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Fundación de Estudios de Economía Aplicada

On Various ways of Measuring Unemployment, with Applications to Switzerland by Joseph Deutsch Yves Flückiger^*** Jacques Silber^** DOCUMENTO DE TRABAJO 2007-20

June 2007

This paper was written while Jacques Silber was visiting the Fundación de Estudios de Economía Aplicada (FEDEA), Madrid. He wishes to thank FEDEA for its warm hospitality. All the authors acknowledge the financial support of SECO, State Secretariat for Economic Affairs, Switzerland, for a research project entitled "Analysis of regional unemployment inequalities in Switzerland".

* Department of Economics, Bar-Ilan University, 52900 Ramat-Gan, Israel. ** Département d'économie politique, Université de Genève, UNI MAIL, 40 bd du Pont d'Arve, 1211 Genève 4, Switzerland.

Los Documentos de Trabajo se distribuyen gratuitamente a las Universidades e Instituciones de Investigación que lo solicitan. No obstante están disponibles en texto completo a través de Internet: http://www.fedea.es. These Working Paper are distributed free of charge to University Department and other Research Centres. They are also available through Internet: http://www.fedea.es.

This paper discusses first various ways of measuring unemployment and, borrowing ideas from the poverty measurement literature, proposes four more general unemployment indices which are parallel to the Sen poverty index, to its generalization by Shorrocks, to the FGT and to the Watts poverty indices. It then presents an empirical illustration based on Swiss data at the level of the “canton”. More precisely, using the so-called Shapley decomposition, it computes the contribution to the difference between the value of each of these four unemployment indices in a given “canton” and in Switzerland as a whole, of three components measuring respectively the impact of differences in the traditional unemployment rate, in the average unemployment duration and in the inequality in the unemployment durations. The paper ends by discussing the impact on the results obtained of assumptions made concerning the maximum unemployment duration.

Abstract

Key Words: FGT poverty index, inequality, Sen poverty index, Shapley decomposition, Switzerland, unemployment duration, Watts poverty index.

J.E.L. Classification: I32 – J64

1. Introduction

Starting with the pathbreaking work of Sen (1976) numerous studies have attempted during the past thirty years to measure the extent of poverty. Part of this work has been theoretical, taking either an ordinal or a cardinal approach to the measurement of poverty (see, Zheng, 1997, for a good survey of the work in this field) but there have been also many empirical studies of the extent of poverty and these works have generally taken a look at what is known as the "Three I's of poverty" (see, Jenkins and Lambert, 1997), that is, its incidence (the percentage of poor in the population), its intensity (how far are the poor from some agreed upon poverty line) and the inequality of poverty (how unequal are the incomes of the poor). Despite the numerous studies that have been published one has to stress the fact that the most popular measure of poverty remains, for both, politicians and the public at large, the headcount ratio which gives the percentage of poor in the population.

This need for a simple index explains probably also why, in another field, unemployment, a simple measure such as the unemployment rate, remains the most popular measure of unemployment. There have been however in recent years some attempts to derive more sophisticated measures of unemployment that would take into account not only the percentage of individuals who are unemployed but also the mean duration of unemployment and even the inequality of these durations (see, for example, the works of Sengupta, 1990, Paul, 1992, Shorrocks, 1992, 1993, Riese and Brunner, 1998, and more recently Basu and Nolen, forthcoming). Some of these works have stressed also the importance of the distinction between the total unemployment duration experienced by an individual and the various spells of unemployment that he experienced.. But clearly the literature on unemployment measurement is much less abundant than that on income inequality or poverty measurement.

The purpose of this paper is to borrow some of the ideas that have appeared in the studies that have just been cited, propose some measures of unemployment that are more sophisticated than the unemployment rate and apply them to data on unemployment in the various Swiss cantons during the period 1993-2005. The paper is organized as follows. Section II discusses various ways of measuring unemployment and, borrowing ideas from the poverty measurement literature, proposes four more general unemployment indices which are parallel to the Sen poverty index, to its generalization by Shorrocks, to the FGT and to the Watts poverty indices. Section III gives then an empirical illustration based on Swiss data at the level of the “canton”. Using the so-called Shapley decomposition it computes the contribution to the difference between the value of each of these four unemployment indices in a given “canton” and in Switzerland as a whole, of three components measuring respectively the impact of differences in the traditional unemployment rate, in the average unemployment duration and in the inequality in the unemployment durations. The paper ends by discussing the impact on the results obtained of assumptions made concerning the maximum unemployment duration.

2. The Methodology:

A) On Various Ways of Measuring Unemployment:

Two indicators are commonly used to measure unemployment. The first one is the unemployment rate which measures total unemployment as a proportion of the total labor force. This measure is obtained by asking individuals at some point of time t whether they are currently unemployed. The second indicator refers to the mean duration of unemployment. However, as stressed by Sengupta (1990) and Shorrocks (1993), there is much less agreement among economists about the way this mean duration should be measured. Several suggestions have in fact been made.

