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Sattinger Assignment Models Pictures

A.D. Roy (28 June 1920–12 March 2003) had a brief but illustrious career as an academic economist (see Sullivan, 2011 for a biography). A Cambridge-educated economist who won the distinguished Tripos award in 1939 in mathematics, he served with the British Army in India and fought in the Battle of Imphal which stemmed the Japanese invasion of India in 1944. After the war, he returned to Cambridge to study economics and history and was awarded a second Tripos award in economics in 1948.

Roy was appointed Faculty Assistant Lecturer in Statistics at Cambridge University in 1949. He completed his Master of Arts in Economics at Cambridge in 1950, reading under Professor Edward Austin Gossage Robinson, who served as editor of this journal. In the same year, he was appointed Director of Economic Studies at Cambridge and then University Lecturer in Economics and Politics at Sidney Sussex College, Cambridge, in 1951. While at Cambridge, Roy wrote the 1950 paper republished here as well as his better known 1951 Oxford Economic Papers article. In addition to the 1950 and 1951 papers reviewed here, Roy made a key contribution to portfolio theory in his paper ‘Safety First and the Holding of Assets’ (1952).

Roy's (1952) paper is widely regarded as a contribution to portfolio theory co-equal with that of the Nobel-Prize winning analysis of Harry Markowitz (1952). Both developed the mean-variance trade-off for a portfolio of correlated assets (Markowitz, 1991). Roy goes one step further than Markowitz in analysing portfolio strategies for worst case situations. He presents methods for estimating the probability of a disaster (i.e. the failure of a portfolio to deliver some minimum rate of return).

Reflecting on his 1952 work and Roy's (1952) paper, Markowitz wrote: ‘On the basis of Markowitz (1952), I am often called the father of modern portfolio theory (MPT), but Roy can claim an equal share of this honor’. (Markowitz, 1999, p. 6) For a more extensive discussion of Roy's contribution to portfolio theory, see Bernstein (1992) and Sullivan (2011).

On secondment from the Economics Faculty, Roy left Sidney Sussex College in 1962 to serve as Economic Consultant and later Senior Economic Advisor to the Treasury. After 1964, Roy held a number of positions as a civil servant at the Department of Trade and Industry, Ministry of Defence, and Department of Health and Social Security. In these positions, Roy addressed issues more closely related to macroeconomics such as labour productivity (Roy, 1982; Roy and Wenban-Smith, 1983).

Roy's papers published in 1950 and 1951 address issues in income distribution that are as relevant now as they were when he wrote them. At the time of their publication, there was a perception–called ‘Pigou's Paradox’ (Pigou, 1932) – of an apparent conflict between the normal, symmetrical distribution of income that one would expect if income were proportional to ability (assumed normally distributed), and the skewed distributions of income that were observed.1 Observed distributions of income are better described by the lognormal distribution than by the normal distribution. Pigou's explanation was that inherited wealth was highly skewed and was an important component of income. Roy observed that labour earnings, excluding income from capital, were themselves right skewed, so Pigou's explanation did not go far enough.

Roy's (1950) paper seeks to explain why the distribution of earnings might follow a lognormal distribution, generating skewed distributions without invoking an unequal distribution of property income that could separately contribute to skewness and inequality. The basic explanation, related to Gibrat's Law of Proportionate Growth, is that the output of a worker in a given period is not the outcome of a normally distributed attribute like that assumed for ability but was the product of several attributes that combine multiplicatively to determine the worker's output. Under conditions more general than the ones assumed in his paper (including correlation among components, Cramér, 1946), the Central Limit Theorem implies that the sum of the logarithms of the several attributes will tend to be normally distributed and the product of the attributes and worker output will be lognormally distributed.

Roy considers reasons why this simple mechanism may not operate. His first explanation is that the attributes or factors contributing to a worker's output may be correlated. Unfortunately, this reasoning is incorrect. The Central Limit Theorem continues to hold even with strongly correlated components.2 His second explanation plants the seeds for his 1951 paper. Roy suggests that the economy could produce a large number of different goods, requiring employment in many occupations. With two occupations, wages would need to be adjusted so that the required number of workers in each occupation equals the number of workers choosing that occupation. Roy comments as follows on the consequences of this equilibrium adjustment:

The shape of the distribution of earnings that results from this adjustment depends on two considerations, the proportions of the labour force required in each occupation and the association existing between any individual's performance in the two occupations.

(Roy, 1950, p. 494)

To avoid this complication, Roy assumes that a worker's outputs in different occupations are highly correlated, an assumption he would relax in his 1951 paper.

