Assignment Models of the Distribution of Earnings

by Michael Sattinger
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Title:
Assignment Models of the Distribution of Earnings
Author:
Michael Sattinger
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1993
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Journal of Economic Literature
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31
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2
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831
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880
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Abstract:

Assignment Models of the Distribution of Earnings

University at Albany,
State University of New York

The author is indebted to Ricardo Burros, James Heckman, Kajal Lahiri, David Lam, Thad Mirer, Lawrence Rafalouich, T. Paul Schultz, and anonymous referees for comments on earlier drafts. Errors and opinions are the responsibility of the author.

I. Introduction

ELATIVE WAGES are changing. Over

the last decade or so, earnings of high school graduates have declined rela- tive to college graduates, and earnings of young adults have declined relative to older adults; as a result, the distribu- tion of earnings has become more unequal. l These relative changes are hard to explain in the context of models where the return to education is fixed by the long run supply behavior of indi- viduals or in which the productivity and earnings of individuals are the result of their education and experience, indepen- dent of the availability of jobs in the econ-

Frank Levy and Richard J. Murnane (1992) ana- lyze recent tends in the distribution of earnings. Us- ing Current Population Survey data, they show that earnings of workers with 16 years schooling have in- creased relative to earnings of workers with 12 years schooling between 1979 and 1987 and that (for work- ers with 12 years schooling) earnings of workers aged 45-54 have increased relative to workers aged 25- 34 during the same time period (Table 7). Levy and Murnane (1992, Table 4) also report results of several authors showing that earnings inequality for all earn- ers and for males increased between 1979 and 1987, using various inequality measures.

omy. While changes in the industrial and occupational mix of the economy are rou- tinely incorporated into ad hoc explana- tions of shifts in the distribution of earn- ings, they are absent from most formal models of the distribution.

This paper reviews models that explain the distribution of earnings as arising from the market economy's solution to the problem of assigning workers to jobs. The amount a worker can contribute to production typically depends on which job the worker performs. This occurs be- cause jobs require many different tasks, and human performances at those tasks are extremely diverse; because industrial sectors use different technologies that rely on different combinations of human skills; or because jobs vary in the amounts of resources combined with la-

bar. the as a output then depends on how workers are assigned to jobs, i.e., which worker performs which job. The existence of an assignment prob- lem implies that workers face a choice in their job Or Their earnings are not determined by their performance in

831

one sector alone: if they do poorly at one job or sector, they can choose another. Choice of job or sector creates an inter- mediate step between individuals' char- acteristics and their earnings. The ob- served relationship is constructed from worker choices.

Income or utility maximization guides workers to choose particular jobs over others. Higher wages for workers with some characteristics then play an alloca- tive role in the economy rather than sim- ply being rewards for the possession of particular characteristics. Workers found in a given sector are not randomly drawn from the population as a whole. Instead workers' locations in sectors or jobs are based on the criterion that their choices maximize their income or utility.

The models discussed here are charac- terized by the presence of an assignment problem, together with the consequences of worker choice and nonrandom selection. Despite outward differ- ences, the models discussed in Section I11 have in common that they specify the jobs or sectors available to workers, the relevant differences among workers, the technology relating worker and job characteristics to output, and the mecha- nism that assigns workers to jobs.

These models generally proceed by first describing the assignment problem present in the economy. Then one can derive the wage differentials that are con- sistent with an equilibrium assignment of workers to jobs. The equilibrium wage differentials are those that yield equality between amounts of labor supplied and demanded in each submarket of labor. By providing a general equilibrium framework for studying inequality, assignment models reveal a rigorous route by which demand factors influence in- equality and correctly specify the relation between the distribution of individual characteristics and inequality. The earn- ings function is no longer a directly ob-

servable relationship but instead is the

equilibrium outcome to the solution of

the assignment problem.

Explicit consideration of the economy's assignment problem provides a unity to seemingly separate topics. Wage differ- entials, occupational choice, organization of hierarchies, unequal skill prices and self-selection bias are topics that have been studied by themselves but which arise as consequences of the assignment problem. The existence of many labor market phenomena, such as search, mo- bility, hierarchy tournaments, unemployment, and specialized labor markets, can be motivated as labor market responses to the problem of assigning work- ers to jobs.

Although not generally recognized as a subcategory of income distribution the- ories, assignment models have a fairly long history. They can be said to begin with Jan Tinbergen's model (1951) wi+h continuous distributions of workers and jobs and A. D. Roy's sectoral model (1951) with workers choosing between two or more occupations. These models differ in a number of ways but share the feature that the distribution of earnings can be explained through the assignment problem.

Empirical modeling of the distribution of earnings requires the econometric specification of worker alternatives, even though only the chosen sector or job is observed. This generates a set of econo- metric problems that have been addressed in applications of Roy's and Tin- bergen's models.

Probably most economists would agree to the basic premises underlying assign- ment models, that both supply and de- mand are relevant and that individual performances vary from job to job. But there may be some disagreement regard- ing the implications of those premises for the conduct of research on the distribu- tion of earnings. This survey emphasizes the implications of assignment models for the earnings functions, the human capital approach and the decomposition of in- equality.

A. Dog Bone Economy

Many distribution theories achieve results by ignoring or trivializing the as- signment problem. This leads to misin- terpretation of empirical relations such as the earnings function. As an example of what can go wrong when the assign- ment problem is ignored, consider the following dog bone economy. The agents in this economy are n dogs kept in a pen. These dogs vary by weight, teeth, mus- cles, and tenacity, all observable. At the beginning of the day, a dump truck ar- rives with n bones, differing in size. The bones are dumped in a neighboring area. The Hicksian Day begins when the gate opens and the dogs go after the bones. At this point a nontatonnement allocation process beings in which each dog can only hold onto one bone, losing the bone to any dog able to take it away. Equilib- rium arises when each dog has a bone that is not wanted by any dog that could take it away, and when each dog prefers its own bone to any bone that it could take away from another dog. Hierarchical ordering of dogs and bones would elimi- nate cycling and guarantee the existence of equilibrium, but this assumption is un- necessary for the story.

After the dust has settled, an econo- mist appears on the scene and collects data on the dogs and their bones. Bones can be rated by their value. The econo- mist then runs a regression of the bone values as a function of dog characteristics and finds a strong relationship (let us say the R' is 0.80). This relationship is an earnings function, with the value of bones as earnings. Pleased with the re- sults, the economist uses the estimated earnings function to predict the distribu- tion of earnings the following day, when the dump truck will bring a new load of bones. For each dog, the economist can predict the dog's bone value on the basis of the dog's characteristics (weight, teeth, muscles, tenacity). From the dis- tribution of dog characteristics, the econ- omist tries to infer the distribution of bone values.

Will the economist succeed in predict- ing the next day's distribution of bone values? After all, the earnings function is fairly accurately estimated.

Of course, the distribution of bone values the next day does not in any way depend on the distribution of dog charac- teristics; it depends simply on what the dump truck brings. But if there is no relation between dog characteristics and the distribution of bone values, how did the economist achieve such an accurate relationship between bone values and dog characteristics? The earnings func- tion estimated by the economist merely describes the assignment that arises be- tween dogs and bones. This assignment is a temporary equilibrium that depends on the given distributions of dogs and bones. The predictive content is limited to identifying the bone that will be found with a given dog, given the distributions of bones and dogs. We can explain why one dog got one of the bones and why another dog got a different bone but we cannot draw any conclusions about the causal or technological relation between dog characteristics and bones.

The most important feature of this story is the illusion created by the success of the regression of bone values against dog characteristics. The nature of the earnings function is not apparent, and the influence of the distributions of bones and dogs on the earnings function is in- visible. The dog bone economy presents an extreme case in that the bones are exogenously determined by what the dump truck brings rather than on any characteristics of the dogs themselves.

Yet the results of this economy conform to the economist's prior beliefs about the determinants of the bone distribution. The bone "earnings function" is consis- tent with a model in which a dog's charac- teristics determine the bone size it "earns." The existence of an assignment problem lying behind the empirically ob- served relationship is completely invisi- ble.

This invisibility provides an example of the fallacy of composition. In thinking about the distribution of earnings, it would seem natural to begin with the explanation of a single individual's earn- ings. Given the economy, including the rewards for education, training, and other characteristics, this individual's earnings will depend only on his or her own characteristics. With the observed relationship between the individual's earnings and his or her characteristics, it would be possible to predict a change in earnings from any change in the indi- vidual's characteris tics. In aggregating individual earnings to get the distribu- tion of earnings, however, the economy, including returns to education and train- ing, cannot be taken as given. The econo- my's rewards for various characteristics are endogenously determined and must themselves be explained by any distribu- tion theory. In particular, the consequences for the earnings distribution of a change in the distribution of worker characteristics cannot in general be pre- dicted from the change for a single worker. What constitutes a theory of the individual's earnings cannot automatically be extended to a theory of the distri- bution of earnings.

A first requirement of an earnings dis- tribution theory is therefore to avoid the fallacy of composition involved in going from the earnings function to the distri- bution of earnings, or else to specify the conditions under which it is legitimate to do so. It is unnecessary to use an as- signment model to avoid the fallacy of composition. But by specifying the deter- minants of the earnings function, assign- ment models accurately represent the in- teraction between supply and demand elements in shaping the distribution of earnings.

B. Relation to Other Approaches

Assignment models are closely related to other approaches to the study of inequality. They are consistent with structuralist theories in sociology, in which wage structures influence the wages associated with particular jobs (Mark Granovetter 1981; Arne L. Kalleberg and Ivar Berg 1987). As in assign- ment models, earnings depend on the characteristics of both the worker and the job. However, the structuralist theories do not assume competitive access to jobs. A major question then concerns how workers are matched to jobs (Aage B. S~rensenand Kalleberg 1981). Noncom- petitive access to jobs, for example through rationing or segmentation, pro- vides a route through which institutional structures can influence the distribution of earnings. Lester Thurow (1975) devel- ops a similar model in which the wage rate is determined mainly by the job. This leads to an assignment in which workers queue for jobs based on train- ability.

Sherwin Rosen (1974) develops a model of the determination of implicit prices of product characteristics. The re- sulting relationship between price and product characteristics, called an hedonic price function, is an envelope of buyer and seller offer curves. As in assignment models, this price function assigns con- sumers to producers in a market with heterogeneous products. The earnings function generated by the assignment problem (e.g., in Tinbergen's model) is essentially an hedonic price function in a labor market context-an hedonic wage function. Studies of hedonic wage func- tions are mainly directed towards esti- mating compensating wage differentials for job characteristics such as risk (Robert

E. B. Lucas 197713; Robert S. Smith 1979; Rosen 1986a). Ronald Ehrenberg and Smith (1991) provide an accessible exposition of how compensating wage dif- ferentials for risk (pp. 266-74) and educa- tion (pp. 314-18) are determined using an hedonic model.

An alternative expression for assigning workers to jobs is matching. Boyan Jova- novic (1979) develops a model in which the output from a specific worker-job match is distributed as a random variable that is initially unknown to the employer or worker. The model is used to explain turnover as information about productiv- ity is revealed during job tenure. The matching literature is primarily concerned with ex post differences in the outputs obtained from worker-firm matches, whereas the assignment models discussed in this article emphasize ex ante differences among workers and firms. As productivities are not explicitly related to ex ante characteristics of work- ers or jobs in matching models, the ap- proach is less useful in explaining the dis- tribution of earnings, although it has been applied to the question of wage growth over a worker's career (Jacob Mincer and Jovanovic 1981), turnover, and unemployment (Jovanovic 1984) and returns to on-the-job training (John M. Barron, Dan A. Black, and Mark A. Loewenstein 1989).

Arguments about mismatches in the labor market, either with respect to loca- tion or skills, are based on simplified forms of assignment models (John D. Kasarda 1988; Levy and Murnane 1992, Section VII. B, review mismatch models). Technological change, in particular the advance of information-based industries, has shifted the skill requirements of jobs. At the same time, entering workers are failing to acquire these skills, leading to a mismatch between supplies and de- mands. The mismatch arguments implic- itly regard skill requirements and sup- plies as unresponsive to economic incentives, at least in the short run, so that planning and intervention are neces- sary. In assignment models, these sup- plies and demands are not rigidly deter- mined but respond to wage differentials. Steeper wage differentials would then re- solve the mismatches that have been ob- served and forecast.

Assignment models tend to be highly abstract and mathematical, often using simplifying and unrealistic assumptions about workers and jobs to achieve analyt- ical results. They have not so far gener- ated a set of easily identifiable questions which can be answered by accessible em- pirical procedures. Further, they may di- vert attention from issues of household composition, income transfers, discrimi- nation and social problems that have a more direct impact on poverty and in- equality. However, they point the way to the steps necessary to incorporate de- mand and job choice in empirical models of earnings.

C. Contents

The next section discusses the extent of the assignment problem in the econ- omy and the way that decisions of work- ers and employers generate assignment patterns. Section I11 presents three basic types of assignment models, depending on whether characteristics of workers and jobs are continuous or discrete. These types are the linear programming opti- mal assignment problem, the differential rents model, and Roy's sectoral model. Section IV compares these models with regard to the choices available to work- ers, wage determination, self-selection, and comparative advantage. Section V considers implications of assignment models for the decompositions that are used to study the distribution of earn- ings. These decompositions include ana- lyzing the distribution by industrial or occupational sector, use of an earnings function, and human capital models. The conclusions in Section VI review the rela- tions between assignment problems, self- selection, and comparative advantage. The section indicates the most important extensions of assignment models and ex- planations for changes in wage differen- tials as well as relevant research ques- tions.

11. The Economy's Assignment Problem

A. Existence

What would the economy be like without an assignment problem? With only a single, observable skill, a worker would be able to get the same wage no matter which job he or she took. No spe- cific training, education, diversity in skills or preferences would limit in any way the jobs that one would seek. Find- ing a job would be reduced to locating a firm with a vacancy.2 Firms would be indifferent as to which workers they em- ployed. Hiring would be reduced to the trivial problem of taking the first worker that came along. Unemployment would only arise if the number of workers ex- ceeded the number of jobs. Wage differ- ences among workers could arise, but all labor could be expressed in terms of the amount of an average or standard worker it was equivalent to. Professors at univer- sities could be replaced by a sufficient number of high school graduates, pre-

Job search by itself does not imply that an assign- ment problem exists. It is conceivable that there is only one skill, with marginal products proportional to the skill. but that this skill is im~erfectlv observed. Workers would search for em~lovkrs whdrated their skills more highly, and firms would search for workers whose skills were underrated. Changes in employ- ment would affect the distribution of earnings but (for a given level of unemployment) not output.

sumably all lecturing at the same time. This very article could have been written by anybody, perhaps in less time.

But of course the economy does have an assignment problem. The size and im- portance of the assignment problem can be seen from the resources expended to solve it. Unemployment imposes large costs through forgone production, nonpecuniary costs, and uncertain incomes. Much of this unemployment arises from workers seeking jobs better than those readily available at the lowest wages, at least when depression conditions are ab- sent. Firms spend substantial amounts through personnel departments in adver- tising positions and interviewing candi- dates. After employment, firms collect information about workers to facilitate later assignment within the firm through internal labor markets. Quits and layoffs by agents seeking better matches impose losses of specific training. Expenditures on screening and signals may arise be- cause of the advantage of some assign- ments over others; they may also inter- fere with efficient assignments. The formation of specialized labor markets may arise in order to reduce the costs of assignment. Occupational segregation and segmentation, by distorting the as- signment, impose efficiency losses on the economy as well as inequities in the treatment of individuals and groups.

Much empirical work supports the ex- istence of an assignment problem. Joop Hartog (1985, 1986a, 198613, 1988) esti- mates earning functions that include both individual and job characteristics, using data from the Netherlands and elsewhere. Hartog compares three models.