The first one is based on answers to a question like “If you are currently unemployed, for how long have you been unemployed?”. When this type of data is taken into account to compute durations of unemployment, one looks in fact at the distribution of “interrupted spells of employment” (Shorrocks, 1993).

A second possibility was suggested by Akerlof and Main (1981). It looks at the distribution of the completed spells of unemployment of those who are currently (at some time t) unemployed. In other words, whereas the first approach looks “backward”, the second one looks “forward”.

A third approach would also take a “backward look” and ask persons who are unemployed at some time t for how long they have been unemployed during a given period in the past (e.g. a year), no matter whether this unemployment duration included one or more spells of unemployment.

Rather than looking at the mean duration of unemployment according to each of the three approaches previously mentioned we could also look at the distribution of these durations and compute some index of inequality of these durations, such as the Gini index.

We can however think of a way of extending the first approach which stresses the concept of “interrupted spell of unemployment”. This approach often assumes that the unemployment rate which serves as reference is that observed in December. It is however possible to base computation of the unemployment rate on data which are available for each of the 12 months and compute the expected monthly unemployment rate over a period of 12 months.

Let be an indicator of unemployed defined as follows:

if individual i was unemployed in month j and otherwise.

The expected monthly unemployment rate (U/N) during year t will then be defined as

\[(\mathrm{U} / \mathrm{N}) = (1 / 1 2) (1 / \mathrm{N}) (\Sigma_ {\mathrm{i} = 1 \text { to N }} \Sigma_ {\mathrm{j} = 1 \text { to 12 }} \mathrm{v} _ {\mathrm{ij}})\tag{1}\]

This is in fact the way an annual unemployment rate is defined.

We may similarly define the mean duration of unemployment (among the unemployed) as the expected mean duration over all of the 12 months for which data are available. Let denote the cumulative number of days of unemployment of individual i in month j. The expected mean duration of unemployment in year t, on the basis of the first approach and assuming we have data for 12 months, may then be defined as

\[\mathrm{D} _ {\mathrm{A}} = (1 / 1 2) (1 / \mathrm{N}) \sum_ {\mathrm{i} = 1 \text {to N}} \sum_ {\mathrm{j} = 1 \text {to 12}} \left(\mathrm{D} _ {\mathrm{ij}} \times \mathrm{v} _ {\mathrm{ij}}\right)\tag{2}\]

The present section has thus shown that depending on the approach adopted one may obtain very different values for the unemployment rate, the mean cumulative unemployment duration as well as for the Gini index of these cumulative unemployment durations.

There is however an additional issue. We have hitherto analyzed separately three types of indicators of unemployment: the traditional unemployment rate, the mean duration of unemployment and a measure of the inequality of these durations. The next section, using some previous work of Shorrocks (1993), will show that it is possible to construct a new measure of unemployment that will take into account all these three aspects of unemployment.

B) Deriving a more complete measure of unemployment:

As mentioned previously the data that are available often give the cumulated number of days at month j during which individual i has been unemployed without interruption (see the hypothetical example presented in table 1).

Let now denote this cumulated number of days of unemployment.

We may therefore write that

\[\mathrm{D} _ {\mathrm{ij}} = \mathrm{D} _ {\mathrm{i}, \mathrm{j} - 1} + \mathrm{d} _ {\mathrm{ij}} \text {if} \mathrm{v} _ {\mathrm{ij}} = 1 \quad \text {and} \mathrm{D} _ {\mathrm{ij}} = 0 \text {if} \mathrm{v} _ {\mathrm{ij}} = 0.\tag{3}\]

Let now be the cumulated value of , that is,

\[\mathrm{V} _ {\mathrm{ij}} = \mathrm{V} _ {\mathrm{i}, \mathrm{j} - 1} + \mathrm{v} _ {\mathrm{ij}} \text {if} \mathrm{v} _ {\mathrm{ij}} = 1 \text {and} \mathrm{V} _ {\mathrm{ij}} = 0 \text {if} \mathrm{v} _ {\mathrm{ij}} = 0\tag{4}\]

To illustrate the inequality in these cumulated days of unemployment we can draw the following graph which has been called unemployment duration profile curve by Shorrocks (1993).

In Figure 1 we plotted on the horizontal axis the cumulative values of the months of unemployment during year t of those who were unemployed in year t, that is, of

. This means in fact that we plotted the cumulative values

On the vertical axis we plotted the cumulative values of the cumulative number of days of unemployment among the unemployed, that is, of (1/12)

On both the horizontal and vertical axes the individuals are ranked by decreasing values of these cumulated durations

Such a plot gives us the curve OHAM. It is easy to see, using (1) and (4), that the horizontal coordinate of A (the length OB) is equal to the annual unemployment rate (U/N) where U is the total number of individuals unemployed in year t and N the size of the labor force.

The vertical coordinate of A (the length AB) will be equal to which is actually equal to the average cumulative duration of unemployment (in days) per individual in the labor force.

