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Equilibrium unemployment-inequality correlation
Journal of Macroeconomics 34 (2012) 454–469
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Equilibrium unemployment-inequality correlation Rubens Penha Cysne ⇑, David Turchick FGV/EPGE, Getulio Vargas Foundation, Graduate School of Economics, Brazil
a r t i c l e
i n f o
Article history: Received 16 February 2011 Accepted 31 December 2011 Available online 25 February 2012 JEL classification: E20 J60 Keywords: Unemployment Income inequality Gini coefficient Job search
a b s t r a c t A vast empirical literature implies that increases in unemployment have an aggravating impact on income inequality, whence international and intertemporal inequality comparisons might be sometimes biased. We show how job-search models can be useful in better understanding this fact. In fact, in the classic Burdett and Mortensen (1998) model, as well as in one of its many possible extensions (Bontemps et al., 2000), search frictions are a force pushing the unemployment-inequality correlation in that direction: provided that the unemployment rate is no larger than 15%, a positive correlation between unemployment and inequality unequivocally emerges. 2012 Elsevier Inc. All rights reserved.
1. Introduction It is a common procedure in economics to compare income inequality measurements of different economies or of the same economy at different points in time. Sometimes, such comparisons are also carried out with the purpose of evaluating the success or failure of policies aimed at reducing inequality. Concomitantly, a vast empirical literature, using data from different countries, has already documented that inequality tends to be positively correlated with unemployment.1 Examples of this literature include Metcalf (1969), Blinder and Esaki (1978), Jäntti (1994) and Mocan (1999), using data from the United States; Nolan (1986), using data from England; Björklund (1991), using data from Sweden; Blejer and Guerrero (1990), using data from the Phillipines; and Cardoso et al. (1995), using data from Brazil. More recently, Castañeda et al. (1998), using data from the US Bureau of the Census during the 1948–86 period, have found that ‘‘the income share earned by the lowest quintile is both the most volatile and the most procyclical’’, and that the procyclicality of the income shares decrease monotonically until one reaches the top 5%. Cysne (2009) has shown that standard search models are able to generate the assessed positive correlation between unemployment and inequality. A limitation of that analysis is that it has drawn upon a class of models (based on the seminal contribution of McCall, 1970) that takes as a given the initial distribution of job offers. We show that this conclusion also holds in the class of on-the-job-search models with firms posting wages and meeting workers in a random, decentralized fashion, but at fixed frequencies.
⇑ Corresponding author. Present address: Praia de Botafogo 190 s, 1100 andar, Rio de Janeiro, RJ 22250-900, Brazil. Tel.: +55 21 3799 5832; fax: +55 21 2553 8821. E-mail address: [email protected] (R.P. Cysne). 1 We refer to those inequality measurements usually reported by researchers, which are based on cross-sectional (rather than long-run) distributions of income. 0164-0704/$ - see front matter 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmacro.2011.12.009
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As representatives of this class, two models were chosen. One is the seminal Burdett and Mortensen (1998) model, which has led to numerous important variations and extensions in several directions (see for example Postel-Vinay and Robin (2002) and Cahuc et al. (2006), in which firms are allowed to renegotiate wages with their employees who are receiving outside offers). The other is a simple case of the model of Bontemps et al. (2000), which extends the most basic Burdett–Mortensen model in the direction of firm-productivity differentiation. Both models generate pure wage dispersion – the adjective pure referring to the fact that worker heterogeneity is not necessary (cf. Acemoglu, 1999; Albrecht and Vroman, 2002; Greiner et al., 2004, for instance, where worker skill heterogeneity may also affect wage dispersion through a change in the very structure/type of the equilibrium). In this way, all the unemployment here comes only from job market frictions represented by the matching technology, and not from the possibility that some firms may endeavor to avoid the hiring of insufficiently skilled workers, or even make adjustments to their level of productivity (this would once more affect wages and their distribution). Also, there is no between-group inequality (since there is only one group of workers), and all inequality could be classified as within group. Job search models with heterogeneous workers may allow for the breaking down of unemployment and inequality in a natural way (in this respect the Theil index could be a natural choice for the inequality measure, due to its decomposability property), and study these partial effects. However, this would be beyond the scope of the present work, which attempts to analytically assess the aggregate effect of unemployment on inequality. As we shall see, a positive correlation stems even from pure wage dispersion models. In the case of heterogeneous workers, one may expect the total effect on inequality to be even larger, since the burden of unemployment is higher for low-skill workers, who endure a longer unemployment spell.2 Our analytical approach to the present issue complements the aforementioned empirical literature by providing closedform expressions directly linking income inequality to unemployment. These expressions may help the macroeconomist in understanding how this link emerges, on which parameters it may depend (unemployment benefits and firm productivity will be shown to be key parameters), and in which cases the correlation may revert its sign. At the same time, our analysis offers a word of caution to policy makers interested in the evaluation of policies aiming at the reduction of income inequality (e.g., through a rise in unemployment insurance benefits): the reported (absolute value) effect on inequality may be biased downward due to technological progress in the production function during the two points in time considered or upward due to a rise in the contact rate between firms and workers (thus bringing the unemployment rate down). In other words, by making explicit a channel through which unemployment (and some parameters of these classical models) affects inequality, this work adds value and information to cross-country and longitudinal studies such as Jaumotte et al. (2008), in the sense that inequality indices at different points in space or time should be expected to naturally differ according to basic differences in the surrounding political, technological and business environment. As an example, in Mortensen and Pissarides (1999, p.26) it is reported: ‘‘Looking at average changes over the OECD, there appears to be a negative correlation between the growth in unemployment and the rise in inequality. The United States, Canada and Sweden experienced the biggest rises in inequality and the smaller rises in unemployment (a fall in the US case). The large European countries experienced small rises or falls in inequality but big rises in unemployment (with the exception of the United Kingdom [. . .]).’’ We shall see how such correlations may be understood, in qualitative terms, in light of the simplest Burdett–Mortensen model, with no skill heterogeneity argument entering the picture (unlike Mortensen and Pissarides, 1999).3 Our presentation can be outlined in the following way. First, the Gini index of income inequality is derived as a function of the model’s parameters – which, in turn, also determine the unemployment rate. Next, we analyze the conditions under which a positive correlation between unemployment and inequality emerges. Our basic conclusion is that such a positive correlation will always be a consequence of the optimizing process assumed by these models, provided that the rate of unemployment is no greater than 28% (in Burdett and Mortensen’s model) or 15% (in Bontemps et al.’s model). The remainder of the paper is organized as follows: Section 2 provides a model-free digression on how unemployment and unemployment benefits may affect the Gini coefficient of income inequality. Section 3 provides the analysis of the unemployment-inequality correlation in the context of Burdett and Mortensen’s model, whereas Section 4 does the same using Bontemps et al.’s model. Section 5 concludes. Calculations are duly detailed in the appendices. 2. Unemployment and inequality This section provides a short digression about theoretical possibilities linking unemployment and the Gini coefficient of income inequality. We start with the case of simple discrete income distributions in order to build the intuition for the type of result to be proven later. The first two examples analyze a one-wage economy with and without unemployment insurance, respectively. It is shown that the correlation between unemployment and inequality is always positive in the first case (except when 2 3
We are indebted to an anonymous referee for making this point. Using income inequality in the place of their preferred notion of wage inequality.
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1
L
0
0
u
1
Fig. 1. Lorenz curve in an economy with one income level and no unemployment insurance.
unemployment reaches 100%), but not necessarily in the second one. A third case shows how easy it is for a negative correlation to emerge, even without unemployment insurance, when there is more than one single wage in the economy. Example four shows that if the wage distribution is independent of the unemployment rate, a positive correlation between unemployment and income should not be necessarily expected a priori. The simplest possible income distribution, here denoted by its cumulative distribution function H : Rþ ! ½0; 1, is the one where a portion u 2 [0,1) of the population is unemployed and receives nothing, while the other 1 u all have the same wage (say, w):
HðwÞ ¼
u; if w < w : 1; if w P w
This yields a Lorenz curve L as shown in Fig. 1. The well-known Gini inequality index I is twice the area between the 45 line and the Lorenz curve. In this very particular case, using the determinant formula to calculate the area of the triangle gives:
0
2
0
0 1
31
B1 6 7C I ¼ 2 @ det 4 u 0 1 5A ¼ u; 2 1 1 1
ð1Þ
precisely the unemployment measure.4 Inspired by the exact relation (1) that the Gini coefficient provides regarding this benchmark distribution – in addition to its widespread usage by governments around the world and international organisms –, we choose to employ Gini’s inequality measure throughout this paper.5 In spite of (1) implying a positive (and perfect) correlation between unemployment and inequality, the assertion that increases in unemployment lead to increases in inequality in this simple economy should be done with due caution: in the limiting case in which unemployment rises to 100%, (1) is no longer valid, and the Gini coefficient must collapse to zero. The second example keeps the hypothesis that there is only one wage in the economy, but now an unemployment insur a common feature of job-search models, enters the picture. The income distribution becomes ance b 2 ð0; wÞ,
8 > < 0; if w < b : HðwÞ ¼ u; if b 6 w < w > : 1; if w P w uÞÞ, the fraction of earnings appropriated by The Lorenz curve L takes on the form shown in Fig. 2, where d :¼ bu=ðbu þ wð1 the unemployed. Here,
2
0
6 I ¼ det 4 u
0 1 d
3
7 bÞ 1 5 ¼ u d ¼ ðw
1 1 1
uð1 uÞ ; uÞ bu þ wð1
4 This distribution is generated, for instance, in the static costly-search model with two types of workers introduced in Acemoglu (1999), provided the relevant parameters are compatible with a separating equilibrium (see Proposition 1 there). 5 Other indices could be used. Burdett pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi et al. (2004), for instance, employ the coefficient of variation in their work on crime. For the simple distribution just considered, it would equal u=ð1 uÞ.
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1
L d
0
0
1
u
Fig. 2. Lorenz curve in an economy with one income level and unemployment insurance.
1 0.9 0.8 0.7
L
1
0.6 0.5
L
2
0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 3. Intersecting Lorenz curves.
so that the sign of the unemployment-inequality correlation is undetermined:
uÞÞ þ uð1 uÞðw bÞ uÞ2 bu2 dI ð1 2uÞðbu þ wð1 wð1 bÞ bÞ ¼ ðw ¼ ðw : 2 du uÞÞ uÞÞ2 ðbu þ wð1 ðbu þ wð1 > ðð1 uÞ=uÞ2 (which is a possibility if u > 0.5), then dI/du < 0, so that the possibility that an increase in Note that if b=w unemployment results in a reduction of inequality is not as extreme as it was in the previous example, where b = 0. For example, if unemployment insurance were half the wage and the unemployment rate were 60%, an increase in unemployment would have a greater effect on moving incomes (weighted according to their corresponding fraction in the population) closer together than farther apart. This effect shall also appear in the job-search models studied in Sections 3 and 4, as a byproduct of our analysis. Our third example shows that even in economies with no unemployment insurance, it can trivially happen that I is not positively correlated with u (provided, given what has been shown above, that there is more than one nonzero wage in the economy). Indeed, consider two stylized economies, with a mass u of unemployed, a ‘‘middle class’’ of size m receiving a > w.6 In the first economy, (u, m, r) = (.20, .75, .05), and yearly wage of w, and a ‘‘rich class’’ of size r = 1 u m earning w ¼ $100; 000, the Gini coefficient for the first economy is in the second, (u, m, r) = (.16, .77, .07).7 Putting w = $20,000 and w I = 0.35, while for the second I = 0.3525 (see formula in Appendix A). That is, a smaller unemployment rate does not necessarily mean a smaller inequality. Fig. 3 shows the corresponding Lorenz curves, naturally intersecting each other. 6
Acemoglu (1999, pooling equilibrium at p. 1263) and Gaumont et al. (2006, p. 835), also generate this type of distribution, with two nonzero wage levels. Note that the distribution of wages, determined by the mass associated with m (or r) in relation to m + r, changes in this case. This will not happen in the case to be analyzed next. 7
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1
L
0
0
u
1
Fig. 4. Generic Lorenz curve in an economy with a continuous distribution of wages.