-

Some versions of human capital models Suggest that only individual characteris-

tics should effect earnings, jobcharacteristics are the maior determinants in segmented labor Aarket theeries. In assignment models, both sets of variables would be significant (in the ab-

sence of an exact correspondence be- tween individual and job characteristics). Hartog finds that both individual and job characteristics affect earnings. Further, there are significant interactions between them, supporting the existence of an as- signment pr~blem.~

Hartog (1977, 1980, 1981b) and R. E. B. Lucas (1974) identify significant ways in which jobs differ. Sat- tinger (1978) establishes the existence of comparative advantage among individu- als using data on mechanical aptitude tests taken by secondary school students. The ratios of performance of pairs of indi- viduals are computed for four tasks. For each pair, these ratios are then ordered from highest to lowest. In the absence of any systematic comparative advantage, one ordering will be as likely as any other. Using a chi-square goodness-of-fit test, the hypothesis of no systematic com- parative advantage is rejected. Heckman and Guilherme Sedlacek (1985), in esti- mating extensions of Roy's model, show that differentials for education and expe- rience are larger in manufacturing than in nonmanufacturing. Also, in a later pa- per (1990), they reject a simpler model with no assignment problem in which worker earnings would be the same in all market sectors.

Unequal wage structures among eco- nomic sectors provide indirect evidence of an assignment problem. Heckman and Jose Scheinkman (1987) establish that worker characteristics receive unequal rewards in different sectors of the econ- omy, so that workers face a nontrivial choice problem. William T. Dickens and Lawrence F. Katz (1987) and Alan B. Krueger and Lawrence H. Summers

3As a specific example of interactions, Hartog (1985) estimates an earnings function with dummy variables for each combination of job level (or re- quired education) and worker education. He tests and rejects a specification in which education and job level contribute independently to earnings. Edu- cational differentials therefore depend upon the job.

(1988) also conclude that wage structures vary among industrial sectors.

Like other major allocative problems of the economy (such as what, how, and for whom), the assignment problem is not apparent to individual agents who are simply solving their own utility or profit maximizing problems. Employed and unemployed workers in an economy en- gage in job search, eliciting job offers un- til they find a satisfactory one. Employers typically interview a number of candi- dates for a job, seeking the most appro- priate candidate. But out of these activi- ties arises an assignment of workers to jobs. An assignment of workers to jobs can be defined as a listing of each worker together with the job he or she per- form~.~

The next section examines how the decisions of individual agents solve the assignment problem facing the econ- omy.

B. Comparative Advantage

Now consider the reasons why some assignments occur instead of others. One reason is comparative ad~antage.~

Consider a fixed-proportions technology in which employers need to have a fixed set of tasks performed to .yield a given level of production. Suppose workers do not have preferences for some tasks over others. Each job is associated with a par- ticular task. Let a,j.be the number of

4The analysis of this paper takes jobs as given. Rosen (1978) considers the subproblem of how em- ployers arrange into jobs the tasks that they need performed.

Application of comparative advantage to the anal- ysis of labor markets is commonly attributed to Roy. Roy does not analyze his model in terms of compara- tive advantage but comments (1951, p. 145), "It should be apparent that the analysis attempted in this article bears some sort of &nity to the theory of comparative advantages. A situation has been ex- amined in which individuals' comp;rative advantages in various activities differ widely. Sattinger (1975) applies comparative advantage to the study of the distribution of earnings and Rosen (1978) develops a general analysis of comparative advantage in labor markets.

times that worker i can perform job j's task per period. If

then worker 1is said to have a compara- tive advantage at job 1and worker 2 has a comparative advantage at job 2 (note that if (1)holds, then a22/a12> azllall).

Comparative advantage determines the assignment in a market system with this technology as follows. Suppose that in equilibrium the wage rate prevailing for worker i is wi.The employer offering job j will seek to minimize the cost of getting the job's tasks performed, taking the wage rate as given.6 Using worker i, the cost would be wilaij.Employer j will prefer to hire worker 1instead of 2 whenever w21a2j> wllalj,or

From (I),it follows that we would never observe employer 2 hiring worker 1 when employer 1 hires worker 2: it is impossible for wllw2to be simultaneously greater than alllazland less than al2/aZ2. Depending on the wage rates, it is possi- ble that both employers would prefer worker 1, or that both prefer worker 2. But the only assignment in which the employers prefer different workers is when employer 1 prefers worker 1and employer 2 prefers worker 2. With this technology, the equilibrium assignment must be consistent with the comparative advantage relations as given by (1).

This example also shows how knowl- edge of the equilibrium assignment can

Alternatively, fees could be offered for the perfor- mance of tasks, and workers could maximize their incomes. The resulting equilibrium wage rates for workers would still satisfy (2) and (3). In models in which an inexact assignment occurs (because of im- perfect information) or in disequilibrium, the wage for a worker may depend on both the worker and job characteristics.

explain wage differences. Suppose in equilibrium worker 1is observed in job 1while worker 2 is in job 2. Then the ratio of wages for the two workers must lie between the workers' trade-offs at the first job (i.e., the ratio of their perfor- mances) and their trade-offs at the second job:

In this way, ratios of performances in the two jobs set limits within which the wage differential must fall.

The term "comparative advantage" is used in different ways by various authors. As defined using (I), comparative advan- tage arises whenever ratios of outputs for two workers are not identically equal in every job. Comparative advantage then establishes the existence of an assignment problem but one would need to know the direction of the inequality in (1)to determine which particular assign- ment comes about. An alternative rela- tion is absolute advantage, which arises when a worker is better at a job than other workers. In terms of the outputs in (I),worker 1has an absolute advantage at job j compared to worker 2 if alj > azj.If each worker has an absolute advan- tage at his or her own job, compared to any other worker and that worker's job, then comparative advantage must also be present in the sense defined in (I).~ The

While in simple economies it is possible that ab- solute advantage determines assignment (with each worker employed at the job at which he or she is best), this becomes unreasonable in large economies. For example, if there are one million workers, there would need to be at least one million different jobs, and in each job a worker would need to be better than nearly a million other workers. Even so, only for a very special set of wages would each worker choose the job at which he or she was best. However, Glenn MacDonald and James T. Markusen (1985) describe a technology with two activities in which absolute advantage (in the form of absolute skill lev- els) results in assignments that are not completely determined by comparative advantage, as in the scale of operations effect in the following section.

significance of comparative advantage is that a worker can still get a job even though he or she is worse at all jobs than other workers, i. e., even though absolute advantage is absent for that worker. Some economists find it useful to restrict comparative advantage to the case where absolute advantage is absent for some worker; this will be referred to as the standard comparative advantage case.'

C. Scale of Operations Eflect

Some economists may believe that comparative advantage is the only pro- duction principle underlying the assign- ment of workers to jobs, but this is incor- rect. As a counterexample, consider an economy in which a job is associated with the use of a particular machine that can be used by only one person at a time. Suppose the possible values of output (price times quantity) obtained per hour from the two workers at two jobs are as

follows:
Job 1 Job 2
Worker 1 $35 $20
Worker 2 $20 $10.

Here, worker 1 has a comparative advan- tage at the second job and worker 2 has a comparative advantage at the first job, because $35/$20 < $20/$10. However, the maximum value of output, $45, is obtained when worker 1 is employed at job 1 and worker 2 is employed at job

2. In an eight hour day with this assign-

'This restricted case is consistent with the example of comparative advantage worked out by David Ri- cardo in the context of trade (1951, p. 135). In that example, Portugal has an absolute advantage over England in the production of both wine and cloth. But there are still gains from trade because England has a comparative advantage in the production of cloth. The importance of comparative advantage is that it explains trade even when one country has an absolute advantage at both goods. If each country had an absolute advantage at one good, trade would be obvious.

ment, output would be $280 at job 1 and $80 at job 2.

Suppose we tried to reallocate labor according to comparative advantage. Suppose we put worker 2 in job 1 for eight hours and worker 1 four hours at job 1and four hours at job 2. If this were possible, it would yield a net increase of $20 from job 1. But it would require twelve hours of labor in job 1during an eight hour day, which is ruled out by the assumption that only one worker at a time can be employed at a job. A worker occupies a job (or the machine associated with the job), preventing the reassignments indicated by comparative advantage.

The reason comparative advantage does not indicate the optimal assignment in this case is that earnings from a job are no longer proportional to physical output at the job. With cooperating fac- tors of production (either explicit in the form of a machine or implicit via a scar- city in the jobs available), an opportunity cost for the cooperating factor must be subtracted from the value of output to yield the earning^.^ Ronald H. Tuck (1954, p. 1) describes the resulting prob- lem facing the economy as one of

. . . assigning each individual member of the economy to work of an appropriate level of re- sponsibility, and of doing this in such a way that the best possible use is made of the avail- able human talent and experience.10

Basically, more resources (in the form of more capital, labor or greater responsi- bility) are allocated to workers with

Some authors determine comparative advantage on the basis of earnings in ditferent jobs or at different educational levels rather than on the basis of physical output. With cooperating factors, this approach is ambiguous.

loTuck (1954) explains the distribution of firm sizes in terms of the distribution of productive resources among entrepreneurs. Robert E. Lucas, Jr. (1978) and Walter Oi (1983) also develop theories of the size distribution of firms that involve the assignment of resources to heterogeneous entrepreneurs.

greater abilities because the resources have a greater effect on output for those workers. In turn, with greater resources, output is more sensitive to the abilities of workers, raising wage differentials for workers with greater abilities.

The principle affecting the distribution of earnings has been developed in a num- ber of contexts. Thomas Mayer (1960) and Melvin Reder (1968) use the term "scale of operations effect" in their mod- els of the distribution of earnings (see also discussions by Reder (1969, pp. 219- 23) and Sattinger (1980, pp. 32-35)). Mayer uses the term to describe the po- tential value of output, while Reder uses it for the value of resources under a per- son's control. Rosen (1981) applies the scale of operations effect to the incomes of superstars. George Akerlof (1981) de- velops an analogy between jobs and dam sites to explain why some workers might be unemployable. A productive dam that does not fully utilize a dam site may not be chosen because it prevents more productive dams from being used at the site. The dam site carries an opportunity cost that must be subtracted from a dam's output to determine whether it is suit- able for the site. Stephen J. Spurr (1987) discusses how the scale of operations ef- fect results in larger claims being assigned to lawyers of higher quality.

The scale of operations effect is also related to theories of compensation within hierarchies developed by Herbert

A. Simon (1957) and Harold Lydall(1959; 1968, pp. 125-29). In a hierarchical model developed by Guillermo A. Calvo and Stanislaw Wellisz (1979), the effect of a supervisor shirking is that workers under the supervisor also shirk. This in- creases the sensitivity of the firm's output to workers' abilities as supervisors and leads a firm to place more able workers at higher levels in the hierarchy. In Calvo and Wellisz' model, the scale of opera- tions effect provides a link between assignment models and efficiency wage models, in which firms pay above market wages in order to influence workers' productivities. Differences in efficiency wages can then be explained in terms of differences in the scale of operations rather than differences in costs of moni- toring workers. l1

With the scale of operations effect, the wage ratio for the two workers will not lie between the ratios of outputs as in the comparative advantage case because of the presence of opportunity costs from the use of a machine or the filling of a position or job. Consider now how wages are determined for workers in the context of a model in which the cooperating fac- tor is capital, in the form of heterogene- ous units called machines. Assume only one worker at a time can be combined with a machine. Let pj be the price of a unit of output from machine j, and let ag be the output produced per period by worker i at machine j. Let wi be the wage rate for worker i. The owner of ma- chine j takes the wage as given and chooses the worker that maximizes the residual pjav -wi instead of the output values pjag appearing in the example at the beginning of this section. If the owner of machine 1is observed to choose worker 1while the owner of machine 2 chooses worker 2, plall -wl r plael -w, and p2a12 -wl Ip2ae2 -w,. Therefore:

The difference in wages must lie between the difference in the value of output pro- duced by the two workers on machine 1, and the corresponding difference on machine 2. The assignment of worker 1

"Rosen (1982), Michael Waldman (1984b), David Grubb (1985) and Peter F. Kostiuk (1990) develop additional models that relate versions of the scale of operations effect to assignment and earnings within hierarchies.

to machine 1 and worker 2 to machine 2 can come about only if p2(a12 -ae2) 5 pl(al1 -~21).If p2(a12 -a22) 'pl(al1

-azl), only the opposite assignment could be observed in equilibrium (i.e., worker 1at machine 2 and worker 2 at machine 1).

Alternatively, one could begin by as- suming workers choose machines. Let rj be the rental cost for machine j. Then worker i chooses j to maximize pjaq rj, Again, if worker 1 is observed to choose machine 1while worker 2 chooses machine 2, plall -rl r p2a12-r2and p,a2, -r1 5 p2az2-r2. Therefore:

In this way, differences in wages and rents are determined symmetrically by the problem of assigning workers to jobs.

The technologies leading to the com- parative advantage and scale of opera- tions cases are very different. In the com- parative advantage case discussed above, the tasks from different jobs are needed in fixed proportions and cannot be substi- tuted for each other, but a particular job could be filled by an indefinite number of workers. Opposite circumstances hold in the scale of operations case. Output from different jobs is simply added to- gether, so that there is perfect substitut- ability among the outputs of different jobs. However, with only one worker per job, more workers cannot be added to a job to make up for low output levels.

Clearly, there are many potential tech- nologies relating worker and job charac- teristics to aggregate production. In these cases, the optimal assignment would not be determined by simple bilat- eral comparisons as in (1)and (2).

D. Preferences

Along a given wage offer curve, the low- est wage acceptable to a worker, woi,OCcurs when the effort requirement equals the effort capability of the worker, i.e., gi = hj. If the effort requirement is higher or lower than g,, the worker must receive a higher wage in order to achieve the same level of utility. Higher values of woi yield higher wage offer curves and higher levels of utility, so that the worker chooses hj to maximize wi -a(gi -h.)2.

With this assumption regarding prefkr- ences, workers with higher effort capabil- ities will always end up in jobs with higher effort requirements. This type of assumption (in which workers and jobs are matched on the basis of distance be- tween characteristics) is useful in gener- ating hierarchical assignments in other contexts, for example marriage (Gary S. Becker 1973; Lam 1988). The assumption in Tinbergen's model can be contrasted

l2 Tinbergen (1956) extends his model to multidi- mensional worker and job characteristics. He further considers a generalization in which the production side of the economy can be incorporated into the determination of the wage function (1956, pp. 170- 71). In this case, wage differentials combine pro- ductivity differences and compensating wage differ- entials. In related work, Tinbergen develops a nor- mative theory of income distribution in which a tax function is found that maximizes social welfare (1970); estimates an empirical model with discrete categories of labor distinguished by educational level (1975a, 1977); and estimates elasticities of substitution among educational levels as a means of explaining educa- tional differentials (1972, 1974, 1975b).

Journal of Economic Literature, Vol. XXXZ (June 1993)

with one in which workers all uniformly prefer jobs with higher values of some characteristic (or else all prefer lower values). For example, in the compensat- ing wage literature, all workers may dis- like a particular job feature such as riski- ness, noise or distance to work but have different valuations of those characteris- tics. The unequal valuations lead to an assignment of workers to jobs.

Wage differentials in Tinbergen's model also differ in an important regard from wage differentials in the compara- tive advantage and scale of operations cases. If the distribution of worker char- acteristics exactly matches the distribu- tion of job characteristics (so that hj = g, if worker i gets job j), wage differences would be eliminated. Further, if workers end up in jobs with effort requirements below their capabilities, wages will need to be a decreasing function of capabilities in order to induce workers to take the jobs. This result would not arise with comparative advantage or the scale of op- erations effect as long as the worker char- acteristic contributes to production.

This section has shown how the profit or utility maximizing decisions of workers or employers generate an assignment of workers to jobs. The aggregate assign- ment problem will typically be invisible to individual agents, but their decisions may lead to a pattern of assignment that prevails throughout the economy. The next step is to examine how the problem of assigning workers to jobs generates wage differentials and the distribution of earnings among workers.

111. Alternative Assignment Models

The three assignment models devel- oped in this section seem very different. The linear programming optimal assign- ment problem is a model of the condi- tions for an efficient assignment. The dif- ferential rents model explains wage differentials. Roy's model explains self- selection into occupations. The point common to all three models is that they explicitly formulate the assignment prob- lem that must be solved in the economy. This problem enters as an intermediate step in the connection between worker characteristics and earnings. Because they are all linked by the explicit pres- ence of an assignment problem, all three models exhibit common phenomena (such as conditions for an efficient assign- ment, wage differentials that depend on job assignments, and self-selection effects), although in different forms and with different emphasis.