It is then easy to derive that the slope of OA will be equal to the ratio

\[\begin{array}{l} (1 / 1 2) (1 / \mathrm{N}) (\Sigma_ {\mathrm{i} = 1 \text {to N}} \Sigma_ {\mathrm{j} = 1 \text {to 12}} \mathrm{D} _ {\mathrm{ij}}) / (1 / 1 2) (1 / \mathrm{N}) (\Sigma_ {\mathrm{i} = 1 \text {to N}} \Sigma_ {\mathrm{j} = 1 \text {to 12}} \mathrm{v} _ {\mathrm{ij}}) \\ = (\Sigma_ {\mathrm{i} = 1 \text {to N}} \Sigma_ {\mathrm{j} = 1 \text {to 12}} \mathrm{D} _ {\mathrm{ij}}) / (\Sigma_ {\mathrm{i} = 1 \text {to N}} \Sigma_ {\mathrm{j} = 1 \text {to 12}} \mathrm{v} _ {\mathrm{ij}}) \end{array}\tag{5}\]

which represents in fact the average cumulative number of days of unemployment among individuals who have been unemployed at some time in year t.

Let now refer to the Gini index of the cumulative unemployment durations and let OPA denote the straight line OA. The area OHAB is therefore equal to the sum of the area OHAP and of the triangle OPAB. The area of this triangle OPAB is clearly equal to since the tangent of the slope AOB is equal to

The area that lies between the curve OHA and the line OPA by construction looks like the area lying between a Lorenz curve and a diagonal and such an area is generally equal to half the Gini index of the variable whose cumulative values have been plotted. However since this "diagonal" OPA does not end at a point whose coordinates are (1,1) but at point A whose coordinates are (U/N) and (U/N) it is easy to derive that the area between the curve OHA and the line OPA is equal to

\[(1 / 2) \mathrm{G} (\mathrm{D} _ {\mathrm{ij}}) (\mathrm{U} / \mathrm{N}) (\mathrm{U} / \mathrm{N}) \mathrm{D} _ {\mathrm{A}} = (1 / 2) \mathrm{G} (\mathrm{D} _ {\mathrm{ij}}) (\mathrm{U} / \mathrm{N}) \mathrm{D} _ {\mathrm{LF}}.\]

The sum M of the two areas OHA and OPBA will therefore be equal to

\[\mathrm{M} = (1 / 2) (\mathrm{U} / \mathrm{N}) (\mathrm{U} / \mathrm{N}) \mathrm{D} _ {\mathrm{A}} (1 + \mathrm{G} (\mathrm{D} _ {\mathrm{ij}})) = (1 / 2) (\mathrm{U} / \mathrm{N}) \mathrm{D} _ {\mathrm{LF}} (1 + \mathrm{G} (\mathrm{D} _ {\mathrm{ij}}))\tag{6}\]

This indicator M may be considered as a generalized measure of unemployment. As may be observed, M is an increasing function of the probability for an individual to be unemployed during a randomly selected month (which is really the way the annual unemployment rate (U/N) is defined). This indicator M increases also with the average value in the whole labor force of the cumulative number of days of unemployment. Finally M will be higher, the more unequal these cumulative numebr of days of unemployment durations are, since it increases with

When these cumulative durations of unemployment are standardized it turns out (see the proof in Appendix 1) that this indicator M is in fact equal to half the value of the product of the unemployment rate (U/N) times "Sen's unemployment index", an index which is obtained by applying to the measurement of unemployment Sen's (1976) index of unidimensional of poverty¹.

PT¹ TP This is in fact only the asymptotic value of Sen's (1976) index, that is, it assumes that the size of the labor force is big enough.

More precisely let be equal to where represents the maximal number of days of cumulative unemployment (it could correspond to one, two ore even more years). represents therefore the number of days during which individual i was employed during the period under consideration (one, two or even more years, depending on how it is defined). Let now be equal to the ratio

where is evidently equal to

Let also be equal to the ratio and to the ratio

When applied to the analysis of unemployment, Sen's (1976) poverty index may therefore be written as

\[\mathrm{S} _ {\mathrm{U}} = (\mathrm{U} / \mathrm{N}) [ \mathrm{f} _ {\mathrm{A}} + (1 - \mathrm{f} _ {\mathrm{A}}) \mathrm{G} (\mathrm{e} _ {\mathrm{ij}}) ]\tag{7}\]

where is the Gini index of the cumulative relative employment duration . Note that, as in the case of Sen's poverty index, the formulation for holds only if U, the total number of unemployed individuals in year t, is big enough.