For the last example, consider an economy as the previous one, but where the unemployed receive b 2 (0,w).8 Additionally, suppose the distribution of wages is independent of u: there exists d 2 (0, 1) such that r = d(1 u). Let d = 0.1, b = $9000, ¼ $100; 000. The math done in Appendix A gives the following Gini inequality function: w = $10,000 and w
IðuÞ ¼
ð1 uÞð0:81 þ 0:19uÞ : 1:9 u
Therefore, in this simple case a 5% unemployment rate yields an approximate inequality of 0.421, while a 10% unemploy can exist such that ment gives a lower one, I 0.415. Moreover, this choice of parameters proves that no upper bound u . In fact, the Gini function above can be shown to be strictly decreasing over the entire [0, 1] interval. I0 (u) > 0 for all u < u In what follows we demonstrate that such an upper bound will in fact exist for some standard equilibrium-search models – specifically, the quite conservative 15% will do. This is to say that the search frictions described by these models act as a force pushing the correlation in the direction assessed by the available empirical evaluations. We end this section with the derivation of a general formula concerning the Gini coefficient of income inequality, given a (where w P b by equilibrium reasoning) and continuous and strictly increasing wage distribution G(w), with support ½w; w ¼ 1). The income distribution is no mass points (so that G(w) = 0 and GðwÞ
HðwÞ ¼
8 0; > > > > > < u;
if w < b if b 6 w < w
> > u þ ð1 uÞGðwÞ; if w 6 w < w > > > : 1; if w P w
:
The corresponding Lorenz function L:[0, 1] ? [0, 1] is given by
8 R wbx > ; if x 6 u > > > < buþ w wdHðwÞ LðxÞ :¼ buþR H1 ðxÞ wdHðwÞ ; > w > > R ; otherwise > w : buþ wdHðwÞ
ð2Þ
w
or, in terms of G,
8 bx R w ; if x 6 u > > > < buþð1uÞ w wdGðwÞ ; LðxÞ ¼ buþð1uÞ R wx wdGðwÞ > > Rww > ; otherwise : buþð1uÞ wdGðwÞ
ð3Þ
w
1
¼ w1 ). It is an increasing and convex function, and a straight line up to where wx:¼G ((x u)/(1 u)) (so that w = wu and w u, as in Fig. 4. 8 This is the type of income distribution corresponding, for example, to the ex-post segmentation equilibrium of Albrecht and Vroman (2002) and Blázquez and Jansen (2008).
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Using integration by parts, (3) can be written alternatively as:
8 bx R w ; if x 6 u > > > GðwÞdw buþwð1uÞð1uÞ > w < R wx LðxÞ ¼ : buþwx ðxuÞð1uÞ GðwÞdw > > w > R ; otherwise > w : buþwð1uÞð1uÞ GðwÞdw
ð4Þ
w
This is the form that shall be used in the following sections. Finally, the Gini inequality index is defined as
I :¼ 1 2
Z
1
LðxÞdx:
ð5Þ
0
3. The Burdett–Mortensen model Burdett and Mortensen (1998) provide a now-classic on-the-job-search model for analyzing wage dispersion. We shall here use their notation and results. The exogenous rate at which employed individuals lose their jobs is d > 0. We assume job offers arrive at the same rate k > 0 for both employed and unemployed.9 In this case, as argued there, the equilibrium reservation wage will equal the unemployment benefit b, the equilibrium unemployment rate will be
u¼
d ; dþk
ð6Þ
and the distribution G of paid wages can be derived from the equilibrium distribution F of wage offers:
GðwÞ ¼
dFðwÞ : d þ kð1 FðwÞÞ
Given (6), which also holds in the model to be considered in the next section, u will be considered from this point on as taking values in the (0, 1) interval. Since we are interested in the correlation between the unemployment rate and an index derived from G, we shall henceforth treat G, F and the Lorenz curve L as functions of both w and u. For instance, the last two equations yield
Gðw; uÞ ¼
Fðw; uÞ : 1 þ 1u ð1 Fðw; uÞÞ u
ð7Þ
The steady-state equilibrium distribution F is found (see Burdett and Mortensen, 1998, pp. 261-263) to be given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi FðwÞ ¼ ððd þ kÞ=kÞð1 ðp wÞ=ðp bÞÞ, that is,
Fðw; uÞ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pw ; 1 1u pb
ð8Þ
where p stands for (exogenous) firm productivity and the real interest rate r is made to approach 0. From (7) and (8) one gets
u Gðw; uÞ ¼ 1u
sffiffiffiffiffiffiffiffiffiffiffiffiffi ! pb 1 : pw
ð9Þ
Also, since G1(v) = p (u/(u + (1 u)v))2(p b), we have (using the notation established in Section 2)10
wx ¼ p ðp bÞ
u2 : x
As for the other integral extrema seen in (4), one has w = wu = b, and
¼ w1 ¼ p ðp bÞu2 : w
ð10Þ
Note from (9) and (10) that a higher unemployment rate moves mass toward lower wages and, concomitantly, lowers the upper bound of the distribution of wages. 9 This assumption, although not taken for granted in the basic (homogeneous workers and homogeneous firms) model in Burdett and Mortensen (1998), is made in the following sections there, as well as in subsequent work of one of those authors, such as Mortensen (2000, 2003). 10 By G1(v) it is meant the smallest wage w such that G(w;u) = v (that is, treating u as a fixed parameter).
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Now (4) takes the specific form (see Appendix B.1)
Lðx; uÞ ¼
8 bx > < bþðpbÞð1uÞ2 ; if x 6 u 2
ðxuÞ > : bxþðpbÞ x 2 ; otherwise bþðpbÞð1uÞ
ð11Þ
:
This leads to the Gini coefficient of income inequality (see Appendix B.2)
I¼
2ðp bÞuð1 u þ u log uÞ b þ ðp bÞð1 uÞ2
:
Calling k the productivity/unemployment-benefit ratio p/b (so that k > 1), we see that the Gini coefficient is a function of u (which, from (6), depends on d and k) and k (which depends on b and p) only:
Iðu; kÞ ¼
2ðk 1Þuð1 u þ u log uÞ 1 þ ðk 1Þð1 uÞ2
ð12Þ
:
Each of the four exogenous variables b, p, d, and k is correlated to inequality in this economy. It is straightforward to see that Ik > 0, so that, since b and p relate to k (but not to u), @I/@b = Ik @k/@b < 0 and @I/@p = Ik @k/@p > 0.11 Thus unemployment benefit cuts and positive productivity growth are accompanied by larger inequality in this model. Similarly, d and k relate to u (but not to k), so that, from (6),
@I @u k ¼ Iu ¼ Iu ; @d @d ðd þ kÞ2 @I @u d ¼ Iu ¼ Iu : @k @k ðd þ kÞ2 Thus, in order to determine the signs of @I/@d and @I/@k, one only needs to know the sign of Iu, which is the objective of our main proposition in this section. It states that, for reasonable values of the unemployment rate, the Burdett–Mortensen model is able to generate the type of correlation between unemployment and inequality observed in the data. Proposition 1. Consider the Burdett–Mortensen job-search model, as described above. If the unemployment rate generated by the model is no larger than 28%, then the Gini coefficient of income inequality is positively correlated to the unemployment rate, regardless of the values of b and p.
Proof. First of all, the denominator in (12) is obviously positive, whence the numerator is positive as well (the Gini measure is positive by construction). The denominator can be seen by simple inspection to decrease with u. As for the numerator, we are interested mainly in the behavior of u(1 u + u log u). Its derivative is 1 u + u log u + u(1 + log u + 1) = 1 u + 2u log u, which (see Appendix B.3) is positive for u 6 0.28, meaning the numerator is increasing in this interval. Therefore Iu > 0 (whence @I/@d > 0 and @I/@k < 0, i.e., income inequality increases with the dismissal rate and decreases with the job-offer rate) in the Burdett–Mortensen model, under the assumption that it does not generate an unemployment rate greater than 28%. h This conclusion is illustrated in Fig. 5, which plots I against u for several values of k. It may be noted that there is no inconsistency between the present model and the quote from Mortensen and Pissarides (1999) in our introduction, which indicates a negative correlation between the growth of unemployment and the rise in inequality. In fact, the increase in American inequality could be related to the consistent rise in US productivity during the second half of the twentieth century (see http://www.bls.gov/lpc/). This increased k, making it possible for income inequality I to rise (even under a limited-decrease-in-unemployment scenario). Across the Atlantic, productivity growth was generally weaker, and at the same time countries such as Austria, Belgium, Denmark, Finland, France, Ireland, Netherlands, Norway, Portugal, Spain, and Switzerland managed to substantially increase their unemployment benefit replacement rates (see Nickell et al., 2002, Table 2), unlike the US and the UK. This decreased k, so that I would naturally tend downward (and, according to the model, could only be made stable because of a simultaneous increase in u). As noted in Section 2, from the very notion of inequality we know that I = 0 when u = 1. This limiting case would correspond to a situation with every worker searching for a job and earning unemployment insurance b. But even if this situation could emerge as a steady-state equilibrium in the Burdett–Mortensen model (it cannot, from (6)), there would be no a priori guarantee that also I(1, ) = 0, as seen in the discussion immediately following (1). This needs to be checked from (12):
Ið1; kÞ ¼ lim
u!1
2ðk 1Þuð1 u þ u log uÞ 1 þ ðk 1Þð1 uÞ2
¼ 2ðk 1Þ
1ð1 1 þ 1 0Þ ¼ 0: 1
As a consequence, there is no k (i.e., no p and no b) capable of making Iu > 0 for any unemployment rate. In fact, if this were the case, I would be strictly increasing in u, and we would necessarily have I(1, k) > 0, a contradiction. 11 Although I was not written explicitly in terms of all the exogenous variables, oI/ob (oI/op,oI/od,oI/ok) should be understood as the marginal effect of unemployment benefits (firm productivity, job destruction rate, job offer rate) on income inequality.
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0.25
I(⋅,3.0)
Gini coefficient
0.2
0.15
I(⋅,2.0)
0.1 I(⋅,1.5) 0.05
0
I(⋅,1.2)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
unemployment rate Fig. 5. Gini coefficient of income inequality as a function of the unemployment rate in the Burdett–Mortensen model, for four different values of the parameter k.
As for the tightness of our result, we may plug (12) into a mathematical software in order to see that I(0.285, 1.005) 6.09961 104 and I(0.286, 1.005) 6.09956 104 < I(0.285, 1.005). This shows that the 28%-limit hypothesis on the unemployment rate is not only sufficient but, within an error of at most 1%, also necessary in order to guarantee Iu > 0 for any b and p. Mortensen (2000) endogenizes the rate at which workers contact firms, k, by assuming a fixed recruiting cost c > 0 per vacancy and a free-entry condition, so that in equilibrium firms’ profits vanish and the (positive) measure v of vacancies can be uniquely determined through (see Eq. (7) there, with r 0):
v ðd þ kðv ÞÞ2 p b ; ¼ c kðv Þ d
ð13Þ
where k is an increasing and concave matching function (in which case v/k(v) is nondecreasing in v) satisfying the Inada conditions. In this case, it is clear that an unemployment benefits raise will yield a lower level of vacancies v and a lower contact rate k(v). From (6), this means more unemployment, more in accordance with our intuitions. But this large b situation may also correspond to a lower income inequality level, as in the ‘‘European’’ scenario. Indeed, if variable contact rates are put into consideration, the fixation of a large unemployment insurance is capable of breaking down the logic behind Proposition 1 and its dI/du > 0 result. Although the Gini coefficient will still be given by (12) (the distribution of wages remains the same, as is shown in Mortensen (2000), again for r 0), it should be noted that now Iu does not correspond to the total differential of I with respect to u anymore. Now one has dI/du = Iu + Ikdk/du, where k and u may be correlated (both vary with b, for instance). Details are given in Appendix C. 4. The Bontemps–Robin–van den Berg model The model in Bontemps et al. (2000) extends Burdett and Mortensen’s (1998) model to allow firms to make counteroffers to workers who have contacted a competitor. We again assume that job opportunities arise indistinctly to both employed and unemployed (at the same rate k), but now there is a continuum of firms whose productivities we assume to be uniformly interval (where p > p P b).12 As in Bontemps et al., w is fixed to p, and given the wage distributed over the exogenous ½p; p pÞ (see Footnote 8 there). Thus offer w = K(p), it follows that K 0 ðpÞ ¼ 2ðk=ðd þ kÞÞðp pÞ=ðp
KðpÞ ¼ p þ
2 d ðp pÞ ; p dþk p
whence K 1 ðwÞ ¼ p þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pÞðw pÞ. That paper also derives a useful relation (Eq. (A.7) in its appendix) between ððd þ kÞ=kÞðp
productivity K1(w) and the distribution of wages G, K1(w) = w + (d + kG(w))/(2kG0 (w)). 12 The unique-k hypothesis is important in the present analysis because it makes it possible to write G as a function of w and u only. The same applies to the homogeneous productivity distribution hypothesis, which makes it possible to obtain a closed-form solution for G. Estimates for k0 and k1 (job offer arrival rates for unemployed and employed, respectively) vary greatly from study to study, obviously depending on the specifics of the model, the data and the estimation procedure. Blau (1992), for instance, has these rates decreasing with time rather than constant, but very similar between unemployed and employed (an average of approximately 0.089 for both arrival rates, see Table 3 there). Van den Berg and Ridder’s (1998) estimates (see Table 10 there) are k0 = 0.033 and k1 = 0.035.
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0.7 0.6
Gini coefficient
0.5 I(⋅,1.46,66.5) 0.4 0.3 0.2 0.1 0
0
0.2
0.4
0.6
0.8
1
unemployment rate Fig. 6. Gini coefficient of income inequality as a function of the unemployment rate in the Bontemps–Robin–van den Berg model, fixing the productivity parameters k and K.