A major difference in the models con- sidered here is in their description of worker and job characteristics. In the lin- ear programming optimal assignment problem, workers and job characteristics take discrete values, and in the differen- tial rents model they are continuously distributed. In Roy's model, jobs are dis- crete (in the form of sectors),while

,.

worker characteristics are continuouslv distributed, so that many workers will end up in the same sector. These differ- ences in modeling: account for the out-

-

ward differences in results.

The starting point will be the linear programming optimal assignment prob- lem. This problem provides a very gen- eral model with which to analvze the economy's assignment of workers to jobs and its results-have many features that are common to all assignment models.13

l3 Dale Mortensen (1988) reviews matching prob- lems related to the assignment problem. In this liter- ature, matches are formed through the voluntary ac- tions of agents rather than as the solution to an aggregate maximization problem. David Gale and Lloyd Shapley (1962)analyze equilibrium in a match- ing market and present an adjustment process that would lead to a stable market structure based on preferences of agents on both sides of the market. Shapley and Martin Shubik (1972)and Becker (1973) analyze the same problem when one agent in the match can compensate the other for forming the match, for example through the wage in a labor con- tract. Alvin Roth and Marilda A. Oliveira Sotomayor (1990) review game-theoretic analysis of two-sided matching problems.

No restrictive hierarchical assumptions are made regarding workers or jobs. That is, there are no explicit parameters de- scribing workers that would allow one to rank them with regard to skills. Fur- ther, no continuity assumptions are made regarding distributions of workers and jobs. Each worker's wage depends in a complex way on the outputs obtained from alternative assignments rather than on the marginal increase in output ob- tained by using more labor or slightly different labor. On the other hand, the linear programming assignment problem imposes some restrictive assumptions: there are equal numbers of workers and jobs, and they must be combined in fixed proportions, with one worker per job. By altering conditions in the model, one can generate the differential rents model that will be discussed in Section 1II.B. or Roy's sectoral model that will be dis- cussed in Section 1II.C. This procedure will facilitate a comparison of various models.

A. Discrete Workers and Jobs: The Linear Programming Optimal Assignment Problem

Tjalling C. Koopmans and Martin Beckmann (1957) consider a linear pro- gramming optimal assignment problem in which economic activities are assigned to locations. The dual prices in the solu- tion of the assignment problem then cor- respond to market determined profits and land rents. By changing the context of the assignment problem, one can con- sider how wages and machine rents (or profits associated with a job) are deter- mined.

A linear programming optimal assign- ment problem arises as follows. Suppose there are n workers and n machines (with each machine corresponding to a job), and let agbe the value of output obtained by worker i at machine j. The problem is to find the assignment, with one worker per machine, that maximizes the sum of output values. The assignment problem is a special case of the general linear programming problem of maximiz- ing a linear objective function subject to inequality constraints. As part of the sim- plex method of solving this problem there are simplex multipliers or dual prices associated with each worker and machine. Let wi be the dual price for worker i and let rj be the dual price for machine j. These dual prices have the following properties. If worker i is assigned to machine j in the optimal solu- tion, then wi + rj = ag;otherwise wi +

> aV With the optimal solution, the 2~ prices exhaust the product. If work- ers obtain their income by renting ma- chines at the prices given by the r;~, then (ruling out ties) they would be led to select the machines they are assigned to in the optimal solution. The maximum income of worker i would be wi. Similarly, if machine owners hire workers at the factor prices wi, they would choose the workers assigned to their machines in the optimal solution, and the maxi- mum income for the owner of machine j would be rj. The dual prices wi and rj distribute income in such a way that the assignment problem is solved through the income maximizing behavior of indi- vidual agents. These dual prices perform as market prices and could arise from a competitive solution.14

In the solution of the optimal assign- ment problem, there is no expression showing the relationship between dual prices and any explicit characteristics of workers or machines, a relationship that would be analogous to an earnings func- tion. However, it is possible to apply fac- tor analysis to the matrix A formed from the outputs agin order to infer character-

l4 Gerald Thompson (1979) investigates the rela- tion between the prices generated by auctioning or bidding and the dual prices of the assignment prob- lem.

istics of workers and machines.15 With this factorization, outputs from matches can be represented as:

where R is the rank of the matrix formed from the outputs aij, pik is the amount of the k-th latent property of worker i, qjkis the amount of the k-th latent prop- erty of machine j, and Ak is the weight for the k-th property. With this factoriza- tion, the k-th property of workers inter- acts only with the k-th property of ma- chines in the determination of outputs. l6

Suppose in the optimal assignment that worker i is matched with machine j and that worker c is matched with ma- chine d. Then from the condition that the owner of machine d would not prefer worker i, aid -wi 5 a,d -w,, and from the condition that the owner of machine j would not prefer worker c, aCj-w, I aij -wi. Combining these inequalities and using (7) yields:

The inequalities in (8) show the upper and lower limits for the wage differences between worker i and worker j. The lim- its depend on the differences between

l5See Sattinger (1984). Factor analysis refers to the factorization of matrices, i.e., representing a ma- trix as the product of other matrices. In the term factor prices, factor refers to factors of production such as labor or land.

l6 It is possible that R is less than the number of workers or machines, in which case the factorization represents the complete data more compactly in terms of the R underlying properties of machines or workers. Alternatively, one could use the factoriza- tion to approximate the data using fewer than R properties. Arrange the Ak in decreasing value, and set

Akpikcljk. Then the matrix B = (bg)is the

bij=

best rank S estimator of A, in the sense that among all rank S matrices with the same numbers of rows and columns, B minimizes the sum of the squares of differences between the entries of A and the entries of B. The main purpose of the factorization used here, however, is that it permits one to relate differences in wages to worker and machine properties.

the latent properties of the two workers, i.e., pik -pck appears on both sides of (8). But the limits also depend on the machine properties qdk and qjk, which enter as weights, increasing the impor- tance of some worker properties and de- creasing the importance of others. The effect of worker properties on wages therefore depends on which jobs are per- formed in equilibrium. This result illus- trates one of the central points about as- signment models: a change in either the workers or jobs in the economy alters the assignment and the wage differentials that are observed.

The determination of limits for machine rents is exactly symmetric to the determination of wage limits:

In this expression, worker properties en- ter as weights for the importance of vari- ous machine properties.

The dual prices from the solution of the optimal assignment problem exhibit two forms of indeterminacy. Because agents choose partners on the basis of relative rewards, it is possible to shift all wages up by a given amount and all rents down by the same amount (or else all wages down and all rents up). In (8) and (9), limits are placed only on differ- ences between wages or rents. In this model, the problem of assigning workers to machines determines relative wages and machine rents but not their absolute levels. The absolute levels of wages and rents are determined outside the assign- ment problem, perhaps by the availabil- ity of idle machines or workers. A second indeterminacy arises because individual wages and rents can increase or decrease within the limits in (8) and (9)while still leading to the same assignment. With continuous distributions of workers and jobs, as in the differential rents model

of the following section, this indetermi- nacy disappears because the bounds for wage differences approach each other in the limit. The particular wages and rents that arise depend on the adjustment pro- cess and institutions that lead the econ- omy to equilibrium. l7

B. Continuous Distributions of Workers and Jobs: The Diferential Rents Model

In Ricardo's analysis of rent (1951, p. 70), the difference in rents for two nearly similar tracts of land with unequal fertil- ity will equal the difference in output on the two tracts, holding labor and capi- tal constant. The absolute level of rents can be calculated from the condition that no rent is paid on marginal land for which cultivation yields only enough to pay for the capital and labor used. These princi- ples can also be applied in the labor mar- ket. The wage differential associated with a particular worker characteristic can be calculated from the increase in output from changing that characteristic, hold- ing everything else the same. While land is heterogeneous in Ricardo's differential rents model, though, both labor and jobs (or capital) are heterogeneous in the labor market. The wage differential therefore depends on the assignment of workers to jobs.

The differential rents model (Sattinger 1979, 1980) arises when the output in the optimal assignment problem depends on a single explicit characteristic of the worker and a single explicit characteristic of the job. l8 Under certain conditions, a

l7Vincent Crawford and Elsie Knoer (1981) and Alexander Kelso and Crawford (1982)analyze an ad- justment mechanism in which employers make offers and workers then accept or reject them. Then the resulting solution is best from the point of view of the employers. Alvin Roth (1984, 1985) shows that the best allocation for one side of the market is the worst for the other.

"The assumption that workers can be described by a single skill or ability is counterfactual. Individu-

hierarchical assignment arises in which more skilled workers perform jobs with greater resources. Hierarchical models of this type are interesting because they di- rect attention to an important feature of market systems, the tendency to rein- force and exaggerate differences among workers. With heterogeneous jobs, more skilled workers (who would perhaps have gotten higher earnings anyway) have their earnings boosted by being assigned to jobs with more capital, responsibility, or subordinates.

By imposing some conditions on the model considered in 'the previous section, it is possible to obtain the differen- tial rents model discussed in this section.l9 Suppose that each job is associ- ated with a unit of capital, called a ma- chine, and suppose that each machine can be described by a single characteris- tic, its size, which measures the amount of resources or capital associated with the job. Let a,j. = f(gi,kj), where g, is a mea- sure of worker i's skill (alternatively, ca- pability, education, or ability), kj is a measure of the size of machine j, and production f(g,k) is an increasing func- tion of g and k and has continuous first and second order derivatives. (It is not necessary for f(g,k) to take the same func- tional form as the linear factorization in (8).) Now suppose that the numbers of workers and machines increase indefi-

als have extremely diverse abilities at various tasks, and these abilities are only partially correlated. A measure of averages like an IQ may be stable but it will be a poor predictor of an individual's performance at any given task. Unfortunately, the simplify- ing assumption of a single worker characteristic may be confused with arguments about the existence of IQ, which are irrelevant to the issues considered here. Tinbergen (1956) constructs a model with con- tinuous distributions of workers and jobs in which workers are described by multiple characteristics.

l9 Tinbergen's models (1951, 1956) and Sattinger's models of comparative advantage (1975)and compen- sating wage differences (1977)also assume continuous distributions of workers and jobs but cannot be de- rived from the same optimal assignment problem dis- cussed in 1II.A.

In addition to production-relevant characteristics, preferences may also guide the assignment. In Tinbergen's original model (1951), workers prefer jobs with effort requirements that are close to their effort capabilities. l2 These requirements and capabilities are unre- lated to production in the model. Let hj be the effort requirement of job j and let gi be the effort capability of worker i. Worker trade-offs between the wage and the effort requirement hj are described by the following family of wage offer or indifference curves:

Journal of Economic Literature, Vol. XXXZ uune 1993)

nitely so that the values of g and k cover intervals. Let G(x)be the proportion of workers with skill levels less than or equal to x, and let K(x)be the proportion of machine sizes that are less than or equal to x."

In this economy, aggregate output is obtained by summing the production from each match of a worker with a ma- chine. In the absence of preferences, the efficient assignment will be the one that maximizes this aggregate production.

Consider now how the production functionf(g, k) together with the distribu- tions G(x) and K(x) determine the rela- tionship between wages and the skill level g. Let this relationship be repre- sented by w(g).The owner of a machine of size k* will attempt to maximize the profits obtained from that machine. If the owner hires a worker of skill g, profits will be given by f(g, k*) -w(g).To decide whether this skill level maximizes profits, the owner would compare the increases in production from using a worker of greater skill with the increase in wages. If the increase in production is greater, the owner would choose a higher skill level. If the increase in production is lower than the wage increase, the em- ployer would choose a less skilled worker. The owner has found the right skill level when the increase in produc- tion equals the increase in wages. For- mally, maximization of profits for the ma- chine owner implies the first order condition

where wl(g) = dwldg. The term wl(g) is simply the wage differential, the in-

''In this model, the distribution of machine sizes is taken as given. Akerlof (1969)considers a model in which capital is allocated to workers. Some workers are then structurally unemployed because their out- put will not cover the cost of capital.

crease in wages from a given increase in the worker's skill level. The term df (g, k*)ldg is the increase in output from using a worker of a higher skill level, holding machine size constant at k*. This method of calculating wage differentials is similar to Ricardo's calculation of differ- ential rents. Also, (10)is analogous to the familiar competitive labor market condi- tion that the wage equals the marginal revenue product, only with an increment in skill replacing an increment in the number of workers.

The first order condition (10)does not by itself determine the wage function w(g). In this economy, the effect of an increase in the worker's skill level, and the size of the wage differential, depend on which job the worker performs. For each value of g at which we wish to calcu- late the wage differential w'(g),we would need to know the size of the machine k* of the employer who hires that labor. This information is contained in the economy's assignment of workers to jobs.

Usually, to find the general equilib- rium of an economy, one must determine simultaneously the prices and quantities that satisfy the equilibrium condition. In the context of an assignment model, this means finding both the wage function w(g) and the assignment at the same time. As employers choose workers on the basis of the wage function w(g), this would be analytically very difficult in the general case.

However, a number of simplifying as- sumptions make it possible to determine the assignment without first knowing the wage function. First, in the time period under consideration, the distribution of jobs or machines does not depend on the wage function w(g).The number of jobs does not increase or decrease in response to a high or low profit. Because workers and jobs are each described by only one variable, the production function f(g, k) may be such that only a simple hierar- chical assignment can arise, i.e., one in which more skilled workers are employed at jobs with larger machines (an alternative would be that more skilled workers are employed at jobs with smaller machines).

The procedure for determining equi- librium is as follows. First, a tentative assignment is assumed (based on what one would expect from the technology). Then this assignment is used to derive the wage function w(g). Finally, the ten- tative assignment together with w(g) are checked to see whether they satisfy the second order conditions and whether any other assignment could arise.

In the model developed here, the ten- tative assignment is that more skilled workers will be employed at jobs with larger machines. With this assumption, the top n jobs will go to the top n work- ers. The n-th worker, in order of decreas- ing skill g, will be employed at the n-th machine, in order of decreasing machine size. The number of workers with skill greater than or equal to some level g is 1 -G(g). Similarly, the number of jobs with machine size greater than or equal to k is 1 -K(k). Setting these two amounts equal yields a relationship k(g) which describes the machine size for the job assigned to a worker of skill g under the tentative assumption. Suppose, for example, that go is such that thirty per- cent of workers have skill levels greater than go. Then thirty percent of jobs will have machine sizes greater than

k(g0). With the assignment determined, it is now possible to use (10) to find the slope of the wage differential wl(g). Suppose we are interested in the slope of the wage differential at skill level go. From the ten- tative assignment, k(go) is the machine size for the owner who chooses to hire the worker with skill level go. From (lo), the slope w '(go) equals the partial deriva-

tive of production with respect to skill, calculated at k* = k(go).21 Formally,

This expression corresponds to the limits for wage differences in the optimal as- signment problem in (8). With continu- ous distributions and only one character- istic for workers, it shows very simply that the wage differential for a worker with skill level go depends on the assign- ment of workers to jobs. One needs to know the machine size k(go) assigned to that worker in order to calculate the wage differential from (11).

One feature of this model is that ma- chine rents are determined simultane- ously with the wage function w(g). Let r(k) be the rent for a machine of size k. The machine rent is given by the residual obtained by subtracting the wage from production: r(k) = f(k,g) -w(g). This factor price is treated as a rent instead of profits as it could be determined in a manner exactly symmetric to the wage function.

The validity of the tentative assump- tion can now be checked. The employer's second order condition for profit (or rent) maximization is that profits should be a concave function of the skill level g, i.e.,

for k* = k(g), where wV(g) = #wldg2. It can be shown that this condition holds if the mixed partial derivative a2f (g, k)l

21 It is important to realize that the partial deriva- tive on the right-hand side in (10) is taken treating k* as a constant. The value of the partial derivative is then found by substituting k(go) for k*. It would be incorrect to substitute k(g) for k* and then take the derivative since the condition (10) is derived for an employer with a fixed machine size k*.

Journal of Economic Literature, Vol. XXXZ Uune 1993)

agak is positive.22 In turn, a positive mixed partial derivative arises when the effect of skill on production is greater at larger machines, i.e., aflag is an increas- ing function of k.

The expression in (11)yields the slope of the wage function w(g).One must inte- grate with respect to g to obtain the wage function itself. The resulting expression includes a constant of integration, an ar- bitrary parameter that determines the absolute level of all wages. The labor market process in which employers choose workers determines only relative wages (i.e., the wages of one worker in relation to wages of another worker) and not their absolute level. For example, in this model all wages could be shifted up by one dollar and the first order condi- tion in (10) would continue to be satis- fied. Because of the fixed proportions technology, in which one worker can only be used in combination with one machine, the marginal products of workers and machines are not defined. The share of output between workers and employ- ers must therefore be determined by other phenomena.