Appendix 1 shows also that another possible measure of unemployment is twice the area lying under the curve OHAM, that is the area OHAMQBO in Figure 1. We then obtain (see Appendix 1) what Shorrocks (1995) called "The Revisited Sen Poverty Index which in the case of unemployment measurement will be expressed as

\[\mathbf {S} _ {\mathrm{UR}} = \left\{ \right.\left[ \right. (\mathrm{U} / \mathrm{N}) ^ {2} \left( \right.(1 - \mathrm{f} _ {\mathrm{A}}) \mathrm{G} (\mathrm{e} _ {\mathrm{ij}}) \left. \right] + \left[ (\mathrm{U} / \mathrm{N}) \mathrm{f} _ {\mathrm{A}} (2 - (\mathrm{U} / \mathrm{N})) \right]\left. \right\}\tag{8}\]

Another very popular poverty index is the so-called FGT index (Foster, Greer and Thorbecke, 1984). When applied to the measurement of unemployment this index will be expressed as

\[\mathrm{FGT} _ {\mathrm{U}} = (1 / 1 2) (1 / \mathrm{N}) \Sigma_ {\mathrm{i} = 1 \text {to U}} \Sigma_ {\mathrm{j} = 1 \text {to 12}} (\mathrm{f} _ {\mathrm{ij}}) ^ {\alpha}\tag{9}\]

It is easy to observe that when

and that when

Let us now take the case of

We may then write that

\[\begin{array}{l} \mathrm {FGT_ {U}} = (1 / 1 2) (1 / \mathrm{N}) \Sigma_ {\mathrm{i} = 1 \text {to U}} (\mathrm {f_ {ij}}) ^ {2} \\ = (1 / 1 2) (\mathrm{U/N}) \{(1 / \mathrm{U}) \Sigma_ {\mathrm{i} = 1 \text {to U}} [ (\mathrm {f_ {ij}} - \mathrm {f_ {A}}) + \mathrm {f_ {A}} ] ^ {2} \} \end{array}\]

\[\mathrm{FGT} _ {\mathrm{U}} = 1 / 1 2) (\mathrm{U} / \mathrm{N}) \left\{\operatorname{Var} \left(\mathrm{f} _ {\mathrm{ij}}\right) + \left(\mathrm{f} _ {\mathrm{A}}\right) ^ {2} = (1 / 1 2) (\mathrm{U} / \mathrm{N}) \left(\mathrm{f} _ {\mathrm{A}}\right) ^ {2} \left\{1 + \text {Coef. Var.} \left(\mathrm{f} _ {\mathrm{ij}}\right) \right. \right\}\tag{10}\]

where Var(.)refers to the variance and Coef. Var (.) to the coefficient of variation of a variable. So in the case where we observe, as in the case of the indices and , that the index is a function of the unemployment ratio (U/N), the average cumulative unemployment duration and of a measure of the dispersion of the relative cumulative unemployment durations, in this case, the coefficient of variation of these relative cumulative durations.

Finally we can also apply to the analysis of unemployment the poverty index defined by Watts (1969). When applied to the measurement of unemployment this index will be written as

\[\mathrm{W} _ {\mathrm{U}} = (1 / \mathrm{N}) \sum_ {\mathrm{i} = 1 \text { to U }} \log (\mathrm{E} _ {\mathrm{M}} / \mathrm{E} _ {\mathrm{ij}})\tag{11}\]

Expression (11) may however be also written as

\[\mathrm{W} _ {\mathrm{U}} = (\mathrm{U} / \mathrm{N}) \left[ \sum_ {\mathrm{i} = 1 \text {to U}} (1 / \mathrm{U}) \log \left(\mathrm{E} _ {\mathrm{M}} / \mathrm{E} _ {\mathrm{A}}\right) + \sum_ {\mathrm{i} = 1 \text {to U}} (1 / \mathrm{U}) \log \left(\mathrm{E} _ {\mathrm{A}} / \mathrm{E} _ {\mathrm{ij}}\right) \right] \tag {12}\]

where is equal to the average of the employment durations .

Note however that the first expression under square brackets on the R.H.S. of (12) may also be written as

\[\mathrm{W} _ {\mathrm{R}} = \log \left(\mathrm{E} _ {\mathrm{M}} / \mathrm{E} _ {\mathrm{A}}\right)\tag{13}\]

so that measures somehow the percentage difference between the maximum cumulative duration of employment and the average cumulative duration of employment. In other words indicates somehow to which percentage of the maximal employment duration the average cumulative unemployment duration corresponds.

The second expression under square brackets on the R.H.S. of (12) may be written as

\[\mathrm {L_ {U}} = \log (\mathrm {E_ {A}}) - \log (\mathrm {E_ {G}})\tag{14}\]

where refers to the geometric mean of the cumulative employment durations . It is then easy to observe that measures the percentage difference between the arithmetic and the geometric means of the cumulative employment durations . Since the gap between the arithmetic and a geometric mean of a variable is usually considered as an indicator of the inequality of the distribution of this variable (see, Champernowne, 1953) the indicator measures in fact the inequality of the cumulative employment durations among the individuals who were unemployed at least part of the year t. This indicator is also known under the name of Bourguignon (1979) – Theil (1967) inequality index.