Roy tests the hypothesis of lognormality of earnings by calculating measures of ‘humpedness’ (the mean deviation divided by the standard deviation) and skewness for a collection of different samples and comparing the measures to what one would expect from a normal versus a lognormal distribution. Criteria for the samples are that workers should be engaged in the same activity and output should depend only on the worker and not on the operation of machines or flow of material. Additionally, each worker in the population should have the same likelihood of choosing the occupation. Commenting on this criterion, Roy observes:

The workers engaged on a particular job have not an equal chance of being drawn from all strata of the working population. They are probably selected from a rather restricted field. If a certain minimum of intelligence, skill, etc., is required, workers below this standard will be rejected by the management at some stage, while workers, whose capacity is not fully used in this activity, will either be promoted or will themselves undertake more suitable sorts of employment.

(Roy, 1950, p. 495)

Roy recognises the seriousness of this point but at the time lacked the tools to do anything but assume that the factors determining a worker's occupation are not correlated with output. He returns to this problem in his more famous 1951 paper.

The activities for the samples collected by Roy include packing chocolates in boxes, packing cigarettes in tins, making springs for electrical equipment, bank wiring, top-stitching shoes, winding armatures, operating phonographic record-presses, packing a variety of objects into cartons and cases, and a psychological test related to work in light industry. Roy shows that the evidence from his tests is not decisive. There is no clear domination of one distributional form over another in the output data for the samples.

The continuing significance of Roy's (1950) paper lies in his response to this empirical evidence. He re-examines the reasons why one would not obtain the expected lognormal result even if the underlying argument, that factors apply multiplicatively to determine worker outputs, is valid. Primary among the reasons that worker outputs do not always follow the lognormal distribution is that worker selection operates to modify the distribution of outputs for workers observed in a given occupation from the distribution of outputs for all workers. This insight, explicitly posed by Roy in the above quotations, is developed systematically in his 1951 paper.

Selection among alternative occupations, as described by Roy, draws on notions of potential outcomes used in the literature on the design of experiments (Neyman, 1923; Fisher, 1952), later applied in Rubin (1974). Like later work by Quandt (1972), Gronau (1974) and Heckman (1974, 1976), he adds to the models of potential outcomes in the statistical literature a choice mechanism determining allocations of individuals across sectors (Heckman, 2008 for discussion of the contribution of economists to the Neyman–Fisher–Rubin literature on potential outcomes). Roy's insight introduces economic decision-making into the determination of occupations and earnings and fundamentally modifies the procedures that should be applied to the analysis of earnings data (see French and Taber (2011) for an up-to-date survey).

In the field of income distribution, consideration of why particular individuals are found in particular occupations is related to the assignment problem in operations research, analysed in an economic context by Koopmans and Beckmann (1957). In this view, a basic problem for any economy is to determine how workers are assigned to jobs. From the perspective of individual workers and firms, the phenomenon of assignment is now more commonly described by matching, a term introduced by Jovanovic (1979) to describe selection through job turnover from the point of view of workers and firms.

A major mechanism guiding assignment is David Ricardo's principle of comparative advantage, recognised by Roy in his 1951 paper. Sattinger (1975) and Rosen (1978) extend the analysis of comparative advantage in international trade to the labour market setting. A worker choosing an occupation will compare the income from one occupation with the income from another occupation. If workers have comparative advantages at different occupations, workers will not all make the same decisions. Using the two occupations of trout fishing and rabbit catching whimsically analysed in his 1951 paper, an individual choosing trout fishing would have a comparative advantage at that activity compared to an individual choosing rabbit catching.

A second consequence of the choice of occupations is that the distribution of earnings is built up from the distributions of earnings in individual occupations. Within an occupation, the distribution of earnings is affected by the selection of workers into that occupation, as determined by ‘the association existing between any individual's performance in the two occupations’ (Roy, 1951, p. 494). Roy considers both positive and negative correlation between performances. As a result of the selection, the aggregate distribution of earnings will not resemble any distribution of abilities and instead will depend on the occupations available to workers and their performances in different occupations. In a review of assignment models of income distribution, Sattinger (1993) shows graphically how the aggregate distribution of earnings is constructed from underlying distributions of earnings within occupations.

An alternative characterisation of Roy's view of earnings as generated by occupational choice is that the earnings of workers are order statistics, which are used to describe the distribution of the n-th largest (e.g. the maximum or minimum) of a set of random variables. In this view, a worker considers the earnings available in different occupations and chooses the occupation that maximises the worker's earnings. The opportunity to choose occupations will in general raise the earnings of the lowest income recipients and reduce inequality. A worker with a low outcome in one occupation will not be forced to accept that income but instead will have a chance to get a higher income in some other occupation. Houthakker (1974) cites Roy's paper in developing a model of the distribution of earnings based on order statistics.

