Is Money Multiplicative?

When I was looking at data on the wealth of UK universities recently I plotted the value of institutions net assets on the y-axis using a log scale. Using a log scale means that equal differences refer to a constant multiplicative factor, rather than, as is more usually the case, a constant additive factor. For example, the distance on the y-axis from 500000 to 1000000 is the same as the distance from 1000000 to 2000000. It is standard to use a log scale when there are a significant number of observations which are more than three standard deviations above the mean. There are quite a few variables which have this type of distribution with lots of low values but only a few high values, including household wealth, city populations and the number of citations for academic papers.

Distributions which have a long tail are almost always the result of processes which are multiplicative rather than additive. Phenomenon which result from multiplicative processes can be modelled using a power law distribution which has the form:


where c and α are constants. In the social sciences, variables that have a power law distribution are often produced by what are termed processes of cumulative advantage. These refer to processes in which patterns of cumulative causation lead to small initial differences between people or between institutions becoming magnified over time. Multiplicative phenomena may also be described, however, using the lognormal distribution. The log-normal distribution is the natural way to model phenomena, like investment funds, which grow by a small multiplicative factor each period.

The difference between the type of process that lead to power law and lognormal distributions can be illustrated with an example. If we put a given sum of money in the bank and reinvest the interest, the amount by which the sum accumulates depends on the initial investment. If the interest rate is 10 percent then a sum of £1000 grows by £100 but a £10000 sum grows by £1000. The difference in the value of the two investments grows exponentially over time from £9900 at the start to £14400 after 5 years and £23300 after 10 years. The ratio of the value of the two investments remains constant over time (at 10:1), however. If instead the interest rate depends on the initial sum (for example, 5 percent interest on £1000 but 10 percent interest on £10000) then both the difference and the ratio of the value of the two investments would grow over time.
One way for determining whether a variable follows a power law or a lognormal distribution is to plot P[X > x] on a log-log scale. Both power law and lognormal distributions should follow a straight line but the power law distribution should have a shallower slope because of its longer tail (see Mitzenmacher 2005). For an empirical example, I decided to look at whether the size of universities assets followed either a power law or lognormal distribution which might tell us something about the kind of processes influencing inequalities in higher education. I used data from 2013 for the value of the endowment funds for US and Canadian higher education institutions ( The US institution with the largest endowment fund is Harvard ($32 billion) followed by Yale ($20 billion) and the University of Texas ($20 billion). In contrast, over 90 percent of institutions have an endowment fund with a value of less than $1 billion. The figure above shows a log-log plot of P[X > x] for the endowment funds of US institutions with fitted lines for both a power law distribution and a lognormal distribution. The figure suggests that the lognormal distribution is a better fit than the power law distribution. Although the universities with the largest endowments are wealthier than other institutions, inequalities in endowments between institutions seem to be described by differences in inherited wealth and constant proportional growth, rather than by the ability of the most wealthy institutions to benefit preferentially from their financial assets.

The Wealth of Universities

The Sunday Times published its annual league table of university rankings a few weeks ago. The league table uses a set of indicators to try to rank universities along a single dimension which can broadly be thought of as prestige. The University of Cambridge and Oxford were placed joint first followed by the University of St. Andrews, Imperial College London and the London School of Economics. Institutions do change their position in the league table from year-to-year although long-term movement in position tends to be uncommon. The top ten institutions were the same this year as last year with only minor changes in the ranking of individual institutions. The single most important factor in the stability of a universities position in the league table is the entry qualifications of the students at each institution. In general, students who have achieved good A level grades want to go to the more prestigious institutions, leaving students who didn’t do so well in their exams with little option other than to apply to institutions where competition for places is less intense. In consequence, there is a strong pattern of cumulative causation influencing an institutions league table position. League table position influences the level of achievement of applicants while the level of achievement of entrants influences league table position.

