Who Gets the Marginal Pound?
The economics of educational allocation, and the objective function nobody will name
Abstract. Public argument about educational spending is conducted as though the question “should we spend equally on all pupils, more on the strongest, or more on the weakest?” admits of an evidential answer, when in fact it admits of none until a maximand has been specified; this essay argues that the allocation problem is jointly determined by two inputs that debate persistently conflates, namely the objective function a society adopts and the shape of the educational production technology, and that neither can be inferred from the other. Working through four candidate objectives — aggregate human capital, welfare under distributional weights, equality of opportunity, and democratic adequacy — I show that each generates a different marginal-value schedule across the attainment distribution, and that the empirical literature, far from adjudicating between them, is largely silent on the question they disagree about. I then argue that the strongest efficiency case for spending on able children is not a case for spending more on those already identified as able, since the causal evidence on gifted programmes and selective schools shows small or null effects at the admission margin, but a case for spending on identification itself, which is cheap, and whose returns are large precisely because the existing system fails at it. Conversely, the strongest case for spending on the weakest is not that remediation is efficient, since often it is not, but that a threshold of civic and economic adequacy has a claim on resources that does not derive from returns at all. Equal spending, I conclude, survives not because it is defensible but because it is the unique allocation that requires no agreement about purpose: it is the fiscal expression of a society that has declined to say what education is for.
Keywords: economics of education; distributive justice; human capital; skill formation; equality of opportunity; educational adequacy; prioritarianism; positional goods; ability grouping; resource allocation
The question that is never actually asked
A finance ministry has an additional hundred million pounds and a decision to make. It can spread the money evenly across every pupil in the system. It can concentrate it on the children who are already flourishing. It can concentrate it on the children who are failing. Every serious participant in the debate believes their answer follows from the evidence. Almost none of them can state the maximand that would make the evidence relevant.
This is not a small omission. It is the whole of the problem. “Which allocation is best?” is a question with no truth conditions until “best at what?” has been settled, and the various answers to that second question are not merely different weights on a common scale. They are different scales. An allocation that maximises the national stock of cognitive skill is not a more aggressive version of one that secures every child a floor of civic competence. It is a different exercise, with a different unit of account, and the two diverge sharply at exactly the margins where the money is actually fought over.
The result is a literature in which people who agree about the facts disagree violently about the policy, and mistake this for a disagreement about the facts. The long quarrel over whether school spending matters is frequently read this way. Read the estimates and it is nothing of the kind. Jackson et al. (2016) find that a ten per cent increase in per-pupil spending sustained across twelve years of schooling raises completed education by around a third of a year and adult wages by roughly seven per cent, with effects markedly larger for children from low-income families. Nothing in that finding tells you whether the money ought to have gone to those children. It tells you what happened when it did.
So let me do the thing that is usually skipped. What follows sets out the two inputs that jointly determine the answer — the objective and the technology — shows what each candidate objective implies, and then asks what the evidence actually constrains. The conclusion is less comfortable than either camp would like.
Four maximands, four different answers
Aggregate human capital. The first candidate treats education as investment and asks for the largest total return. On this view the marginal pound belongs wherever its marginal product is highest, and questions about who receives it are questions of efficiency rather than desert. The macroeconomic case is not trivial. Hanushek and Woessmann (2012) find that cross-country differences in measured cognitive skill are strongly, and they argue causally, associated with differences in growth; Lucas (1988) and Moretti (2004) supply the mechanism by which private returns understate social ones, since human capital generates spillovers the individual cannot capture. If skill has externalities, we systematically underinvest, and the case for public money follows directly.
Welfare under distributional weights. The second candidate concedes the investment framing but denies that a pound of benefit counts the same wherever it lands. Prioritarianism holds that a given benefit matters more the worse off its recipient (Parfit, 1997), converting an efficiency calculation into a weighted one. Schouten (2012) develops the position for education specifically, arguing for a prioritarian principle governing the distribution of resources across children of differing natural ability. Note the structure: this objective can recommend spending on the weakest even where doing so produces less total skill, and it can be defeated if the return gap is large enough. It is a thumb on the scale, not a lexical rule.