One thus obtains the linear differential equation
G0 ðwÞ ¼
d þ kGðwÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dþk ð p pÞðw pÞ ðw pÞ 2k k
ð14Þ
! ½0; 1 given by with initial condition G(p) = 0. The function G : ½w; w
0
1 pffiffiffiffiffiffiffiffiffiffiffiffi pp dB C GðwÞ ¼ @pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1A; k k p p dþk ðw pÞ
ð15Þ
¼ ðdp þ kp Þ=ðd þ kÞ, can be checked (see Appendix D) to be the solution. Or, from (6) (which is the expreswhere w = p and w sion stemming from Proposition 2 in Bontemps et al. with k0 = k),
0
1 pffiffiffiffiffiffiffiffiffiffiffiffi p p u B C Gðw; uÞ ¼ @ ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1A: 1 u pffiffiffiffiffiffiffiffiffiffiffi p ð1 uÞðw pÞ p
Application of (4) here gives (see Appendix E.1)
Lðx; uÞ ¼
8 bx if x 6 u > u 1uþ2 log u ; < buþpð1uÞþðppÞ1u ðu Þ > :
pÞ u ðux uxþ2 loguxÞ buþpðxuÞþðp 1u pÞ u ð1uuþ2 log uÞ buþpð1uÞþðp 1u
:
ð16Þ
; otherwise
In order to find the Gini coefficient, we need to integrate this expression with respect to x and then use (5). As shown in Appendix E.2, this leads to
I¼
u pÞ 1u ðp bÞuð1 uÞ þ 2ðp ð2u 2 ðu þ 1Þ log uÞ : u pÞ 1u bu þ pð1 uÞ þ ðp ð1u u þ 2 log uÞ
=b (so that K > k P 1), we then have Making k:¼p/b and K :¼ p
Iðu; k; KÞ ¼
u ðk 1Þuð1 uÞ þ 2ðK kÞ 1u ð2u 2 ðu þ 1Þ log uÞ 1 : u u þ 2 log u u þ kð1 uÞ þ ðK kÞ 1u u
ð17Þ
At this point we are able to provide a counterpart to Proposition 1, this time concerning the Bontemps–Robin–van den Berg model. Proposition 2. Consider the Bontemps–Robin–van den Berg job-search model, as described above. If the unemployment rate generated by the model is no larger than 15%, then the Gini coefficient of income inequality is positively correlated to the . unemployment rate, regardless of the values of b, p and p Proof. See Appendix E.3. h
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Gini coefficient
0.4
0.3
0.2
0.1
0 10 8
K
6 4 2
0
0.2
0.4
0.8
0.6
1
u
Fig. 7. Gini coefficient as a function of the unemployment rate and the high-productivity parameter K in the Bontemps–Robin–van den Berg model, for k = 1.
The Bontemps et al. (2000) framework also allows for (endogenous) differentiation between a formal and an informal sector of the economy. In fact, Meghir et al. (2010) incorporate this possibility into the Bontemps et al. model and show that firms with high enough productivities will choose formality in equilibrium, whereas the less productive ones will prefer informality (whence their laid-off employees should not expect to receive unemployment benefits). Having estimated their model using data from Brazil, they find that informal-sector wages may include a considerable compensating differential (this being the typical case, for instance, for low-education males living in the cities of São Paulo and Salvador). This makes it difficult to assert that the positive correlation between u and I will still hold in this case: once a highly paid worker in the informal sector of one of those cities gets fired and starts receiving 0 until he/she goes back into the labor force, that may actually bring incomes closer together, instead of further apart. The following picture illustrates Proposition 2. It plots the Gini coefficient of income inequality as a function of the unemployment rate with k = 1.46 and K = 66.5, which are the values stemming from Bontemps et al.’s (2000) analysis of the French labor market.13 As in the Burdett–Mortensen model, there is no vector (k, K) generating Iu > 0 for every u. The reasoning is exactly the same as before, and is based on the fact that, from (17),
Ið1; k; KÞ ¼ lim
u!1
¼ lim
u!1
u ðk 1Þuð1 uÞ þ 2ðK kÞ 1u ð2u 2 ðu þ 1Þ log uÞ 1 u u þ 2 log u u þ kð1 uÞ þ ðK kÞ 1u u log u ðk 1Þuð1 uÞ þ 2ðK kÞuð2 þ log uÞ 4ðK kÞ u1u
u þ kð1 uÞ þ ðK kÞð1 þ uÞ þ 2ðK
log u kÞ u1u
¼
2ðK kÞð2Þ 4ðK kÞð1Þ ¼ 0; 1 þ 2ðK kÞ þ 2ðK kÞð1Þ
where in the third equality l’Hôpital’s rule was employed to conclude that limu?1((u log u)/(1 u)) = limu?1((log u + 1)/ (1)) = 1. Fig. 7 shows, for k = 1, the joint dependence of I on u and K. It may be noted that, for a fixed K, I(, k, K) presents the same inverted U shape as featured in the Burdett–Mortensen model. Finally, a note on the tightness of this result. Plugging (17) into a mathematical software yields I(0.15,1,1.05) 8.3010 103 and I(0.16, 1, 1.05) 8.2997 103 < I(0.15, 1, 1.05), showing the 15%-limit hypothesis on the unemployment rate to be, within an error of at most 1%, also necessary so that Iu > 0. Note, however, that in the specific case where the parameters assume the values used to plot Fig. 6, the positive correlation between u and I emerges for rates of unemployment as high as 58%. This suggests that, for real-world parameter values, the condition u 6 0.15 may be too stringent. 5. Conclusion A vast literature has documented a positive correlation between unemployment and inequality in several different economies. Understanding how such a correlation may emerge and on which parameters it may depend is crucial for the correct interpretation of empirical evaluations of inequality. We have shown that job-search frictions as modeled in Burdett and Mortensen (1998) and Bontemps et al. (2000) generate positive correlations between the Gini coefficient of income inequality and the unemployment rate whenever this rate is no larger than 28% and 15%, respectively. 13 ¼ 299; 166. Here we base ourselves on the upper left figure on page 334 of Bontemps et al. (2000), which presents the figures b = w = 4500, p = 6563, and p Also, as noted earlier, the reader should bear in mind that the model used here, with k0 = k1 and a uniform distribution of firm productivities, is a particular instance of the broader model introduced in Bontemps et al.
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Also, as can be implied from (12), both time-series and cross-sectional studies featuring inequality comparisons could also benefit from taking a closer look at changes in the surrounding political and technological environment (here represented by parameters b and p), in addition to the job-market parameters determining the level of unemployment. Taking as a benchmark the classic Burdett–Mortensen model, not only larger (and reasonable) unemployment rates directly imply greater income dispersion as measured by the Gini coefficient, but also do higher productivity levels and lower unemployment insurance. Models with worker skill heterogeneity (e.g., Acemoglu, 1999; Albrecht and Vroman, 2002; Greiner et al., 2004) may be used in future research, and in that case a most interesting analysis should try to separate the effect of unemployment on within-group inequality (the one appearing in this paper) from its effect on between-group inequality, or even treating structural unemployment (that present here) separately from that stemming from skill inadequacy of some workers for some jobs. Appendix A We are interested in finding a formula for the Gini coefficient of income inequality in the case of the stylized economy as in the third and fourth examples in Section 2. The Lorenz curve will described by u, r(m:¼1 u r), and b, w, and w, be as in Fig. A1, where e:¼bu/W, d:¼(bu + w(1 r u))/W, and W ¼ bu þ wð1 r uÞ þ wr: Considering the triangles marked A and B in this figure, the Gini index can be calculated as:
0
I ¼ 2ðA þ BÞ ¼ 1 r
1
0 1
0
d 1 þ u
1 1 1 r
0 1
e 1 ¼ 1 r d þ du eð1 rÞ ¼ ð1 eÞð1 rÞ dð1 uÞ:
d 1
1
d
A
B
e 0
0
u
L
1−r
1
Fig. A1. Lorenz curve in an economy with two income levels and unemployment insurance.
Appendix B B.1 Here we show that (4) takes the form (11) in the Burdett and Mortensen (1998) model. Using (9):
Z
wx
w
sffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! pffiffiffiffiffiffiffiffiffiffiffiffi Z wx 1 pb u pb ðp wÞ2 dw ðwx wÞ 1 dw ¼ pw 1u w w " ! 2
u pffiffiffiffiffiffiffiffiffiffiffiffi 1 pð x Þ ðpbÞ u u 2 2 p b 2ðp wÞ
p ðp bÞ b ¼ 1u x b pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi u2 u u pb pb 1 ðp bÞ 2 p b ¼ 1u x x u2 u u u u u ¼ ðp bÞ 2 1 1 ðp bÞ 1 2 1þ ¼ 1u x x 1u x x u u2 ¼ ðp bÞ 1 : 1u x
u Gðw; uÞdw ¼ 1u
Z
wx
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Thus, the numerator in the second line in (4) is
bu þ wx ðx uÞ ð1 uÞ
Z
wx
w
u2 u2 Gðw; uÞdw ¼ bu þ p ðp bÞ ðx uÞ ðp bÞu 1 x x u2 u2 u ¼ bu þ pðx uÞ ðp bÞðx uÞ ðp bÞu þ 2ðp bÞu ðp bÞu x x x u2 u ¼ bu þ pðx uÞ ðp bÞx ðp bÞu þ 2ðp bÞu x x u2 u ¼ bu þ pðx uÞ ðp bÞu þ ðp bÞ ¼ bu þ pðx uÞ ðp bÞ ðx uÞ x x u ¼ bu þ pðx uÞ ðp bÞðx uÞ ðp bÞðx uÞ 1 x ðx uÞ2 ¼ bx þ ðp bÞ ; x
while the denominators there correspond to the special case x = 1 of this numerator:
uÞ ð1 uÞ bu þ wð1
Z
w
Gðw; uÞdw ¼ b þ ðp bÞð1 uÞ2 :
w
B.2 We now use (11) and (5) in order to calculate the Gini coefficient stemming from the Burdett–Mortensen model:
Ru I ¼12 ¼1 ¼
0
bxdx þ
R1 2 bx þ ðp bÞ ðxuÞ dx u x
b þ ðp bÞð1 uÞ2
R1 ¼1
b þ ðp bÞð1 u2 4uð1 uÞ 2u2 log uÞ b þ ðp bÞð1 uÞ2
2ðp bÞuð1 u þ u log uÞ b þ ðp bÞð1 uÞ2
0
2bxdx þ ðp bÞ
R1 2 2x 4u þ 2 ux dx u
b þ ðp bÞð1 uÞ2 ¼
ðp bÞðð1 uÞ2 ð3u2 4u þ 1 2u2 log uÞÞ b þ ðp bÞð1 uÞ2
:
B.3 Here we show that 1 u + 2ulogu is positive for u 2 (0, 0.28] (used in the proof of Proposition 1). The derivative of 1 1 u + 2ulogu is 1 + 2logu, which is negative as long as u < e2 0:607. That is, up to 0.60,1 u + 2ulog u decreases with 3 u. Also, at u = 0.28, 1 u + 2ulogu 7.1 10 , i.e., it is still positive (at u = 0.29 it would already be negative, approximately 8.0 103). Therefore 1 u + 2ulogu is positive for u 2 (0, 0.28]. Appendix C Here we deal with the case of an endogenous k as in Mortensen (2000), and analyze the sign of dI/du = Iu + Ikdk/du within that framework. Since (12) still holds, Iu is the same as in the Burdett–Mortensen model:
Iu ðu; kÞ ¼ 2ðk 1Þ ¼ 2ðk 1Þ ¼ 2ðk 1Þ
ð1 u þ u log u þ uð1 þ log u þ 1ÞÞð1 þ ðk 1Þð1 uÞ2 Þ þ 2ðk 1Þuð1 u þ u log uÞð1 uÞ
ð1 þ ðk 1Þð1 uÞ2 Þ2 ð1 uÞð1 þ ðk 1Þð1 uÞ2 þ 2ðk 1Þuð1 uÞÞ þ 2uð1 þ ðk 1Þð1 uÞ2 þ ðk 1Þuð1 uÞÞ log u ð1 þ ðk 1Þð1 uÞ2 Þ2
ð1 uÞð1 þ ðk 1Þð1 u2 ÞÞ þ 2uð1 þ ðk 1Þð1 uÞÞ log u ð1 þ ðk 1Þð1 uÞ2 Þ2
:
One also has