The choices made by students with different levels of academic achievement are not the only factor influencing a universities league table position, however. The figure below shows the relationship between the league table items that measure the inputs of different resources into education: the entry qualifications of students but also student/staff ratio and spending on student services (£). The Russell group institutions which are usually considered to be the elite universities are in blue. The figure shows that there is a positive relationship between entry qualifications and spending on services (r = 0.71) while the staff/student ratio shows a negative relationship to both entry qualifications (r = -0.77) and spending on services (r = -0.60). The league table of the more prestigious institutions are not therefore due to a single factor but are due to cumulative advantages including the level of financial and staff resources which they have available. pairs

The magnitude of the variation in student spending and staff resources across institutions seems to be quite large. For example, the spending on services varies by a factor of around three from over £3000 per student each year to less than £1000 per student. It would seem therefore that universities do not all have the same financial resources per student, despite the majority of institutions charging the same tuition fee of £9000 per year. The financial resources available to universities to fund their teaching activities include tuition fees but also income from endowment funds that the more prestigious institutions, in particular, have accumulated over time. The plot on the left in the figure below shows the relationship between the net assets of universities (on a log scale) and their position in the Sunday Times league table. While the majority of universities have relatively modest assets, the endowment funds of the most prestigious institutions are significant. For example, Cambridge has net assets of over £3 billion and Oxford assets of £2.4 billion while, at the other end of the scale, the University of Bolton has net assets of around £50 million. The figure also shows that there is a relationship between the financial position of an institution and its league table position with those institutions with greater assets achieving better positions in league tables than the less wealthy institutions. The financial advantages that prestigious institutions have over other institutions seem important therefore in attracting the best students and maintaining their position at the top of league tables.


The most important question about economic inequalities tends to be whether inequalities are increasing over time rather than the absolute size of differences between those at the top and those at the bottom. The plot on the right in the figure above shows the growth in net assets (on a log scale) for Russell group institutions over the period since 2005/06, adjusted for inflations using the CPI. The figure shows that the net assets of most institutions have grown at a rate of somewhere between 6 and 8 percent per year. The net assets of the most well-off institutions, such as Cambridge and Oxford, do not appear to have increased at a significantly faster rate than those of other institutions in the Russell group. The common rate of growth in the value of net assets across institutions would seem to suggest that investment returns have been the main factor in the increased value of assets. The extent to which the variation in wealth across institutions can be explained by donations from alumni is unclear. At least over the period considered, however, the variation in the donative wealth of alumni would not seem to be the prime factor in the variation of wealth across institutions.

Destinations of Students Leaving Higher Education

In the UK, research on higher education has tended to focus on factors associated with admission to university while, only a handful of studies have examined the labour market outcomes of students following university. The majority of studies which have examined the labour market outcomes of students following university have focused on the returns to having a degree and how this varies by factors such as gender, subject of study and institution. The results of these studies have shown that graduates receive a significantly higher wage than non-graduates. The returns to having a degree vary with the subject of study and type of institution attended, however, with students who studied medicine, mathematical and computer sciences or law and those who attended more elite institutions obtaining the highest returns to having a degree.

A further dimension of labour market inequality among graduates relates to the type of job obtained following graduation. In particular, there has been increasing concern about the number of students leaving higher education who do not find jobs for which a degree is necessary. The extent of over-education may be measured using the occupations of the jobs obtained by graduates and using this criterion around 20 percent of graduates do not find jobs for which a degree is usually necessary. Although some studies have sought to explain over-education as a result of individual differences in skills, the extent of over-education among graduates can be related to wider patterns of economic change. Research has shown that over the last several decades there has been a polarisation in the nature of employment with an increase in the number of jobs at both the top and the bottom of the occupational hierarchy and a decline in intermediate occupations. Occupational changes in the labour market have been accompanied by a rise in wage inequality. Although wages vary within occupations to a greater degree than they do between occupations, all of the rise in wage inequality in the UK over the last several decades can be explained by changes taking place between occupations (Williams 2012).

Although the nature of the labour market that students enter after graduation has changed significantly over the last several decades little research has examined changes in the type of jobs obtained by students leaving higher education. The Destination of Leavers from Higher Education (DLHE) survey can be used, however, to describe changes in the types of job obtained by students leaving higher education. The DLHE is carried out with all students leaving higher education approximately six months after they complete their course. The survey asks respondents for their current activity and, if employed, collects information on the type of job.

