# Exact unemployment rate indices.

I. INTRODUCTION

Few economic statistics command more widespread attention than the rate of unemployment. It is viewed as a key measure of a country's economic performance, and its release often makes the headlines and the evening news. It plays a central role in many key macroeconomic relationships, such as Okun's law and the Phillips curve, and it is fundamental to such concepts as the nonaccelerating inflation rate of unemployment. It is therefore essential to design it and to measure it in a way that is consistent with its purpose.

To economists, the unemployment rate is a measure of the degree of underutilization of the nation's labor endowment. The conventional rate of unemployment (the number of unemployed persons divided by the number of persons in the labor force) may seem reasonable to some, but adding up workers (e.g., physicians and dishwashers) without regard to their skills and without taking the relative price of their services into account makes little economic sense. There are few examples in economics in which aggregation is routinely carried out in so crude a fashion. To draw a blunt analogy, it is as if one measured the nation's capital stock by adding up the numbers of trucks, office buildings, and screwdrivers.

During the past 40 years, concerns in the United States about unemployment concepts and measures have led to the appointment of two blue-ribbon groups--the Gordon Committee during the 1960s and the National Commission on Employment and Unemployment Statistics (or the Levitan Commission) during the 1970s. Over the years, numerous criticisms have been directed at conventional unemployment rate statistics (Cain, 1980). These criticisms have ranged from the accuracy of the surveys that underlie the unemployment statistics to the proper concept to be measured (e.g., unemployed persons versus unemployed person-hours) to various adjustments that ought to better reflect the changing demographic composition of the labor force (e.g., an age adjustment because unemployment rates tend to vary by age). Yet the formula used to construct the index seems to have never been seriously questioned.

The productive capacity of the labor force is certainly a function of the skills and education embodied in it. This observation is presumably one reason that earnings functions based on human capital concepts include as key arguments age (a proxy for experience) and years of schooling. If the unemployment rate is relatively higher among the less skilled than among the more skilled, simply summing up the number of unemployed and dividing by the labor force will overstate the true proportion of labor's productive capacity that is unutilized. How large is the resulting bias? The answer is entirely an empirical matter on which the authors try to shed some light in this article.

In what follows, the authors propose several measures of the rate of unemployment. Each measure is exact (in the sense of Diewert, 1976) under certain conditions, but the authors believe that each one can serve as a good approximation under more general circumstances. Moreover, this article constructs lower and upper bounds of the "true" rate of labor underutilization, and it turns out that the conventional unemployment rate does not fall within these bounds. Finally, using March Current Population Survey data for 1964-95, the authors compare the various unemployment rate measures for the United States.

II. AGGREGATION, LABOR CAPACITY UTILIZATION, AND UNEMPLOYMENT

Assume that aggregate production can be modeled by a multiple-input multiple-output transformation function. Assume I outputs, J nonlabor inputs, and H categories of labor. The authors denote output quantities by [y.sub.i], the quantities of nonlabor inputs by [x.sub.j], and the number of workers in each category by [n.sub.h]. Let the aggregate transformation function be given by

(1) t([y.sub.1]...[y.sub.1], [x.sub.1]...[x.sub.j], [n.sub.1]...[n.sub.H]) = 0.

Assume that this transformation function is weakly separable between labor and the other inputs and outputs, so that it can be written as

(2) v[[y.sub.1]...[y.sub.I], [x.sub.1]...[x.sub.J], h([n.sub.1]...[n.sub.H])] = v([y.sub.1]...[y.sub.I], [x.sub.1]...[x.sub.J], n) = 0,

where

(3) n = h ([n.sub.1]...[n.sub.H]);

n is an index of aggregate labor and h(*) is the labor aggregator function. The authors assume that h(*) is increasing, quasi-concave, and linearly homogeneous.

Weak separability means that the various categories of labor can be consistently aggregated. Assuming optimizing behavior, the profit-maximization or cost-minimization problem can be decomposed into two steps. The first step involves determining the optimum labor mix, given the level of aggregate labor n and the wage rates of the different labor categories. The second step involves determining the optimum level of aggregate labor and of the other inputs and outputs, given all input and output prices. The authors (and readers) need only concern themselves with the first step of this twin decision process.

Let [w.sub.h] be the wage rate of the employed workers in the hth category. First-step optimization requires that the marginal products of each category of labor be equal to their marginal cost:

(4) [[partial]h(*)/[partial][n.sub.h]] = [w.sub.h]/w,

where w is the Lagrange multiplier and can be interpreted as the aggregate wage rate.

The H categories of labor can be identified by skills, sex, age, and so on. The authors assume that the workers within these categories are perfectly homogeneous. This is not as restrictive as it may sound because the number of categories can be increased almost at will. This article denotes the total number of workers, employed and unemployed, of the hth category by [l.sub.h]. If all workers were employed, the aggregate labor quantity index (l) would be equal to

(5) l = h([l.sub.1], [l.sub.2]...[l.sub.H]).

The rate of labor capacity utilization is therefore given by n/l, and we can define the rate of unemployment u as

(6) u [equivalent to] 1 - ([h([n.sub.1], [n.sub.2]...[n.sub.H])]/[h([l.sub.1], [l.sub.2]...[l.sub.H])]).

The true rate of unemployment therefore depends on h(*), the labor aggregator function. Unfortunately, this function is generally unknown. There are ways to get around this difficulty, however. One possibility, if the appropriate data are available, is to estimate the labor aggregator using econometric techniques. An alternative approach, one that may seem somewhat less controversial (because there is no need then to specify a stochastic specification and to select an estimation method), is based on index-number theory.

III. INDEX-NUMBER APPROACH

Probably the simplest functional form for h(*) that one can chose is the linear one:

(7) h(*) = [summation over (H/h=1)][b.sub.h][n.sub.h], [b.sub.h] [greater than or equal to] 0.

The first-order conditions (4) then imply

(8) [b.sub.h] = [w.sub.h]/w.

Making use of (7) and (8) in (6), one can express the linear version of the rate of unemployment ([u.sup.LN]) as:

(9) [u.sup.LN] = 1 - ([[summation over (H/h=1)] [w.sub.h][n.sub.h]]/[[summation over (H/h=1)] [w.sub.h][l.sub.h]]).

The conventional measure of the unemployment rate (u) can be viewed as a special case of (7), that where all [b.sub.h]'s are arbitrarily set to unity:

(10) h(*) = [summation over (H/h=1)] [n.sub.h],

in which case

(11) u = 1 - ([[summation over (H/h=1)][n.sub.h]]/[[summation over (H/h=1)] [l.sub.h]]).

To the extent that wages vary across labor categories, the first-order conditions can clearly not be met in the case of (10).