Ik ¼ 2uð1 u þ u log uÞ
1 þ ðk 1Þð1 uÞ2 ðk 1Þð1 uÞ2 ð1 þ ðk 1Þð1 uÞ2 Þ2
¼
2uð1 u þ u log uÞ ð1 þ ðk 1Þð1 uÞ2 Þ2
:
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Now, from (6) and (13):
v uð1 uÞ
b ¼ ðk 1Þ: c
pffi Let kðÞ ¼ L ; L > 0. Then (6) yields v = (d/L)2((1 u)/u)2. Substitution in the above expression gives k = 1 + (c/b)(d/L)2(1 u)/ 3 u , whence
2 dk c d u3 2u 3 ¼ ð3u4 þ 2u3 Þ ¼ ðk 1Þ ð3u4 þ 2u3 Þ ¼ ðk 1Þ du b L uð1 uÞ 1u in equilibrium. So 2u3 ð1 u þ u log uÞ dI ð1 uÞð1 þ ðk 1Þð1 u2 ÞÞ þ 2uð1 þ ðk 1Þð1 uÞÞ log u ¼ 2ðk 1Þ þ 2ðk 1Þ 1u ; 2 2 du ð1 þ ðk 1Þð1 uÞ2 Þ2 ð1 þ ðk 1Þð1 uÞ Þ
and sgn dI/du = sgn E(u, k), where
Eðu; kÞ :¼ ð1 uÞ2 ð1 þ ðk 1Þð1 u2 ÞÞ þ 2uð1 uÞð1 þ ðk 1Þð1 uÞÞ log u þ ð2u 3Þð1 u þ u log uÞ ¼ ð1 uÞðð1 uÞð1 þ ðk 1Þð1 u2 ÞÞ þ 2u 3Þ þ uð2u 3 þ 2ð1 uÞð1 þ ðk 1Þð1 uÞÞÞ log u ¼ ð1 uÞð1 þ ð1 uÞð1 þ ðk 1Þð1 u2 ÞÞÞ þ uð1 þ 2ðk 1Þð1 uÞ2 Þ log u: Since Ek(u,k) = (1 u)2(1 u2) + 2u(1 u)2logu = (1 u)2(1 u2 + 2ulogu) is positive for u 2 (0,1) ðlimu!1 ð1 u2 þ 2u log uÞ ¼ 0 and d(1 u2 + 2ulogu)/du = 2(1 u + logu) < 0), lower levels of k tend to bring dI/du down. Now, from (13), we could replace the k’s in the above expression for 1 + (c/b)v/(u(1 u)). Since a large b yields a lower v and a higher u (and u(1 u), at least for u < 1/2), we thus get a lower k and a lower dI/du. For instance, it may be seen that, if k = 1.46 (the infimum of the k’s in Bontemps et al.’s analysis of the French labor market, see Section 4), dI/du < 0,"u 2 (0, 1). If k = 66.5 (the supremum of those k ’s), one has dI/du > 0,"u 2 (0,0.61], while the intermediate value k = 5.4 would lead to dI/du > 0,"u 2 (0,0.15]. Appendix D We check that the distribution G given in (15) is a proper solution to (14). Differentiating (15), one gets
G0 ðwÞ ¼ while
rffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi p p p p d k 1 d 1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ ffiffiffiffiffiffiffiffiffiffiffiffi ffi p q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ; p ffiffiffiffiffiffiffiffiffiffiffi ffi p ffiffiffiffiffiffiffiffiffiffiffi ffi 2 kð p d þ k 2 w p 2k dþk ðw pÞ k k p dþk ðw pÞÞ p dþk ðw pÞ p k ! pffiffiffiffiffiffi p p ffi d þ d pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k
pffiffiffiffiffiffiffiffiffiffiffiffi pp p p dþkðwpÞ d þ kGðwÞ d 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2k dþk dþk dþk k p dþk 2k ðp pÞðw pÞ ðw pÞ ðp pÞðw pÞ ðw pÞ ðw pÞ ð p ðw pÞÞ2 2k k k k so that the differential Eq. (14) is satisfied. Furthermore, G(p) = 0, as wanted. We also have G0 > 0 and
0 1 ! pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi p B C d p p p 1 dB d C ¼G Þ ¼ @ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi k pffiffiffiffiffiffiffiffiffiffiffi 1 ¼ ðdp þ kp GðwÞ ffi 1A ¼ k pffiffiffiffiffiffiffiffiffiffi p dþk p dþk k pffiffiffiffiffiffiffiffiffiffiffi k p p k 1 Þ p p dþk dþk ðdp þ kp p
Appendix E E.1 In order to derive (16) from (15), we must calculate wx and The first is achieved by writing
R wx w
0 1 pffiffiffiffiffiffiffiffiffiffiffiffi p p xu u B C ¼ Gðwx ; uÞ ¼ @ ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1A; 1u 1 u pffiffiffiffiffiffiffiffiffiffiffi p ð1 uÞðw pÞ p x
Gðw; uÞdw.
1 d dþk
! 1 ¼ 1:
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467
whence:
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p u p p x u qffiffiffiffiffiffiffiffiffiffiffiffi p p ð1 uÞðwx pÞ) p x ¼ u pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) ¼ p x x p ð1 uÞðwx pÞ p 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 u Þ2 1 ux x pÞ: ðp pÞ ¼ wx p)wx ¼ p þ ðp ¼ ð1 uÞðwx pÞ) 1u 1u Now for the second:
Z
wx
w
u Gðw; uÞdw ¼ 1u
0
Z
wx
w
B @
1 1 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1Adw: 1u 1 pp ðw pÞ
pÞ, this equals Making the substitution z2 ¼ ð1 uÞðw pÞ=ðp
u 1u
Z
1ux
Z 1ux 2ðp pÞ 1 u z2 pÞ 1 dz: zdz ¼ 2ðp 2 1z 1z 1u ð1 uÞ 0
0 2
Since z /2 z log(1 z) is a primitive of z2/(1 z), we have
Z
wx
u 1u ½z2 þ 2z þ 2 logð1 zÞj0 x ð1 uÞ2 u u2 u u pÞ þ 2 log : ¼ ðp 1 þ2 1 2 x x x ð1 uÞ
pÞ Gðw; uÞdw ¼ ðp
w
Therefore, the numerator in the second line in (4) is
bu þ wx ðx uÞ ð1 uÞ
Z
wx
Gðw; uÞdw
w
u 2
!
1x u u2 u u pÞ pÞ ðx uÞ þ ðp 1 þ 2 log þ2 1 ðp 1u x x x 1u 2 2 u u u u xu u pÞ 1 þ 2 log þ 1 þ2 1 ¼ bu þ pðx uÞ þ ðp 1u x x x u x u u u2 u x u u u2 u pÞ ¼ bu þ pðx uÞ þ ðp 12 þ þ22 þ 12þ2 þ þ 2 log 1u x x x u x x x x u x u u pÞ ¼ bu þ pðx uÞ þ ðp þ 2 log ; 1u u x x ¼ bu þ p þ
while the denominators there correspond to the special case x = 1 of this numerator:
uÞ ð1 uÞ bu þ wðx
Z
w
pÞ Gðw; uÞdw ¼ bu þ pð1 uÞ þ ðp w
u 1 u þ 2 log u : 1u u
E.2 Here we use (16) and (5) in order to calculate the Gini coefficient originating from the Bontemps–Robin–van den Berg model. The second equality below uses the fact that x + xlog(u/x) is a primitive of log(u/x).
x u R1 u pÞ 1u bu þ pðx uÞ þ ðp x þ 2 log ux dx u u 1 I ¼1 u pÞ 1u u þ 2 log u bu þ pð1 uÞ þ ðp u 2 2 u 1u2 pÞ 1u 2 b u2 þ buð1 uÞ þ p ð1uÞ þ ðp þ u log u þ 2ð1 u þ log uÞ 2 2u 1 ¼1 u pÞ 1u u þ 2 log u bu þ pð1 uÞ þ ðp u 1 2 u pÞ 1u bu þ 2bu þ pð1 uÞ2 þ ðp þ 4 5u þ ð2u þ 4Þ log u u ¼1 u 1 pÞ 1u u þ 2 log u bu þ pð1 uÞ þ ðp u 2 u 1 pÞ 1u u u þ 2 log u 1u 4 þ 5u ð2u þ 4Þ log u bu þ bu þ puð1 uÞ þ ðp ¼ u 1 pÞ 1u bu þ pð1 uÞ þ ðp u þ 2 log u u 2
¼
R u 0
bxdx þ
u pÞ 1u ðp bÞuð1 uÞ þ 2ðp ð2u 2 ðu þ 1Þ log uÞ : u bu þ pð1 uÞ þ ðp pÞ 1u 1u u þ 2 log u
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E.3 Proof of Proposition 2. Write
Nðu; k; KÞ :¼ ðk 1Þuð1 uÞ þ 2ðK kÞðu=ð1 uÞÞð2u 2 ðu þ 1Þ log uÞ and
Dðu; k; KÞ :¼ u þ kð1 uÞ þ ðK kÞðu=ð1 uÞÞð1=u u þ 2 log uÞ; so that I = N/D. D is positive since 1/u u + 2logu is positive (its derivative is 1/u2 1 + 2/u = ((1 u)/u)2 < 0 and at u = 1 it takes on the value 1/1 1 + 2 0 = 0), and since I is a Gini coefficient, N is positive too. In Appendix E.4, N and D are shown to be, respectively, increasing and decreasing functions of u over the (0,0.15] interval. Therefore Iu > 0 (whence again @I/ @d > 0 and @I/@k < 0) in the Bontemps-Robin-van den Berg model, under the assumption that it does not generate an unemployment rate greater than 15%. E.4 We first show that D (defined in Appendix E.3) decreases with u. Its derivative is:
! 1 u 1 2 u þ 2 log u þ 1 þ 1u u2 u ð1 uÞ2 u K k 1 1 K k u þ 2 log u þ ð1 uÞ u þ 2 ¼1kþ ðu2 4u þ 3 þ 2 log uÞ: ¼1kþ u u ð1 uÞ2 ð1 uÞ2
Du ðu; k; KÞ ¼ 1 k þ ðK kÞ
1
Since u2 4u + 3 + 2logu < 0 (its derivative is 2u 4 + 2/u = 2(1 u)2/u > 0 and at u = 1 it takes on the value 1 4 + 3 + 2 0 = 0), we have Du decreasing with K. Since at the minimum value for K (which is k), Du already takes on a negative value (1 k), we conclude that Du < 0. We now show that N (also defined in Appendix E.3) increases with u for u 2 (0, 0.15]. We may start by imposing u < 0.50, so that u(1 u) increases with u. Therefore, if Q(u) :¼(u/(1 u))(2u 2 (u + 1)logu) increases with u, so does N. The derivative of Q is:
Q u ðuÞ ¼ ¼
1 2
ð1 uÞ 1
ð1 uÞ2
ð2u 2 ðu þ 1Þ log uÞ þ
u uþ1 2 log u 1u u
ðð2u 2 ðu þ 1Þ log uÞ þ ð1 uÞðu 1 u log uÞÞ ¼
1 ð1 uÞ2
ðu2 þ 4u 3 þ ðu2 2u 1Þ log uÞ:
We are left to verify that S(u):¼ u2 + 4u 3 + (u2 2u 1)log u is positive. In fact, its derivative is 2u + 4 + (2u 2)log u + (u2 2u 1)/u = u + 2 1/u 2(1 u)logu = ((1 u)/u)(1 u + 2ulogu), which is negative for u 6 0.28 (incidentally, this is shown in Appendix B.3, which refers to the Burdett–Mortensen model). So, in the whole (0,0.28] interval, S decreases with u. Unfortunately, at u = 0.28, S is already negative (approximately 7.2 102), which makes the Iu > 0 result necessarily weaker than in the original Burdett–Mortensen model. Nevertheless, S(0.15) 1.07 103 (and S(0.16) 1.35 102), whence S (and Qu) is positive for u 6 0.15, making N an increasing function of u in the required interval. References Acemoglu, D., 1999. Changes in unemployment and wage inequality: an alternative theory and some evidence. American Economic Review 89, 1259–1278. Albrecht, J., Vroman, S., 2002. A matching model with endogenous skill requirements. International Economic Review 43, 283–305. Björklund, A., 1991. Unemployment and income distribution: time-series evidence from Sweden. Scandinavian Journal of Economics 93, 457–465. Blau, D.M., 1992. An empirical analysis of employed and unemployed job search behavior. Industrial and Labor Relations Review 45, 738–752. Blázquez, M., Jansen, M., 2008. Search, mismatch and unemployment. European Economic Review 52, 498–526. Blejer, M.I., Guerrero, I., 1990. The impact of macroeconomic policies on income distribution: an empirical study of the Philippines. Review of Economics and Statistics 72, 414–423. Blinder, A.S., Esaki, H.Y., 1978. Macroeconomic activity and income distribution in the postwar United States. Review of Economics and Statistics 60, 604– 609. Bontemps, C., Robin, J.-M., van den Berg, G.J., 2000. Equilibrium search with continuous productivity dispersion: theory and nonparametric estimation. International Economic Review 41, 305–358. Burdett, K., Mortensen, D.T., 1998. Wage differentials, employer size, and unemployment. International Economic Review 39, 257–273. Burdett, K., Lagos, R., Wright, R., 2004. An on-the-job search model of crime, inequality, and unemployment. International Economic Review 45, 681–706. Cahuc, P., Postel-Vinay, F., Robin, J.-M., 2006. Wage bargaining with on-the-job search: theory and evidence. Econometrica 74, 323–364. Cardoso, E., Paes de Barros, R., Urani, A., 1995. Inflation and unemployment as determinants of inequality in Brazil: the 1980s. In: Dornbusch, R., Edwards, S. (Eds.), Reform, Recovery, and Growth: Latin America and the Middle East. University of Chicago Press, Chicago, pp. 37–64. Castañeda, A., Dı´az-Giménez, J., Ríos-Rull, J.-V., 1998. Exploring the income distribution business cycle dynamics. Journal of Monetary Economics 42, 93– 130. Cysne, R.P., 2009. On the positive correlation between income inequality and unemployment. Review of Economics and Statistics 91, 218–226. Gaumont, D., Schindler, M., Wright, R., 2006. Alternative theories of wage dispersion. European Economic Review 50, 831–848.