The figure below shows the variation in the proportion of first degree respondents in employment who were working in each of the nine major occupational groups, separately by gender and age group for the cohorts of students who left university between 2006 and 2012. The figure shows that in each year between 50 and 60 percent of 21 to 24 year old and around 70 percent of 25 to 29 year old graduates were employed in either professional or associate professional occupations. The most notable change in the type of occupations in which graduates were working is the steady decline in the proportion finding jobs in administrative and secretarial occupations. The decline in the number of graduates working in administrative and secretarial occupations over the period has been concentrated mainly among women, with the percentage of 21 to 24 year old women employed in administrative and secretarial occupations falling from 20 percent in 2006 to around 12 percent in 2012.


Change from 2006 to 2012 in the occupations in which first degree leavers were employed (as a percentage of all those in employment) separately by gender and age group.

The corresponding figure for postgraduate students is shown below. For postgraduates, the figure shows that at least 90 percent of postgraduate leavers who were in employment were employed in jobs in the top three occupational categories (managers, professional, associate professional and technical). The figure also shows that there has been a steady decline, however, in the proportion of postgraduate students working in professional jobs and a rise in the proportion working in associate professional and technical occupations. The rise in postgraduates working in associate professional jobs suggests either that postgraduates are ‘bumping down’ in the labour market because they can’t get professional jobs or that ‘up-skilling’ has increased the skills needed to do associate professional jobs. What is clear, however, is that the chances of getting a professional job are much higher for postgraduate students in comparison to undergraduates. The introduction of higher tuition fees for undergraduate students has been accompanied by a number of steps aiming to ensure that young people from disadvantaged backgrounds are not excluded from going to university. In contrast, there has been no attempt by government to offset the effect on students from different socioeconomic backgrounds of rising fees for many postgraduate courses. Given that postgraduate degrees are now needed for many professional jobs, this is likely to limit the extent to which the expansion of higher education will increase levels of social mobility in the UK.


Change from 2006 to 2012 in the occupations in which postgraduate degree leavers were employed (as a percentage of all those in employment) separately by age group and gender

Higher Education League Tables and Performance Indicators

The Guardian university league tables for 2015 were published earlier this month.

Cambridge was in first place followed by Oxford and, in third place, St. Andrews. The league tables are constructed using information on a range of university characteristics including ratings of student satisfaction from the National Student Survey, the resources available to students at an institution (the average entry tariff of students, expenditure per student and student-staff ratio) and student’s career destinations following university. The indicators are first standardised and then added-up using a series of weights which reflect the importance of the different indicators. The use of the entry tariff as an indicator of university quality is often criticised because the educational achievement of students prior to starting university would seem not to have a clear link to anything universities actually do themselves. It is clear, however, that students with higher achievement tend to choose to go to the more prestigious Russell group institutions. The entry tariff measures the number of points students have been awarded based on their A level grades. The figure below shows the average entry tariff for the top twenty universities in the league table ordered by their ranking. The average entry tariff of students at Oxbridge is equivalent to more than four A* grades while further down the hierarchy three A* grades are the average entry requirement for students at universities such as Lancaster, York and Southampton.


League tables are not the approach used by the government bodies responsible for regulating universities, however. For this purpose, the Higher Education Statistics Agency (HESA) produce an annual set of performance indicators. The performance indicators are chosen to reflect important aspects of how higher education functions. For example, the proportion of students from low social class backgrounds is used as an indicator of widening participation. The performance indicators are published individually rather than aggregated into an overall index and are also distinguished from league tables in using indirect standardisation to calculate a benchmark value of each indicator for each institution. The standardisation used in the calculation of the performance indicators aims to adjust for factors related to performance but which institutions can’t do much about, including the entry tariff of students. The benchmarks for institutions indicate what the national value of the relevant indicator would be if all universities in the country had a particular institutions students. The benchmarks therefore indicate roughly what an effective level of performance would be on an indicator for each institution.