Another natural choice for the labor aggregator function is the Leontief form:

(12) h(*) = min{([n.sub.1]/[a.sub.1]), ([n.sub.2]/[a.sub.2]) ...([n.sub.H]/[a.sub.H])}, [a.sub.h] > 0.

Under cost minimization, one can set

(13) n = [n.sub.h]/[a.sub.h], [for all]h.

The corresponding rate of unemployment ([u.sup.LF]) can then be obtained as:

(14) [u.sup.LF] = 1 - ([n.sub.k]/[l.sub.k]),

where

(15) [l.sub.k] = [a.sub.k] min {([l.sub.1]/[a.sub.1]), ([l.sub.2]/[a.sub.2]) ... ([l.sub.H]/[a.sub.H])}.

This implies that

(16) [u.sup.LF] = min {[u.sub.1], [u.sub.2]...[u.sub.H]},

where [u.sub.h] [equivalent to] 1 - [n.sub.h]/[l.sub.h] is the unemployment rate for the hth category of labor.

The Leontief aggregator function assumes that no substitution whatsoever is possible between the different categories of labor. If one category of labor is in short supply, this will severely restrict the maximum amount of aggregate labor that is available. Put in other words, actual employment would be closer to full employment than one might think, and [u.sup.LF] can therefore be viewed as a lower bound of the true rate of unemployment. The linear functional form, on the other hand, assumes that the different categories of labor are perfect substitutes for each other. This would imply that total employment could be stretched further than it first appears: [u.sup.LN] can therefore be viewed as an upper bound for the true unemployment rate.

The linear functional form assumes that the Allen-Uzawa elasticity of substitution between the different types of labor is infinitely large, whereas the Leontief function assumes that it is zero. A less restrictive assumption would be to assume that the aggregator function has the following Cobb-Douglas form:

(17) h(*) = [[PI].sup.H.sub.h=1][n.sup.[[alpha].sub.h].sub.h], [[alpha].sub.h] [greater than or equal to] 0,

where the [[alpha].sub.h]'s are unknown nonnegative parameters. The first-order conditions then can be written as

(18) [[alpha].sub.h] = ([w.sub.h][n.sub.h])/([summation over (k)][w.sub.k][n.sub.k]).

Making use of (17) and (18) in (6), we obtain the Cobb-Douglas measure of the rate of unemployment ([u.sup.CD]) as:

(19) [u.sup.CD] = 1 - [([[PI].sup.H.sub.h=1] [n.sup.[s.sub.h].sub.h]/ [[PI].sup.H.sub.h=1] [l.sup.[s.sub.h].sub.h])],

where [s.sub.h] [equivalent to] [w.sub.h][n.sub.h]/([SIGMA][w.sub.k][n.sub.k]), an observable variable.

Finally, one may want to consider the constant elasticity of substitution case. Let the aggregator function be given by: (1)

(20) h(*) = [([[delta].sub.1] [n.sup.[rho].sub.1] + [[delta].sub.2] [n.sup.[rho].sub.2] +...+ [[delta].sub.H] [n.sup.[rho].sub.1]).sup.1/[rho]],

[[delta].sub.h] > 0, [rho] [less than or equal to] 1.

The first order conditions then can be written as

(21) [[delta].sub.h] [n.sup.[rho]-1.sub.h][([summation over (k)] [[delta].sub.k] [n.sup.[rho].sub.k]).sup.(1-[rho])/[rho]] = [w.sub.h]/w.

This can be rewritten as

(22) [[delta].sub.h] [n.sup.1-[rho].sub.h]([w.sub.h]/w)[([summation oer (k)][[delta].sub.k] [n.sup.[rho].sub.k]).sup.([rho]-1)/[rho]].

Making use of this result in (20) and (6) yield the CES measure of the rate of unemployment:

(23) [u.sup.CES] = 1 - [[([summation over (h)][n.sub.h][w.sub.h]).sup.1/[rho]]/[([summation over (h)][n.sup.1-[rho].sub.h][l.sup.[rho].sub.h][w.sub.h]).sup.1/[rho]]].

IV. ESTIMATES FOR THE UNITED STATES

The article now examines the empirical significance of the problem discussed. The data are for the United States, 1964 to 1995, and are drawn from the March Current Population Survey. The focus is on educational attainment, but the authors could have focused on age (experience), or on both age and schooling, or on still more categories to which the Bureau of Labor Statistics has given credence (such as sex and marital status). (2) Because this article is concerned with educational attainment, the authors restrict the sample to persons 25 years of age and older and disaggregate labor into ii levels of educational attainment. For 1995, Table 1 reports the number of labor force members, annual earnings, and the unemployment rate by level of education. Clearly, the unemployment rate falls sharply with increased education. (3)

The first column of Table 2 shows the conventional measure of the rate of unemployment (u) computed with the help of (11).

It varies between 2.40% (in 1969) and 8.62% (in 1983). The second column shows the estimates of [u.sup.LN], which is exact if the aggregator function is linear. This index is systematically lower than u. The difference between the two indices varies between 0.23% (in 1969) and 0.87% (in 1983). The third column shows the estimate of [u.sup.CD], which is exact if the aggregator function is Cobb-Douglas. As expected, it is lower still than [u.sup.LN] and it can be seen that the difference between [u.sup.CD] and u exceeds one percentage point in some years. The fourth column shows the values of [u.sup.CES] (for p = -4); it is marginally lower than the Cobb-Douglas estimate. The fifth column reports the values of [u.sup.LF], which is exact if h(*) has the Leontief form. As expected, this index is by far the lowest. It is noteworthy that not for a single year does the conventional measure of the unemployment rate (u) fall within the upper and lower bounds as determined by [u.sup.LN] and [u.sup.LF].

The authors also report at the bottom of the table the coefficients of variation of the different series. Because the various indices are highly correlated, the coefficients of variation are of similar magnitude, although they are somewhat smaller in the case of the Cobb-Douglas and the CES indices.

Although the U.S. unemployment rate as conventionally measured drifted upward throughout much of the latter half of the twentieth century, beginning in the 1980s a structural shift appears to have occurred as the unemployment and wage gaps between skilled and unskilled workers widened dramatically. This shift impacts on the differential between the present measures and the traditional unemployment rate, and it is apparent in Table 2. The mean conventional unemployment rate increased from 4.20% during the 1964--79 period to 5.72% during the 1980-95 period. The mean Cobb-Douglas index increased by considerably less, from 3.74 to 4.95, SO that the mean difference between the two indices rose from 0.46 to 0.77 percentage points. The divergence is even larger in the case of the CBS index. Thus the indices proposed here do not perfectly mimic the conventional measure, although they obviously tend to move with it.