Equality of opportunity. The third candidate is not about totals at all. It asks that a child’s prospects not be determined by circumstances they did not choose — a demand articulated by Rawls (1999) through fair equality of opportunity and formalised by Roemer (1998) as the equalisation of outcomes across those exerting comparable effort. Applied to schooling it has a distinctive and uncomfortable implication. It targets not the low-attaining but the unluckily placed, and those are not the same set. A bright child in a failing school has a claim under this objective that a struggling child of attentive and wealthy parents does not.
Democratic adequacy. The fourth candidate abandons comparison. Anderson (2007) and Satz (2007) argue that what we owe children is not an equal share but an education sufficient to function as an equal citizen — sufficient to hold office, serve on a jury, be heard. Gutmann’s (1987) earlier formulation makes the same move. On this view, inequality above the threshold is not automatically objectionable, and the allocation question below it is already settled: the money goes to whoever has not yet reached the line, because the claim is not a claim about returns.
These four are not variations on a theme. Figure 1 sets out what each implies about the social value of the next pound as a function of where the child already stands.
Figure 1. Stylised marginal-value schedules under three objectives. Curve (a) assumes aggregate output maximisation where prior skill raises the productivity of further investment; curve (b) applies prioritarian weights; curve (c) applies a near-lexical adequacy threshold. Shapes are illustrative, not estimated.
The figure plots three of the four, and the omission is instructive. Equality of opportunity cannot be drawn on these axes at all, because it is not defined over attainment: it is defined over circumstance. Two children at the same percentile have entirely different claims under it depending on how they arrived there. That is not a defect of the diagram but a property of the objective, and it is the reason equality of opportunity generates administrative demands the other three do not — it requires the state to know things about a child’s household that the other objectives can remain indifferent to.
The schedules cross. That is the entire difficulty. Any evidence about effects that does not also tell you which curve to weight by is evidence that cannot settle the argument, and almost all of the available evidence is of that kind.
The fifth maximand, which nobody puts in a manifesto
There is a fifth objective, and it is the one that in practice governs a great deal of the spending. On this view the primary output of an education system is not skill, welfare or citizenship. It is a ranking — and a society that must allocate scarce and desirable positions requires a ranking it can defend when challenged.
Nobody writes this into a manifesto, but it is the reading under which a good deal of otherwise inexplicable institutional behaviour becomes rational. Spence (1973) supplied the formal apparatus: where ability is unobservable, costly education can function as a credible signal without raising productivity at all. Caplan (2018) pushes the claim to its limit, arguing that the signalling share of educational return is very large. One need not accept the strong version to accept that some of what schooling delivers is sorting.
The sorting objective has an allocation implication that differs sharply from the other four. Under it, the marginal pound belongs wherever it most improves the discriminating power of the ranking — and that is typically neither the top nor the bottom, but the region of the distribution where the classifier is least certain. Resources flow to where the sorting decision is close, because that is where sorting errors are made and where they are most expensive to defend.
This also explains two features of real systems that the other objectives cannot. It explains why so much money and instructional time concentrates at exactly the ages where selection occurs rather than the ages where learning is most tractable. And it explains the extraordinary institutional resistance to the finding that examinations measure the wrong thing. If the objective is a defensible ranking, then a measurement error that is stable, legible and widely accepted is not a defect. It is the product.
I am not endorsing this objective. I am observing that it is doing work, that it is never stated, and that a great many allocation decisions become intelligible the moment you assume it.
The technology is not what you think it is
The second input is the production function, and here the modern literature has produced one genuinely powerful idea that is very widely misapplied.
Cunha and Heckman’s (2007) model of skill formation rests on self-productivity — skills already acquired raise the productivity of the acquisition process itself — and on dynamic complementarity, whereby investment at an early stage raises the return to investment at a later one. Heckman (2006) drew the familiar policy inference: returns to investment in disadvantaged children decline steeply with the age at which the investment is made.
The misapplication is this. Complementarity is a claim about stages within a life. It says that a pound spent on a five-year-old raises the productivity of a pound spent on the same child at fifteen. It does not say that a pound spent on a fifteen-year-old at the ninetieth percentile buys more than a pound spent on a fifteen-year-old at the tenth. The first is a statement about intertemporal complementarity within a single production process; the second is a statement about the cross-sectional curvature of the production function at a point in time. They are different claims with different evidence bases, and the second does not follow from the first. Yet the second is routinely asserted with the first as its warrant — and it is the second that the “spend on the top” position actually requires.