The table below gives a simplified example of how the benchmark is calculated using the proportion of students from low social class backgrounds as an example. In the table, students are grouped into three tariff score categories (lower tariff, middle tariff and higher tariff). In the country as a whole, it is assumed that the proportion of students from low social class backgrounds varies from 70 percent for lower tariff students to 50 percent for middle tariff and 30 percent for higher tariff students. The benchmark at our hypothetical University X is calculated by applying the national rates to the number of students in each tariff score category. For example, if the proportion of students from low social class backgrounds was the same at our hypothetical institution as in England as a whole, we would expect that among students with lower tariff scores we would have 1750 students (0.7 x 2500) from low social class backgrounds. Repeating the calculation for the other tariff score groups shows that while 44.2 percent of students at this institution come from low social class backgrounds, we would expect a benchmark of 58.8 percent of students to have been from low social class backgrounds given their tariff scores. Students from low social backgrounds seem therefore to be underrepresented at our hypothetical institution.


Table 1 Illustration of indirect standardisation

Because there are large differences in the tariff scores of students at different institutions standardisation can make a big difference to the value of an indicator. The figure below shows the benchmark (or standardised) and actual figures for 2012 for the institutions with the highest and lowest proportion of students from low social class backgrounds. The figure shows that the proportion of students from low social class backgrounds at the most selective universities are well below their benchmarks. For example, while around ten percent of students admitted to Oxford come from low social class backgrounds we would expect the figure to have been around 18 percent based on student’s tariff scores. The difference between the benchmark and the actual proportion of students can be used to estimate the number of students with specific characteristics who are missing at the top universities. The Sutton Trust have estimated that there are around 3000 fewer students from state schools than expected at the top dozen universities suggesting that factors other than tariff score play a part in influencing admissions at these institutions.








Statistics and Public Policy

In common use, the term statistic is used to refer to a summary of a set of data (e.g. total rainfall in February was 15 cm), but in a more technical sense, a statistic refers to a function of a number of random variables. In the first use, no concept of probability is involved in the calculation of a statistic while, in the second, a statistic refers to a feature of a model which includes assumptions about the population and sampling process. Researchers often use summary statistics in an informal way to judge the importance of between group differences and to check for outliers. In principle, calculating a summary statistic is straightforward and in many areas of public policy summary statistics are sufficient to answer important questions (e.g. is GDP higher this year than last year). In other areas, in particular where questions of causation are important, it is usually not possible, however, to provide direct answers to policy questions by simply summarising observed data. In these situations, researchers need to calculate measures of uncertainty and conduct statistical tests and therefore need to resort to using statistical models.

For example, an article in Monday’s Guardian by Madeleine Bunting discussed whether ethnic diversity was a source of social cohesion or of social conflict. In order to examine the association between ethnic diversity and social cohesion, studies need to use statistical models rather than descriptive statistics partly because there are big differences in the characteristics of areas that are related to ethnic diversity and that might also influence social cohesion, in particular, area deprivation. For example, if we wanted to compare the level of trust that people in, say, Richmond-upon-Thames (where around 30 percent of the population are from non-White backgrounds) and people in Brent (where over 80 percent of the population are from non-White backgrounds) have for their neighbours there is little point in simply comparing the responses that people give to survey questions. We know that people living in more deprived neighbourhoods have lower levels of trust in their neighbours than people in better-off neighbourhoods, so we would expect that people in Richmond-upon-Thames had much higher levels of trust in their neighbours than people in Brent. Instead of relying on a summary statistic we can construct a statistical model of the relationship between diversity and cohesion, however, in which the assumptions of the model allow us to identify the effect of diversity on cohesion after adjusting for differences in deprivation between areas. There might be, however, only limited variation in ethnic diversity which is independent of deprivation (i.e. all areas with high proportions of the population from ethnic minority backgrounds tend to be deprived). If this is the case, we might still not be able to say anything meaningful about the relationship between ethnic diversity and social cohesion because the division between ethnically mixed and ethnically homogeneous areas is also a division between deprived and more affluent areas. Modelling might still produce estimates of the effect of diversity on cohesion but, in this situation, they would largely reflect the structure imposed on the data from the model rather than information in the data itself.