The results are summarized in Figure 1, which shows the path of three unemployment-rate indices: the conventional measure (u), the linear measure ([u.sup.LN]), and the Cobb-Douglas measure ([u.sup.CD]). The extent of the upward bias contained in the conventional unemployment rate measure is clearly visible. The cyclical pattern of the different series is, however, similar.

It is perhaps noteworthy that the bias appears to be greater when the unemployment rate is rising. The difference between the conventional measure and the Cobb-Douglas measure averages 0.73 for 10 years when the conventional measure rose, but 0.55 for 21 years when it fell. One possible reason for this observation is that during periods when the economy is sluggish, the least productive workers suffer disproportionate unemployment, but when the economy expands they are drawn disproportionately back to jobs.

V. ECONOMETRIC APPROACH

As suggested earlier, aggregator function h(*) can be estimated in principle. Assume that it has the CBS form. For empirical purposes, to allow for technological change and other shifts in the technology, it is useful to allow the [delta]'s to change over time. The authors therefore rewrite the function as follows:

(24) h(*) = [[([[delta].sub.1] + [[delta].sub.1T]t)[n.sup.[rho].sub.1] + ([[delta].sub.2] + [[delta].sub.2T]t)[n.sup.[rho].sub.2] +...+ ([[delta].sub.H] + [[delta].sub.HT]t)[n.sup.[rho].sub.H]].sup.1/[rho]], [[delta].sub.h] + [[delta].sub.hT]t > 0, [rho] [less than or equal to] 1,

where t is a time trend. It is convenient to express the first-order conditions in share form

(25) [s.sub.h] = ([[delta].sub.h]+[[delta].sub.hT]t)[n.sup.[rho].sub.h][[[summation over (k)] ([[delta].sub.k] + [[delta].sub.kT]t)[n.sup.[rho].sub.k]].sup.-1],

h = 1...H,

where [s.sub.h] again is the share of the hth type of labor in total labor costs.

The authors have estimated the system of share equations (25) using the nonlinear estimation procedure as implemented in SHAZAM, version 8.0, which yields estimates of the [delta]'s, the [[delta].sub.hT]'s, and [rho] (whose estimate is 0.362 with a standard error of 0.046). The authors then introduce these estimates into (24) and (6) to get an econometric estimate of the rate of unemployment, denoted [u.sup.CES]. The resulting figures are shown in Table 2. The authors find that they are significantly smaller than the conventional measure (u) suggests. In fact, they are quite close to the Cobb-Douglas estimates.

By constraining all Hicksian elasticities of complementarity to be constant, the CBS function is rather restrictive. It is therefore tempting to reestimate the aggregator function using a flexible functional form. Assume that the aggregator function has the following Translog form:

(26) h(*) = [[beta].sub.0] + [SIGMA][[beta].sub.h] ln [n.sub.h]

+ (1/2)[SIGMA][SIGMA][[phi].sub.hk] ln [n.sub.h] ln [n.sub.k]

+ [SIGMA][[delta].sub.hT]ln[n.sub.h]t+[[beta].sub.T]t

+ (1/2)[[phi].sub.TT][t.sup.2],

where [SIGMA][[beta].sub.h] = 1, [[phi].sub.hk] = [[phi].sub.kh], [SIGMA][[phi].sub.hk] = 0, and [SIGMA][[delta].sub.hT] = 0. Again, expressing the first-order conditions in share form, one gets

(27) [s.sub.h] = [[beta].sub.h] + [SIGMA] [[phi].sub.hk] ln [n.sub.k]+[[delta].sub.hT]t,

h = 1...H.

Estimation of large-size Translog aggregator functions (remember that H = 11 in this case) almost invariably leads to curvature condition problems. The authors have therefore imposed concavity using the procedure of Jorgenson and Fraumeni (1981). (4) Estimation of (27) subject to this reparameterization, again using the nonlinear estimation procedure implemented in SHAZAM, makes it possible to get a Translog measure of the rate of unemployment. Estimates of this index ([u.sup.TL]) are reported in the last column of Table 2. (5) The authors find that the Translog measure comes quite close to the Cobb-Douglas index, and it is substantially less than the conventional rate.

VI. CONCLUSION

This article proposes a number of alternative measures of the unemployment rate based on index number theory as well as on econometric techniques. These measures are exact for specific labor aggregator functions. The results for the United States show that the conventional unweighted unemployment rate yields an overestimate of about 0.6 percentage point, on average, or about 13.9% relative to the Cobb-Douglas measure. The bias appears to have increased together with the widening unemployment and wage gaps between skilled and unskilled workers starting in the 1980s.

The conventional unemployment rate for persons 25 years of age and over is considerably lower than the official unemployment rate for persons age 16 and over. (6) This lower rate is obviously due to the procedure of cutting Out the young end of the labor force age distribution, where experience and education tend to be lower and unemployment rates higher. Based on the population used by the Bureau of Labor Statistics to calculate unemployment rates, the unemployment rate measures proposed here would undoubtedly result in even greater measures of upward bias. Because it is frequently argued that the unemployment rate is biased downward due to the elimination of discouraged workers from the unemployment total, the estimates of bias reported here obviously run in the opposite direction.

Of all the unemployment indices reported here, the authors favor the Cobb-Douglas index. It is simple to calculate; it is based on index number theory rather than on more controversial econometric techniques; it allows for some, but only some, substitution between the different types of labor; and it conveys essentially the same message as the more sophisticated CES and Translog indices. A still more conservative measure would be given by the linear index. In any case, it is quite clear that the conventional measure of the rate of unemployment grossly overstates the true rate of labor underutilization, by probably more than one half of one percentage point for most years.

Some caveats are in order. The measures of labor underutilization suggested by index-number theory as discussed require that the marginal product of each type of labor equal its marginal cost. If discrimination is present at the lower end of the wage distribution (e.g., directed against minorities who tend to have less education and higher unemployment rates than average workers), the alternative measures would bias upward the differential with the conventional measure. Monopoly conditions in the provision of labor services (e.g., trade unions) or monopsony conditions in the hiring of workers could also drive the wedge between marginal productivity and marginal cost. Moreover, working conditions might differ across jobs and might be reflected by relative wages. However, the authors feel that as a first approximation the assumption that labor receives its marginal product is reasonable, and it has a long tradition as a starting point for many analyses of economic phenomena.

Like many other economic concepts, the measures proposed here may not be easily understandable for an average non-economist. Furthermore, they are unlikely in the foreseeable future to be calculable on a monthly basis because they require more information than is routinely gathered in labor force surveys. Nonetheless, they can easily be calculated annually and should prove to be useful for economists engaged in empirical work, trying to estimate potential gross domestic product or attempting to assess wage and price pressures, for instance. Many recent empirical studies in this area use sophisticated econometric and time-series techniques, which makes it even more important not to use patently biased indicators when avoidable. In any case, the objective in this article was not to propose that the conventional rate of unemployment be abandoned or replaced, but to draw the profession's attention to the fact that it is a poor measure of the degree of underutilization of a country's labor endowment.