In the model developed here, reserve prices of labor and capital determine ab-

22 From (ll),

so rearranging yields:

The right side of this expression should be positive for the employer's second order condition for profit maximization to be satisfied. If a2f(g,k)lagak is posi- tive for all workers and machines, then the tentative assignment with dkldg > 0 satisfies the employer's second order condition. Also, the second order condi- tion would not hold with the reverse assignment, in which dkldg < 0.

solute levels of wages and rents.23 The reserve price of labor, p,, is the mini- mum amount that workers must receive in order to be willing to work. If wages are below p,, workers choose to remain idle or engage in some other activity rather than work. Similarly, owners of machines must receive p, or else they will withhold their machines from pro- duction.

As with Ricardo's differential rents, the absolute levels of the wage and rent func- tions are determined by the conditions that hold for the last or marginal match. As one moves down the list of workers in order of decreasing skill, the machine size assigned to that worker in equilib- rium declines, along with the level of production from the match, f(g,k).In one possible outcome, the level of production declines to the sum of the reserve wage and reserve rent, p, + p,, while there are still workers with lower skill levels and machines of smaller size. Suppose the skill level when this occurs is gm and the corresponding machine size is k, = k(gm).If the wage w(gm)were greater than p,, unemployed workers would bid the wage down to p,. If the wage were less than p,, then the rent r(k,) would be greater than p,. Employers with idle machines would offer higher wages and accept lower rents until the wage again equaled p,. The outcome of this adjust- ment process is that w (g,) = p, and r(km) = p,. These conditions are sufficient to determine the absolute levels of the wage and rent functions.24 The conditions

Reserve prices of labor could be represented in the linear programming optimal assignment prob- lem by the presence of extra "null machines" for which output equals labor's reserve price. A surplus of such machines would force their rents to zero. Machine reserve prices could be represented in a similar manner.

24 TWO other possible cases could arise. First, sup- pose that there are more workers than machines, and that production f(g,k) is sufficient so that all ma- chines can be used (i. e., f(g,k) is always greater than the sum of the reserve prices, p, + p,). Suppose the smallest machine size is k, and the corresponding

guarantee that output equals the sum of the wage and machine rent for the last or marginal match. It can then be shown that wages and rents exhaust the product for the nonmarginal, more productive matches.25

By making assumptions regarding the functional forms for production and the distribution of workers and machine sizes, it is possible to draw specific con- clusions regarding the shape of the wage function and earnings inequality. For ex- ample, suppose f (g,k) takes the Cobb-

worker's skill level is g,. Then unemployed workers bid the lowest wage rate down to p,, so w(g,) = p,. The owner of machine size k, gets the residual, so dk,) =f(g,, k,) -P,. These two conditions deter- mine the absolute levels of w(g) and dk). In the other case, there are more machines than workers. At the last match, the machine owner gets p, while the worker gets the residual.

"For any level of skill go, the total differential of output is:

af dk

ago dg Using (11)and the analogous expression for r(k),

df = -dg0 + -dg0 [z][k=k,)].

so that the change in output equals the change in factor payments. With output equal to factor pay- ments for the marginal matches, output will also be exhausted for matches involving higher values of go. Some other technical questions related to the differ- ential rents model are whether the equilibrium ex- ists, is unique and is efficient, in the sense of maxi- mizing production net of reserve prices of labor and capital. Existence and uniqueness are established by construction: given the production function and dis- tributions of workers and machines, the equilibrium is actually found, including the assignment k(g)and the wage function w(g).The efficiency of the assign- ment can be established from the assumption that production from a match is an increasing function of skill and machine size and that the mixed partial derivative is positive. The proof proceeds by suppos- ing that another assignment maximizes production net of reserve prices. Then because the assignment is different, two workers and their machines can be found such that the more skilled worker is using the smaller machine. By switching machines for these two workers, even greater production can be ob- tained, contradicting the assumption that the alterna- tive assignment maximizes production net of reserve prices. The contradiction proves that the hierarchical assignment is efficient.

Douglas form gakP, and suppose skills and machine sizes are lognormally dis- tributed with variances of logarithms ui and ui, respectively. Then using (11)the wage function w(g) takes the form

where A is a constant and C, is the con- stant of integration obtained when wl(g) is integrated. This function will be con- cave, linear, or convex depending on whether (amg + puk)/ug is greater than, equal to or less than one. If a + p = 1 and if uk > ug (i.e., machine sizes are more unequally distributed than skills), then w(g) will be convex.26 The quantity

-C, will be lognormally distributed with variance of logarithms aUg f puk, a linear combination of the inequalities in skill and machine size distributions. If both workers and machines are unem- ployed, then (from the condition that la- bor and capital factor prices equal their reserve prices in the marginal match), C, can be calculated as

This amount could be positive or nega- tive. A larger value of C, reduces earn- ings inequality. In this way, the separate influences of the production function, the distributions of skills and machine sizes, and the reserve prices can be found. As mentioned in the introduction, these in- fluences will be unapparent. Only the wage function w(g) will be observed, so that wages will appear to depend only on g.

The differential rents model and related hierarchical models explain why the

26 The wage function derived by Tinbergen (1951, 1956, 1970) displays the same flexibility. The slope of the wage function and whether it is concave or convex can be related to the distributions of worker and job characteristics. John Pettengill (1980) uses a similar methodology to examine the effects of union- ization on earnings inequality.

distribution of earnings differs in shape from the distribution of abilities. A. C. Pigou (1952, p. 650) raises the question (now known as Pigou's paradox) of why the distribution of earnings is positively skewed when abilities are symmetrically distributed. This paradox presumes that earnings ought to be proportional to some single-dimensional measure of abil- ities. It can be resolved by recognizing that workers are engaged in many differ- ent activities: there is no single measure of ability that determines a worker's earn- ings. In the context of the differential rents model, one obtains a different dis- tribution of abilities among workers de- pending on which machine is used. If every worker used the same type of ma- chine, the distribution of earnings would take the same shape as the distribution of abilities, defined by worker outputs.27 With unequal machine sizes, however, workers with greater skill levels are as- signed to larger machines. This boosts their earnings above what they would be if everyone used the same machine. Be- cause of the positive mixed partial deriva- tive d2fldgdk, differentials for skill (i.e., dfldg) will be greater for more skilled workers. The distribution of earnings will not resemble the distribution of outputs at any one machine and instead will be positively skewed relative to any such distribution.

The model presented here incorporates some important elements of the economy. These include an explicit as- signment problem and the role of cooper- ating factors, in this case capital as repre- sented by separate machines. The main point is the expression for the wage dif- ferential in (11).But there are also sev-

"Each worker's earnings would be the value of output minus the machine rent, which would be the same for all workers. The distribution of earnings would be the same as the distribution ofoutput values but shifted to the left by the amount of the machine rent. These conclusions can be derived directly by setting uk = 0 in (13)and (14).

eral shortcomings of the approach. It is essentially a short-run model, taking the distributions of workers and jobs as given.28 It assumes a very restrictive pro- duction technology in which only one worker can be combined with a machine. More generally, one would expect a pro- duction function in which there could be variable numbers of workers combined with the capital. The model relies heavily on calculus and on continuity assump- tions that allow one to work with deriva- tives. Most importantly, because of the absence of any stochastic element, the model predicts an exact correspondence between skill and machine size, which is inconsistent with observed assignments.

C. Discrete Jobs or Occupations: Roy's Sectoral Model

The model developed in this section (commonly called Roy's model) differs from the previous section's model in that workers choose among only a few jobs or occupations instead of a continuum of jobs. Rather than each job being filled by only one worker, a subset of all work- ers can be found in a given job. Because of worker self-selection of jobs, the distri- bution of workers in a given job will differ systematically from the labor force as a whole.29 These effects of worker choice

28 The assumption of rigid distributions of workers and jobs is the same for both mismatch arguments and differential rents models. However, unlike mis- match arguments, there are no excess supplies or demands in differential rents models, as these are eliminated by the wage differentials.

29 In an earlier article, Roy (1950)attempts to show that abilities combined multiplicatively, so that out- puts and hence incomes are lognormally distributed. He uses data from manual occupations such as choco- late packing to test this hypothesis. The 1951 article arises from the realization that workers ". . . engaged in a particular occupation tend to be selected in a purposive manner from the working population as a whole" (1951, p. 135). Then the workers found in a specific occupation would not have the same distribution of abilities as the population as a whole would have in that occupation.

or self-selection make Roy's model par- ticularly interesting to econometricians dealing with self-selectivity issues.

In the basic two-sector model devel- oped by Roy, members of a simple econ- omy must choose between catching rab- bits or fishing for trout. A worker's income in a sector is proportional to the number of rabbits or trout caught. Unlike the model developed in the previous sec- tion, there is no restriction on the num- ber of workers in a job or occupation. Workers can move from one job to an- other depending on the price of trout in terms of rabbits. The model is very rich in yielding a wide range of outcomes depending upon the relation between abilities in the two sectors. The predic- tion of observable distributions renders the model more applicable to empirical work than the previous section's model.

Roy's model can be represented as a special case of the linear programming optimal assignment problem discussed in Section 1II.A. With two sectors, the out- put value entries ay for worker i would be the same for all jobs in a given sector. In order for all output to go to the worker, with no subtraction of rent for the machine or job, it is necessary for the opportunity cost of filling a given job to be zero. This can be accomplished by assuming that there are more jobs in each sector than there are workers. ("Null workers," with zero outputs in each sec- tor, can be added so that the number of workers equals the number of jobs.) Then the rent for each job will be zero and the wage rate for a worker choosing a job would equal the value of output. The

matrix of output ('v) have rank

2. This demonstrates an important difference between Roy's and the differential rents models: with no scarcity of jobs in either sector, taking a job in either sector entails no opportunity cost so that assignment is based on comparative advantage.

In Roy's model, there is no simple ex- pression for the wage differential as in (ll), showing the relevance of the assign- ment. However, as in that model, a worker's income is not a simple function of a skill measure but depends on which job the worker performs (i. e., rabbits or trout). In both models, aggregate output depends on the assignment of workers to jobs, and the assignment problem is solved by the income-maximizing deci- sions of agents. A shift in demand (e.g., an increase in the price of trout in terms of rabbits) can i~~fluence

the distribution of earnings by increasing the relative earnings of trout fishers and leading some workers to move from catching rabbits to fishing for trout.

This section presents a graphical treat- ment of Roy's model.30 Suppose there are two sectors, rabbits and trout. Sup- pose that if the entire population chose one of the sectors, the distribution of out- puts (in terms of rabbits or trout caught) would be lognormal. Let a: and a; be the variances of logarithms of outputs in the rabbit and trout sectors, respectively, and let p be the correlation between the logarithms of a worker's outputs in the two sectors. Without loss of generality, suppose that amounts are more unequally distributed in the second sector (trout), so that a: <a:. This corresponds to Roy's assumption that trout are more difficult to catch than rabbits. Three basic cases arise depending on the correlation between the two sectors.

i. Case of a,lu2 5 p 5 u2/u1.31

30 Heckman and Sedlacek (1985, 1990), James

Heckman and Bo Honore (lggo), and G, S, Maddala (1977, 1983) develop statistical versions of Roy's model. According to comments made to this author by Michael Farrell, Roy had developed a complete statistical model as the basis for his conclusions but did not include it in the 1951 paper. This is consistent with the detailed conclusions he reaches. 31 Cases can also be described in terms of the cova,.iance, given by u12 = pulu2, Case i, u! 5 u125 u?j, while in Case ii a125 0 and in Case iii 0 5u125a: 5a;.

Log Trout Proportion Hunting Rabbits

I.

Log Rabbits

Figure 1. Contour Plot of Density of Sector Performances

This is the standard comparative ad- vantage case. In this case, outputs are highly correlated, so that workers with higher levels of output in one occupation are also very likely to have higher levels of output in the other sector.

The effect of selection on the distribu- tion of outputs in the two sectors and on the distribution of income can be seen in Figures 1 through 5. Figure 1 shows a contour plot of the distribution of worker performances in the two sectors. In this figure, it is assumed that the vari- ance of logarithms of rabbits caught by the population, a:, is 1, while the vari- ance of logarithms of trout, a;, is 4. The means of the logarithms of rabbits and trout are both 4.32 Also, the correlation between sector outputs is assumed to be

0.75. The points on a given contour line in Figure 1 correspond to combinations of rabbits and trout such that the density of workers is the same.

Assume that the price of a rabbit is

32 Although the logarithmic means are equal to 4 for both trout and rabbit skills, the means themselves are unequal. The mean of a lognormal distribution is given by ek+uu2, where p and u2are the mean and variance of logarithms (John Aitchison and

J. A. C. Brown 1957). The means for rabbit and trout skills are thus 90 and 403.

Log Rabbits

Figure 2. Proportion of Workers Choosing to Hunt Rabbits

1.2 while the price of a trout is 1, so that one rabbit is worth 1.2 trout. A worker chooses to hunt rabbits whenever 1.2times his or her rabbit catch is greater than one times his or her trout catch. The 45-degree line sloping upward from the logarithm of the price ratio, 0.182, on the vertical axis shows all combina- tions of rabbits and trout that yield the same income. Any worker with a combi- nation of rabbits and trout below this line would make greater income hunting rab- bits. Any worker with a combination above this line would choose to fish for trout.

Figures 2 through 4 show how this as- signment mechanism affects the distribu- tions of workers observed hunting rabbits and fishing for trout. Figure 2 shows, from among workers who can catch a par- ticular number of rabbits, the proportion that choose to hunt rabbits. In this case, the proportion hunting rabbits declines as the number of rabbits increases. The mean logarithm of rabbits caught among those choosing to hunt rabbits will there- fore be less than the population mean of 4. Workers who can catch many rabbits are likely to have a comparative advan- tage at trout fishing, and therefore choose that A better rabbit hunter would get only a small income advantage from his or her superior catch because the in- equality in number of rabbits caught is relatively small. The number of trout

Density of
Workers

Log Rabbits

Figure 3. Rabbit Hunting Abilities Among All
Workers and Among Rabbit Hunters

caught are more unequally distributed, so an above average performance in that occupation will yield a much higher in- come.

In Figure 3, the upper curve shows the distribution of rabbits caught by all workers (the vertical axis is the density of workers who catch a given number of rabbits). This upper curve is a lognormal distribution, so that the logarithm of rab- bits is normally distributed with mean 4 and variance 1. The lower curve shows the density of workers by rabbits caught for those choosing the rabbit sector (this curve is not normalized; the area under the curve is 0.55, equal to the proportion of all workers choosing the rabbit sector).33 At higher numbers of rabbits, workers are less likely to choose the rab- bit sector because their income may be larger in the trout sector. Workers catch-

'' This density is obtained by multiplying (for each number of rabbits caught) the proportion choosing the rabbit sector, given in Figure 2, times the density of all workers who can catch a given number of rab- bits, given by the upper curve in Figure 3. Let n(xl,xz;al,ap,p) be the joint probability density func- tion for the bivariate lognormal distribution, where xl is the number of rabbits, x2 is the number of trout, and a,,u2and p are the standard deviations for rab- bits and trout and the correlation, respectively. Those who choose rabbits are workers for whom q 5 1.2x1, so that at xl the height of the lower curve in Figure 3 is given by

Log Trout

-.

Figure 4. Trout Fishing Abilities Among All
Workers and Among Trout Fishers

ing low numbers of rabbits, however, are likely to choose the rabbit sector.

The distribution of income in the econ- omy can be found by combining the lower curves in Figures 3 and 4, setting

1.2 trout equal to one rabbit and express- ing income in rabbits. This is done in Figure 5. The upper curve is the distri- bution of incomes in the economy, ob- tained by summing the densities of work- ers by income for the two sectors. The lower curve on the left arises from the rabbit sector, while the lower curve on the right arises from the trout sector. This figure shows that the upper tail of the income distribution comes from workers in the trout sector, while the lower tail comes from workers in the rab- bit sector. There is, however, substantial overlap in incomes from the two sectors: the assignment of workers to sectors is not entirely hierarchical, in the sense that some workers in the trout sector are

Figure 4 makes the same comparison with respect to the number of trout caught. The upper curve shows the dis- tribution of trout caught by all workers. The logarithms of trout are normally dis- tributed with mean 4 and variance 4. The lower curve shows the density of workers by trout caught for workers choosing the trout sector. Nearly all those with high trout catches choose the trout sector, while those with low trout catches select the rabbit sector.