In order to illustrate the difficulty of separating ethnicity and deprivation at the area level, the figure below plots estimates of the social capital in neighbourhoods (measured using questions from the Citizenship Survey about trust, co-operation, shared values and belonging) against the proportion of the population from ethnic minority backgrounds separately for four English regions. Social capital in neighbourhoods was estimated for neighbourhoods grouped into deciles of the index of multiple deprivation giving ten observation in each region with the most affluent neighbourhoods in decile 1 and the most deprived neighbourhoods in decile 10 (the approach is described here). The figure shows that overall there is a strong negative relationship between the level of social capital in different types of area and the proportion of the population from an ethnic minority background. The points are coloured, however, according to the decile of the deprivation index and show that in each region there is also a significant negative relationship between social capital and neighbourhood deprivation. Studies have concluded that after adjusting for neighbourhood deprivation the association between ethnic diversity and social capital is actually positive (i.e. the reverse of the association actually observed in the data). The extent to which this result isolates the effect of varying ethnic diversity between neighbourhoods with similar levels of deprivation or is based on a comparison of what are fundamentally different types of area having different levels of both diversity and deprivation needs to be considered, however.


My point here is that although whether diversity is positively associated with social conditions in neighbourhoods is an important question, researchers can’t answer that or similar questions without recourse to using modelling techniques which require a number of assumptions. The reliability of the conclusion that diversity is positively associated with social cohesion does depend on whether those assumptions are reasonable. In most cases, modelling assumptions are made for convenience rather than realism and moving from the results of statistical models to public policy conclusions is something that we should only do with caution.

Do private schools educate children beyond their intelligence?

Michael Gove recently said that he wants state schools to do more of the things that private schools do. What struck me most about the ensuing hoo-hah was the idea that state schools might be able to provide the advantages of a private education if only the government provided more money. In most cases, parents send their children to a private school as a means of passing on privilege to their children rather than as a way of rescuing their children from the horrors of a state school. Because privilege can only exist if some people don’t have it, state schools can’t provide the same benefits to pupils as private schools. Politicians may see education as promoting social mobility, but it is also used by parents to try to make sure that their children get the advantages they need over other children to get on in life.

One argument (there are many) for how private schools help better-off families secure advantages for their children can be sketched as follows.

Firstly, children from private schools do achieve significantly better results than those from state schools at A level. The figure below (using data from the Department of Education) shows the proportion of A-levels awarded to pupils at private and state schools in 2013 that were in different grades. The difference in achievement of children at private and at state schools is clearly significant with around 50 percent of the grades awarded to pupils at private schools in the A* or A category in comparison to less than 25 percent for pupils at state schools. One consequence of the better A-level grades of children from private schools is that they are more likely to get into a higher status university (e.g. Oxford or Cambridge) than children from state schools. Figures from HESA show that in 2011 around 42 percent of students admitted to Cambridge and Oxford had been to a private school, while only around 12 percent of all children admitted to university in England had been to private schools.


Secondly, although private schools outperform state schools in terms of A-level results, students who have been to a private school are less likely to get a good degree when compared to those from state schools with the same entrance qualifications. On average, a student from a state school with ABB grades at A-level can be expected to get the same degree as a student from a private school with AAA grades at A level (HEFCE 2003). The reason that students from state schools outperform those from private schools at university is not well-established. The most reliable evidence suggests, however, that students from private schools benefit from teaching effects which increase their A-level grades in comparison to students from state schools but these then wash-out during university. The best evidence for teaching effects (and not student ability) as the cause of the better A-level grades at private schools comes from Ogg et. al. (British Educational Research Journal 2009) which used the results of an aptitude test administered to students who applied for admission to Oxford in 2002. The study shows that, on the basis of the aptitude test, students from private schools achieved the level of degree expected but, on the basis of A-level grades, they underperformed relative to students from state schools. If we assume that the aptitude test provides a true measure of student’s ability, then the better A-level grades of students from private schools can be interpreted as being due to the temporary influence of teaching conditions in private schools.