The Gordon Committee, appointed by President Kennedy to study unemployment and other labor market statistics, suggested that the various concepts should be objectively measured, obtainable at reasonable cost, readily understood, and easily interpretable, among other criteria (Cain, 1980). In large parts because the unemployment rate as conventionally measured satisfies these criteria, it has become the basis of much economic policy. Historically, various transfer payments have been tied to the national unemployment rate, and federal allocations to state and substate areas have been linked to its local counterparts. One would certainly view the measures of labor underutilization as inappropriate guides for the conduct of social policy. The conventional rate of unemployment unquestionably gives a better reading of the human toll of unemployment. One could even argue that when assessing the human, social, and political cost of unemployment, one should give more weight to the less privileged members of society. O n the other hand, these measures could be useful for macroeconomic policy purposes. In the case of monetary policy, for instance, policy makers often rely on Taylor rules, where the unemployment rate might be used as a proxy for the gross domestic product gap. A finer measure of the degree of labor underutilization would give them a better feel about the degree of slack in the economy.

[FIGURE 1 OMITTED]

(1.) This functional form implies that the Hicksian elasticities of complementarity arc constant; they are given by 1 - [rho]. As to the Allen-Uzawa elasticities of substitution, they are equal to 1/(1 - [rho]), and they are constant as well.

(2.) For example, Flaim (1979) has proposed adjusting the unemployment rate to account for the changing age distribution of the labor force. This adjustment would account for the maturing of the baby boom, first into young labor force ages with typically high unemployment rates and later into older ages with typically low unemployment rates. He attributes between 0.6 and 1.0 percentage point of the change in the unemployment rate between 1957 and 1977 to the changed demographic composition of the labor force. (The March rate increased from 4.3% to 7.9% over this period.)

(3.) Over the entire period, the correlation coefficient between [w.sub.h]/[w.sub.1] and [u.sub.h]/[u.sub.1] is -0.689.

(4.) A sufficient condition, for the Translog function to be globally concave is that the matrix [PHI] [equivalent to] [[[PHI].sub.hk]] be negative semi-definite. This can be imposed by setting [PHI] = -TT', where T [equivalent to] [[[tau].sub.hk]] is a lower triangular matrix.

(5.) Because ([u.sup.TL]) is exact for what is a flexible functional form, it is a superlative index in the sense of Diewert (1976).

(6.) The authors' average figure is 5.0% compared to a Bureau of Labor Statistics average of 6.5%.

REFERENCES

Cain, G. G. "Labor Force Concepts and Definitions in View of Their Purposes," in Concepts and Data Needs, vol. 1, Counting the Labor Force, National Commission on Employment and Unemployment Statistics. Washington, DC: U.S. Government Printing Office, 1980, 3-46.

Diewert, W E. "Exact and Superlative Index Numbers." Journal of Econometrics, 4, 1976, 115-45.

Flaim, P. O. "The Effect of Demographic Changes on the Nation's Unemployment Rate." Monthly Labor Review, 102, 1979, 13-23.

Jorgenson, D. W, and B. M. Fraumeni. "Relative Prices and Technical Change," in Modeling and Measuring Natural Resource Substitution, edited by E. R. Berndt and B. C. Field. Cambridge, MA: MIT Press, 1981, 17-47.

ULRICH KOHLI *

* The authors wish to thank Ramon Key for assisting with data preparation in the early stages of this study. They are also grateful to W Erwin Diewert, Kevin J. Fox, and two anonymous referees for a number of helpful comments and suggestions, but they are obviously not responsible for any errors or omissions. Much of this research was done while Kohli was at the University of Geneva, and it was partially supported by the Swiss National Science Foundation under grant #12-45777.95.

Greenwood: Professor of Economics, Center for Economic Analysis, Campus Box 257, University of Colorado, Boulder, CO 80309-0257. Phone 1-303-492-7413, Fax 1-303-492-8960, E-mail greenwood@spot.colorado.edu

Kohli: Chief Economist, Swiss National Bank, Borsenstrasse 15, P.O. Box 2800, CH-8022 Zurich, Switzerland. Phone: +41-1-631-3233/3234, Fax: +41-1-631-3188, E-mail ulrich.kohli@snb.ch

Few economic statistics command more widespread attention than the rate of unemployment. It is viewed as a key measure of a country's economic performance, and its release often makes the headlines and the evening news. It plays a central role in many key macroeconomic relationships, such as Okun's law and the Phillips curve, and it is fundamental to such concepts as the nonaccelerating inflation rate of unemployment. It is therefore essential to design it and to measure it in a way that is consistent with its purpose.

To economists, the unemployment rate is a measure of the degree of underutilization of the nation's labor endowment. The conventional rate of unemployment (the number of unemployed persons divided by the number of persons in the labor force) may seem reasonable to some, but adding up workers (e.g., physicians and dishwashers) without regard to their skills and without taking the relative price of their services into account makes little economic sense. There are few examples in economics in which aggregation is routinely carried out in so crude a fashion. To draw a blunt analogy, it is as if one measured the nation's capital stock by adding up the numbers of trucks, office buildings, and screwdrivers.

During the past 40 years, concerns in the United States about unemployment concepts and measures have led to the appointment of two blue-ribbon groups--the Gordon Committee during the 1960s and the National Commission on Employment and Unemployment Statistics (or the Levitan Commission) during the 1970s. Over the years, numerous criticisms have been directed at conventional unemployment rate statistics (Cain, 1980). These criticisms have ranged from the accuracy of the surveys that underlie the unemployment statistics to the proper concept to be measured (e.g., unemployed persons versus unemployed person-hours) to various adjustments that ought to better reflect the changing demographic composition of the labor force (e.g., an age adjustment because unemployment rates tend to vary by age). Yet the formula used to construct the index seems to have never been seriously questioned.

The productive capacity of the labor force is certainly a function of the skills and education embodied in it. This observation is presumably one reason that earnings functions based on human capital concepts include as key arguments age (a proxy for experience) and years of schooling. If the unemployment rate is relatively higher among the less skilled than among the more skilled, simply summing up the number of unemployed and dividing by the labor force will overstate the true proportion of labor's productive capacity that is unutilized. How large is the resulting bias? The answer is entirely an empirical matter on which the authors try to shed some light in this article.