Density of Density of
Workers Workers

workers tend to have a comparative ad- vantage.

ii. Case of p < 0.

This case arises when performances in the two sectors are negatively corre- lated, i.e., the better rabbit hunters tend to be the worse trout fishers. Those with worse performances in an occupation are more likely to choose the other occupa- tion to earn their living. In this case, the assignment is roughly described by abso- lute advantage, which arises when work- ers in an occupation are better at that occupation than workers choosing the other occupation. Workers in an occupa- tion tend to have higher outputs in that occupation than workers choosing the other occupation, although there will be exceptions. Worker choices lead to a sim- ple assignment pattern: each occupation tends to be filled with the best workers in that occupation. The workers with the highest incomes will tend to be those with extreme performances, good and bad, rather than those with average or above average performances in both oc- cupations.

Figure 6, corresponding to Figure 5, shows the aggregate and sectoral distri- butions of income in the case where the correlation between sector performances is -0.5, everything else the same.

Density of Workers Income

Figure 7. Aggregate and Sectoral Distributions of Income, Case iii with 0 < plz < ul/uz

The mean logarithm of income is 4.97 and the variance of logarithms is 1.31. Compared to case i, there are virtually no workers with logarithms of income be- low 2. As in Figure 5, though, the upper tail is dominated by workers in the high variance trout sector.

iii. Case of 0 5 p < a,/a,

This intermediate case arises when outputs in the two occupations are posi- tively correlated but not as much as in the standard comparative advantage case in i above. Workers with better perfor- mances in the first sector are more likely to choose that sector, even though they also tend to be slightly better in the sec- ond sector. The importance of this case is that a positive correlation between sec- tor performances does not necessarily generate the standard comparative ad- vantage case.

Figure 7, corresponding to Figures 5 and 6, shows the aggregate and sectoral distributions of income for this case, as- suming p = 0.25. In this case, the mean logarithm of income is 4.71 and the vari- ance of logarithms is 1.76.

Comparison of Figures 5 through 7 for the three cases reveals a number of com- mon features. In all three cases, the up- per tail is dominated by workers in the high variance sector, trout. This effect stands out clearly because the variances in the two sectors were arbitrarily chosen to be so far apart. The aggregate distribu- tion of income takes the same general shape in all three cases, with the largest inequality (as measured by the variance of logarithms) in the case where p is the highest. The lower tail is dominated by workers in the low variance rabbit sector. However, in the case with p < 0, some low income workers are also in the trout sector. Despite the assumption that a trout is worth less than a rabbit and that mean logarithms of performances are the same, average incomes are higher in the trout sector. The higher average income arises because of the high incomes going to workers in the upper tail of the trout ability distribution. The unequal variance between the sectors appears to play at least as strong a role as correlation be- tween sector performances in shaping the distribution of income.

The listing of cases in this section shows that a variety of outcomes is possi- ble depending on the correlation p. In particular, the standard comparative ad- vantage case in i is not inevitable and is a special case of Roy's model.

Roy's model can also be used to illus- trate how demand can influence the dis- tribution of earnings and the division of workers between sectors. Table 1 shows the effects of changing the price of trout in terms of rabbits in case i. As the price of trout goes up, the proportion of work- ers selecting the rabbit sector declines, meal1 earnings increases and the variance of logs increases. Workers originally in the trout sector find their earnings boosted by the price increase, relative to workers in the rabbit sector, who have lower earnings on average. The effects can be seen in Figure 8, showing the distribution of earnings for two prices of trout, 0.5 and 1. The lower tail is unaf- fected because it arises from workers who stay in the rabbit sector. However, the upper tail is shifted to the right as work-

TABLE I EFFECTSOF TROUTPRICE CHANGES ON SECTORS

Price of Trout Proportion
in Terms of Hunting Mean
Rabbits Rabbits Earnings

ers from the trout sector, who account for the upper tail, experience an increase in earnings from the higher trout price. In this case, the increase in the price of trout raises earnings inequality as mea- sured by the variance of logarithms of earnings.

As the price of trout doubles from 0.5 to 1.0, average earnings of workers in the trout sector will not double. The rea- son sector earnings are not proportional to prices is that a nonrandom selection of workers moves from the rabbit sector to the trout sector in response to a trout price increase. This is demonstrated for Case i in Figures 9 and 10, which show worker movements in response to an in- crease in the price of trout from 0.83 rab- bits (corresponding to 1.2 trout per rab- bit) to 1.0 rabbits. This price change would be represented in Figure 1 by a shift in the "Equal Incomes" line down- ward from a vertical intercept of log (1.21 1)to log (Ill), through the origin. Figure

Density of workerS Pnce of trout equals  
0.35 ' one-half rabblt  
0.3..    
0.25    
0.2..    
0.15..    
0.1..    
0.05..    
    Income
  2 4 6 8 1 0  

Figure 8. Shift in Earnings Distribution from Change in Price of Trout

Variance Mean Mean
of Rabbit Trout
Logarithms Skill Skill

9 shows the proportion ofworkers leaving the rabbit sector as a function of the num- ber of rabbits they can catch. As shown, workers with greater rabbit skills are more likely to leave that sector, thereby lowering the average skill level in the rabbit sector. Figure 10 shows how the movers compare with the workers already in the trout sector. The ratio of the number of entrants to current work- ers in the trout sector is greater at lower numbers of trout caught. After the price change, average skill levels in the trout sector will be lower. The average wage in the trout sector increases less than the price of trout.

The response of average skill levels to changes in rabbit or trout prices demon- strates an important feature of Roy's model, the aggregation bias that arises because of movements between sectors (Heckman and Sedlacek 1985, pp. 1107- 10). Changes in average wage rates do not accurately reflect changes in the wage

Proportion

Leaving

0.3'

0.25..

0.2 ..

: Log Income 246810 Figure 9. Proportions of Workers Leaving Rabbit Sector

Entrants' Ratio

Figure 10. Ratios of New Entrants to Current
Trout Fishers

rates for workers with given skill levels.

Heckman and Sedlacek (1985) estimate an empirical version of Roy's model using Current Population Survey data from 1968 to 1981. They assume that workers choose between two sectors, manufactur- ing and nonmanufacturing, justifying this division because manufacturing has been the focus of so much previous empirical work. This model is rejected by two test criteria that they propose. If wages are the product of tasks and task prices, and if the relation between tasks and skills does not change over time, the coeffi- cients of skills in estimates of the loga- rithms of wages in successive cross-sec- tions should be the same. This is referred to as the proportionality hypothesis and fails to hold for the estimated model. The second test is whether the residuals in the loglinear wage equation are distrib- uted normally as assumed, and this is rejected using a chi-square goodness-of- fit test.

In response, Heckman and Sedlacek consider a multi-sector generalization but reject it because of the expense of estimating such models. Instead they ex- tend Roy's model in four other ways. First, individuals are assumed to maxi- mize utility instead of money incomes. Second, earnings are decomposed into hourly wage rates and hours of work that are freely chosen. Third, Heckman and Sedlacek deveIop a general non-normal model for the distribution of residuals which has Roy's lognormal assumption as a special case. Fourth, individuals are assumed to have a nonmarket or house- hold production sector as an alternative to market

With the assumption of utility maxi- mization, preferences influence assign- ment and earnings in the extended model by leading workers to choose sectors that do not necessarily maximize earnings. In this way, the distribution of utilities, which generates sectoral choice, can dif- fer from the distribution of task perfor- mances, which generates the earnings distribution.

One cautionary note concerning pref- erences in Heckman and Sedlacek's model is that their presence may be nec- essary to mimic sector-specific training. In the model, workers at each point in time are assumed to choose between the manufacturing, nonmanufacturing and nonmarket sectors. It seems likely that workers who have experience and train- ing in a sector would have their pro- ductivity raised in that sector relative to what it would be in another sector. The training would be specific to the sector, just as specific training raises productiv- ity only in the firm that gives it. An expe- rienced worker in a sector could expect to get less in the other sector than indi- cated by the estimated task-skills rela- tionship. The worker would then be much less likely to switch sectors than on the basis of predicted earnings alone. A suitable distribution of preferences would correct for the absence of assumed sector specific training.

Preferences could also be present in

34 In a later article, Heckman and Sedlacek (1990) try to determine which extensions to Roy's model are most important in improving its goodness of fit. They find that the existence of a nonmarket sector is more important than allowing for departures from lognormality.

lieu of search by workers. With sectoral choice made through search (and with many sectors), workers do not usually end up in the sector that absolutely maxi- mizes their earnings. In the absence of assumptions that workers find sectors through search, the distribution of pref- erences could substitute for the random wage outcomes of search.

Heckman and Sedlacek find that edu- cation has twice as strong an effect in manufacturing as in nonmanufacturing. Wages grow much more rapidly with work experience in manufacturing than in nonmanufacturing. These results show that it would be incorrect to apply one earnings function to all workers indepen- dent of sector.35 The model leaves unex- plained the shapes of the wage functions in the two sectors. What aspect of the manufacturing sector causes its wage function to differ from the wage function in the nonmanufacturing sector? The wage functions in each sector could themselves be hedonic wage functions, arising from assignment problems within each sector. It seems unlikely that all workers in the manufacturing sector are perfect substitutes for each other, as re- quired for the efficiency units assump- tion. Also, it is not clear that workers choose manufacturing versus nonmanu- facturing sectors instead of occupations.

The variance of the error term in non- manufacturing is greater than in manu- facturing. Heckman and Sedlacek find this to be consistent with greater hetero- geneity in the industries classified as non- manufacturing. Because preferences and not earnings determine sectoral choice, the results of the extended model do not

35 Heckman and Sedlacek (1990, p. S353) explicitly test whether the observed wage distribution could be explained by a model with only a single production market, so that workers only decide whether to work or not and get the same wage no matter where they work. Such a model is rejected.

conform exactly to one of the cases dis- cussed earlier in this section. In particu- lar, the correlation p is not identified in this model because of the presence of unobservables. However, education and experience positively affect both manufacturing and nonmanufacturing tasks, so performances in the two market sectors would be positively correlated in the ab- sence of other skill related variables.

Heckman and Sedlacek estimate the effect of self-selection on inequality in Roy's model by comparing the observed earnings inequality with the level that would arise if workers chose sectors ran- domly. They find that self-selection de- creases the variance of logarithms of wages within each sector, moves the mean wages in the two sectors closer to- gether, and reduces the variance of loga- rithms of wages in the economy by 11.6 percent.36

In a later article, Heckman and Sedla- cek (1990) examine extensions to the Roy model in more detail. Heckman and Honore (1990) analyze statistical proper- ties and the empirical content of Roy's model. Although the statistical analysis used in all of these models appears to be very difficult, the work establishes Roy's model and its extensions as a practi- cal way to apply assignment models em- pirically.

IV. Comparisons and Extensions

A. Choices

The three models discussed in the previous section share a number of fea- tures in common. First, of course, is the existence of multiple sectors. Workers therefore face a choice of sectors, or else employers in each sector face a choice of workers. From the point of view of

James Heckman and Bo Honore (1990) prove that in the Roy model, self-selection reduces inequal- ity compared to random assignment to sectors.

the economy as a whole, the existence of multiple sectors entails an assignment problem.

In the simplest case, where workers are described by a single characteristic, multiple sectors arise because the output in some jobs is more sensitive to that characteristic than others. In the differ- ential rents model, the effect of worker skill on output, aflag, is an increasing function of the size of the machine. As in other scale of operations models, an efficient assignment requires that more skilled workers be assigned to jobs with more resources, either capital or respon- sibility. Alternatively, jobs may differ by a single parameter of difficulty, which measures the sensitivity of the job to the worker skill. This sensitivity to worker skill is also present in Roy's model. If performances in the two sectors were perfectly correlated (so that one could predict performance in one sector from the performance in the other sector), workers would still face a choice between rabbits and trout. With unequal vari- ances of performances, better rabbit catchers would choose to fish for trout as that sector is more sensitive to their skill levels.

Even if jobs did not differ in their sen- sitivity to worker skills, multiple sectors would arise because of the great variety of tasks performed in the production of goods and services and the diversity of human performances at those tasks. Mul- tiple sectors provide workers with choices. Both the existence of choice and the features of those choices affect the distribution of earnings. The discussion of Roy's model in Section 1II.C. empha- sizes the features of those choices, such as the unequal variances and correlation between the two occupational perfor- mances. An alternative way to view Roy's model is by comparing its outcome with the outcome of a single sector model, in which the distribution of outputs would be identical in shape to the distri- bution of earnings. Suppose the second sector has the same distribution of out- puts as the first sector (equal variances and zero correlation). Then the addition of a second sector provides workers with a choice and alters the distribution of earnings from its shape in the single sec- tor case. It is clear that the addition of more sectors would alter the distribution further. But the effect of additional choice is not apparent in Roy's model as in the alternative cases considered in Section III.C., the number of sectors stays the same at two. Benoit Mandelbrot (1962) and Hendrik S. Houthakker (1974) develop models that partly explain how choice among many sectors, in the ab- sence of unequal variances, affects the distribution of earnings.

Mandelbrot seeks to show how several occupations can have different exponents for Pareto upper tails. As a simplified ver- sion of Mandelbrot's model, suppose that workers have vectors of aptitudes given by (yl, . . , yn). Assume each aptitude follows a Pareto distribution, with proba- bility (ylyo)-" that income is greater than or equal to y, where yo is the lowest apti- tude and a is the same for all aptitudes.37 If the sector's offers are proportional to only a single aptitude, then the offers will follow the Pareto distribution. Accepted offers will differ from the Pareto distribution because lower offers are likely to be rejected. But the higher of- fers are more likely to be accepted, so in the upper tail accepted offers will again resemble the Pareto distribution. Now suppose a sector weights two aptitudes heavily. To be specific, suppose a firm offers a wage equal to A0 whenever the lower of the two aptitudes is 0, where

37 Instead of the Pareto distribution, Mandelbrot adopts a somewhat weaker assumption regarding ap- titudes. They are assumed to follow the weak law of Pareto, so they asymptotically resemble the Pareto distribution in the upper tail.

A is some constant.38 Multiplying the two randomly with cumulative density func- cumulative distribution functions to-tion F (yl, y2, . . . , yn). An individual gether, the likelihood that an offer equal selects the occupation that maximizes in- to or exceeding A0 is made is (~/y,,)-~~. come, z. If the prices of aptitudes are

The distribution of wages for this occupa- tion will asymptotically resemble a Pa- reto distribution but with parameter 2a instead of a. Similarly, a sector for which the offer depends on c aptitudes being simultaneously large will have a wage dis- tribution which in the upper tail will re- semble a Pareto distribution with param- eter ca. In this way different sectors can have wage distributions which are asymptotically Paretian but with different parameters.

In this economy, the very high offers go to workers in sectors that weight a single aptitude highly. Sectors requiring two or more aptitudes do not make many high offers. The workers getting the high- est wages are those who are extremely good at a single skill that is crucial to a sector rather than workers who have a high average of aptitudes. The distribu- tion of offers with the lowest Pareto coef- ficient (corresponding to the greatest in- equality) will come to dominate the upper tail of the entire earnings distribu- tion, which will then resemble a Pareto distribution with coefficient a.

Houthakker's model is similar to Man- delbrot's in that a worker has a vector of n occupational aptitudes and chooses the occupation that yields the highest in- come. Houthakker shows how the distri- bution of earnings can be derived from the joint distribution of aptitudes. The individual has a vector of occupational aptitudes (yl, yz, . . . , y,) which varies

38 This functional form for the offer would arise if the worker's productivity is a fixed-proportions func- tion of the worker's aptitudes. For example, if a job required a worker to perform two tasks simultane- ously, the productivity would be proportional to the minimum of the two aptitudes. Mandelbrot derives his results under much more general conditions and assumes that offers are a nonhomogeneous form of the independent aptitude factors (1962, p. 61).

each unity, the cumulative density func- tion of incomes will be given by F (z,z, . . . , z), the probability that each aptitude will be less than or equal to z. The resulting distribution of income will differ in form from the distributions of individual aptitudes, and Houthakker illustrates this general result with exam- ples using the bivariate exponential and bivariate Pareto distributions.