Thirdly, studies such as Naylor et al. (Bulletin of Economic Research 2002) have shown that the after leaving university graduates who have been to a private school get paid significantly more in comparison to graduates from state schools. I’m not aware of studies which show why students from private schools get better jobs than those who went to state schools. It seems likely to be due largely to social factors, however, such as the status of the school and university attended and other people’s perceptions of the abilities of students from different backgrounds. State schools can’t deliver the same benefits as private schools because the benefits of private schools run through social factors and not through education.

I’m not complaining that parents try to help their children get ahead in life. Whether it is acceptable for parents to try to give their children an unfair advantage over other children is something that it would be useful to have a public debate about, however, particularly given politicians current passion for social mobility. The debate would be improved if we could recognise that private schools do give some children unfair advantages over others, rather than allowing them to fulfill what seems to be their natural potential.

University Admissions in London

Over the New Year, I listened to a debate on the Radio 4 news about the future of cities involving Saskia Sassen and Ed Glaeser. It reminded me of how different the views of a geographer and an economist can be regarding the importance and origins of social inequalities in cities. Like most economists, Glaeser professed little concern about inequalities arguing that they were largely the consequences of the choices people make and of little public interest. Sassen, on the other hand, saw inequalities as being an important public issue and went out of her way to stress the relationship between inequalities and the distribution of power between different groups within the city.

In the UK, there has been a long debate about the origins of inequalities in education in urban areas. Although the proportion of young people who go to university has increased significantly over the last 20 years with around 40 percent of 18 year olds now entering higher education, inequalities in participation in higher education between children from different socioeconomic backgrounds has changed little. Research suggests that this is because differences in educational achievement at secondary school level between children from different backgrounds are very durable. Children from less well-off families have not had the same ability as those from better-off families to take advantage of the expansion of higher education. The numbers of children from less well-off families going to university has increased; however, the expansion of higher education has done little to close the relative gap in participation between children from different socioeconomic backgrounds.

Although inequalities in admission to higher education appear to be durable there are significant regional differences in the proportion of young people who go to university. In particular, London has a higher rate of participation in higher education than remaining regions and a higher rate of participation for young people from deprived areas which seems unusual given the extent of deprivation in many areas of the city. The finding that a higher proportion of young people from London are admitted to higher education is an observation about local areas, however, and it doesn’t necessarily follow that young people from less well-off families in London have been getting into university in higher numbers than in other regions. London is a very mixed city with rich people and poor people living in close proximity to each other. The observation that the rate of participation in higher education among children from poor areas is higher in London could therefore be a result of young people from affluent families who live in poor areas managing to get into university.

Recently it has become possible to link individual data from the school census to participation in higher education and the Department of Education has started to use this data to examine how the rate of participation in higher education varies between children from different socioeconomic backgrounds across local education authorities. The school census contains information on whether children received free school meals and receipt of free school meals has been commonly used in research as an indicator of family poverty. Overall, the figures for young people who took A levels and who left school in 2010 show that the rate of participation in higher education for children who did and did not receive free school meals was 48 percent and 46 percent, respectively. The regional differences in the rate of participation of children from different backgrounds are, however, quite striking. The figure below shows how the proportion of young people from different backgrounds going to university varies across regions. The figure shows in all regions except London children in receipt of free school meals have lower rates of participation in higher education, in comparison to children who did not receive free school meals. In London, however, the picture is dramatically different with children receiving free school meals having a slightly higher rate of participation in higher education than children who did not receive free school meals. If they manage to get to their A-levels, young people from poor families in London have a nearly 60 percent chance of going to university.


One possible for why children from poor families who take A levels have much higher rates of participation in higher education in London might be that they go to different types of institution in comparison to children from better-off families. The availability of linked administrative data also allows examination of the type of university young people from different backgrounds go to and the Department of Education distinguish between institutions in the top third and remaining institutions. The figure below shows the proportion of young people from different backgrounds going to different types of institution (highly ranked vs. other) separately for Inner and Outer London. The figure shows that there is little difference in the type of university attended by children from Inner and Outer London who received free school meals with only around 13 percent going to a highly ranked institution. Among children who did not receive free school meals, however, around 20 percent of children living in Outer London went to a highly ranked institution in comparison to around 13 percent of children living in Inner London. Location does still matter for children’s education.