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Greiner, A., Rubart, J., Semmler, W., 2004. Economic growth, skill-biased technical change and wage inequality: a model and estimations for the US and Europe. Journal of Macroeconomics 26, 597–621. Jäntti, M., 1994. A more efficient estimate of the effects of macroeconomic activity on the distribution of income. Review of Economics and Statistics 76, 372–378. Jaumotte, F., Lall, S., Papageorgiou, C., 2008. Rising income inequality: technology, or trade and financial globalization? IMF Working paper 08/185. McCall, J.J., 1970. Economics of information and job search. Quarterly Journal of Economics 84, 113–126. Meghir, C., Narita, R., Robin, J.-M., 2010. Wages and Informality in Developing Countries. University College London, Mimeo. Metcalf, C.E., 1969. The size distribution of personal income during the business cycle. American Economic Review 59, 657–668. Mocan, H.N., 1999. Structural unemployment, cyclical unemployment, and income inequality. Review of Economics and Statistics 81, 122–134. Mortensen, D.T., 2000. Equilibrium unemployment with wage posting: Burdett–Mortensen meet Pissarides. In: Bunzel, H., Christianesen, B.J., Keifer, N.M., Mortensen, D.T. (Eds.), Panel Data and Structural Labour Market Models. Elsevier, Amsterdam, pp. 281–292. Mortensen, D.T., 2003. How monopsonistic is the (Danish) labor market? In: Aghion, P., Frydman, R., Stiglitz, J.E., Woodford, M. (Eds.), Knowledge, Information, and Expectations in Modern Macroeconomics: In Honor of Edmund S. Phelps. Princeton University Press, Princeton, pp. 285–308. Mortensen, D.T., Pissarides, C.A., 1999. Unemployment responses to ‘skill-biased’ technology shocks: the role of labour market policy. Economic Journal 109, 242–265. Nickell, S., Nunziata, L., Ochel, W., Quintini, G., 2002. The Beveridge curve, unemployment and wages in the OECD from the 1960s to the 1990s. Centre for Economic Performance, London School of Economics and Political Science. Nolan, B., 1986. Unemployment and the size distribution of income. Economica 53, 421–445. Postel-Vinay, F., Robin, J.-M., 2002. The distribution of earnings in an equilibrium search model with state-dependent offers and counteroffers. International Economic Review 43, 989–1016. Van den Berg, G.J., Ridder, G., 1998. An empirical equilibrium search model of the labor market. Econometrica 66, 1183–1221.
Contents lists available at SciVerse ScienceDirect
Journal of Macroeconomics journal homepage: www.elsevier.com/locate/jmacro
Equilibrium unemployment-inequality correlation Rubens Penha Cysne ⇑, David Turchick FGV/EPGE, Getulio Vargas Foundation, Graduate School of Economics, Brazil
a r t i c l e
i n f o
Article history: Received 16 February 2011 Accepted 31 December 2011 Available online 25 February 2012 JEL classification: E20 J60 Keywords: Unemployment Income inequality Gini coefficient Job search
a b s t r a c t A vast empirical literature implies that increases in unemployment have an aggravating impact on income inequality, whence international and intertemporal inequality comparisons might be sometimes biased. We show how job-search models can be useful in better understanding this fact. In fact, in the classic Burdett and Mortensen (1998) model, as well as in one of its many possible extensions (Bontemps et al., 2000), search frictions are a force pushing the unemployment-inequality correlation in that direction: provided that the unemployment rate is no larger than 15%, a positive correlation between unemployment and inequality unequivocally emerges. 2012 Elsevier Inc. All rights reserved.
1. Introduction It is a common procedure in economics to compare income inequality measurements of different economies or of the same economy at different points in time. Sometimes, such comparisons are also carried out with the purpose of evaluating the success or failure of policies aimed at reducing inequality. Concomitantly, a vast empirical literature, using data from different countries, has already documented that inequality tends to be positively correlated with unemployment.1 Examples of this literature include Metcalf (1969), Blinder and Esaki (1978), Jäntti (1994) and Mocan (1999), using data from the United States; Nolan (1986), using data from England; Björklund (1991), using data from Sweden; Blejer and Guerrero (1990), using data from the Phillipines; and Cardoso et al. (1995), using data from Brazil. More recently, Castañeda et al. (1998), using data from the US Bureau of the Census during the 1948–86 period, have found that ‘‘the income share earned by the lowest quintile is both the most volatile and the most procyclical’’, and that the procyclicality of the income shares decrease monotonically until one reaches the top 5%. Cysne (2009) has shown that standard search models are able to generate the assessed positive correlation between unemployment and inequality. A limitation of that analysis is that it has drawn upon a class of models (based on the seminal contribution of McCall, 1970) that takes as a given the initial distribution of job offers. We show that this conclusion also holds in the class of on-the-job-search models with firms posting wages and meeting workers in a random, decentralized fashion, but at fixed frequencies.
⇑ Corresponding author. Present address: Praia de Botafogo 190 s, 1100 andar, Rio de Janeiro, RJ 22250-900, Brazil. Tel.: +55 21 3799 5832; fax: +55 21 2553 8821. E-mail address: [email protected] (R.P. Cysne). 1 We refer to those inequality measurements usually reported by researchers, which are based on cross-sectional (rather than long-run) distributions of income. 0164-0704/$ - see front matter 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmacro.2011.12.009
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As representatives of this class, two models were chosen. One is the seminal Burdett and Mortensen (1998) model, which has led to numerous important variations and extensions in several directions (see for example Postel-Vinay and Robin (2002) and Cahuc et al. (2006), in which firms are allowed to renegotiate wages with their employees who are receiving outside offers). The other is a simple case of the model of Bontemps et al. (2000), which extends the most basic Burdett–Mortensen model in the direction of firm-productivity differentiation. Both models generate pure wage dispersion – the adjective pure referring to the fact that worker heterogeneity is not necessary (cf. Acemoglu, 1999; Albrecht and Vroman, 2002; Greiner et al., 2004, for instance, where worker skill heterogeneity may also affect wage dispersion through a change in the very structure/type of the equilibrium). In this way, all the unemployment here comes only from job market frictions represented by the matching technology, and not from the possibility that some firms may endeavor to avoid the hiring of insufficiently skilled workers, or even make adjustments to their level of productivity (this would once more affect wages and their distribution). Also, there is no between-group inequality (since there is only one group of workers), and all inequality could be classified as within group. Job search models with heterogeneous workers may allow for the breaking down of unemployment and inequality in a natural way (in this respect the Theil index could be a natural choice for the inequality measure, due to its decomposability property), and study these partial effects. However, this would be beyond the scope of the present work, which attempts to analytically assess the aggregate effect of unemployment on inequality. As we shall see, a positive correlation stems even from pure wage dispersion models. In the case of heterogeneous workers, one may expect the total effect on inequality to be even larger, since the burden of unemployment is higher for low-skill workers, who endure a longer unemployment spell.2 Our analytical approach to the present issue complements the aforementioned empirical literature by providing closedform expressions directly linking income inequality to unemployment. These expressions may help the macroeconomist in understanding how this link emerges, on which parameters it may depend (unemployment benefits and firm productivity will be shown to be key parameters), and in which cases the correlation may revert its sign. At the same time, our analysis offers a word of caution to policy makers interested in the evaluation of policies aiming at the reduction of income inequality (e.g., through a rise in unemployment insurance benefits): the reported (absolute value) effect on inequality may be biased downward due to technological progress in the production function during the two points in time considered or upward due to a rise in the contact rate between firms and workers (thus bringing the unemployment rate down). In other words, by making explicit a channel through which unemployment (and some parameters of these classical models) affects inequality, this work adds value and information to cross-country and longitudinal studies such as Jaumotte et al. (2008), in the sense that inequality indices at different points in space or time should be expected to naturally differ according to basic differences in the surrounding political, technological and business environment. As an example, in Mortensen and Pissarides (1999, p.26) it is reported: ‘‘Looking at average changes over the OECD, there appears to be a negative correlation between the growth in unemployment and the rise in inequality. The United States, Canada and Sweden experienced the biggest rises in inequality and the smaller rises in unemployment (a fall in the US case). The large European countries experienced small rises or falls in inequality but big rises in unemployment (with the exception of the United Kingdom [. . .]).’’ We shall see how such correlations may be understood, in qualitative terms, in light of the simplest Burdett–Mortensen model, with no skill heterogeneity argument entering the picture (unlike Mortensen and Pissarides, 1999).3 Our presentation can be outlined in the following way. First, the Gini index of income inequality is derived as a function of the model’s parameters – which, in turn, also determine the unemployment rate. Next, we analyze the conditions under which a positive correlation between unemployment and inequality emerges. Our basic conclusion is that such a positive correlation will always be a consequence of the optimizing process assumed by these models, provided that the rate of unemployment is no greater than 28% (in Burdett and Mortensen’s model) or 15% (in Bontemps et al.’s model). The remainder of the paper is organized as follows: Section 2 provides a model-free digression on how unemployment and unemployment benefits may affect the Gini coefficient of income inequality. Section 3 provides the analysis of the unemployment-inequality correlation in the context of Burdett and Mortensen’s model, whereas Section 4 does the same using Bontemps et al.’s model. Section 5 concludes. Calculations are duly detailed in the appendices. 2. Unemployment and inequality This section provides a short digression about theoretical possibilities linking unemployment and the Gini coefficient of income inequality. We start with the case of simple discrete income distributions in order to build the intuition for the type of result to be proven later. The first two examples analyze a one-wage economy with and without unemployment insurance, respectively. It is shown that the correlation between unemployment and inequality is always positive in the first case (except when 2 3
We are indebted to an anonymous referee for making this point. Using income inequality in the place of their preferred notion of wage inequality.
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1
L
0
0
u
1
Fig. 1. Lorenz curve in an economy with one income level and no unemployment insurance.
unemployment reaches 100%), but not necessarily in the second one. A third case shows how easy it is for a negative correlation to emerge, even without unemployment insurance, when there is more than one single wage in the economy. Example four shows that if the wage distribution is independent of the unemployment rate, a positive correlation between unemployment and income should not be necessarily expected a priori. The simplest possible income distribution, here denoted by its cumulative distribution function H : Rþ ! ½0; 1, is the one where a portion u 2 [0,1) of the population is unemployed and receives nothing, while the other 1 u all have the same wage (say, w):
HðwÞ ¼
u; if w < w : 1; if w P w
This yields a Lorenz curve L as shown in Fig. 1. The well-known Gini inequality index I is twice the area between the 45 line and the Lorenz curve. In this very particular case, using the determinant formula to calculate the area of the triangle gives:
0
2
0
0 1
31
B1 6 7C I ¼ 2 @ det 4 u 0 1 5A ¼ u; 2 1 1 1
ð1Þ
precisely the unemployment measure.4 Inspired by the exact relation (1) that the Gini coefficient provides regarding this benchmark distribution – in addition to its widespread usage by governments around the world and international organisms –, we choose to employ Gini’s inequality measure throughout this paper.5 In spite of (1) implying a positive (and perfect) correlation between unemployment and inequality, the assertion that increases in unemployment lead to increases in inequality in this simple economy should be done with due caution: in the limiting case in which unemployment rises to 100%, (1) is no longer valid, and the Gini coefficient must collapse to zero. The second example keeps the hypothesis that there is only one wage in the economy, but now an unemployment insur a common feature of job-search models, enters the picture. The income distribution becomes ance b 2 ð0; wÞ,
8 > < 0; if w < b : HðwÞ ¼ u; if b 6 w < w > : 1; if w P w uÞÞ, the fraction of earnings appropriated by The Lorenz curve L takes on the form shown in Fig. 2, where d :¼ bu=ðbu þ wð1 the unemployed. Here,
2
0
6 I ¼ det 4 u
0 1 d
3
7 bÞ 1 5 ¼ u d ¼ ðw
1 1 1
uð1 uÞ ; uÞ bu þ wð1
4 This distribution is generated, for instance, in the static costly-search model with two types of workers introduced in Acemoglu (1999), provided the relevant parameters are compatible with a separating equilibrium (see Proposition 1 there). 5 Other indices could be used. Burdett pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi et al. (2004), for instance, employ the coefficient of variation in their work on crime. For the simple distribution just considered, it would equal u=ð1 uÞ.
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1
L d
0
0
1
u
Fig. 2. Lorenz curve in an economy with one income level and unemployment insurance.