Figure 5. Aggregate and Sectoral Distributions of Income

less productive than some workers in the rabbit sector.

In this case, workers are assigned to sectors on the basis of comparative ad- vantage. Workers who do well in a sector (i.e., rabbits) do not necessarily select that sector; instead they may select the other sector because they have a compar- ative advantage in it. Workers may select a sector (rabbits) even though they do badly in it because they have a compara- tive advantage in that sector. There is an implicit ranking of the sectors in that trout fishing is the sector at which better

; Income

Figure 6. Aggregate and Sectoral Distributions of Income, Case ii with p,, < 0

The aggregate income distribution tends to be more skewed to the right than a lognormal distribution as the right tail, from the trout sector, is more elon- gated than the left tail. The mean loga- rithm of income is 4.48, greater than the mean would be if there were no choice and all workers had to stay in one sector. Overall, income inequality, as measured by the variance of logarithms, is 2.03, greater than the population inequality in rabbit catches, a: = 1,but less than the population inequality in trout catches, oi = 4. The distribution of income resem- bles neither the distribution of abilities in catching rabbits nor the distribution of abilities in catching trout. As illus- trated, it will tend to be more highly skewed than either of the ability distribu- tions.

B. Wage Diflerentials

In a single competitive labor market, the wage rate is determined by the famil- iar condition that quantity supplied equals quantity demanded. The supply and demand curves for a single labor market, however, do not show their de- pendence on substitution of labor or jobs from other labor markets. With many closely related labor markets, as in as- signment models, the demand curve for a particular type of labor is determined by the cost of hiring alternative types of labor. Wage determination in assignment models generally takes the form of condi- tions imposed on wage differentials. These conditions are expressed differently in the three assignment models that have been considered but are all essen- tially generated by trade-offs in produc- tion that arise from varying skill levels. The conditions are shown in (3) for the comparative advantage case and (4) for the scale of operations case. The role of trade-offs in production is clearest in the differential rents model. In (1l), the wage differential equals the effect of an increase in the skill level g on output, hold- ing the size of machine constant at the level corresponding to the equilibrium assignment. In the optimal assignment problem, the limits in (8)are the differ- ences in the two workers' outputs at ma- chines d and j.

The role of trade-offs in production is less clear in Roy's two-sector model as there are no wages associated with work- ers. However, by defining workers' wages as equal to their incomes, it is pos- sible to derive an analogous result.3g

An important conclusion arising from the determination of wage differentials in all three assignment models is that prices or values of worker characteristics are not uniform across the economy. Finis Welch (1969) raises the issue of uni- form skill prices and develops a model in which production costs depend on ag- gregate combinations of skills, so that skill prices can be equalized across sec- tors. Rosen (1983) and Heckman and Scheinkman (1987) describe conditions under which uniform skill pricing will not arise.

Assignment models provide a direct explanation of unequal skill prices in an economy. For example, in Roy's model, suppose the two skills are rabbit hunting and trout fishing. If a worker chooses to fish for trout, the price of the rabbit skill is zero and the price of the trout skill is given by the trout price. A worker catch- ing rabbits similarly receives a zero re- ward for any trout fishing skill. In the differential rents model, the value of an increment in the worker skill g depends

39 Let pj be the price of output in sector j and let aV be the output of worker i in sector j. Let wu = p,a. the amount that worker i would get if he or she%hose sector j. If worker 1 chooses sector 1, then wll = plall and wlz = pza12Iwll. If worker 2 chooses sector 2, then wzz = pzaz2and wzl = plazl Iwzz Then

Plall -Wll >9>wlz -Pzalz plapl Wp2 -u;p2 w2z pnap2

so that %2 wll 2 %

azl wpp ap2

This is the same relation as in (3) in Section II.B., even though Roy's model uses a different technology and assumes workers choose sectors instead of em- ployers choosing workers.

on the size of the machine assigned to that worker in equilibrium. In the opti- mal assignment problem, the value placed on differences in worker charac- teristics in (8)depends on the characteris- tics of workers' machines. Unequal wage structures in sectors of the economy are therefore a direct outcome of the exis- tence of an assignment problem.

C. Self-selection

In all three of the assignment models considered here, self-selection is the mechanism that is used to bring about the assignment. Workers select a sector or job, and thereby assign themselves to it, when it offers them greater income or utility than any other sector. This se- lection criterion results in a distribution of performances within the sector that differs systematically from the distribu- tion for the population as a whole. These selection corrections have been worked out for the two-sector Roy model but not for the general multisector case. Self-selection is not a necessary feature of as- signment models, however, and is only one of a few mechanisms that may oper- ate in the economy to assign workers to jobs. Self-selection requires that a worker have complete information about potential earnings or utility in each sec- tor. This is reasonable when there are only two sectors. But the assumption of complete information becomes unreason- able when there are many sectors with little guide to the worker as to which one is most suitable. Once one abandons full information and self-selection, the in- formation structure plays a role in deter- mining the feasible assignment mecha- nism, the resulting assignment and wage differences.

Studies of worker behavior suggest that workers engage in search to find jobs. In the standard search model, work- ers need to know only the distribution of wage offers among jobs and not the wage corresponding to each job. A worker selects a reservation wage and ac- cepts the first job offer with a wage that equals or exceeds it. The selection crite- rion under search would appear to be much simpler than under self-selection. A worker observed to be in a sector (or job) gets a wage that exceeds the reserva- tion wage but that does not necessarily exceed the wage in every other sector (or job). Then the income or utility in a given sector needs to satisfy only one bi- lateral comparison, i. e., with the work- er's reservation income or An alternative approach is to assume that in each industry or sector, workers make a choice between that sector and a compos- ite sector consisting of all other sectors. Maddala (1983, pp. 275-78) considers ad- ditional methods that rely on multiple binary-choice rules or on applications of order statistics.

With search present, the process of as- signing workers to jobs itself generates some additional sources of inequality (Sattinger (1985) analyzes the distribu- tional consequences of job search). Workers suffer unequal amounts of un- employment in their search, contributing to inequality in earnings and welfare. Workers with identical characteristics may get unequal wage rates because of the random outcomes of search, further contributing to inequality. Workers can influence their expected wage rates and likelihood of unemployment through their choice of reservation wages, provid- ing an alternative source of inequality. On the other hand, search alters the as- signment in such a way that higher

40The degree of simplification depends on how the reservation wage is determined. If it is exogenous (an unknown function of the worker's characteristics, the same for all workers, plus a random element), then the simplification is straightfonvard. If the reser- vation wage is determined endogenously, as the solu- tion to the worker's optimizing problem, then the econometrics could be as complicated as the self- selection procedures.

skilled workers end up, on average, with fewer resources than in an exact, self- selection assignment, possibly reducing their earnings and overall inequality.

A further consequence of departing from the mechanism of full information self-selection is that there will be a de- mand for information about worker or job characteristics. MacDonald (1980) devel- ops a model of person-specific informa- tion in which there are two types of work- ers and two types of firms. The type of a worker cannot be directly observed, but workers can invest in generating in- formation about their own types. Neither worker type has an absolute advantage at both types of firms, so any information leads to improved worker-firm matches. The value of information in the labor market leads firms to offer higher wages to workers who can provide information about their types. All workers then re- ceive a return to information investment. In later articles, MacDonald (1982a, 1982b) considers how information enters the production function through the as- signment. A continuum of tasks is assigned to workers of two types based on the quality of information, and output rises when the quality of information in- creases.

Models of signaling, filters, and screening (Kenneth Arrow 1973; A. Michael Spence 1973) show that worker in- vestment in information about themselves could simply give them a competitive advantage in the labor mar- ket. Workers get a private return to infor- mational investments that exceeds the social returns. With an assignment prob- lem in the economy, however, informa- tional investment can yield a social return that equals or exceeds the private return, even though it does not change worker productivity (e. g., Arrow's model with two types of jobs; 1973, p. 202). In Mac- Donald's model (1980), there are no ex- ternalities from information about worker types as in signaling or screening models so that the social returns equal private returns. Waldman (1984a) and Joan E. Ricart i Costa (1988) develop models in which the worker's assignment itself acts as a signal to other firms, leading to possi- ble inefficient assignments.

Investment in information generates a pattern of mobility over time as well as effects on the life-cycle earnings profile. Hartog (1981a) develops a two-period model in which wages in the first period are based on signals and in the second period on capabilities. He shows that dis- persion in signal classes increases over time, and more capable individuals expe- rience higher earnings growth.

The structure of information also affects the amount of earnings inequality. Michael Rothschild and Joseph Stiglitz (1982) develop a model in which a work- er's output is maximized when placed in a job level corresponding to the worker's ability. However, the worker's ability de- pends on both observed and unobserved characteristics, so the job placement de- pends on the worker's expected ability level. A given observed characteristic is then related both directly to production and indirectly, through its correlation with unobserved characteristics. Firms may be unable to distinguish direct and indirect effects. When few characteristics are observable, expected ability levels do not vary greatly, and workers are placed in narrowly varying job levels. When many characteristics are observable, ex- pected ability levels vary greatly and workers are placed in widely varying job levels. With more observable character- istics, both the expected wage and the variance of wages are greater.

Tournaments may be regarded as a mechanism of assigning workers to hier- archical levels in a context in which worker abilities are revealed through competition (Edward Lazear and Rosen 1981; Rosen 1986b). As performances de- pend on effort, large prizes are required for workers in the top ranks to maintain incentives to compete.

Labor markets provide another mecha- nism for assigning workers to jobs. Instead of one big labor market, sub-markets arise based on observable characteristics of workers and jobs. With- out an assignment problem (at least spa- tially), the existence of submarkets would serve no economic purpose. Screening, job market signaling, dual labor markets, and occupational segregation may deter- mine assignment through restricted ac- cess to jobs for some workers, resulting in assignments that are less efficient than possible given the constraints of costly and incomplete information (Dickens and Kevin Lang 1985; Insan Tunali 1988;

T. Magnac 1991).

D. Comparative Advantage

Comparative advantage is not neces- sarily present in the three models consid- ered here, and it does not necessarily determine the assignment. In the linear programming optimal assignment prob- lem, comparative advantage will be ab- sent if the matrix of output values (av) has rank one. Then each column will be a scalar multiple of any other column. The ratios of output values for any two workers, aUlazj, will be the same no mat- ter which machine they use. Despite the absence of comparative advantage, an op- timal assignment will arise in which more productive workers are assigned to more productive machines, according to the scale of operations effect. If the matrix has rank two or more, then comparative advantage must arise but it will not be the only determinant of the assignment.

In the differential rents model, comparative advantage will be absent when- ever f (g,k) is multiplicatively separable (i.e., it can be written as a function of g times a function of k). For example, sup- posef (g,k) = gakP as in the Cobb-Doug- las production function. Then the ratio of output values for workers 1and 2 will be (g~lkP)l(g~lkP) (gl/g2)Q, an amount

=

that does not depend on the machine size. Comparative advantage will there- fore be absent. The optimal assignment will require larger machines to be com- bined with more skilled workers as in the scale of operations effect. Iff (g,k) is not separable (e.g., as in the constant elasticity of substitution production func- tion with elasticity unequal to one), then comparative advantage will arise.

Only in Roy's model does comparative advantage determine the assignment of workers to jobs. Because the value of out- put in each sector equals the worker's earnings, worker self-selection leads to an assignment that is consistent with comparative advantage in the sense de- fined in (1). With this general definition, no restrictions are placed on the correla- tion between performances in one sector and in the other. The standard compara- tive advantage case, in which absolute advantage is absent (case i in Section III.C.), is only one of three possible cases.

The basic reason comparative advan- tage determines assignment in Roy's model but not in the others is that coop- erating factors such as capital play no sig- nificant role. In the linear programming optimal assignment problem and the dif- ferential rents model, the value of output is divided between labor and the employer so that wages are no longer pro- portional to output. In Roy's model, how- ever, the value of output goes entirely to the worker.

The absence of any role for cooperating factors of production is not an inherent feature of Roy's model. Capital can be incorporated into Roy's model as follows. Suppose that within a sector, workers perform tasks that are an input together with capital. Suppose output in a sector is given by

Ire, Vol. XXXZ Uune 1993)

where Q is total output per period in the sector, n is the number of workers, K is the amount of capital, and T is the total number of tasks performed. Assume Q has continuous first and second order derivatives. If n does not appear as an argument of Q in (15), then the marginal ~roduct of a worker will be tidQldT, where ti is the number of tasks performed by the worker (this is essentially Heck- man and Sedlacek's assumption (1985, p. 1080) in their derivation of Roy's model). Potential earnings in a sector will con- tinue to be proportional to tasks as in Roy's standard model.

However, suppose n appears in (15) and suppose further that Q has constant returns to scale in all three arguments. Then

where p = Kln = capital per worker, y = Tln = average tasks per worker, and h(p,y) = Q(l,Kln,Tln) = output per worker. Then the marginal product of a worker who performs titasks per period is

where MPi is the marginal product, h, = dhldp and h, = dhldy.

Unless h(p,y) is homogeneous of de- gree one, the intercept in (17) will be positive or negative so that the marginal ~roduct is no longer proportional to the number of tasks performed.41 It can be shown that inequality in marginal prod- ucts and wages in a sector will be greater or less than inequality in tasks performed

41 If h(p,y)is homogeneous of degree one, h(p,y) = ph, + yh, by Euler's Theorem, so that MPj = tih, and potential earnings in a sector are proportional to tasks performed. This reduces to the case where Q is a function of K and T only as Q = nh(p,y) = h(np, ny) = h(K, TI.

depending on whether the intercept is negative or positive. Depending on the functional form for h(p,y), a change in either p or y will alter the relation be- tween wages and tasks within a sector. Movement of workers from one sector to another, with no movement of capital, raises the capital to labor ratio in the sec- tor they move from and lowers it in the sector they move into. These changes al- ter relative wages between and within sectors.

In this way, capital can be incorporated into Roy's model. This results in a much more complex model that would be more difficult to estimate econometrically. But this extension is necessary in order to investigate how growth, capital accumu- lation, and business cycles affect the dis- tribution of earnings and why wage struc- tures vary from sector to sector.

V. Strategies in Studying the Distribution of Earnings

In analyzing any complex research question, a standard approach is to de- compose the problem by breaking it up into smaller questions that can be more easily explained. Approaches to the dis- tribution of earnings can be understood in terms of the decompositions used to analyze it. These decompositions include breaking the economy up into sectors, use of an earnings function, or perfectly elastic supply or demand curves. Assign- ment models demonstrate the invalidity of the ceteris paribus assumptions which lie behind these decompositions. This section discusses the decompositions used in various approaches, the problems revealed by analyzing the economy's as- signment problem, the solutions suggested by existing models, and strategies for further work.

A. Sectoral Decompositions

A seemingly natural way to study the distribution of earnings is to break the population down into subgroups based on demographic, occupational, or indus- trial categories. At any point in time, one can then study earnings inquality in terms of differences within and between groups. With this disaggregation or de- composition, one would expect to be able to calculate the consequences for inequality of a change within a sector, e.g., from the number in that sector or the distribution of earnings within that sec- tor. However, this calculation requires the ceteris paribus assumption that the composition of the other sector remains the same, and Roy's model shows directly why this assumption does not hold.

In Roy's model, a change in the num- ber of workers, mean earnings or vari- ance of logarithms of earnings within a sector does not occur in isolation. Shifts of workers from one sector to another occur because of changes in the relative prices of output in the two sectors. When the price of output of one of the sectors increases, a nonrandom selection of workers in the second sector move into that sector. This movement alters the means and distributions in both sectors. Figures 9 and 10 and Table 1in Section 1II.C. demonstrate the consequences of changes in relative sector prices. In Table 1, as the price of trout in terms of rabbits goes up from 0.83 to 1.00, the proportion hunting rabbits declines from 0.551 to

0.5. If one used this result to predict the effects of the price change on inequality, though, one would miss the self-selection effects of the shift on the composition of workers within sectors. As shown, both mean rabbit skills and mean trout skills decline, and other changes occur within the sectors. Thus the number or distribu- tion in one sector cannot be taken as given as the other sector changes. From the assignment perspective, the source of the change in distribution lies in the reassignment of workers from one sector to another in response to sector price changes, rather than in the separate changes within each sector. By incorpo- rating the selection decision, Roy's model allows one to predict the conse- quences of changes in sector prices.