In what follows, the authors propose several measures of the rate of unemployment. Each measure is exact (in the sense of Diewert, 1976) under certain conditions, but the authors believe that each one can serve as a good approximation under more general circumstances. Moreover, this article constructs lower and upper bounds of the "true" rate of labor underutilization, and it turns out that the conventional unemployment rate does not fall within these bounds. Finally, using March Current Population Survey data for 1964-95, the authors compare the various unemployment rate measures for the United States.

II. AGGREGATION, LABOR CAPACITY UTILIZATION, AND UNEMPLOYMENT

Assume that aggregate production can be modeled by a multiple-input multiple-output transformation function. Assume I outputs, J nonlabor inputs, and H categories of labor. The authors denote output quantities by [y.sub.i], the quantities of nonlabor inputs by [x.sub.j], and the number of workers in each category by [n.sub.h]. Let the aggregate transformation function be given by

(1) t([y.sub.1]...[y.sub.1], [x.sub.1]...[x.sub.j], [n.sub.1]...[n.sub.H]) = 0.

Assume that this transformation function is weakly separable between labor and the other inputs and outputs, so that it can be written as

(2) v[[y.sub.1]...[y.sub.I], [x.sub.1]...[x.sub.J], h([n.sub.1]...[n.sub.H])] = v([y.sub.1]...[y.sub.I], [x.sub.1]...[x.sub.J], n) = 0,

where

(3) n = h ([n.sub.1]...[n.sub.H]);

n is an index of aggregate labor and h(*) is the labor aggregator function. The authors assume that h(*) is increasing, quasi-concave, and linearly homogeneous.

Weak separability means that the various categories of labor can be consistently aggregated. Assuming optimizing behavior, the profit-maximization or cost-minimization problem can be decomposed into two steps. The first step involves determining the optimum labor mix, given the level of aggregate labor n and the wage rates of the different labor categories. The second step involves determining the optimum level of aggregate labor and of the other inputs and outputs, given all input and output prices. The authors (and readers) need only concern themselves with the first step of this twin decision process.

Let [w.sub.h] be the wage rate of the employed workers in the hth category. First-step optimization requires that the marginal products of each category of labor be equal to their marginal cost:

(4) [[partial]h(*)/[partial][n.sub.h]] = [w.sub.h]/w,

where w is the Lagrange multiplier and can be interpreted as the aggregate wage rate.

The H categories of labor can be identified by skills, sex, age, and so on. The authors assume that the workers within these categories are perfectly homogeneous. This is not as restrictive as it may sound because the number of categories can be increased almost at will. This article denotes the total number of workers, employed and unemployed, of the hth category by [l.sub.h]. If all workers were employed, the aggregate labor quantity index (l) would be equal to

(5) l = h([l.sub.1], [l.sub.2]...[l.sub.H]).

The rate of labor capacity utilization is therefore given by n/l, and we can define the rate of unemployment u as

(6) u [equivalent to] 1 - ([h([n.sub.1], [n.sub.2]...[n.sub.H])]/[h([l.sub.1], [l.sub.2]...[l.sub.H])]).

The true rate of unemployment therefore depends on h(*), the labor aggregator function. Unfortunately, this function is generally unknown. There are ways to get around this difficulty, however. One possibility, if the appropriate data are available, is to estimate the labor aggregator using econometric techniques. An alternative approach, one that may seem somewhat less controversial (because there is no need then to specify a stochastic specification and to select an estimation method), is based on index-number theory.

III. INDEX-NUMBER APPROACH

Probably the simplest functional form for h(*) that one can chose is the linear one:

(7) h(*) = [summation over (H/h=1)][b.sub.h][n.sub.h], [b.sub.h] [greater than or equal to] 0.

The first-order conditions (4) then imply

(8) [b.sub.h] = [w.sub.h]/w.

Making use of (7) and (8) in (6), one can express the linear version of the rate of unemployment ([u.sup.LN]) as:

(9) [u.sup.LN] = 1 - ([[summation over (H/h=1)] [w.sub.h][n.sub.h]]/[[summation over (H/h=1)] [w.sub.h][l.sub.h]]).

The conventional measure of the unemployment rate (u) can be viewed as a special case of (7), that where all [b.sub.h]'s are arbitrarily set to unity:

(10) h(*) = [summation over (H/h=1)] [n.sub.h],

in which case

(11) u = 1 - ([[summation over (H/h=1)][n.sub.h]]/[[summation over (H/h=1)] [l.sub.h]]).

To the extent that wages vary across labor categories, the first-order conditions can clearly not be met in the case of (10).

Another natural choice for the labor aggregator function is the Leontief form:

(12) h(*) = min{([n.sub.1]/[a.sub.1]), ([n.sub.2]/[a.sub.2]) ...([n.sub.H]/[a.sub.H])}, [a.sub.h] > 0.

Under cost minimization, one can set

(13) n = [n.sub.h]/[a.sub.h], [for all]h.

The corresponding rate of unemployment ([u.sup.LF]) can then be obtained as:

(14) [u.sup.LF] = 1 - ([n.sub.k]/[l.sub.k]),

where

(15) [l.sub.k] = [a.sub.k] min {([l.sub.1]/[a.sub.1]), ([l.sub.2]/[a.sub.2]) ... ([l.sub.H]/[a.sub.H])}.

This implies that

(16) [u.sup.LF] = min {[u.sub.1], [u.sub.2]...[u.sub.H]},

where [u.sub.h] [equivalent to] 1 - [n.sub.h]/[l.sub.h] is the unemployment rate for the hth category of labor.

The Leontief aggregator function assumes that no substitution whatsoever is possible between the different categories of labor. If one category of labor is in short supply, this will severely restrict the maximum amount of aggregate labor that is available. Put in other words, actual employment would be closer to full employment than one might think, and [u.sup.LF] can therefore be viewed as a lower bound of the true rate of unemployment. The linear functional form, on the other hand, assumes that the different categories of labor are perfect substitutes for each other. This would imply that total employment could be stretched further than it first appears: [u.sup.LN] can therefore be viewed as an upper bound for the true unemployment rate.

The linear functional form assumes that the Allen-Uzawa elasticity of substitution between the different types of labor is infinitely large, whereas the Leontief function assumes that it is zero. A less restrictive assumption would be to assume that the aggregator function has the following Cobb-Douglas form:

(17) h(*) = [[PI].sup.H.sub.h=1][n.sup.[[alpha].sub.h].sub.h], [[alpha].sub.h] [greater than or equal to] 0,

where the [[alpha].sub.h]'s are unknown nonnegative parameters. The first-order conditions then can be written as

(18) [[alpha].sub.h] = ([w.sub.h][n.sub.h])/([summation over (k)][w.sub.k][n.sub.k]).