1 0.9 0.8 0.7
L
1
0.6 0.5
L
2
0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 3. Intersecting Lorenz curves.
so that the sign of the unemployment-inequality correlation is undetermined:
uÞÞ þ uð1 uÞðw bÞ uÞ2 bu2 dI ð1 2uÞðbu þ wð1 wð1 bÞ bÞ ¼ ðw ¼ ðw : 2 du uÞÞ uÞÞ2 ðbu þ wð1 ðbu þ wð1 > ðð1 uÞ=uÞ2 (which is a possibility if u > 0.5), then dI/du < 0, so that the possibility that an increase in Note that if b=w unemployment results in a reduction of inequality is not as extreme as it was in the previous example, where b = 0. For example, if unemployment insurance were half the wage and the unemployment rate were 60%, an increase in unemployment would have a greater effect on moving incomes (weighted according to their corresponding fraction in the population) closer together than farther apart. This effect shall also appear in the job-search models studied in Sections 3 and 4, as a byproduct of our analysis. Our third example shows that even in economies with no unemployment insurance, it can trivially happen that I is not positively correlated with u (provided, given what has been shown above, that there is more than one nonzero wage in the economy). Indeed, consider two stylized economies, with a mass u of unemployed, a ‘‘middle class’’ of size m receiving a > w.6 In the first economy, (u, m, r) = (.20, .75, .05), and yearly wage of w, and a ‘‘rich class’’ of size r = 1 u m earning w ¼ $100; 000, the Gini coefficient for the first economy is in the second, (u, m, r) = (.16, .77, .07).7 Putting w = $20,000 and w I = 0.35, while for the second I = 0.3525 (see formula in Appendix A). That is, a smaller unemployment rate does not necessarily mean a smaller inequality. Fig. 3 shows the corresponding Lorenz curves, naturally intersecting each other. 6
Acemoglu (1999, pooling equilibrium at p. 1263) and Gaumont et al. (2006, p. 835), also generate this type of distribution, with two nonzero wage levels. Note that the distribution of wages, determined by the mass associated with m (or r) in relation to m + r, changes in this case. This will not happen in the case to be analyzed next. 7
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1
L
0
0
u
1
Fig. 4. Generic Lorenz curve in an economy with a continuous distribution of wages.
For the last example, consider an economy as the previous one, but where the unemployed receive b 2 (0,w).8 Additionally, suppose the distribution of wages is independent of u: there exists d 2 (0, 1) such that r = d(1 u). Let d = 0.1, b = $9000, ¼ $100; 000. The math done in Appendix A gives the following Gini inequality function: w = $10,000 and w
IðuÞ ¼
ð1 uÞð0:81 þ 0:19uÞ : 1:9 u
Therefore, in this simple case a 5% unemployment rate yields an approximate inequality of 0.421, while a 10% unemploy can exist such that ment gives a lower one, I 0.415. Moreover, this choice of parameters proves that no upper bound u . In fact, the Gini function above can be shown to be strictly decreasing over the entire [0, 1] interval. I0 (u) > 0 for all u < u In what follows we demonstrate that such an upper bound will in fact exist for some standard equilibrium-search models – specifically, the quite conservative 15% will do. This is to say that the search frictions described by these models act as a force pushing the correlation in the direction assessed by the available empirical evaluations. We end this section with the derivation of a general formula concerning the Gini coefficient of income inequality, given a (where w P b by equilibrium reasoning) and continuous and strictly increasing wage distribution G(w), with support ½w; w ¼ 1). The income distribution is no mass points (so that G(w) = 0 and GðwÞ
HðwÞ ¼
8 0; > > > > > < u;
if w < b if b 6 w < w
> > u þ ð1 uÞGðwÞ; if w 6 w < w > > > : 1; if w P w
:
The corresponding Lorenz function L:[0, 1] ? [0, 1] is given by
8 R wbx > ; if x 6 u > > > < buþ w wdHðwÞ LðxÞ :¼ buþR H1 ðxÞ wdHðwÞ ; > w > > R ; otherwise > w : buþ wdHðwÞ
ð2Þ
w
or, in terms of G,
8 bx R w ; if x 6 u > > > < buþð1uÞ w wdGðwÞ ; LðxÞ ¼ buþð1uÞ R wx wdGðwÞ > > Rww > ; otherwise : buþð1uÞ wdGðwÞ
ð3Þ
w
1
¼ w1 ). It is an increasing and convex function, and a straight line up to where wx:¼G ((x u)/(1 u)) (so that w = wu and w u, as in Fig. 4. 8 This is the type of income distribution corresponding, for example, to the ex-post segmentation equilibrium of Albrecht and Vroman (2002) and Blázquez and Jansen (2008).
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Using integration by parts, (3) can be written alternatively as:
8 bx R w ; if x 6 u > > > GðwÞdw buþwð1uÞð1uÞ > w < R wx LðxÞ ¼ : buþwx ðxuÞð1uÞ GðwÞdw > > w > R ; otherwise > w : buþwð1uÞð1uÞ GðwÞdw
ð4Þ
w
This is the form that shall be used in the following sections. Finally, the Gini inequality index is defined as
I :¼ 1 2
Z
1
LðxÞdx:
ð5Þ
0
3. The Burdett–Mortensen model Burdett and Mortensen (1998) provide a now-classic on-the-job-search model for analyzing wage dispersion. We shall here use their notation and results. The exogenous rate at which employed individuals lose their jobs is d > 0. We assume job offers arrive at the same rate k > 0 for both employed and unemployed.9 In this case, as argued there, the equilibrium reservation wage will equal the unemployment benefit b, the equilibrium unemployment rate will be
u¼
d ; dþk
ð6Þ
and the distribution G of paid wages can be derived from the equilibrium distribution F of wage offers:
GðwÞ ¼
dFðwÞ : d þ kð1 FðwÞÞ
Given (6), which also holds in the model to be considered in the next section, u will be considered from this point on as taking values in the (0, 1) interval. Since we are interested in the correlation between the unemployment rate and an index derived from G, we shall henceforth treat G, F and the Lorenz curve L as functions of both w and u. For instance, the last two equations yield
Gðw; uÞ ¼
Fðw; uÞ : 1 þ 1u ð1 Fðw; uÞÞ u
ð7Þ
The steady-state equilibrium distribution F is found (see Burdett and Mortensen, 1998, pp. 261-263) to be given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi FðwÞ ¼ ððd þ kÞ=kÞð1 ðp wÞ=ðp bÞÞ, that is,
Fðw; uÞ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pw ; 1 1u pb
ð8Þ
where p stands for (exogenous) firm productivity and the real interest rate r is made to approach 0. From (7) and (8) one gets
u Gðw; uÞ ¼ 1u
sffiffiffiffiffiffiffiffiffiffiffiffiffi ! pb 1 : pw
ð9Þ
Also, since G1(v) = p (u/(u + (1 u)v))2(p b), we have (using the notation established in Section 2)10
wx ¼ p ðp bÞ
u2 : x
As for the other integral extrema seen in (4), one has w = wu = b, and
¼ w1 ¼ p ðp bÞu2 : w
ð10Þ
Note from (9) and (10) that a higher unemployment rate moves mass toward lower wages and, concomitantly, lowers the upper bound of the distribution of wages. 9 This assumption, although not taken for granted in the basic (homogeneous workers and homogeneous firms) model in Burdett and Mortensen (1998), is made in the following sections there, as well as in subsequent work of one of those authors, such as Mortensen (2000, 2003). 10 By G1(v) it is meant the smallest wage w such that G(w;u) = v (that is, treating u as a fixed parameter).
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Now (4) takes the specific form (see Appendix B.1)
Lðx; uÞ ¼
8 bx > < bþðpbÞð1uÞ2 ; if x 6 u 2
ðxuÞ > : bxþðpbÞ x 2 ; otherwise bþðpbÞð1uÞ
ð11Þ
:
This leads to the Gini coefficient of income inequality (see Appendix B.2)
I¼
2ðp bÞuð1 u þ u log uÞ b þ ðp bÞð1 uÞ2
:
Calling k the productivity/unemployment-benefit ratio p/b (so that k > 1), we see that the Gini coefficient is a function of u (which, from (6), depends on d and k) and k (which depends on b and p) only:
Iðu; kÞ ¼
2ðk 1Þuð1 u þ u log uÞ 1 þ ðk 1Þð1 uÞ2
ð12Þ
:
Each of the four exogenous variables b, p, d, and k is correlated to inequality in this economy. It is straightforward to see that Ik > 0, so that, since b and p relate to k (but not to u), @I/@b = Ik @k/@b < 0 and @I/@p = Ik @k/@p > 0.11 Thus unemployment benefit cuts and positive productivity growth are accompanied by larger inequality in this model. Similarly, d and k relate to u (but not to k), so that, from (6),
@I @u k ¼ Iu ¼ Iu ; @d @d ðd þ kÞ2 @I @u d ¼ Iu ¼ Iu : @k @k ðd þ kÞ2 Thus, in order to determine the signs of @I/@d and @I/@k, one only needs to know the sign of Iu, which is the objective of our main proposition in this section. It states that, for reasonable values of the unemployment rate, the Burdett–Mortensen model is able to generate the type of correlation between unemployment and inequality observed in the data. Proposition 1. Consider the Burdett–Mortensen job-search model, as described above. If the unemployment rate generated by the model is no larger than 28%, then the Gini coefficient of income inequality is positively correlated to the unemployment rate, regardless of the values of b and p.
Proof. First of all, the denominator in (12) is obviously positive, whence the numerator is positive as well (the Gini measure is positive by construction). The denominator can be seen by simple inspection to decrease with u. As for the numerator, we are interested mainly in the behavior of u(1 u + u log u). Its derivative is 1 u + u log u + u(1 + log u + 1) = 1 u + 2u log u, which (see Appendix B.3) is positive for u 6 0.28, meaning the numerator is increasing in this interval. Therefore Iu > 0 (whence @I/@d > 0 and @I/@k < 0, i.e., income inequality increases with the dismissal rate and decreases with the job-offer rate) in the Burdett–Mortensen model, under the assumption that it does not generate an unemployment rate greater than 28%. h This conclusion is illustrated in Fig. 5, which plots I against u for several values of k. It may be noted that there is no inconsistency between the present model and the quote from Mortensen and Pissarides (1999) in our introduction, which indicates a negative correlation between the growth of unemployment and the rise in inequality. In fact, the increase in American inequality could be related to the consistent rise in US productivity during the second half of the twentieth century (see http://www.bls.gov/lpc/). This increased k, making it possible for income inequality I to rise (even under a limited-decrease-in-unemployment scenario). Across the Atlantic, productivity growth was generally weaker, and at the same time countries such as Austria, Belgium, Denmark, Finland, France, Ireland, Netherlands, Norway, Portugal, Spain, and Switzerland managed to substantially increase their unemployment benefit replacement rates (see Nickell et al., 2002, Table 2), unlike the US and the UK. This decreased k, so that I would naturally tend downward (and, according to the model, could only be made stable because of a simultaneous increase in u). As noted in Section 2, from the very notion of inequality we know that I = 0 when u = 1. This limiting case would correspond to a situation with every worker searching for a job and earning unemployment insurance b. But even if this situation could emerge as a steady-state equilibrium in the Burdett–Mortensen model (it cannot, from (6)), there would be no a priori guarantee that also I(1, ) = 0, as seen in the discussion immediately following (1). This needs to be checked from (12):
Ið1; kÞ ¼ lim
u!1
2ðk 1Þuð1 u þ u log uÞ 1 þ ðk 1Þð1 uÞ2
¼ 2ðk 1Þ
1ð1 1 þ 1 0Þ ¼ 0: 1
As a consequence, there is no k (i.e., no p and no b) capable of making Iu > 0 for any unemployment rate. In fact, if this were the case, I would be strictly increasing in u, and we would necessarily have I(1, k) > 0, a contradiction. 11 Although I was not written explicitly in terms of all the exogenous variables, oI/ob (oI/op,oI/od,oI/ok) should be understood as the marginal effect of unemployment benefits (firm productivity, job destruction rate, job offer rate) on income inequality.
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0.25
I(⋅,3.0)
Gini coefficient
0.2
0.15
I(⋅,2.0)
0.1 I(⋅,1.5) 0.05
0
I(⋅,1.2)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
unemployment rate Fig. 5. Gini coefficient of income inequality as a function of the unemployment rate in the Burdett–Mortensen model, for four different values of the parameter k.