Combining expressions (11) to (14) we end up with

\[\mathbf {W} _ {\mathrm{U}} = (\mathbf {U} / \mathbf {N}) \left(\mathbf {W} _ {\mathrm{R}} + \mathbf {L} _ {\mathrm{U}}\right)\tag{15}\]

Like the indices and defined earlier the index is a function of three components measuring respectively the unemployment rate, the gap between maximal employment duration and the average cumulative unemployment duration and finally the inequality in the employment durations among those who were unemployed at least part of the time in year t.

C) Comparing unemployment measures in different areas:

The four indices and that have been defined previously may be computed for each area j in a given country as well as for the whole country. The difference between the value that a given index (one of the four previously mentioned) takes for the whole country and for a given area may then be decomposed, using the so-called Shapley decomposition procedure, into three components (see, Appendix 2) that measure respectively the extent of differences between the country as a whole and a given area in the unemployment rate, in the gap between the maximal and average values of the cumulative number of days of unemployment and finally in the inequality of the cumulative number of days of unemployment (employment) among those who were unemployed at least part of the year.

3. An Empirical Illustration:

The concepts that have been previously presented have been applied to data on unemployment in the 26 Swiss areas which are called "cantons" for the period 1993 to 2005. To illustrate these concepts we have used the approach where unemployment is measured via the information on the expectancy of the interrupted spells of unemployment over the whole year. But we clearly could have used one of the three other approaches.²

² In fact computations similar to those presented in this section but based on the other three approaches are available from the authors upon request.

As was just mentioned we look at the values of the cumulative durations of unemployment as they are given each month of the year for the various unemployed individuals. More precisely we work with the expectancy of these cumulative duration of unemployment on the basis of data on cumulative unemployment for each of the 12 months of the year. As maximal value for these cumulative durations we assumed again that it was equal to 365 days.

In Table 1 we give data on the unemployment rate (the expectancy of the monthly unemployment rates which is also the value of the annual unemployment rate), on the average value observed during the year of the cumulative unemployment durations and finally on the Gini index of these cumulative durations of unemployment, for Switzerland as a whole and for each canton in 2005. It appears that the highest rates of unemployment are observed in the cantons of Geneva, Vaud and Tessin and the lowest in the cantons of Uri, Appenzell Ai and Obwalden. As far as average durations of unemployment are concerned the highest averages are observed in the cantons Geneva and Vaud. The lowest average durations are observed in the cantons of Graubünden, Uri and Obwalden. Finally the highest levels of unemployment duration inequality are observed in the cantons of Valais, Graubünden and Obwalden and the lowest in the cantons of Geneva, Neuchâtel and Vaud.

In Table 2 we give the values in 2005, for Switzerland as a whole as well as for each canton, of the three indices of unemployment which we have defined previously, the Sen index, , Shorrocks´ generalization of the Sen Index, and the Foster, Greer and Thorbecke Index . It appears that the highest values of the unemployment indices are observed in the cantons of Geneva, Vaud, Tessin and Neuchâtel and the lowest in the cantons of Uri, Appenzell AI, Obwalden and Graubünden.

In Tables 3 to 5 we give, for each of the three unemployment indices previously mentioned, the results of the Shapley decomposition of the gap between the national value of these indices and that observed in each canton. Such a breakdown allows one to identify the respective contributions to this gap of differences in the unemployment rate, in the average durations of unemployment as well as in the inequality of unemployment (or employment, depending on the index selected) durations.

It appears that in the four cantons with the highest positive (Geneva and Vaud) or negative (Uri and Obwalden) gap, the contribution of differences between the unemployment rate in these cantons and that in Switzerland account for 60% to 78% of the overall gap, depending on the index of unemployment which is used. Note however that for these four cantons the contribution of the two other factors (differences in the average unemployment durations and differences in the inequality of unemployment durations) cannot be ignored, this being specially true for the average unemployment duration.

Analyzing the Impact of the Maximum Unemployment:

In this section we want to analyze the impact that the choice of a maximum duration of unemployment may have on the results of the “Shapley decomposition”. For simplicity we will only consider the case where we take a “backward looking” approach and measure unemployment via the information on the interrupted spells of unemployment as they are observed in the month of December. Here also we limit the analysis to the year 2005.

We will compare three cases:

- the maximum duration is assumed to be 365 days (as in section C)

- the maximum duration of unemployment is that actually observed in December 2005

the maximum duration of unemployment is 5000 days, which is slightly above the greatest unemployment duration observed in all years for which data are available (1994 to 2005)

We present the results only for the decomposition for the Sen index of unemployment S_U.

Finally in each table we give first the actual gap between the value of the index in Switzerland and its value in a given canton and then the contributions, in percentage terms, of the three determinants of the indices of unemployment, namely the actual unemployment rate, the average duration of unemployment and the inequality of unemployment (or employment) durations. All these results are presented in Tables 6 to 8.