Making use of (17) and (18) in (6), we obtain the Cobb-Douglas measure of the rate of unemployment ([u.sup.CD]) as:

(19) [u.sup.CD] = 1 - [([[PI].sup.H.sub.h=1] [n.sup.[s.sub.h].sub.h]/ [[PI].sup.H.sub.h=1] [l.sup.[s.sub.h].sub.h])],

where [s.sub.h] [equivalent to] [w.sub.h][n.sub.h]/([SIGMA][w.sub.k][n.sub.k]), an observable variable.

Finally, one may want to consider the constant elasticity of substitution case. Let the aggregator function be given by: (1)

(20) h(*) = [([[delta].sub.1] [n.sup.[rho].sub.1] + [[delta].sub.2] [n.sup.[rho].sub.2] +...+ [[delta].sub.H] [n.sup.[rho].sub.1]).sup.1/[rho]],

[[delta].sub.h] > 0, [rho] [less than or equal to] 1.

The first order conditions then can be written as

(21) [[delta].sub.h] [n.sup.[rho]-1.sub.h][([summation over (k)] [[delta].sub.k] [n.sup.[rho].sub.k]).sup.(1-[rho])/[rho]] = [w.sub.h]/w.

This can be rewritten as

(22) [[delta].sub.h] [n.sup.1-[rho].sub.h]([w.sub.h]/w)[([summation oer (k)][[delta].sub.k] [n.sup.[rho].sub.k]).sup.([rho]-1)/[rho]].

Making use of this result in (20) and (6) yield the CES measure of the rate of unemployment:

(23) [u.sup.CES] = 1 - [[([summation over (h)][n.sub.h][w.sub.h]).sup.1/[rho]]/[([summation over (h)][n.sup.1-[rho].sub.h][l.sup.[rho].sub.h][w.sub.h]).sup.1/[rho]]].

IV. ESTIMATES FOR THE UNITED STATES

The article now examines the empirical significance of the problem discussed. The data are for the United States, 1964 to 1995, and are drawn from the March Current Population Survey. The focus is on educational attainment, but the authors could have focused on age (experience), or on both age and schooling, or on still more categories to which the Bureau of Labor Statistics has given credence (such as sex and marital status). (2) Because this article is concerned with educational attainment, the authors restrict the sample to persons 25 years of age and older and disaggregate labor into ii levels of educational attainment. For 1995, Table 1 reports the number of labor force members, annual earnings, and the unemployment rate by level of education. Clearly, the unemployment rate falls sharply with increased education. (3)

The first column of Table 2 shows the conventional measure of the rate of unemployment (u) computed with the help of (11).

It varies between 2.40% (in 1969) and 8.62% (in 1983). The second column shows the estimates of [u.sup.LN], which is exact if the aggregator function is linear. This index is systematically lower than u. The difference between the two indices varies between 0.23% (in 1969) and 0.87% (in 1983). The third column shows the estimate of [u.sup.CD], which is exact if the aggregator function is Cobb-Douglas. As expected, it is lower still than [u.sup.LN] and it can be seen that the difference between [u.sup.CD] and u exceeds one percentage point in some years. The fourth column shows the values of [u.sup.CES] (for p = -4); it is marginally lower than the Cobb-Douglas estimate. The fifth column reports the values of [u.sup.LF], which is exact if h(*) has the Leontief form. As expected, this index is by far the lowest. It is noteworthy that not for a single year does the conventional measure of the unemployment rate (u) fall within the upper and lower bounds as determined by [u.sup.LN] and [u.sup.LF].

The authors also report at the bottom of the table the coefficients of variation of the different series. Because the various indices are highly correlated, the coefficients of variation are of similar magnitude, although they are somewhat smaller in the case of the Cobb-Douglas and the CES indices.

Although the U.S. unemployment rate as conventionally measured drifted upward throughout much of the latter half of the twentieth century, beginning in the 1980s a structural shift appears to have occurred as the unemployment and wage gaps between skilled and unskilled workers widened dramatically. This shift impacts on the differential between the present measures and the traditional unemployment rate, and it is apparent in Table 2. The mean conventional unemployment rate increased from 4.20% during the 1964--79 period to 5.72% during the 1980-95 period. The mean Cobb-Douglas index increased by considerably less, from 3.74 to 4.95, SO that the mean difference between the two indices rose from 0.46 to 0.77 percentage points. The divergence is even larger in the case of the CBS index. Thus the indices proposed here do not perfectly mimic the conventional measure, although they obviously tend to move with it.

The results are summarized in Figure 1, which shows the path of three unemployment-rate indices: the conventional measure (u), the linear measure ([u.sup.LN]), and the Cobb-Douglas measure ([u.sup.CD]). The extent of the upward bias contained in the conventional unemployment rate measure is clearly visible. The cyclical pattern of the different series is, however, similar.

It is perhaps noteworthy that the bias appears to be greater when the unemployment rate is rising. The difference between the conventional measure and the Cobb-Douglas measure averages 0.73 for 10 years when the conventional measure rose, but 0.55 for 21 years when it fell. One possible reason for this observation is that during periods when the economy is sluggish, the least productive workers suffer disproportionate unemployment, but when the economy expands they are drawn disproportionately back to jobs.

V. ECONOMETRIC APPROACH

As suggested earlier, aggregator function h(*) can be estimated in principle. Assume that it has the CBS form. For empirical purposes, to allow for technological change and other shifts in the technology, it is useful to allow the [delta]'s to change over time. The authors therefore rewrite the function as follows:

(24) h(*) = [[([[delta].sub.1] + [[delta].sub.1T]t)[n.sup.[rho].sub.1] + ([[delta].sub.2] + [[delta].sub.2T]t)[n.sup.[rho].sub.2] +...+ ([[delta].sub.H] + [[delta].sub.HT]t)[n.sup.[rho].sub.H]].sup.1/[rho]], [[delta].sub.h] + [[delta].sub.hT]t > 0, [rho] [less than or equal to] 1,

where t is a time trend. It is convenient to express the first-order conditions in share form

(25) [s.sub.h] = ([[delta].sub.h]+[[delta].sub.hT]t)[n.sup.[rho].sub.h][[[summation over (k)] ([[delta].sub.k] + [[delta].sub.kT]t)[n.sup.[rho].sub.k]].sup.-1],

h = 1...H,

where [s.sub.h] again is the share of the hth type of labor in total labor costs.

The authors have estimated the system of share equations (25) using the nonlinear estimation procedure as implemented in SHAZAM, version 8.0, which yields estimates of the [delta]'s, the [[delta].sub.hT]'s, and [rho] (whose estimate is 0.362 with a standard error of 0.046). The authors then introduce these estimates into (24) and (6) to get an econometric estimate of the rate of unemployment, denoted [u.sup.CES]. The resulting figures are shown in Table 2. The authors find that they are significantly smaller than the conventional measure (u) suggests. In fact, they are quite close to the Cobb-Douglas estimates.