The age claim itself is also less secure than its ubiquity suggests. Rea and Burton (2020), using benefit–cost ratios compiled by the Washington State Institute for Public Policy, find no support for the proposition that programmes targeted early in the life course systematically outperform those targeted later. Heckman has replied that the curve was never a claim about the average return of programmes classified by recipient age. The reply may well be right; it also concedes that the figure carries far less policy weight than it has been made to bear. Figure 2 shows the two claims side by side, which is the honest way to present a proposition that is contested rather than settled.
Figure 2. Schematic comparison of the declining-return schedule associated with Heckman (2006) and the approximately age-invariant pattern reported by Rea and Burton (2020). Vertical scale is ordinal.
What is better established is that early and later investment interact. Johnson and Jackson (2019) find that the benefits of Head Start participation are substantially larger for children who subsequently attended better-funded schools — dynamic complementarity observed directly rather than assumed. The implication is not “spend early instead of late” but that an early pound and a late pound are worth more together than the sum of their separate worths. That is an argument for sequencing and continuity, not for front-loading and abandonment.
The case for spending on the top, and what is wrong with it
The strongest version of the elite case is not about fairness or desert. It is about tails. Scientific and technological progress is produced by very few people, the distribution of contribution is extraordinarily skewed, and if the marginal social product of an exceptional mind exceeds that of an ordinary one by orders of magnitude, then even a modest probability of converting a talented child into a productive researcher dominates almost any competing use of the money. Agarwal and Gaulé (2020), studying International Mathematical Olympiad participants, find that high-scoring students from low-income countries are markedly less likely to become active researchers than equally talented students from wealthier ones, and argue that the knowledge frontier advances more slowly as a result. Bell et al. (2019) document the same structure domestically: exposure to innovation, not measured aptitude alone, predicts who becomes an inventor.
This is a serious argument, and I think it is substantially right about the size of the prize. What it does not establish is the conclusion usually drawn from it.
The causal evidence on giving more resources to children already identified as able is remarkably thin. Bui et al. (2014) exploit a discontinuity in gifted-programme eligibility and find no discernible achievement effect for marginal students, despite substantially higher peer quality and a more advanced curriculum. Abdulkadiroğlu et al. (2014), using admissions cutoffs at Boston and New York exam schools, find little evidence that exposure to extremely high-achieving peers raises test scores, advanced-placement outcomes or college quality — a result they name, pointedly, the elite illusion. The children who look like the system’s greatest successes are largely succeeding for reasons the system did not supply.
Set that beside Card and Giuliano’s work and a pattern emerges. Their study of a district creating separate high-achiever classrooms finds gains of around half a standard deviation, concentrated entirely among black and Hispanic participants, with no detectable effect for white students (Card & Giuliano, 2016a). Their companion paper shows why. When the same district replaced parent and teacher referral with universal screening, the number of poor and minority children meeting the gifted threshold rose sharply, implying that the referral system had been failing systematically to see them (Card & Giuliano, 2016b).
The inference is not that resources for able children are wasted. It is that the binding constraint is identification, not provision. Where a child’s ability was already visible and already being served, the marginal programme adds little; where it was invisible, the same programme is transformative. This is a cheap constraint to relax — universal screening costs a rounding error against the provision it feeds — and it is one that the aggregate-output objective and the equality-of-opportunity objective both, unusually, agree should be relaxed. Genuine convergences between objectives are rare enough to be worth banking.
The case for spending on the bottom, and what is wrong with it
The efficiency case for remediation is weaker than its advocates admit and stronger than its critics allow. Banerjee et al. (2007) evaluated a programme placing young tutors with children lagging in basic literacy and numeracy in urban India and found substantial gains concentrated among the weakest students. Duflo et al. (2011) found that tracking in Kenyan primary schools benefited lower-achieving pupils — not by improving their peers, which it necessarily worsened, but by allowing teachers to pitch instruction at their level. Chetty et al. (2011) traced Project STAR participants into adulthood and found that classroom assignment in kindergarten showed up in earnings and college attendance decades later. Hendren and Sprung-Keyser (2020), comparing 133 US policy changes on a common welfare metric, find that policies investing in children tend to sit at the top of the distribution of returns.