S_UR

3 Results relative to the decomposition of the Shorrocks´ generalization of Sen´s unemployment index and of the Foster, Greer and Thorbecke FGT index of unemployment are available upon request from the authors.

Table 1: Looking at the “expected” interrupted spells of unemployment in 2005. Value of the Unemployment Rate (U/N), of the average value of the cumulative unemployment durations and of the Gini Index of unemployment durations for Switzerland and the various cantons ⁴

4 Similar tables for the other years are available upon request from the authors.

Table 2: Looking at the “expected” interrupted spells of unemployment in 2005. Value of the three indices of unemployment (the Sen Index Shorrocks extension of the Sen Index and the index for Switzerland and the various cantons ⁵

⁵ Similar tables for the other years are available upon request from the authors.

Table 3: Looking at the “expected” interrupted spells of unemployment in 2005. Shapley Decomposition of the difference between the value of the Sen Index for Switzerland as a whole and for each canton

⁶ Similar tables for other years are available upon request from the authors.

Table 4: Looking at the “expected” interrupted spells of unemployment in 2005. Shapley Decomposition of the difference between the value of Shorrocks Generalization of the Sen Index of Unemployment for Switzerland as a whole and for each canton

⁷ Similar tables for other years are available upon request from the authors.

Table 5: Looking at the “expected” interrupted spells of unemployment in 2005. Shapley Decomposition of the difference between the value of the Foster, Greer and Thorbecke Index FGT_U of Unemployment for Switzerland as a whole and for each canton 8

⁸ Similar tables for other years are given in Appendix 3D.

Let us first consider the first case where the maximum duration of unemployment is still 365 days (table 6). It appears that although the greatest relative contribution to the unemployment indices is that of the unemployment rates, there are cases where the contribution of the average unemployment duration is quite high in comparison to that of the unemployment rate. When using the index this is, for example, the case of the cantons of Zurich (ZH) and of Valais (VS).

If we now take the case where the maximum unemployment duration is that actually observed in December 2005 (table 7) we observe a somehow different picture. The cantons of Zurich (ZH) and Valais (VS) are not anymore the only ones, for the index , for which the contribution of the average unemployment duration (in percentage term) is important. This is now also the case for the cantons of Basel Stadt (BS), Graubünden (GR) and even Jura (JU).

Finally the results of the cases where we take as maximum unemployment duration 5000 days (table 8) are very close to those where the maximum unemployment duration is that actually observed in December 2005 so that we will not analyze them separately.

In all the results what however is quite striking is the growing role of the average unemployment duration when a longer maximal unemployment duration is selected and even to some degree the more important impact of the inequality of unemployment durations. Take, for example, the case of the canton of Geneva (GE) for which the value of the index of unemployment, whatever the index, is always the highest of all cantons. Here we have observed that whereas 60% to 74% (depending on the index) of the gap between the value of the index in Switzerland as a whole and in the canton of Geneva was the consequence of differences in unemployment rates, when the maximum unemployment duration is 365 days, this impact of the unemployment rate goes down to 45% to 60% (depending on the index) when a longer maximal unemployment duration is selected (either the maximal duration observed or 5000 days). Moreover even the impact of the inequality in unemployment (or employment) durations is now greater. For the canton of Geneva, depending on the index, it varied from 0% to 11% when the maximum unemployment duration is 365 days. With a greater maximal duration the impact of this inequality in unemployment durations for the canton of Geneva varies now from 0% to 23%.

4. Concluding Comments:

This paper attempted to borrow some ideas from the poverty measurement literature to propose some more sophisticated measures of unemployment which take into account not only the unemployment rate but also the average duration of unemployment and the inequality in the distribution of these durations. It also applied the so-called Shapley decomposition to decompose the difference between the value of an unemployment index at the national and at the regional level into three contributions reflecting the three aspects of unemployment that have just been mentioned.

An empirical illustration based on Swiss data for the period 1993-2005 seems to confirm the usefulness of such an approach. It also showed the relative sensibility of the decomposition results to the maximum unemployment duration that has been selected.

Table 6: Looking at the interrupted spells of unemployment in 2005. Shapley Decomposition in percentage terms of the difference between the value of the Sen Index for Switzerland as a whole and for each canton, under the assumption that the maximum duration of unemployment is 365 days.

Table 7: Looking at the interrupted spells of unemployment in 2005. Shapley Decomposition in percentage terms of the difference between the value of the Sen Index for Switzerland as a whole and for each canton, under the assumption that the maximum duration of unemployment is that observed in 2005.

Table 8: Looking at the interrupted spells of unemployment in 2005. Shapley Decomposition in percentage terms of the difference between the value of the Sen Index for Switzerland as a whole and for each canton, under the assumption that the maximum duration of unemployment is 5000 days.

