no?.

This is difficult and time-consuming, but certainly possible. I would start by introducing other distinctions: your career path for the next 27 years, your current salary, salary growth rates for different careers, etc. I could then assess my probabilities for these (or add new distinctions until I could). Then simple Bayesian calculations would yield a probability distribution that represents exactly what I know right now about your salary in 27 years.

If it was a really important decision, I could hire an expert who (perhaps tacitly, perhaps explicitly) knew lots about people’s incomes and careers and get the expert’s probability distribution to use instead of my own.

I have to do this kind of assessment all the time in business and medical decision consulting; most of the time you don’t have data, and you need to rely on expert judgment.

How long will the patient live? How long until Alzheimer’s symptoms develop? How many barrels of oil can we pump out of this well? How long will phase II of this clinical trial take? What cash flows will result from this product going to market? What will the NPV be?

To all of these questions, the answer “I don’t know” is identical to the answer “I can assess a probability distribution for it.”

“In other words, I’ve got no problem with discounting, NPV and risk-adjusted rates per se, it’s just that there are a number of contexts where I don’t think they can sensibly be applied (basically, anything that an actuary would refuse to give you an estimate for).”

People use these methods to make business decisions all the time, do they not? Most of these decisions have uncertainties that no actuary would want to deal with (success rate of new drug, profit margins for new product, manufacturing costs, winner of the next presidential election, etc.).

You could argue that all these people are mistaken. I think that in many cases people use risk-adjusted discounting when they shouldn’t, for example. However, you can always go back to the basic decision-theoretic models that work much more generally, and don’t require an actuary, just someone who needs to make an important decision with incomplete information, and an analyst who can translate this imperfect information into numbers.

37

Mark 01.26.04 at 7:54 pm

Daniel — aside from cracks about buying from catalogues, what’s your take on the figures Chris cites? Are the figures wrong, somehow? Are they misleading? Do you think the government will change the terms of the agreement after the bill passes? If none of the above are true, what objections do you have, given those numbers?

38

james 01.26.04 at 9:06 pm

“Nope, though if the terms were as good as these: zero rate real interest and repayments tied to my income and not the size of the loan, I might.”

Is this not a good reason to assume the terms won’t remain this good? And once the principle is breached…

39

Wuffly 01.26.04 at 9:59 pm

Of course the terms will change. Admitting the breach of the principle leaves the door wide open for future tinkering. Also, the differential effect is going to be minimal as most universities will charge the maximum as they need the cash. The only way variability will work is if there is more convergence between the fee and the true cost. If this goes through, I’ll bet my mortgage that the cap will be lifted after the next two terms (and half my mortgage that it will happen before) and we will then be looking at fees of 4-10k.

40

Wuffly 01.26.04 at 10:00 pm

good post by the way.

41

ahem 01.27.04 at 5:00 am

On the other hand the bill seems to create an incentive to emigrate – get the free university education, emigrate and refuse to pay afterwards.

You’d be surprised at the lengths that the current Student Loans Company goes to address the rising number of people who do just that.

42

ahem 01.27.04 at 5:04 am

The obvious free-market solution is to bring back the old-fashioned custom of indentured servitude.

I’m sure certain Oxford colleges would jump at the chance to reintroduce the old-style ‘Commoner’, whose fees were offset by performing service duties (a university version of ‘fagging’) during their degrees.

43

dsquared 01.27.04 at 7:10 am

To all of these questions, the answer “I don’t know” is identical to the answer “I can assess a probability distribution for it.”

Absolutely disagree. I don’t think you can sensibly assess even a mean and variance. Business decisions get made on the basis of animal spirits.

44

Ruth Hoffmann 01.27.04 at 8:01 am

they’re are being asked to stump up the price of 2 pints of lager or one packet of cigarettes on a weekly basis. Most students of my acquaintaince — even “scruffy kids” — do not seem deterred from such expenditure.

But you don’t take into account the other point D^2 raised: the full amount of the debt will still count against that person’s borrowing ability. From personal experience, I can tell you that “sure I’ve got $22K in student loans, but it’s on an income-contingent repayment plan” made no difference to the bank counting the whole debt against me in my attempt at mortgage qualification.

And they did the same thing to the MN lottery that they did in MI. So not to be snarky, but you may want to take a good hard look at the kind of debt load US students are carrying these days, and how much (and how fast) it’s increased since the program was implemented, before you start down that road yourselves.

45

Rich 01.27.04 at 11:53 am

> aside from cracks about buying from catalogues, what’s your take on the figures Chris cites? Are the figures wrong, somehow? Are they misleading?

If you’re attempting to repay a £20,000 loan at £5.30 per week – even assuming no interest at all – it’ll take you 72 years to repay the whole lot. If you take into account any interest rate over 1.3% pa, your capital just increases and you get further into debt. So I, at least, think they’re misleading.

46

Chris Bertram 01.27.04 at 11:56 am

Except, Rich, that another part of the proposal is to write off any remaining debt after 25 years.

47

dsquared 01.27.04 at 1:12 pm

They write the debt off after 25 years. Otoh, £5.30/week is £275.60/year which is about 2% of your disposable income if your salary is £15k, which isn’t nothing.

48

Rich 01.27.04 at 3:07 pm

> They write the debt off after 25 years. Otoh, £5.30/week is £275.60/year which is about 2% of your disposable income if your salary is £15k, which isn’t nothing.

So what’s the point in making someone pay so little towards the debt? If it doesn’t cover the interest/inflation, and is going to be written off in 25 years, why not write it off straight away and save the poor hypothetical bugger 25 years of being £20,000 in debt? Or at least start the payback at a level that pays the debt off in a finite time.