As for the tightness of our result, we may plug (12) into a mathematical software in order to see that I(0.285, 1.005) 6.09961 104 and I(0.286, 1.005) 6.09956 104 < I(0.285, 1.005). This shows that the 28%-limit hypothesis on the unemployment rate is not only sufficient but, within an error of at most 1%, also necessary in order to guarantee Iu > 0 for any b and p. Mortensen (2000) endogenizes the rate at which workers contact firms, k, by assuming a fixed recruiting cost c > 0 per vacancy and a free-entry condition, so that in equilibrium firms’ profits vanish and the (positive) measure v of vacancies can be uniquely determined through (see Eq. (7) there, with r 0):
v ðd þ kðv ÞÞ2 p b ; ¼ c kðv Þ d
ð13Þ
where k is an increasing and concave matching function (in which case v/k(v) is nondecreasing in v) satisfying the Inada conditions. In this case, it is clear that an unemployment benefits raise will yield a lower level of vacancies v and a lower contact rate k(v). From (6), this means more unemployment, more in accordance with our intuitions. But this large b situation may also correspond to a lower income inequality level, as in the ‘‘European’’ scenario. Indeed, if variable contact rates are put into consideration, the fixation of a large unemployment insurance is capable of breaking down the logic behind Proposition 1 and its dI/du > 0 result. Although the Gini coefficient will still be given by (12) (the distribution of wages remains the same, as is shown in Mortensen (2000), again for r 0), it should be noted that now Iu does not correspond to the total differential of I with respect to u anymore. Now one has dI/du = Iu + Ikdk/du, where k and u may be correlated (both vary with b, for instance). Details are given in Appendix C. 4. The Bontemps–Robin–van den Berg model The model in Bontemps et al. (2000) extends Burdett and Mortensen’s (1998) model to allow firms to make counteroffers to workers who have contacted a competitor. We again assume that job opportunities arise indistinctly to both employed and unemployed (at the same rate k), but now there is a continuum of firms whose productivities we assume to be uniformly interval (where p > p P b).12 As in Bontemps et al., w is fixed to p, and given the wage distributed over the exogenous ½p; p pÞ (see Footnote 8 there). Thus offer w = K(p), it follows that K 0 ðpÞ ¼ 2ðk=ðd þ kÞÞðp pÞ=ðp
KðpÞ ¼ p þ
2 d ðp pÞ ; p dþk p
whence K 1 ðwÞ ¼ p þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pÞðw pÞ. That paper also derives a useful relation (Eq. (A.7) in its appendix) between ððd þ kÞ=kÞðp
productivity K1(w) and the distribution of wages G, K1(w) = w + (d + kG(w))/(2kG0 (w)). 12 The unique-k hypothesis is important in the present analysis because it makes it possible to write G as a function of w and u only. The same applies to the homogeneous productivity distribution hypothesis, which makes it possible to obtain a closed-form solution for G. Estimates for k0 and k1 (job offer arrival rates for unemployed and employed, respectively) vary greatly from study to study, obviously depending on the specifics of the model, the data and the estimation procedure. Blau (1992), for instance, has these rates decreasing with time rather than constant, but very similar between unemployed and employed (an average of approximately 0.089 for both arrival rates, see Table 3 there). Van den Berg and Ridder’s (1998) estimates (see Table 10 there) are k0 = 0.033 and k1 = 0.035.
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0.7 0.6
Gini coefficient
0.5 I(⋅,1.46,66.5) 0.4 0.3 0.2 0.1 0
0
0.2
0.4
0.6
0.8
1
unemployment rate Fig. 6. Gini coefficient of income inequality as a function of the unemployment rate in the Bontemps–Robin–van den Berg model, fixing the productivity parameters k and K.
One thus obtains the linear differential equation
G0 ðwÞ ¼
d þ kGðwÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dþk ð p pÞðw pÞ ðw pÞ 2k k
ð14Þ
! ½0; 1 given by with initial condition G(p) = 0. The function G : ½w; w
0
1 pffiffiffiffiffiffiffiffiffiffiffiffi pp dB C GðwÞ ¼ @pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1A; k k p p dþk ðw pÞ
ð15Þ
¼ ðdp þ kp Þ=ðd þ kÞ, can be checked (see Appendix D) to be the solution. Or, from (6) (which is the expreswhere w = p and w sion stemming from Proposition 2 in Bontemps et al. with k0 = k),
0
1 pffiffiffiffiffiffiffiffiffiffiffiffi p p u B C Gðw; uÞ ¼ @ ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1A: 1 u pffiffiffiffiffiffiffiffiffiffiffi p ð1 uÞðw pÞ p
Application of (4) here gives (see Appendix E.1)
Lðx; uÞ ¼
8 bx if x 6 u > u 1uþ2 log u ; < buþpð1uÞþðppÞ1u ðu Þ > :
pÞ u ðux uxþ2 loguxÞ buþpðxuÞþðp 1u pÞ u ð1uuþ2 log uÞ buþpð1uÞþðp 1u
:
ð16Þ
; otherwise
In order to find the Gini coefficient, we need to integrate this expression with respect to x and then use (5). As shown in Appendix E.2, this leads to
I¼
u pÞ 1u ðp bÞuð1 uÞ þ 2ðp ð2u 2 ðu þ 1Þ log uÞ : u pÞ 1u bu þ pð1 uÞ þ ðp ð1u u þ 2 log uÞ
=b (so that K > k P 1), we then have Making k:¼p/b and K :¼ p
Iðu; k; KÞ ¼
u ðk 1Þuð1 uÞ þ 2ðK kÞ 1u ð2u 2 ðu þ 1Þ log uÞ 1 : u u þ 2 log u u þ kð1 uÞ þ ðK kÞ 1u u
ð17Þ
At this point we are able to provide a counterpart to Proposition 1, this time concerning the Bontemps–Robin–van den Berg model. Proposition 2. Consider the Bontemps–Robin–van den Berg job-search model, as described above. If the unemployment rate generated by the model is no larger than 15%, then the Gini coefficient of income inequality is positively correlated to the . unemployment rate, regardless of the values of b, p and p Proof. See Appendix E.3. h
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Gini coefficient
0.4
0.3
0.2
0.1
0 10 8
K
6 4 2
0
0.2
0.4
0.8
0.6
1
u
Fig. 7. Gini coefficient as a function of the unemployment rate and the high-productivity parameter K in the Bontemps–Robin–van den Berg model, for k = 1.
The Bontemps et al. (2000) framework also allows for (endogenous) differentiation between a formal and an informal sector of the economy. In fact, Meghir et al. (2010) incorporate this possibility into the Bontemps et al. model and show that firms with high enough productivities will choose formality in equilibrium, whereas the less productive ones will prefer informality (whence their laid-off employees should not expect to receive unemployment benefits). Having estimated their model using data from Brazil, they find that informal-sector wages may include a considerable compensating differential (this being the typical case, for instance, for low-education males living in the cities of São Paulo and Salvador). This makes it difficult to assert that the positive correlation between u and I will still hold in this case: once a highly paid worker in the informal sector of one of those cities gets fired and starts receiving 0 until he/she goes back into the labor force, that may actually bring incomes closer together, instead of further apart. The following picture illustrates Proposition 2. It plots the Gini coefficient of income inequality as a function of the unemployment rate with k = 1.46 and K = 66.5, which are the values stemming from Bontemps et al.’s (2000) analysis of the French labor market.13 As in the Burdett–Mortensen model, there is no vector (k, K) generating Iu > 0 for every u. The reasoning is exactly the same as before, and is based on the fact that, from (17),
Ið1; k; KÞ ¼ lim
u!1
¼ lim
u!1
u ðk 1Þuð1 uÞ þ 2ðK kÞ 1u ð2u 2 ðu þ 1Þ log uÞ 1 u u þ 2 log u u þ kð1 uÞ þ ðK kÞ 1u u log u ðk 1Þuð1 uÞ þ 2ðK kÞuð2 þ log uÞ 4ðK kÞ u1u
u þ kð1 uÞ þ ðK kÞð1 þ uÞ þ 2ðK
log u kÞ u1u
¼
2ðK kÞð2Þ 4ðK kÞð1Þ ¼ 0; 1 þ 2ðK kÞ þ 2ðK kÞð1Þ
where in the third equality l’Hôpital’s rule was employed to conclude that limu?1((u log u)/(1 u)) = limu?1((log u + 1)/ (1)) = 1. Fig. 7 shows, for k = 1, the joint dependence of I on u and K. It may be noted that, for a fixed K, I(, k, K) presents the same inverted U shape as featured in the Burdett–Mortensen model. Finally, a note on the tightness of this result. Plugging (17) into a mathematical software yields I(0.15,1,1.05) 8.3010 103 and I(0.16, 1, 1.05) 8.2997 103 < I(0.15, 1, 1.05), showing the 15%-limit hypothesis on the unemployment rate to be, within an error of at most 1%, also necessary so that Iu > 0. Note, however, that in the specific case where the parameters assume the values used to plot Fig. 6, the positive correlation between u and I emerges for rates of unemployment as high as 58%. This suggests that, for real-world parameter values, the condition u 6 0.15 may be too stringent. 5. Conclusion A vast literature has documented a positive correlation between unemployment and inequality in several different economies. Understanding how such a correlation may emerge and on which parameters it may depend is crucial for the correct interpretation of empirical evaluations of inequality. We have shown that job-search frictions as modeled in Burdett and Mortensen (1998) and Bontemps et al. (2000) generate positive correlations between the Gini coefficient of income inequality and the unemployment rate whenever this rate is no larger than 28% and 15%, respectively. 13 ¼ 299; 166. Here we base ourselves on the upper left figure on page 334 of Bontemps et al. (2000), which presents the figures b = w = 4500, p = 6563, and p Also, as noted earlier, the reader should bear in mind that the model used here, with k0 = k1 and a uniform distribution of firm productivities, is a particular instance of the broader model introduced in Bontemps et al.
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Also, as can be implied from (12), both time-series and cross-sectional studies featuring inequality comparisons could also benefit from taking a closer look at changes in the surrounding political and technological environment (here represented by parameters b and p), in addition to the job-market parameters determining the level of unemployment. Taking as a benchmark the classic Burdett–Mortensen model, not only larger (and reasonable) unemployment rates directly imply greater income dispersion as measured by the Gini coefficient, but also do higher productivity levels and lower unemployment insurance. Models with worker skill heterogeneity (e.g., Acemoglu, 1999; Albrecht and Vroman, 2002; Greiner et al., 2004) may be used in future research, and in that case a most interesting analysis should try to separate the effect of unemployment on within-group inequality (the one appearing in this paper) from its effect on between-group inequality, or even treating structural unemployment (that present here) separately from that stemming from skill inadequacy of some workers for some jobs. Appendix A We are interested in finding a formula for the Gini coefficient of income inequality in the case of the stylized economy as in the third and fourth examples in Section 2. The Lorenz curve will described by u, r(m:¼1 u r), and b, w, and w, be as in Fig. A1, where e:¼bu/W, d:¼(bu + w(1 r u))/W, and W ¼ bu þ wð1 r uÞ þ wr: Considering the triangles marked A and B in this figure, the Gini index can be calculated as:
0
I ¼ 2ðA þ BÞ ¼ 1 r
1
0 1
0
d 1 þ u
1 1 1 r
0 1
e 1 ¼ 1 r d þ du eð1 rÞ ¼ ð1 eÞð1 rÞ dð1 uÞ:
d 1
1
d
A
B
e 0
0
u
L
1−r
1
Fig. A1. Lorenz curve in an economy with two income levels and unemployment insurance.