Appendix 1: On the link between the unemployment duration profile curve and Sen's index of poverty (when applied to the measurement of unemployment)

Recall (see expression (6) ) that the area OHAB may be expressed as

\[\mathrm{M} = (1 / 2) (\mathrm{U} / \mathrm{N}) (\mathrm{U} / \mathrm{N}) \mathrm{D} _ {\mathrm{A}} (1 + \mathrm{G} (\mathrm{D} _ {\mathrm{ij}}))\tag{A-1}\]

If we now normalize th durations and by dividing them by their maximal value we may write that

\[\left(\mathrm{M} / \mathrm{D} _ {\mathrm{M}}\right) = (1 / 2) (\mathrm{U} / \mathrm{N}) (\mathrm{U} / \mathrm{N}) \left(\mathrm{D} _ {\mathrm{A}} / \mathrm{D} _ {\mathrm{M}}\right) \left(1 + \mathrm{G} \left(\mathrm{D} _ {\mathrm{ij}} / \mathrm{D} _ {\mathrm{M}}\right)\right)\tag{A-2}\]

or as

\[\left(\mathrm{M} / \mathrm{D} _ {\mathrm{M}}\right) = (1 / 2) (\mathrm{U} / \mathrm{N}) (\mathrm{U} / \mathrm{N}) \left(\mathrm{f} _ {\mathrm{A}}\right) (1 + \mathrm{G} \left(\mathrm{f} _ {\mathrm{ij}}\right))\tag{A-3}\]

where and

But where

We may therefore write that

(A-4)

Using however the well-known formulas for expressing the Gini index of a sum of components (see, for example, Silber, 1989) we may rewrite as

\[\mathrm{G} (1 - \mathrm{e} _ {\mathrm{ij}}) = (1 / (1 - \mathrm{e} _ {\mathrm{A}})) (\mathrm{Ps}. \mathrm{G} (1) + ((- \mathrm{e} _ {\mathrm{A}}) / (1 - \mathrm{e} _ {\mathrm{A}})) (\mathrm{Ps}. \mathrm{G} (\mathrm{e} _ {\mathrm{ij}}))\tag{A-5}\]

where Ps. refers to the Pseudo-Gini and is equal to the average value of the standardized employment durations

Note however that the Pseudo-Gini of a vector of the constant 1 is 0 so that the first expression on the R.H.S. of is zero. Since (see, Silber, 1989) in the second expression on the R.H.S. of (A-2) the Pseudo-Gini of implies that the elements are ranked by decreasing values of the expressions we easily derive that

\[\mathrm{Ps.} \mathrm{G} \left(\mathrm{e} _ {\mathrm{ij}}\right)) = - \mathrm{G} \left(\mathrm{e} _ {\mathrm{ij}}\right)\]

(A-6)

Combining (A-4) and (A-6) we obtain

\[\mathrm{G} \left(\mathrm{f} _ {\mathrm{ij}}\right)) = \mathrm{G} \left(1 - \mathrm{e} _ {\mathrm{ij}}\right) = \left(\left(- \mathrm{e} _ {\mathrm{A}}\right) / \left(1 - \mathrm{e} _ {\mathrm{A}}\right)\right) \left(- \mathrm{G} \left(\mathrm{e} _ {\mathrm{ij}}\right)\right) = \left(\mathrm{e} _ {\mathrm{A}} / \left(1 - \mathrm{e} _ {\mathrm{A}}\right)\right) \mathrm{G} \left(\mathrm{e} _ {\mathrm{ij}}\right)\tag{A-7}\]

and since by definition we conclude that

\[\mathbf {G} \left(\mathrm{f} _ {\mathrm{ij}}\right)) = \left(\left(1 - \mathrm{f} _ {\mathrm{A}}\right) / \mathrm{f} _ {\mathrm{A}}\right) \mathbf {G} \left(\mathrm{e} _ {\mathrm{ij}}\right)\tag{A-8}\]

Combining now (A-3) and (A-8) we end up with

\[\begin{array}{r l} & (\mathbf {M} / \mathbf {D} _ {\mathrm{M}}) = (1 / 2) (\mathbf {U} / \mathbf {N}) (\mathbf {U} / \mathbf {N}) f _ {\mathrm{A}} (1 + \mathbf {G} (f _ {\mathrm{ij}})) \\ & \quad = (1 / 2) (\mathbf {U} / \mathbf {N}) (\mathbf {U} / \mathbf {N}) f _ {\mathrm{A}} (1 + (((1 - f _ {\mathrm{A}}) / f _ {\mathrm{A}}) G (e _ {\mathrm{ij}}))) \\ & \quad = (1 / 2) (\mathbf {U} / \mathbf {N}) (\mathbf {U} / \mathbf {N}) f _ {\mathrm{A}} (1 / f _ {\mathrm{A}}) (f _ {\mathrm{A}} + ((1 - f _ {\mathrm{A}}) G (e _ {\mathrm{ij}}))) \\ & \quad = (1 / 2) (\mathbf {U} / \mathbf {N}) (\mathbf {U} / \mathbf {N}) (f _ {\mathrm{A}} + ((1 - f _ {\mathrm{A}}) G (e _ {\mathrm{ij}}))) \\ & \leftrightarrow (\mathbf {M} / \mathbf {D} _ {\mathrm{M}}) = (1 / 2) (\mathbf {U} / \mathbf {N}) S _ {\mathrm{U}} \end{array}\tag{A-9}\]

where is the application to unemployment measurement of the Sen index of poverty. (see expression (7)).