Roy's model is appropriate any time the population is divided into two groups based on individual choice. For example, consider the decision to participate in the labor market. This context provided much of the early work on self-selection corrections (Reuben Gronau 1974;

H. Gregg Lewis 1974; Heckman 1974, 1979), and Heckman and Sedlacek esti- mate models with a nonmarket or house- hold sector (1985, 1990). The decision to participate in the labor market divides the population into two sectors, the paid labor market and the nonmarket or household sector. A decomposition in which one looked at only the paid labor force would be misleading. From the perspective of Roy's model, workers in the paid labor market are a selection of all potential workers. As the task price in the paid labor market goes up, a selec- tion of individuals will move from the nonmarket to the paid labor market sec- tor. Changes in average wages will not be proportional to changes in task prices, and average worker productivity will be affected, depending on the parameters of the task distributions. An empirical version of Roy's model can be used to examine these and other effects which occur along with changes in labor force participation.

Measures of earnings inequality may be used to compare alternative distribu- tions among a fixed population but may inaccurately indicate changes in inequal- ity when the number of earners grows or contracts. For example, an increase in the paid labor market task price could draw in predominantly low wage work- ers, making them better off while raising measured earnings inequality. The ap- propriate correction is to include workers outside the paid labor market in the mea- sure of inequality, so that movement of a worker in or out of the labor force would not by itself cause changes in inequality. Heckman and Honore (1990, Theorem 6, p. 1135) derive results relating in- equality in one of the sectors in Roy's model to overall inequality under the as- sumption of lognormality.

A related application arises in studying the effect of development on inequality. One sector in Roy's model would be the market (perhaps urban) sector and the other a nonmarket sector (perhaps rural, agricultural, or subsistence). Develop- ment generates a higher task price for labor producing the market good relative to the nonmarket good. The effects on the selection of workers in the market sector could then be derived and related to average earnings, productivity, and observed inequality as development pro- ceeds.

International trade provides another potential empirical application of Roy's model. A classic question in trade theory, reflected in the Hecksher-Ohlin and Samuelson-Stolper theorems, is the ef- fect of trade on factor payments and in- come distribution. Roy's model could be applied to this question by assuming that within a country, workers (or producing units) are divided between export and import-competing sectors. Results from Roy's model could then be used to exam- ine effects of terms of trade on production in the two sectors, average earnings and productivity in each sector, and the over- all distribution of earnings.

In addition to their own work, Maddala (1983, p. 289) and Heckman and Honore (1990, p. 1121) discuss further applica- tions of Roy's self-selection corrections, which include labor force participation, returns to education, retirement, union wage differentials, migration, occupational choice, movement between regions, and marital status. More recently, George Borjas (1990) examines the effects of self-selection on the skill composition of immigrants. Charles Brown (1990) ana- lyzes the operation of self-selection in army retention. Also, Gary Solon (1988) and Robert Gibbons and Katz (1992) con- sider the earnings of industry changers, Lazear (1986) examines choice of piece rate versus salary pay structures as a con- sequence of sorting between firms, and Tunali (1988) and Magnac (1991) extend Roy's model to examine segmented labor markets.

Further applications of Roy's model are possible if it can be extended to more than two sectors. A current research question concerns the reasons why wages differ among industrial sectors or estab- lishments. Levy and Murnane (1992, Section V1.C) review this literature in relation to the distribution of earnings, and James D. Montgomery (1991), Lang (1991), and Sattinger (1991) analyze capi- tal intensity as a source of wage differ- ences. Other possible reasons include ef- ficiency wages, unobserved abilities, union threats, and involuntary unemployment. A first step in analyzing the relation between wages and industrial characteristics is to correct for sectoral selection biases within industries. How- ever, the econometrics of Roy's model with self-selection would appear to place a barrier of only two or three sectors that can be estimated as a worker's income or utility in a sector must exceed the in- come or utility in every other sector. The extension to many sectors would appear to be possible if workers are assumed to engage in search instead of self-selection to find jobs, as discussed in Section 1V.C.

B. Earnings Function

A second approach to decomposing the distribution of earnings is to express separately the prices and quantities of worker characteristics that contribute to earnings. This approach uses an earnings function to describe the prices for worker characteristics, i. e., the relation between worker characteristics and earnings. This earnings function can be combined with the distribution of worker characteristics to generate the distribution of earnings. As discussed in the introduction, this ap- proach neglects that what is exogenous to the determination of an individual's earnings is endogenous to the determina- tion of the distribution of earnings. The approach therefore involves a form of the fallacy of composition.

The problems that arise from using an earnings function can be demonstrated using the following simple model. Sup- pose at any one point in time that work- ers differ by a single characteristic x and that the logarithm of earnings is related to x by the following earnings function:

where ei is a mean zero random error term uncorrelated with xi.From the per- spective of supply and demand models, it seems reasonable to suppose that the return to the skill variable x depends on both demand and supply variables. The demand for x, according to the capital- skill complementarity hypothesis, would depend on the economy's capital to labor ratio p. Suppose therefore that the coeffi- cient b depends positively on the capital to labor ratio p and negatively on the average population value of the worker characteristic x.

Now taking variances on both sides of (18),

Var(1n y) = b2var(ln x) + Var(e). (19)

This relationship will hold tautologically whenever (18) holds. If a single worker experiences a change in his or her own characteristic from xito xi, the expected logarithm of earnings for that worker would change from a + b In xito a + b In xi,But suppose all worker character- istics increase by 10 percent. Then (19) will incorrectly predict the consequences of this change for earnings inequality, as measured by the variance of the loga- rithms of earnings. Var(1n x) will increase by 1.l2= 1.21, but Var(1n y) will not increase by 1.21 b2: as b depends nega- tively on the average worker characteris- tic, b2 will decline. Further, if one at- tempts to estimate (19) directly (using for example time series data), the estimated coefficient of Var(1n x) will not equal b2; it will instead confound changes in Var(1n x) with changes in b2.42

Use of an earnings function obscures the influence of demand on the distribu- tion of earnings. Demand variables such as the capital to labor ratio appear to play no role in the determination of individual earnings in (18) but that is an illusion. The influence of the aggregate capital to labor ratio on earnings would be invisible in any single period empirical estimation of (18) as it would be the same for each observation. Then in the expression for earnings inequality in (19), demand vari- ables do not appear explicitly, suggesting that earnings inequality does not depend on them. But while the coefficient b can be regarded as a constant in a single pe- riod estimate of (18), it will vary from one period to another in (19).

With an assignment problem present in the economy, the earnings function is no longer a direct relationship arising from the contributions of worker charac- teristics to production. The assignment problem introduces an intermediate step between worker characteristics and earn- ings. The observed earnings function is generated from the supply and demand decisions of workers and firms. The he- donic wage and price literature develops

42 In approaches derived from a solution to the assignment problem, the coefficient b in (20) is en- dogenously determined, and the influence of the dis- tribution of individual characteristics, Var(1n x) in (21), on earnings inequality is correctly specified (for example in the earnings function solved by Tinbergen 1956, p. 168).

econometric procedures for estimating how the wage varies with worker attri- butes and for the identification of supply and demand functions for worker charac- teristics (Rosen 1974; Dennis Epple 1987; Timothy Bartik 1987). An immedi- ate application of this approach in the area of the distribution of earnings is compensating wage differentials for job characteristics (R. E. B. Lucas 1977b; Smith 1979; Greg J. Duncan and Bertil Holmlund 1983; John H. Goddeeris 1988; Mark Killingsworth 1986). In Kill- ingsworth's analysis, workers have heter- ogeneous preferences for a given job characteristic so that the compensating wage differential will depend on the dis- tribution of worker preferences. Killings- worth applies the analysis to differentials between white and blue collar labor. Causes of compensating wage differen- tials are relevant to issues of segmenta- tion, discrimination and comparable worth (Killingsworth 1987).

Pettengill (1980) applies related proce- dures to the question of how labor unions affect skill differentials and inequality. Unionized firms adjust to higher negoti- ated wages by employing higher quality workers. The effect of unionization is then to shift demands for workers to higher quality levels, leading to greater skill differentials and inequality in the economy. Pettengill also extends the analysis to credentialism, discrimination, absenteeism, cyclical changes in productivity and wages, and minimum wages.

A major reason for interest in the earn- ings function is that it describes the earn- ings that a worker with a given set of characteristics can obtain in the labor market. With an assignment problem, however, there will no longer exist a sin- gle expected wage associated with a given set of worker characteristics, as implied by the traditional earnings function. In- stead, the worker will face a distribution of potential wages and job characteristics from alternative jobs or sectors. The pur- pose of describing the alternatives facing workers would be better served by esti- mating the wage offer distributions for workers with given sets of characteris- tic~.~~

If job characteristics such as risk or satisfaction vary from sector to sector, the joint distribution of wages and job characteristics should be estimated.

Tinbergen (1975a, 1977) and Hartog (1986a, 1986b, 1988) estimate models in which earnings depend on both worker and job characteristics. Such models are relevant to questions of overeducation (Mun C. Tsang and Henry M. Levin 1985; Russell W. Rumberger 1987; Na- chum Sicherman 1991) and mismatches. An important outcome from such models is that the return to education varies ac- cording to the job placement of the worker, e.g., whether the job requires more or less education than the worker has.

C. Human Capital Models

Human capital models of the distribu- tion of earnings also rely on decomposi- tions. These models structure the deter- mination of earnings in such a way that the influences of supply and demand can be separated. The decompositions used are inconsistent with the existence of an assignment problem but are not essential features of human capital models of indi- vidual behavior (as distinct from models of the distribution of earnings). Robert

J. Willis and Rosen's (1979) intertempo- ral extension of Roy's model shows that the human capital and assignment mod- els can be combined. 44

In the model developed by Mincer (1958, 1974), workers choose a level of

*Christopher J. Flinn and Heckman (1982) and Nicholas Kiefer and George Neumann (1979a, 1979b) estimate wage offer distributions facing workers en- gagd in search.

Mark Blaug (1976), Lucas (1977a), Rosen (1977), Sattinger (1980) and Willis (1986) discuss alternative interpretations of human capital earnings models.

schooling based on the maximization of the present discounted value of lifetime earnings. With continuous discounting, the earnings hnction generated by this assumption in the long run is

where Y, is the yearly income with s years of schooling beyond the minimum, Yo is yearly income with the minimum schooling level, r is the discount rate and s is the number of years of schooling be- yond the minimum. If Y, and s satisfy the relation in (20), the present discounted values will be equalized for each level of schooling.

At each level of schooling, there is in the long run a perfectly elastic supply of labor at the yearly income determined by (20). If the yearly income for a given level of schooling yields a higher present discounted value than other levels, new workers will choose that schooling level. The amount of labor will continue to in- crease until the yearly income is pushed down to a level which yields the same present discounted value as all other schooling levels.

With horizontal supply curves, the lo- cation of demand curves cannot influence the yearly incomes of workers in the long run. Under the assumptions of Mincer's model, the equilibrium earnings function is invariant to changes in the demands for labor. The coefficient of the schooling variable in (20) is the discount rate and does not depend on demand variables.

However, in this model, the distribu- tion of earnings depends both on the yearly incomes for workers at each schooling level and the numbers at each schooling level. With a horizontal supply curve for each level of schooling, the number of workers with that level de- pends on the location of the demand curve. Although demand does not enter explicitly anywhere in the model, it plays a central role in determining the distribu- tion of earnings.

In Y, = In Yo + rs, (20)

This human capital model provides a

simple decomposition of the determina- tion of the earnings distribution. Worker supply behavior completely determines the earnings function, which remains the same in the long run as long as the dis- count rate is the same. Demands for workers do not influence this earnings function. With horizontal supply curves, the location of demand curves completely determine the numbers of work- ers at each schooling level.

This decomposition is possible because of the absence of any assignment prob- lem. All workers are identical, so any worker can obtain any schooling level. In the long run each worker is indifferent as to which schooling level to obtain. If instead workers had preferences for some schooling levels or if they faced different costs of obtaining schooling, the decom- position would break down. The supply of workers to a given schooling level would no longer be horizontal. As the demand for workers at a particular schooling level increases, the yearly in- come would need to be higher to com- pensate for cost and utility differences. Shifts in demand would then alter the earnings function.

The terms in the earnings function (20) can be reinterpreted to yield a very dif- ferent model, one that yields an explicit decomposition of earnings inequality. From a worker's point of view, a rate of return to an investment in schooling can always be calculated. In Mincer's model, this rate of return is exactly equal to the discount rate in the long run equilibrium. Now suppose that the rate of return is instead a random variable.45 The earn- ings function can then be written as

45 Lucas (1977a) analyzes different interpretations of the interest rate in human capital models. Mincer (1974, p. 27), Becker and Barry Chiswick (1966), and Chiswick and Mincer (1972) use human capital mod- els of the earnings function in which the rate of return to schooling is a random variable.

where Var(1n Y) is the variance of loga- rithms of earnings, and ? and i are the average rate of return and level of school- ing for all workers.

This procedure appears to provide a neat decomposition of earnings inequal- ity into rate of return and schooling sources, but unless restrictive assump- tions hold, the decomposition has no pre- dictive content.

Willis and Rosen (1979) develop a model that can be used to show why the existence of an assignment problem in- terferes with the use of the decomposi- tion in (22). In their development, the assignment problem is to allocate indi- viduals to schooling levels on the basis of tastes, talents, expectations, and pa- rental wealth. Individuals base their de- cision on human capital considerations but because of heterogeneous tastes and talents they are not indifferent among schooling levels. Rates of return calculated from observed relationships between schooling and wages will not reflect the rates of return facing individu- als. Willis and Rosen argue that individu- als who stop with a high school education earn more than college educated individ- uals would if they had stopped with a high school education. On average, then, individuals have an absolute advantage in terms of earnings at the schooling level they choose: they earn more than the rest of the population would at that

In response to a wage increase for a particular schooling level, only a subset of individuals switch to that level. The supply of individuals to a schooling level is therefore upward sloping rather than perfectly elastic as in Mincer's model. The wages and rates of return for workers at a schooling level depend on both the supply and demand for work- ers with that amount of schooling as well as observed and unobserved worker skills.

Now consider what happens when the distribution of schooling shifts. We may suppose that changes in parental wealth or tastes lead more workers to get a col- lege education. Moving down the de- mand curve for college-educated labor, the rates of return for college educated labor declines. The change in the distri- bution of schooling causes a change in the distribution of rates of return.47

This connection between schooling

46 According to Willis and Rosen (1979), this result argues against a large positive correlation between a worker's productivity with a high school education and with a college education, ruling out simple rank- ings of individuals. This corresponds to cases ii and iii in the Roy model (1II.C) and is inconsistent with the one-dimensional ability or skill models discussed in 1II.B. In addition to absolute advantage, compara- tive advantage is also ~resent in the sense defined in g). "

To be caused by shifts in the distribution of schooling, the recent increases in the returns to schooling (noted in Section I) would require that fewer workers choose to get a college education.

and rates of return makes (22) unusable. To keep things simple, suppose 3 goes up while Var(s) stays the same. The ex- pression in (22) would predict that earn- ings inequality would go up by 2SVar(r) times the increase in S. This prediction would be incorrect as and Var(r) also change. The ceteris paribus assumption that is needed to use comparative statics does not hold. Instead of earnings in- equality increasing as workers get more education, it may decline if the rate of return falls enough. 48

Early assignment models trivialize hu- man capital issues by assuming that worker characteristics such as schooling levels are exogenously determined. Early human capital models trivialize as- signment issues by assuming that work- ers all face the same investment returns independent of sector. But assignment and human capital models are not inher- ently competing theories of the distribu- tion of earnings. In both, behavior is motivated by wealth maximization (or, in extensions, utility maximization). In both, workers are assigned to sectors or educational levels on the basis of their own choices rather than rationing. The

This daculty can be avoided by supposing that a worker's rate of return does not depend on the sector or job chosen. Each worker's labor could be expressed as a multiple of some standard worker's labor, independent of occupation or job. This is known as the efficiency units assumption as all labor can be expressed in terms of a common measure (see discussions by Rosen 1977; Sattinger 1980, pp. 15-20; and Paul Taubman 1975, pp. 3-6).It is equiva- lent to the absence of any assignment problem. With wages proportional to a worker's efficiency units, em- ployers are indifferent as to which labor to use in a job. For example, an employer is indifferent between hiring a worker with two efficiency units at a wage rate w and hiring two workers of one efficiency unit apiece at a wage rate of wI2 each. This assumption would insure that the distribution of rates of return is unaffected by shifts in the distribution of schooling, so that the standard comparative static analysis can be applied to (22). However, the efficiency units as- sumption seems an unreasonable restriction in mod- els designed to explain the economic consequences of human diversity.