By constraining all Hicksian elasticities of complementarity to be constant, the CBS function is rather restrictive. It is therefore tempting to reestimate the aggregator function using a flexible functional form. Assume that the aggregator function has the following Translog form:

(26) h(*) = [[beta].sub.0] + [SIGMA][[beta].sub.h] ln [n.sub.h]

+ (1/2)[SIGMA][SIGMA][[phi].sub.hk] ln [n.sub.h] ln [n.sub.k]

+ [SIGMA][[delta].sub.hT]ln[n.sub.h]t+[[beta].sub.T]t

+ (1/2)[[phi].sub.TT][t.sup.2],

where [SIGMA][[beta].sub.h] = 1, [[phi].sub.hk] = [[phi].sub.kh], [SIGMA][[phi].sub.hk] = 0, and [SIGMA][[delta].sub.hT] = 0. Again, expressing the first-order conditions in share form, one gets

(27) [s.sub.h] = [[beta].sub.h] + [SIGMA] [[phi].sub.hk] ln [n.sub.k]+[[delta].sub.hT]t,

h = 1...H.

Estimation of large-size Translog aggregator functions (remember that H = 11 in this case) almost invariably leads to curvature condition problems. The authors have therefore imposed concavity using the procedure of Jorgenson and Fraumeni (1981). (4) Estimation of (27) subject to this reparameterization, again using the nonlinear estimation procedure implemented in SHAZAM, makes it possible to get a Translog measure of the rate of unemployment. Estimates of this index ([u.sup.TL]) are reported in the last column of Table 2. (5) The authors find that the Translog measure comes quite close to the Cobb-Douglas index, and it is substantially less than the conventional rate.

VI. CONCLUSION

This article proposes a number of alternative measures of the unemployment rate based on index number theory as well as on econometric techniques. These measures are exact for specific labor aggregator functions. The results for the United States show that the conventional unweighted unemployment rate yields an overestimate of about 0.6 percentage point, on average, or about 13.9% relative to the Cobb-Douglas measure. The bias appears to have increased together with the widening unemployment and wage gaps between skilled and unskilled workers starting in the 1980s.

The conventional unemployment rate for persons 25 years of age and over is considerably lower than the official unemployment rate for persons age 16 and over. (6) This lower rate is obviously due to the procedure of cutting Out the young end of the labor force age distribution, where experience and education tend to be lower and unemployment rates higher. Based on the population used by the Bureau of Labor Statistics to calculate unemployment rates, the unemployment rate measures proposed here would undoubtedly result in even greater measures of upward bias. Because it is frequently argued that the unemployment rate is biased downward due to the elimination of discouraged workers from the unemployment total, the estimates of bias reported here obviously run in the opposite direction.

Of all the unemployment indices reported here, the authors favor the Cobb-Douglas index. It is simple to calculate; it is based on index number theory rather than on more controversial econometric techniques; it allows for some, but only some, substitution between the different types of labor; and it conveys essentially the same message as the more sophisticated CES and Translog indices. A still more conservative measure would be given by the linear index. In any case, it is quite clear that the conventional measure of the rate of unemployment grossly overstates the true rate of labor underutilization, by probably more than one half of one percentage point for most years.

Some caveats are in order. The measures of labor underutilization suggested by index-number theory as discussed require that the marginal product of each type of labor equal its marginal cost. If discrimination is present at the lower end of the wage distribution (e.g., directed against minorities who tend to have less education and higher unemployment rates than average workers), the alternative measures would bias upward the differential with the conventional measure. Monopoly conditions in the provision of labor services (e.g., trade unions) or monopsony conditions in the hiring of workers could also drive the wedge between marginal productivity and marginal cost. Moreover, working conditions might differ across jobs and might be reflected by relative wages. However, the authors feel that as a first approximation the assumption that labor receives its marginal product is reasonable, and it has a long tradition as a starting point for many analyses of economic phenomena.

Like many other economic concepts, the measures proposed here may not be easily understandable for an average non-economist. Furthermore, they are unlikely in the foreseeable future to be calculable on a monthly basis because they require more information than is routinely gathered in labor force surveys. Nonetheless, they can easily be calculated annually and should prove to be useful for economists engaged in empirical work, trying to estimate potential gross domestic product or attempting to assess wage and price pressures, for instance. Many recent empirical studies in this area use sophisticated econometric and time-series techniques, which makes it even more important not to use patently biased indicators when avoidable. In any case, the objective in this article was not to propose that the conventional rate of unemployment be abandoned or replaced, but to draw the profession's attention to the fact that it is a poor measure of the degree of underutilization of a country's labor endowment.

The Gordon Committee, appointed by President Kennedy to study unemployment and other labor market statistics, suggested that the various concepts should be objectively measured, obtainable at reasonable cost, readily understood, and easily interpretable, among other criteria (Cain, 1980). In large parts because the unemployment rate as conventionally measured satisfies these criteria, it has become the basis of much economic policy. Historically, various transfer payments have been tied to the national unemployment rate, and federal allocations to state and substate areas have been linked to its local counterparts. One would certainly view the measures of labor underutilization as inappropriate guides for the conduct of social policy. The conventional rate of unemployment unquestionably gives a better reading of the human toll of unemployment. One could even argue that when assessing the human, social, and political cost of unemployment, one should give more weight to the less privileged members of society. O n the other hand, these measures could be useful for macroeconomic policy purposes. In the case of monetary policy, for instance, policy makers often rely on Taylor rules, where the unemployment rate might be used as a proxy for the gross domestic product gap. A finer measure of the degree of labor underutilization would give them a better feel about the degree of slack in the economy.