Finally since the area BAMQ is equal to it is easy to conclude, using (A-9), that the area OHAMQBO is equal to

\[\begin{array}{l} \mathrm {D_ {M}} [ (1 / 2) (\mathrm{U/N}) (\mathrm{U/N}) (\mathrm {f_ {A}} + ((1 - \mathrm {f_ {A}}) \mathrm{G(e} _ {\mathrm{ij}}))) + (1 - (\mathrm{U/N})) ((\mathrm{U/N}) \mathrm {f_ {A}} ] \\ = \mathrm {D_ {M}} [ (1 / 2) \{[ (\mathrm{U/N}) ^ {2} ((1 - \mathrm {f_ {A}}) \mathrm{G(e} _ {\mathrm{ij}})) ] + [ (\mathrm{U/N}) \mathrm {f_ {A}} ((\mathrm{U/N}) + 2 (1 - (\mathrm{U/N}))) ] \} ] \\ = \mathrm {D_ {M}} [ (1 / 2) \{[ (\mathrm{U/N}) ^ {2} ((1 - \mathrm {f_ {A}}) \mathrm{G(e} _ {\mathrm{ij}})) ] + [ (\mathrm{U/N}) \mathrm {f_ {A}} (2 - (\mathrm{U/N})) ] \} ] \end{array} \tag {A-10}\]

Expression (A-10) is in fact equal the application to unemployment measurement of what Shorrocks (1995) has called "The Revisited Sen Poverty Index" which has better properties than Sen's (1976) original index.

Appendix 2: On the Concept of Shapley Decomposition

The concept of Shapley (1953) decomposition is a technique borrowed from game theory but extended to applied economics by Shorrocks (1999) and Sastre and Trannoy (2002). Let us explain it briefly.

Assume an indicator I is a function of three determinants and is written as . I could be an index of inequality but more generally any function of variables, this function being linear or not.

There are obviously 3!=6 ways of ordering these three determinants b and c:

\[(a, b, c), (a, c, b), (b, a, c), (b, c, a), (c, a, b), (c, b, a)\tag{B-1}\]

Each of these three determinants may be eliminated first, second or third. The respective (marginal) contributions of the determinants will hence be a function of all the possible ways in which each of these determinants may be eliminated. Let for example be the marginal contribution of to the indicator

If a is eliminated first its contribution to the overall value of the indicator I will be expressed as where corresponds to the case where a is equal to zero. Since expression (1) indicates that there are two cases in which a appears first and may thus be eliminated first we will give a weight of (2/6) to this possibility.

If is eliminated second, it implies that another determinant has been eliminated first (and been assumed to be equal to 0). Expression (A-1) indicates that there are two cases in which this possibility occurs, the one denoted in (1) as and the one denoted . In the first case the contribution of a will be written as while in the second it is expressed as . To each of these two cases we evidently give a weight of (1/6).

Finally if is eliminated third, it implies that both b and c are assumed to be equal to 0. Expression (31) indicates that there are two such cases, the one denoted and the one denoted . Since we may assume that when each of the three determinants is equal to 0, the indicator I is equal to 0, we may write that the contribution of a in this case will be equal to and evidently we have to give a weight of (2/6) to such a possibility since there are two such cases.

We may therefore summarize what we have just explained by stating that the marginal contribution of the determinant a to the overall value of the indicator I may be written as

\[\begin{array}{l} C (a) = (2 / 6) [ I (a, b, c) - I (b, c) ] + (1 / 6) [ I (a, c) - I (c) ] + (1 / 6) [ I (a, b) - I (b) ] \\ + (2 / 6) I (a) \end{array}\tag{B-2}\]

One can similarly determine the marginal contribution C(b) of b and of c and then find out that

\[\mathrm{I} (\mathrm{a}, \mathrm{b}, \mathrm{c}) = \mathrm{C} (\mathrm{a}) + \mathrm{C} (\mathrm{b}) + \mathrm{C} (\mathrm{c})\tag{B-3}\]

This Shapley decomposition may be also applied in a similar way to the case where one wants to understand the respective contributions to the change over time in the value of the indicator I, this change being written as of the variations over time in the values of the three determinants b and these variations being expressed as and

In our case ∆I would refer, for example, to the difference between the value of the Sen index in a given canton and its value in the whole of Switzerland, to the difference between the unemployment rate K in the canton and in Switzerland, ∆b to the difference between the average unemployment duration in the canton and in Switzerland and finally to the difference between the inequality in unemployment durations in the canton and in Switzerland.

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