49

bill carone 01.27.04 at 3:18 pm

“Absolutely disagree. I don’t think you can sensibly assess even a mean and variance. Business decisions get made on the basis of animal spirits.”

I guess I’m not sure what you mean by “sensibly” or “animal spirits”

Let’s say I want to assess your information about, say, the weight of your desk in your office. You don’t know this, right?

Let’s say I’d like to give you a chance to win $10,000 from me, and I give you a choice:

A. Win $10,000 if you can call a coin flip correctly, or
B. Win $10,000 if the weight of your desk is over 100 kilograms.

Which would you prefer?

If you prefer A, then your probability of your desk being over 100 kilos is less than 50%. If you prefer B, it is greater than 50%.

So, I have at least put bounds on a probability that represents your personal information about the weight of your desk. You can imagine how I could zero in on your probability of “greater than 100 kilos” and how I could get a complete probability distribution for any weight.

What part of this is “unsensible” or based on “animal spirits”?

50

Conrad barwa 01.27.04 at 6:15 pm

Well, less than an hour to go before the vote and I wonder whether anyone would care for a wager?

My bet is that Blair will squeeze through at worst, with a few votes majority and stroll by with a safe margin if he is lucky?

51

Anurag 01.28.04 at 4:20 am

Bill,

On your point on D^2’s desk, I think the difference is that we can expect that D^2 has good information on what it’s weight is. That’s pretty different from estimating what 10% of his income 27 years from now might be. The future is uncertain in ways that are different from his personal information about the present.

Would you follow a similar path to placing a bet on what the desk he’s using in 27 years might weigh?

52

Idiot/Savant 01.28.04 at 4:25 am

New Zealand has had a similar scheme for 12 years now. Universities get to set their own tuition fees to make up for being underfunded by the government; the government provides loans to cover fees, course costs and living expenses; repayment kicks in at the minimum wage. What’s it done? Here’s some factoids:

* Fees have increased from NZ$1000/yr when introduced to on the order of NZ$4500 today (and a lot more for med students and dentists). Having given institutions the power to set their own fees, the government is now trying to stop further increases, by threatening to deny increased funding to any institution that does.
* Vastly more people are going to uni, and from a greater range of social backgrounds. Part of this is because the government is funding more places (at a lower rate per head), and partly it is because of a growing realisation that you need a degree if you want to be anything other than a McProle (hinted at by Conrad B above).
* Graduates face enormous debt burdens (mostly due to living expenses – the introduction of fees was combined with the elimination of student allowances for all but the very poor, meaning many students have to borrow for food); as a result they are deferring getting a mortgage and having kids until much later in life. This is a dramatic change in the kiwi lifestyle.
* More graduates are working overseas to repay their debts. This is an attractive option for Kiwis (who have a long OE tradition as well as a weak currency), but we’re not sure yet how many are coming back.
* Fewer people are training as teachers and nurses, because the wages in those professions will never repay the loans. Instead, they’re training as lawyers and accountants.
* Even high-paid government policy analysts are looking at not being able to repay their loans until after they retire.
* On the plus side, because the loan is forgiven if you die, more pensioners are spending their golden years in study.

53

bill carone 01.28.04 at 3:28 pm

Anurag,

“The future is uncertain in ways that are different from his personal information about the present.”

I’m not sure I follow this; he has imperfect information about the present, imperfect information about the past, and imperfect information about the future. There is a lot known about the future, and a lot not known about the past and present, no?

“Would you follow a similar path to placing a bet on what the desk he’s using in 27 years might weigh?”

I think so; the resulting distribution would have to be wider, implying that he has much less information about his future desk than his present desk. But he still has some (it probably doesn’t weigh 4000 kilos; I’d bet against it, so I have some information about it).

We might have to define “desk” more carefully (what if the buildings of the future simply use nanotech to create whatever furnishings you need at the moment you need them? Impossible? Maybe, but I certainly don’t know enough to claim it would be impossible.).

We might also need to add new distinctions: for example, we could assess his probability that he is working in academia in 27 years. We could then assess a distributions given “yes” and “no” answers to that question, then calculate the overall distribution, not given that information. We could add any number of distinctions to help him think more carefully about the future. This happens a lot in business; the price of oil in 27 years might be of great importance, and we might add 20 or 30 new distinctions in order to get a high-quality assessment.

Note that the questions I ask (“Would you prefer A or B?”) do not allow the answer “I don’t know.” I’m asking, right now, given everything you know about the world, which deal you choose; you don’t have to know anything about probabililty for me to encode your information. Sometimes I use the following way to get someone’s mind engaged.

“We have access to a clairvoyant, and she knows exactly how much your desk will weigh in 27 years. She has given me this piece of paper (I show it) with the answer written on it. Now choose, and we will look at the paper to see, right now, if you win or not. Note: you will immediately forget this after our conversation (you can’t alter the future based on this information).”

So, short answer, I would use very similar techniques in both cases; the uncertainty is higher about the future than the past, but it is essentially the same kind of uncertainty, and can be encoded into a probability distribution.

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Gavin 01.30.04 at 8:50 am

A couple of minor points: since the actual impact of the plan is 9k of extra for those with family incomes above a certain level, then 9k is the appropriate wager for the Kelly bet. Also, since the plan improves the situation of people from the lowest income families, that also takes a lot of people with no assets out of the betting pool. And when it comes to assessing assets for the Kelly bet, I’d be inclined to think about including parental assets too, as well expected future (non-university education) incomes.

This is all just an empirical question really – so why don’t we all just wait and see what happens to student enrollments? If Daniel is right, then enrollments will fall dramatically. If not, they won’t. I’m inclined to the latter myself.

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