These are real effects. They are also, the Indian results aside, expensive per unit of gain, and they arrive in a literature whose central tendency is sobering: across 747 randomised trials of education interventions measured on standardised achievement outcomes, Kraft (2020) reports a median effect of 0.10 standard deviations, with the 0.05 SD mark he proposes as the boundary of “small” falling at roughly the thirty-seventh percentile of the distribution. Anyone who tells you that money reliably fixes low attainment is selling something.
But the efficiency case is not where the weight sits. The serious argument for spending on the weakest does not run through returns at all. It runs through the claim that a person who cannot read a tenancy agreement, follow a public argument or compute a compound interest rate is excluded from citizenship in a way no amount of aggregate growth compensates. Anderson (2007) and Satz (2007) make this case as a matter of democratic standing; Frankfurt (1987) makes the general version, that what matters morally is whether people have enough rather than whether they have as much as others.
The objection to threshold views is well known and forceful. Casal (2007) argues that sufficiency without an accompanying distributive principle licenses indifference to inequalities above the line, which is implausible where the good is contested. Brighouse and Swift (2006, 2009) press the sharpest version: education is partly positional, its value to a person depending on how much of it others have, so a threshold leaving the distribution above it untouched has not secured what the threshold was meant to secure. If the labour market rewards relative rank, adequacy defined in absolute terms is a promise the economy will not honour.
There is a further complication that neither camp handles well. The allocation problem is not separable, because pupils are inputs into one another’s production. Move a child and you have changed the peer composition of two classrooms, not one. This is what makes the Kenyan tracking result so useful: Duflo et al. (2011) show that separating pupils by prior attainment worsened the peer environment of the weakest group and improved their outcomes anyway, because the dominant channel was not peer quality but the ability of a teacher to pitch instruction at a narrower band. The lesson generalises badly in one direction and well in another. It does not license tracking wherever teachers are unconstrained; it does establish that “peer effects” and “instructional match” are distinct mechanisms with opposite signs, and that policy which treats a change in class composition as a single-valued intervention is measuring a sum of two things it has not separated.
That objection is correct, and it is also why the argument cannot be settled inside education policy. If a substantial fraction of what schooling delivers is a sorting signal rather than a productive capability — the possibility opened by Spence (1973) and pressed to its limit by Caplan (2018) — then spending intended to raise capability partly buys reshuffled queue positions instead. Positional goods have the property that everyone’s investment can rise while nobody’s relative position improves. The competitive dynamic that makes parents demand more spending is precisely the dynamic that makes the extra spending worth less than they think. Hirsch (1976) named this half a century ago, and education policy has been ignoring it ever since.
The strange emptiness of equal spending
Equal spending has an enormous constituency and almost no defenders on the merits. Notice that none of the four objectives above recommends it except by coincidence. Aggregate output recommends equality only if marginal products happen to be equal across children, which nothing suggests. Prioritarian welfare recommends progressivity. Equality of opportunity recommends compensation, which is progressivity with respect to circumstance rather than attainment. Democratic adequacy recommends whatever gets everyone over the line, which is progressivity below it and indifference above.
Nor is equality what any developed system actually does. Special educational provision consumes a share of budgets far exceeding the share of pupils it serves; compensatory funding formulae in most jurisdictions weight by deprivation. The equal-spending baseline is a rhetorical position, not a fiscal one.
What equality does possess is a property no other rule has. It requires no agreement about purpose. It can be defended to a parent who thinks education is investment, to one who thinks it is a right, and to one who thinks it is a civic apprenticeship, without conceding anything to any of them. It is the allocation of a society that cannot agree on the maximand and would rather not discover that it cannot. That is a genuine political virtue and it should not be mistaken for a justification.
There is a second, less charitable reading. Equal spending is stable because it is legible. Deviations from it must be argued for, and the argument exposes the arguer. A minister who says “we are spending more on able children” is saying something about what the state values; so is one who says the reverse. Equality permits the state to spend without saying. This is not a failure of nerve peculiar to any one government. It is the equilibrium that emerges whenever a decision procedure has no agreed objective and every deviation is a target.