If the problems of extending Roy's model to many sectors can be solved, then an empirical general equilibrium method of studying the distribution of earnings can be developed in which the conditions in each sector can be consid- ered separately. However, it is difficult to describe how to formulate such an em- pirical, multi-sector model without actu- ally doing so.

VI. Conclusions

advantage of viewing investment in edu- cation as an assignment to schooling lev- els is that it more accurately describes the choices available to individuals. As in Willis and Rosen's analysis (1979), a model that specifies the unobserved al- ternatives for workers allows one to ex- amine the issue of ability bias and esti- mate the real return to individuals for investment in schooling.

D. Assignment Models

In their structuring of the determina- tion of earnings, assignment models also rely on a decomposition of the distribu- tion of earnings. The existence of an as- signment problem introduces an intermediate decision stage between workers' characteristics and their earnings. In many of the theoretical models, the as- signment problem can be solved first and then used to determine the wage differ- entials. This occurs, for example, in Tin- bergen's model, the differential rents model, and the linear programming opti- mal assignment problem. In general, however, assignment and the determina- tion of earnings occur simultaneously.

In empirical studies, the information that would be needed to solve the econo- my's assignment problem is unavailable. The current assignment can be observed but not the alternatives facing individual agents. The relevant characteristics of workers and firms may not even be ob- servable. Complete specification of the assignment problem, prior to the deter- mination of earnings, is therefore unreal- istic in empirical estimates.

Tinbergen's empirical study of the dis- tribution of earnings structures the prob- lem using a supply and demand approach (1975a, pp. 29-30). In a single labor mar- ket, the structural supply and demand equations can be solved to yield two re- duced form equations, one for the price (wage) and the other for the quantity (em- ployment). These reduced form equa- tions are functions of both supply and demand factors. Extending this approach to multiple labor markets, the earnings function and measures of inequality should be regarded as reduced form equations that depend on the distribu- tion of both supply and demand factors. This approach does not incorporate self- selection corrections but carries the sim- ple, practical injunction that both supply and demand factors need to be included in empirical estimates of the earnings function or inequality measures. Tinber- gen (1975a, 1977) applies an empirical decomposition of the determination of earnings in which the labor market is bro- ken down into compartments based on discrete values of some worker character- istic such as education. In this approach, the separate effects of supply and de- mand (in the form of job requirements) can be estimated. Tinbergen's separate estimates of labor market compartments suggest a very pragmatic, disaggregated, partial equilibrium approach to studying the distribution of earnings. First, one isolates a sector or segment of the econ- omy of interest (for example, wholesale and retail trade, which employs many younger workers at low wages). Then one analyzes the market in terms of supply and demand, fully specifying the alterna- tives that are available to employers and workers (in order to account for self- selection phenomena) and institutions in the labor market. This approach is often followed in practice but without refer- ence to the income distribution develop- ments that lie behind it.

where Ysiis the yearly income for worker i, riis the average rate of return to school- ing for worker i, and si is the number of years of schooling beyond the mini- mum for worker i. In this model, the variables riand si are determined exoge- nously so that a separate model is needed to explain a worker's level of schooling and rate of return (see Becker 1975, pp. 94-117, for a theory of investment behav- ior that explains the rate of return and schooling level). So far, the earnings function in (21) describes the generation of a worker's earnings, taking the econ- omy as given. The next step is to extend the earnings function in (21) to an expla- nation of earnings inequality by taking variances on both sides of (21). Under the assumption that the rate of return ri and level of schooling si are independently distributed,

A. The Importance of Choice

This survey has examined three po- tential reasons why the existence of an assignment problem affects the distribu- tion of earnings. The first reason is com- parative advantage, which is measured by bilateral comparisons of output of two workers at two jobs. In technologies with cooperating factors of production, as in the linear programming optimal assign- ment problem and differential rents model (Sections 1II.A and III.B), com- parative advantage does not necessarily determine the assignment; instead, the scale of operations effect may influence the assignment. In technologies where comparative advantage does determine the assignment, as in Roy's model, it is consistent with all cases, in the trivial sense that the ratios of outputs for two workers varies from sector to sector. The standard comparative advantage case, in which there is a large positive correlation between sectoral outputs, yields predict- able consequences for the distribution of earnings but does not necessarily arise in Roy's model. In short, saying there is comparative advantage tells us very lit- tle about the distribution of earnings other than that it doesn't resemble the distribution of abilities in any one sector.

Self-selection is another phenomenon that is often closely associated with as- signment models. Although all three models in Section I11 use self-selection as a means of choosing among sectors or jobs, assignment models do not require the self-selection mechanism. Instead, workers could engage in search to find the sector that maximizes their income or utility. This is particularly appropriate when there are many sectors or jobs to choose from, making the assumption of full information about jobs less reason- able.

The major point common to all assign- ment models is the existence of choice among jobs, occupations, or sectors for a worker. With choice, a worker's earn- ings or utility are not determined by per- formance within a single area of endeavor. Instead, the worker can avoid the consequences of a bad performance in one sector by choosing another sector. Comparative advantage is significant in describing relationships among opportu- nities in different sectors. Self-selection describes how decisions may be made. But the underlying feature is the variabil- ity of worker output among sectors or jobs. This variability arises from the dif- ferent sensitivity of jobs to worker abili- ties (i.e, the difficulty of jobs), the large dispersion in tasks performed in different jobs throughout the economy, and the diversity and lack of correlation among individuals' performances of those tasks. Variability of output from worker-firm matches generates a problem in choosing jobs or sectors, from the worker's point of view, and a problem of assigning work- ers to jobs, from the perspective of the economy as a whole.

B. Extensions

Through the work of Heckman and Sedlacek (1985, 1990), Roy's model pro- vides the most promising route to esti- mate assignment models. They find that two extensions of Roy's model are essen- tial to explain the distribution of wages in the U.S. economy. First, workers choose sectors on the basis of utility maxi- mization instead of income maximization. Workers have sector-specific preferences that alter the assignment of workers to jobs relative to what one would expect on the basis of income maximization. Sec- ond, Heckman and Sedlacek find that nonparticipation in the labor market is an important choice facing workers.

Journal of Economic Literature, Vol. XXXI (June 1993)

Workers participating in either the man- ufacturing or nonmanufacturing sectors are therefore a selection from all poten- tial workers in the population.

This article suggests that two additional extensions are important. First, the interactions between capital and de- mands for labor should be adequately specified and estimated. In Heckman and Sedlacek's models, changes in capital intensity or energy prices affect demands for workers and the division of labor through task prices. With their assump- tions, earnings are proportional to tasks, which serve as measures of sector specific abilities. Then wage differentials for worker characteristics within a sector would not change over time. However, if capital intensity affects marginal prod- ucts as in (17), earnings will not be pro- portional to tasks. Then changes in capi- tal per worker or tasks per worker could influence the demand for workers in more complicated ways. Incorporation of the role of capital will provide a link be- tween observable conditions of demand and the distribution of earnings.

The second extension concerns the number of sectors. With self-selection, the econometrics appear to place a bar- rier of only two or three sectors that can be estimated because a worker's income or utility in a sector must exceed the in- come or utility in every other sector. The extension to many sectors would appear to be possible if workers are assumed to engage in search instead of self-selection to find jobs. Then the income or utility in a given sector needs only to satisfy one bilateral comparison, i. e., with the worker's reservation income or utility.

With many sectors, the relation be- tween sector characteristics and demands for workers can be examined. Character- istics of industries lead firms to seek dif- ferent mixes of workers. In the context of a search model firms must pay higher wages to gain more acceptances from workers with desired characteristics. Such wage premia are consistent with ab- sence of monopsony when search assigns workers to jobs. Wage differences be- tween industries can persist without all workers going to the higher wage indus- try. With search assigning workers to jobs, the extension of assignment models to industrial sectors can be used to ex- plain interindustry differences in wage structures.

C. Relation to Other Distribution Theories

The distribution theory arising from assignment models is distinct from estab- lished theories. In perfectly competitive neoclassical models, wages in a labor market are determined from intersecting supply and demand curves, where the demand curve is derived from firm pro- duction functions. In equilibrium, wages equal the marginal revenue product, the product price times the marginal product of labor. The expression for the wage dif- ferential in the differential rents model, (ll), provides a comparable condition. In neoclassical models, wages are found by varying the quantity of labor, whereas in assignment models, the wage differen- tial is found by varying the assignment (i.e., changing the type of labor used in a particular job). In cases where workers and capital are combined in fixed propor- tions (the differential rents model and the linear programming assignment problem), the marginal product of labor will not be defined. Then the wage differen- tials will be consistent with alternative absolute levels of wages and profits. This indeterminacy is absent in neoclassical models but instead characterizes classical models, in which factor prices are deter- mined exogenously. Classical models, however, are usually concerned with fac- tor shares and do not explain relative wage rates.

The major difference between assign- ment and human capital models is in the interpretation of the earnings function. In the earnings function in Mincer's model, the coefficient of schooling is the discount rate. This coefficient would re- main unchanged in response to changes in the distributions of jobs or workers. In the alternative approach, as developed by Becker and Chiswick, the coefficient of schooling is a rate of return which does not explicitly depend on supply or de- mand variables. In these models, auxil- iary assumptions combined with the human capital model of investment be- havior allow one to decompose the distri- bution of earnings. The existence of an assignment problem is inconsistent with these auxiliary assumptions. In assign- ment models, the earnings function is in- terpreted as an hedonic wage function, a reduced form relationship instead of a structural relationship. A change in the distribution of either workers or jobs would lead to a new equilibrium, with a new coefficient of schooling. With an assignment problem, the earnings func- tion cannot be used to predict the conse- quences of changes in jobs or workers. There is no reason, however, why the human capital model of individual behav- ior cannot be combined with sectoral choice or set in the context of an assign- ment problem, as in Willis and Rosen's model.

Because abilities enter so importantly, assignment models may appear to be akin to theories that relate the distribution of earnings to the distribution of abilities. This link is perhaps fostered by the sim- plifying assumption in some models of a single parameter describing workers. However, the abilities considered in as- signment models are the outputs or per- formances in various jobs, not IQ's or other measures of innate ability. Further, the point of assignment models is just the opposite of ability models: the distribution of abilities cannot by itself explain the distribution of earnings. In particular, using one task to identify abili- ties, the distributions of earnings and abilities will not have the same shape. Assignment models emphasize the diver- sity of human performances from one task to another and the roles of choice and demand in placing higher or lower values on particular abilities. Earnings inequal- ity depends both on differences among workers and the extent to which the economy exaggerates or moderates those differences through the assignment of workers to different tasks.

D. Relative Wage Changes

Assignment models provide several explanations for why relative wages change. In the linear programming as- signment problem, a shift in jobs would produce changes in wage differentials. In (8), if new jobs have machine properties which are greater for those jobs that have higher values of the X's, then wage differ- entials would increase. However, this model is too abstract to relate its results to observed changes in skill and age dif- ferentials.

The differential rents model of Section 1II.B provides a realistic explanation of how differentials can change over time. In (13), the wage function derived using specific assumptions about functional forms can be expressed by w(g) = ~g(~''g+~"k)'~g where A is a con-

+ C,, stant and C, is a constant determined by the reserve prices of workers and ma- chines. The exponent of the skill variable g in this expression is an increasing func- tion of uk,which measures the inequality in the distribution of capital among jobs. As capital becomes more unequally dis- tributed among jobs, the wage function w(g) becomes more concave. With more capital per worker among the most skilled workers, their wage differentials increase. With less capital per worker among the less skilled, their wage differ- entials decline. The quantity w -C, is lognormally distributed with variance of logarithms aug + Puk,and this measure of inequality is an increasing function of uk.*'

Using an extended version of Roy's model, Heckman and Sedlacek (1985, p. 1107) estimate that the price of the manu- facturing task declined 22 percent from 1976 to 1980, while the price of the non- manufacturing task rose 21 percent in the same time period. In their analysis, this produces a movement of workers from manufacturing to nonmanufacturing, which is the sector with the lower educa- tion and experience differentials. This movement alone would tend to reduce aggregate differentials. However, in the nonrnanufacturing sector, the greater task price raises skill and experience dif- ferentials, while in the manufacturing sector these differentials decline.50 Workers moving from manufacturing to nonrnanufacturing tend to be at the bot- tom of the task distributions jn the two sectors. Their movement raises average worker quality in manufacturing while reducing it in nonmanufacturing. The av- erage wage in manufacturing declines both because of the lower task price and the lower average worker skill levels. In nonmanufacturing, the task price and av- erage worker quality have opposite ef- fects on the average wage and the net effect is ambiguous. Heckman and Sedla- cek do not indicate what the net effect

49 Sattinger (1980) finds that inequality in the dis- tribution of capital is significantly related to earnings inequality. The measure of inequality in the distribu- tion of capital is derived from industry capital to labor ratios. Dickens and Katz (1987, p. 66) review studies that have found a positive relationship between capi- tal to labor ratios and average industry wage rates. Thev also find a ~ositive relations hi^ in their own study (p. 78).

A

That is, the slope of the skill-earnings relation- ship increases in nonrnanufacturing and decreases in manufacturing. Note, however, that under the as- sumptions used by Heckman and Sedlacek, elastici- ties of earnings with respect to skills are constant.

of all these movements would be on ag- gregate educational and experience dif- ferentials. If the economy is not described by the manufacturing versus nonrnanufacturing dichotomy, then movements within sectors will also con- tribute to changes in relative wages. The extensions to Roy's model to incorporate capital, discussed in Section IV. D, would explain how differentials could change within sectors independently of task price changes.

E. Research Questions

Each approach to the distribution of earnings suggests a set of relevant research questions. In the human capital approach, for example, important phe- nomena are decisions to invest and the returns to those investments. The exis- tence of an assignment problem suggests a different set of questions.

First, how do workers differ in ways that are relevant to employment? What choices do workers face between occupa- tions and industrial sectors? How do the choices facing workers differ by educa- tional level? How many sectors do work- ers choose among? How do worker pref- erences affect choice of sector, wage differentials and earnings inequality?

On the employer side, how does the technology in each sector relate worker characteristics to output? How do these technologies generate different demands for workers? How are demands for work- ers related to features of the industry or occupation such as capital intensity or hi- erarchical span of control? Do workers occupy capital in the sense discussed in Section II.C? How has the mix of jobs changed over time, how is it related to shifts between manufacturing and non- manufacturing, and how does trade affect the mix? What wage differentials do firms need to offer to attract a labor force with a given set of characteristics?

What mechanisms operate in the econ- omy to assign workers to jobs? What combinations of self-selection, search, and separated labor markets do workers use to find jobs? What are the costs of these assignment mechanisms, what are the efficiency properties of the assign- ments they bring about, and how do they affect the distribution of earnings? What are the relative contributions of unequal variances of worker performances among sectors, correlation among sector perfor- mances and number of sectors to earn- ings inequality?

A final set of questions is related to explaining observed changes in the dis- tribution of earnings. How are shifts in the mix of jobs and workers related to changes in the wage rates for high school and college graduates and the returns to education? How do wage or unemploy- ment differentials reconcile so-called mismatches between supplies and de- mands for workers?

Assignment models offer the promise of incorporating the influence of demand on the distribution of earnings, accurately representing the relation between worker characteristics and earnings, and rigorously explaining changes in earnings inequality and wage differentials over time. This promise has been met only partially through applications of assign- ment models to many aspects of the dis- tribution of earnings. While assignment models indicate the shortcomings of earnings function and human capital ap- proaches, empirical work has only begun to provide a comprehensive alternative.

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