[FIGURE 1 OMITTED]

TABLE 1 Labor Force, Average Annual Earnings, and Unemployment Rate, by Level of Education: March 1995 Number of Labor Average Annual Force Members Earnings Level of Education ([l.sub.h]) ([w.sub.h]) ($) 1. Less than 1st grade 166 13,662 2. 1st-4th grade 435 14,418 3. 5th and 6th grade 986 14,886 4. 7th and 8th grade 1,266 17,133 5. 9th grade 1,011 17,132 6. 10th grade 1,442 18,780 7. 11th grade 1,552 19,287 8. 12th grade 21,704 24,043 9. 1st-3rd year college 17,555 28,660 10. 4th and 5th year college 11,721 40,355 11. 6th and more years college 6,101 58,321 Unemployment Rate ([u.sub.h]) (%) 1. Less than 1st grade 12.05 2. 1st-4th grade 10.81 3. 5th and 6th grade 7.40 4. 7th and 8th grade 8.21 5. 9th grade 8.61 6. 10th grade 9.15 7. 11th grade 9.86 8. 12th grade 5.16 9. 1st-3rd year college 4.24 10. 4th and 5th year college 2.62 11. 6th and more years college 2.15 Source: Current Population Survey, March, 1995. TABLE 2 Alternative Unemployment Rate Indices (Percentages) Year u [u.sup.LN] [u.sup.CD] [u.sup.CES] [u.sup.LF] 1964 4.59 4.08 4.03 3.99 0.84 1965 3.87 3.45 3.38 3.38 0.69 1966 3.18 2.86 2.78 2.81 0.94 1967 2.98 2.65 2.57 2.60 0.59 1968 2.76 2.49 2.42 2.45 0.72 1969 2.40 2.17 2.10 2.14 0.32 1970 3.82 3.50 3.47 3.44 0.80 1971 4.54 4.20 4.08 4.14 1.28 1972 4.30 4.02 3.91 3.98 1.45 1973 3.69 3.42 3.33 3.37 1.29 1974 3.66 3.42 3.34 3.38 1.34 1975 6.90 6.27 6.06 6.07 1.83 1976 5.85 5.38 5.23 5.25 1.80 1977 5.74 5.32 5.20 5.21 2.12 1978 4.49 4.14 4.02 4.07 1.79 1979 4.37 4.06 3.96 4.00 1.87 1980 4.79 4.37 4.26 4.27 1.56 1981 5.61 5.10 4.94 4.95 1.42 1982 7.38 6.63 6.35 6.38 1.90 1983 8.62 7.75 7.51 7.39 2.54 1984 6.39 5.71 5.54 5.51 1.99 1985 6.06 5.42 5.17 5.24 2.10 1986 5.78 5.14 4.98 4.96 1.78 1987 5.35 4.76 4.65 4.61 1.77 1988 4.47 3.96 3.85 3.84 1.17 1989 4.15 3.69 3.61 3.61 1.58 1990 4.41 3.89 3.81 3.78 1.49 1991 5.85 5.13 5.04 4.96 1.76 1992 6.47 5.70 5.54 5.50 2.18 1993 6.06 5.37 5.25 5.22 2.62 1994 5.55 4.89 4.78 4.75 2.17 1995 4.56 4.01 3.98 3.92 2.15 Coefficient 0.205 0.208 0.203 0.202 0.212 of variation Year [u.sup.CES] [u.sup.TL] 1964 4.07 3.91 1965 3.40 3.29 1966 2.78 2.70 1967 2.59 2.54 1968 2.41 2.38 1969 2.10 2.07 1970 3.45 3.44 1971 4.09 4.05 1972 3.91 3.90 1973 3.31 3.33 1974 3.32 3.34 1975 6.09 6.16 1976 5.21 5.25 1977 5.16 5.24 1978 3.99 4.08 1979 3.93 4.02 1980 4.22 4.31 1981 4.91 4.98 1982 6.45 6.59 1983 7.55 7.69 1984 5.58 5.68 1985 5.25 5.30 1986 5.01 5.07 1987 4.63 4.67 1988 3.84 3.87 1989 3.61 3.58 1990 3.79 3.76 1991 5.04 4.98 1992 5.59 5.42 1993 5.30 5.15 1994 4.81 4.66 1995 4.01 3.85 Coefficient 0.207 0.216 of variation

(1.) This functional form implies that the Hicksian elasticities of complementarity arc constant; they are given by 1 - [rho]. As to the Allen-Uzawa elasticities of substitution, they are equal to 1/(1 - [rho]), and they are constant as well.

(2.) For example, Flaim (1979) has proposed adjusting the unemployment rate to account for the changing age distribution of the labor force. This adjustment would account for the maturing of the baby boom, first into young labor force ages with typically high unemployment rates and later into older ages with typically low unemployment rates. He attributes between 0.6 and 1.0 percentage point of the change in the unemployment rate between 1957 and 1977 to the changed demographic composition of the labor force. (The March rate increased from 4.3% to 7.9% over this period.)

(3.) Over the entire period, the correlation coefficient between [w.sub.h]/[w.sub.1] and [u.sub.h]/[u.sub.1] is -0.689.

(4.) A sufficient condition, for the Translog function to be globally concave is that the matrix [PHI] [equivalent to] [[[PHI].sub.hk]] be negative semi-definite. This can be imposed by setting [PHI] = -TT', where T [equivalent to] [[[tau].sub.hk]] is a lower triangular matrix.

(5.) Because ([u.sup.TL]) is exact for what is a flexible functional form, it is a superlative index in the sense of Diewert (1976).

(6.) The authors' average figure is 5.0% compared to a Bureau of Labor Statistics average of 6.5%.

REFERENCES

Cain, G. G. "Labor Force Concepts and Definitions in View of Their Purposes," in Concepts and Data Needs, vol. 1, Counting the Labor Force, National Commission on Employment and Unemployment Statistics. Washington, DC: U.S. Government Printing Office, 1980, 3-46.

Diewert, W E. "Exact and Superlative Index Numbers." Journal of Econometrics, 4, 1976, 115-45.

Flaim, P. O. "The Effect of Demographic Changes on the Nation's Unemployment Rate." Monthly Labor Review, 102, 1979, 13-23.

Jorgenson, D. W, and B. M. Fraumeni. "Relative Prices and Technical Change," in Modeling and Measuring Natural Resource Substitution, edited by E. R. Berndt and B. C. Field. Cambridge, MA: MIT Press, 1981, 17-47.

ULRICH KOHLI *

* The authors wish to thank Ramon Key for assisting with data preparation in the early stages of this study. They are also grateful to W Erwin Diewert, Kevin J. Fox, and two anonymous referees for a number of helpful comments and suggestions, but they are obviously not responsible for any errors or omissions. Much of this research was done while Kohli was at the University of Geneva, and it was partially supported by the Swiss National Science Foundation under grant #12-45777.95.

Greenwood: Professor of Economics, Center for Economic Analysis, Campus Box 257, University of Colorado, Boulder, CO 80309-0257. Phone 1-303-492-7413, Fax 1-303-492-8960, E-mail greenwood@spot.colorado.edu

Kohli: Chief Economist, Swiss National Bank, Borsenstrasse 15, P.O. Box 2800, CH-8022 Zurich, Switzerland. Phone: +41-1-631-3233/3234, Fax: +41-1-631-3188, E-mail ulrich.kohli@snb.ch

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Author: | Greenwood, Michael J.; Kohli, Ulrich |
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Publication: | Contemporary Economic Policy |

Geographic Code: | 1USA |

Date: | Apr 1, 2003 |

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