What the measurement problem does to all of this
Every argument above has been conducted in the currency of measured attainment, and that currency is weaker than the arguments require.
The pupils we call “the top” and “the bottom” are identified by tests whose relationship to the capacities we claim to care about is loose. Jackson (2018) shows that teachers who raise test scores and teachers who improve behavioural outcomes are substantially different sets of people, and that the latter effects better predict high-school completion and longer-run outcomes. Any allocation rule optimised against test scores is therefore optimising against a proxy that omits much of its object. The Coleman report (Coleman et al., 1966) is still, sixty years on, the origin of the modern habit of treating standardised achievement as the outcome variable, and the habit has outlived its justification.
This matters for allocation specifically, not merely in general. If measured attainment at age eleven is a noisy signal of underlying capability, a rule concentrating resources on the measured top concentrates them partly on measurement error — and worse, on measurement error correlated with family background, since well-resourced families purchase test performance directly. Card and Giuliano’s screening result is the cleanest demonstration available that the identification technology, not the underlying distribution of talent, was generating the observed pattern. An allocation rule is only ever as good as the classifier it runs on, and ours is poor and biased in a known direction.
What actually follows
Four propositions survive the argument. They are less satisfying and more useful than a slogan.
First, identification dominates provision. The highest-return educational expenditure available to most systems is not a programme at all but a screening technology, because the evidence shows small effects from giving more to children already identified as able and large effects from identifying children who were not. This is the one recommendation that survives under every objective considered here, and it is cheap.
Second, a threshold has priority, and it is a claim of a different kind. The case for funding the weakest to a defined level of literacy, numeracy and civic capability does not rest on rate of return and should stop being defended as though it did. It rests on the proposition that below that level a person is not a participant. Defend it on efficiency grounds and you invite refutation on efficiency grounds — a refutation that will sometimes succeed.
Third, above the threshold, allocate by marginal product rather than by rank. The relevant question about a child at the ninetieth percentile is not whether they are able but whether the next pound does more for them than for anyone else. The exam-school and gifted-programme literature suggests that where ability is already recognised and served, frequently it does not. Rank is not a proxy for marginal product, and treating it as one is the central technical error of the elite case.
Fourth, sequence rather than front-load. Complementarity means early and late investment are complements, not substitutes: an early intervention followed by an underfunded school is a wasted early intervention (Johnson & Jackson, 2019). The policy that follows is continuity of funding across a child’s life, which is precisely the policy electoral cycles make hardest to deliver.
It is worth seeing what the rule does to concrete cases. Take three children. The first is at the fifth percentile in reading at age eleven; under the second proposition she has a near-lexical claim on resources until she clears the threshold, and the claim does not weaken if the intervention turns out to be expensive. The second is at the ninety-fifth percentile in a well-resourced school with attentive parents and an existing place in an advanced programme; under the third proposition his claim on the marginal pound is weak, not because he is undeserving but because the evidence says the pound will do little there. The third is at the seventieth percentile in a school that has never referred a child to an advanced programme, whose measured attainment is a floor rather than a ceiling on her capability; under the first proposition she is the highest-value target in the system, and under the existing arrangements she is invisible. Note that the second and third children are the ones a rank-based rule cannot tell apart, and that the difference between them is the entire argument.
None of this yields “spend equally”, “spend on the top” or “spend on the bottom”. It yields a structured rule behaving differently in different regions of the distribution, because the objectives governing those regions are different objectives.
Conclusion
The question — equal, top, or bottom — is not answerable and never was, and the reason is not that the evidence is inadequate. The evidence is considerably better than the argument it is asked to support. The question is unanswerable because it presupposes a single maximand, and there is no single maximand. There are at least four, they generate marginal-value schedules that cross, and the crossings fall precisely where the money is contested.
What we do instead is spend approximately equally and quarrel at the margins. That is not a compromise between the positions; it is an evasion of the choice between them, and the evasion has a cost. It means we cannot say why any particular pound went where it went, cannot evaluate whether it did what we wanted, and cannot learn — because learning requires an objective against which to score. A system that will not say what it is for cannot discover whether it is working.
Say what the money is for. The allocation will follow, and it will not be equal.
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