Roy's selection phenomenon has been applied and extended to a wide range of other contexts, beginning with choice of market versus non-market work and wage comparisons (Gronau, 1974; Heckman, 1974, 1976; Lewis, 1974). Decision rules more general than simple income maximisation have been developed and applied under the rubric of the generalised Roy model (Heckman and Vytlacil, 2007a). Applications of the Roy and generalised Roy models include choice of union versus non-union sectors (Lee, 1978), levels of education (Willis and Rosen, 1979), geographical region (Robinson and Tomes, 1982), marital status (McElroy and Horney, 1981), occupational choice (Miller, 1984), piece rate versus salary pay structures (Lazear, 1986), industry changers (Solon, 1988; Gibbons and Katz, 1992), immigration (Borjas, 1990), army retention (Brown, 1980) and segmented labour markets (Magnac, 1991). Flinn and Heckman (1982) and Moscarini (2001) extend Roy's selection model by incorporating choices based on search.

By moving beyond choice rules based on simple comparisons of outcomes across occupations, Roy's selection insight has generated progressively wider ranging results in econometrics, beginning with self-selection biases and extending to evaluation of social policies, treatment effects, and causality. Implications of selection for econometrics have been reviewed by Heckman and Vytlacil (2007a, b), Heckman and Taber (2010), Heckman (2010) and French and Taber (2011). Heckman (1976, 1979) and Heckman and Sedlacek (1985) describe the statistical problems that arise from the selection problem identified by Roy. In a simple regression model of a behavioural relationship using ordinary least squares applied to a random sample from a larger population, the error terms are assumed to be randomly distributed with an expected value of zero. With selection, not all potential observations have the same likelihood of appearing in the data set, with the result that the expected value of the error terms for the included observations may not be zero. For example, in an early application, women participate in the paid labour force if their market wage exceeds what they could earn in home production. Observations of market wages for working women would only be observed for those women whose market wages were higher than their alternatives out of the labour force and would therefore tend to be higher than the market wage that would be offered to all women (Gronau, 1974; Heckman, 1974). Alternatively, data analysts may apply some selection criteria to exclude incomplete observations or panel data that do not extend over the full sample period. Heckman provides methods to eliminate the bias generated by selection so that simple regression techniques can be employed. These methods have been substantially extended in the later literature in econometrics (French and Taber, 2011).

Heckman and Sedlacek (1985) estimate a Roy model of earnings in manufacturing and non-manufacturing incorporating selection of workers into the two sectors. In the initial specification of the model, workers choose between the two sectors based on income maximisation. Since the results for the initial specification fail test criteria that they develop, they extend the model by adding a third sector for non-market or household production and assume that individuals choose sectors on the basis of utility maximisation instead of income maximisation. The results explain the distribution of earnings within the manufacturing and non-manufacturing sectors, aggregation bias in wages, and the effect of selection on earnings inequality. The contributions of the non-market sector and utility maximisation to improvement in goodness-of-fit tests are analysed further in Heckman and Sedlacek (1990).

Heckman and Honore (1990) derive further statistical implications of Roy's selection process. They demonstrate that worker selection of sectors based on comparative advantage reduces inequality compared to random assignment to sectors. Further, the aggregate distribution of the logarithms of earnings will be skewed to the right even when one of the sectors is skewed to the left. They also consider conditions which would permit identification of parameters in the model.

Analysis of selection bias has led to improvement in methods used to evaluate social programs (Heckman et al., 1996, 1997; Abbring and Heckman, 2007; Heckman and Vytlacil, 2007a, b). A further extension considers general equilibrium models of income distribution and treatment effects (Heckman et al., 1998; Abbring and Heckman, 2007).

Roy's 1950 paper has received steady citations from the time of its publication. Although early citations arose from its results for earnings distributions, the increase in citations in recent decades arises from its relevance to fundamental problems in observational analyses and the econometrics that addresses them. In empirical social sciences, including sociology, political science and criminal justice, current trends towards increasing the validity of causal inferences are rooted in Roy's problem of explaining why the workers observed in an occupation are not representative of the population as a whole.


1Roy (1950) correctly observes that the normal distribution of ability was a convention of psychology and not a fundamental law of nature. He thus implicitly questions whether Pigou's paradox was paradoxical.

2Gary Becker's resolution of Pigou's Paradox is also based on the skew-inducing effect of multiplying random variables. In his case human capital multiplied by the rate of return, both assumed to be random variables, produce skewed earnings distributions (Becker, 1975).


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