Appendix B B.1 Here we show that (4) takes the form (11) in the Burdett and Mortensen (1998) model. Using (9):
Z
wx
w
sffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! pffiffiffiffiffiffiffiffiffiffiffiffi Z wx 1 pb u pb ðp wÞ2 dw ðwx wÞ 1 dw ¼ pw 1u w w " ! 2
u pffiffiffiffiffiffiffiffiffiffiffiffi 1 pð x Þ ðpbÞ u u 2 2 p b 2ðp wÞ
p ðp bÞ b ¼ 1u x b pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi u2 u u pb pb 1 ðp bÞ 2 p b ¼ 1u x x u2 u u u u u ¼ ðp bÞ 2 1 1 ðp bÞ 1 2 1þ ¼ 1u x x 1u x x u u2 ¼ ðp bÞ 1 : 1u x
u Gðw; uÞdw ¼ 1u
Z
wx
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Thus, the numerator in the second line in (4) is
bu þ wx ðx uÞ ð1 uÞ
Z
wx
w
u2 u2 Gðw; uÞdw ¼ bu þ p ðp bÞ ðx uÞ ðp bÞu 1 x x u2 u2 u ¼ bu þ pðx uÞ ðp bÞðx uÞ ðp bÞu þ 2ðp bÞu ðp bÞu x x x u2 u ¼ bu þ pðx uÞ ðp bÞx ðp bÞu þ 2ðp bÞu x x u2 u ¼ bu þ pðx uÞ ðp bÞu þ ðp bÞ ¼ bu þ pðx uÞ ðp bÞ ðx uÞ x x u ¼ bu þ pðx uÞ ðp bÞðx uÞ ðp bÞðx uÞ 1 x ðx uÞ2 ¼ bx þ ðp bÞ ; x
while the denominators there correspond to the special case x = 1 of this numerator:
uÞ ð1 uÞ bu þ wð1
Z
w
Gðw; uÞdw ¼ b þ ðp bÞð1 uÞ2 :
w
B.2 We now use (11) and (5) in order to calculate the Gini coefficient stemming from the Burdett–Mortensen model:
Ru I ¼12 ¼1 ¼
0
bxdx þ
R1 2 bx þ ðp bÞ ðxuÞ dx u x
b þ ðp bÞð1 uÞ2
R1 ¼1
b þ ðp bÞð1 u2 4uð1 uÞ 2u2 log uÞ b þ ðp bÞð1 uÞ2
2ðp bÞuð1 u þ u log uÞ b þ ðp bÞð1 uÞ2
0
2bxdx þ ðp bÞ
R1 2 2x 4u þ 2 ux dx u
b þ ðp bÞð1 uÞ2 ¼
ðp bÞðð1 uÞ2 ð3u2 4u þ 1 2u2 log uÞÞ b þ ðp bÞð1 uÞ2
:
B.3 Here we show that 1 u + 2ulogu is positive for u 2 (0, 0.28] (used in the proof of Proposition 1). The derivative of 1 1 u + 2ulogu is 1 + 2logu, which is negative as long as u < e2 0:607. That is, up to 0.60,1 u + 2ulog u decreases with 3 u. Also, at u = 0.28, 1 u + 2ulogu 7.1 10 , i.e., it is still positive (at u = 0.29 it would already be negative, approximately 8.0 103). Therefore 1 u + 2ulogu is positive for u 2 (0, 0.28]. Appendix C Here we deal with the case of an endogenous k as in Mortensen (2000), and analyze the sign of dI/du = Iu + Ikdk/du within that framework. Since (12) still holds, Iu is the same as in the Burdett–Mortensen model:
Iu ðu; kÞ ¼ 2ðk 1Þ ¼ 2ðk 1Þ ¼ 2ðk 1Þ
ð1 u þ u log u þ uð1 þ log u þ 1ÞÞð1 þ ðk 1Þð1 uÞ2 Þ þ 2ðk 1Þuð1 u þ u log uÞð1 uÞ
ð1 þ ðk 1Þð1 uÞ2 Þ2 ð1 uÞð1 þ ðk 1Þð1 uÞ2 þ 2ðk 1Þuð1 uÞÞ þ 2uð1 þ ðk 1Þð1 uÞ2 þ ðk 1Þuð1 uÞÞ log u ð1 þ ðk 1Þð1 uÞ2 Þ2
ð1 uÞð1 þ ðk 1Þð1 u2 ÞÞ þ 2uð1 þ ðk 1Þð1 uÞÞ log u ð1 þ ðk 1Þð1 uÞ2 Þ2
:
One also has
Ik ¼ 2uð1 u þ u log uÞ
1 þ ðk 1Þð1 uÞ2 ðk 1Þð1 uÞ2 ð1 þ ðk 1Þð1 uÞ2 Þ2
¼
2uð1 u þ u log uÞ ð1 þ ðk 1Þð1 uÞ2 Þ2
:
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Now, from (6) and (13):
v uð1 uÞ
b ¼ ðk 1Þ: c
pffi Let kðÞ ¼ L ; L > 0. Then (6) yields v = (d/L)2((1 u)/u)2. Substitution in the above expression gives k = 1 + (c/b)(d/L)2(1 u)/ 3 u , whence
2 dk c d u3 2u 3 ¼ ð3u4 þ 2u3 Þ ¼ ðk 1Þ ð3u4 þ 2u3 Þ ¼ ðk 1Þ du b L uð1 uÞ 1u in equilibrium. So 2u3 ð1 u þ u log uÞ dI ð1 uÞð1 þ ðk 1Þð1 u2 ÞÞ þ 2uð1 þ ðk 1Þð1 uÞÞ log u ¼ 2ðk 1Þ þ 2ðk 1Þ 1u ; 2 2 du ð1 þ ðk 1Þð1 uÞ2 Þ2 ð1 þ ðk 1Þð1 uÞ Þ
and sgn dI/du = sgn E(u, k), where
Eðu; kÞ :¼ ð1 uÞ2 ð1 þ ðk 1Þð1 u2 ÞÞ þ 2uð1 uÞð1 þ ðk 1Þð1 uÞÞ log u þ ð2u 3Þð1 u þ u log uÞ ¼ ð1 uÞðð1 uÞð1 þ ðk 1Þð1 u2 ÞÞ þ 2u 3Þ þ uð2u 3 þ 2ð1 uÞð1 þ ðk 1Þð1 uÞÞÞ log u ¼ ð1 uÞð1 þ ð1 uÞð1 þ ðk 1Þð1 u2 ÞÞÞ þ uð1 þ 2ðk 1Þð1 uÞ2 Þ log u: Since Ek(u,k) = (1 u)2(1 u2) + 2u(1 u)2logu = (1 u)2(1 u2 + 2ulogu) is positive for u 2 (0,1) ðlimu!1 ð1 u2 þ 2u log uÞ ¼ 0 and d(1 u2 + 2ulogu)/du = 2(1 u + logu) < 0), lower levels of k tend to bring dI/du down. Now, from (13), we could replace the k’s in the above expression for 1 + (c/b)v/(u(1 u)). Since a large b yields a lower v and a higher u (and u(1 u), at least for u < 1/2), we thus get a lower k and a lower dI/du. For instance, it may be seen that, if k = 1.46 (the infimum of the k’s in Bontemps et al.’s analysis of the French labor market, see Section 4), dI/du < 0,"u 2 (0, 1). If k = 66.5 (the supremum of those k ’s), one has dI/du > 0,"u 2 (0,0.61], while the intermediate value k = 5.4 would lead to dI/du > 0,"u 2 (0,0.15]. Appendix D We check that the distribution G given in (15) is a proper solution to (14). Differentiating (15), one gets
G0 ðwÞ ¼ while
rffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi p p p p d k 1 d 1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ ffiffiffiffiffiffiffiffiffiffiffiffi ffi p q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ; p ffiffiffiffiffiffiffiffiffiffiffi ffi p ffiffiffiffiffiffiffiffiffiffiffi ffi 2 kð p d þ k 2 w p 2k dþk ðw pÞ k k p dþk ðw pÞÞ p dþk ðw pÞ p k ! pffiffiffiffiffiffi p p ffi d þ d pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k
pffiffiffiffiffiffiffiffiffiffiffiffi pp p p dþkðwpÞ d þ kGðwÞ d 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2k dþk dþk dþk k p dþk 2k ðp pÞðw pÞ ðw pÞ ðp pÞðw pÞ ðw pÞ ðw pÞ ð p ðw pÞÞ2 2k k k k so that the differential Eq. (14) is satisfied. Furthermore, G(p) = 0, as wanted. We also have G0 > 0 and
0 1 ! pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi p B C d p p p 1 dB d C ¼G Þ ¼ @ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi k pffiffiffiffiffiffiffiffiffiffiffi 1 ¼ ðdp þ kp GðwÞ ffi 1A ¼ k pffiffiffiffiffiffiffiffiffiffi p dþk p dþk k pffiffiffiffiffiffiffiffiffiffiffi k p p k 1 Þ p p dþk dþk ðdp þ kp p
Appendix E E.1 In order to derive (16) from (15), we must calculate wx and The first is achieved by writing
R wx w
0 1 pffiffiffiffiffiffiffiffiffiffiffiffi p p xu u B C ¼ Gðwx ; uÞ ¼ @ ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1A; 1u 1 u pffiffiffiffiffiffiffiffiffiffiffi p ð1 uÞðw pÞ p x
Gðw; uÞdw.
1 d dþk
! 1 ¼ 1:
R.P. Cysne, D. Turchick / Journal of Macroeconomics 34 (2012) 454–469
467
whence:
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p u p p x u qffiffiffiffiffiffiffiffiffiffiffiffi p p ð1 uÞðwx pÞ) p x ¼ u pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) ¼ p x x p ð1 uÞðwx pÞ p 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 u Þ2 1 ux x pÞ: ðp pÞ ¼ wx p)wx ¼ p þ ðp ¼ ð1 uÞðwx pÞ) 1u 1u Now for the second:
Z
wx
w
u Gðw; uÞdw ¼ 1u
0
Z
wx
w
B @
1 1 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1Adw: 1u 1 pp ðw pÞ
pÞ, this equals Making the substitution z2 ¼ ð1 uÞðw pÞ=ðp
u 1u
Z
1ux
Z 1ux 2ðp pÞ 1 u z2 pÞ 1 dz: zdz ¼ 2ðp 2 1z 1z 1u ð1 uÞ 0
0 2
Since z /2 z log(1 z) is a primitive of z2/(1 z), we have
Z
wx
u 1u ½z2 þ 2z þ 2 logð1 zÞj0 x ð1 uÞ2 u u2 u u pÞ þ 2 log : ¼ ðp 1 þ2 1 2 x x x ð1 uÞ
pÞ Gðw; uÞdw ¼ ðp
w
Therefore, the numerator in the second line in (4) is
bu þ wx ðx uÞ ð1 uÞ
Z
wx
Gðw; uÞdw
w
u 2
!
1x u u2 u u pÞ pÞ ðx uÞ þ ðp 1 þ 2 log þ2 1 ðp 1u x x x 1u 2 2 u u u u xu u pÞ 1 þ 2 log þ 1 þ2 1 ¼ bu þ pðx uÞ þ ðp 1u x x x u x u u u2 u x u u u2 u pÞ ¼ bu þ pðx uÞ þ ðp 12 þ þ22 þ 12þ2 þ þ 2 log 1u x x x u x x x x u x u u pÞ ¼ bu þ pðx uÞ þ ðp þ 2 log ; 1u u x x ¼ bu þ p þ
while the denominators there correspond to the special case x = 1 of this numerator:
uÞ ð1 uÞ bu þ wðx
Z
w
pÞ Gðw; uÞdw ¼ bu þ pð1 uÞ þ ðp w
u 1 u þ 2 log u : 1u u
E.2 Here we use (16) and (5) in order to calculate the Gini coefficient originating from the Bontemps–Robin–van den Berg model. The second equality below uses the fact that x + xlog(u/x) is a primitive of log(u/x).
x u R1 u pÞ 1u bu þ pðx uÞ þ ðp x þ 2 log ux dx u u 1 I ¼1 u pÞ 1u u þ 2 log u bu þ pð1 uÞ þ ðp u 2 2 u 1u2 pÞ 1u 2 b u2 þ buð1 uÞ þ p ð1uÞ þ ðp þ u log u þ 2ð1 u þ log uÞ 2 2u 1 ¼1 u pÞ 1u u þ 2 log u bu þ pð1 uÞ þ ðp u 1 2 u pÞ 1u bu þ 2bu þ pð1 uÞ2 þ ðp þ 4 5u þ ð2u þ 4Þ log u u ¼1 u 1 pÞ 1u u þ 2 log u bu þ pð1 uÞ þ ðp u 2 u 1 pÞ 1u u u þ 2 log u 1u 4 þ 5u ð2u þ 4Þ log u bu þ bu þ puð1 uÞ þ ðp ¼ u 1 pÞ 1u bu þ pð1 uÞ þ ðp u þ 2 log u u 2
¼
R u 0
bxdx þ
u pÞ 1u ðp bÞuð1 uÞ þ 2ðp ð2u 2 ðu þ 1Þ log uÞ : u bu þ pð1 uÞ þ ðp pÞ 1u 1u u þ 2 log u
468
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E.3 Proof of Proposition 2. Write
Nðu; k; KÞ :¼ ðk 1Þuð1 uÞ þ 2ðK kÞðu=ð1 uÞÞð2u 2 ðu þ 1Þ log uÞ and
Dðu; k; KÞ :¼ u þ kð1 uÞ þ ðK kÞðu=ð1 uÞÞð1=u u þ 2 log uÞ; so that I = N/D. D is positive since 1/u u + 2logu is positive (its derivative is 1/u2 1 + 2/u = ((1 u)/u)2 < 0 and at u = 1 it takes on the value 1/1 1 + 2 0 = 0), and since I is a Gini coefficient, N is positive too. In Appendix E.4, N and D are shown to be, respectively, increasing and decreasing functions of u over the (0,0.15] interval. Therefore Iu > 0 (whence again @I/ @d > 0 and @I/@k < 0) in the Bontemps-Robin-van den Berg model, under the assumption that it does not generate an unemployment rate greater than 15%. E.4 We first show that D (defined in Appendix E.3) decreases with u. Its derivative is:
! 1 u 1 2 u þ 2 log u þ 1 þ 1u u2 u ð1 uÞ2 u K k 1 1 K k u þ 2 log u þ ð1 uÞ u þ 2 ¼1kþ ðu2 4u þ 3 þ 2 log uÞ: ¼1kþ u u ð1 uÞ2 ð1 uÞ2
Du ðu; k; KÞ ¼ 1 k þ ðK kÞ
1
Since u2 4u + 3 + 2logu < 0 (its derivative is 2u 4 + 2/u = 2(1 u)2/u > 0 and at u = 1 it takes on the value 1 4 + 3 + 2 0 = 0), we have Du decreasing with K. Since at the minimum value for K (which is k), Du already takes on a negative value (1 k), we conclude that Du < 0. We now show that N (also defined in Appendix E.3) increases with u for u 2 (0, 0.15]. We may start by imposing u < 0.50, so that u(1 u) increases with u. Therefore, if Q(u) :¼(u/(1 u))(2u 2 (u + 1)logu) increases with u, so does N. The derivative of Q is:
Q u ðuÞ ¼ ¼
1 2
ð1 uÞ 1
ð1 uÞ2
ð2u 2 ðu þ 1Þ log uÞ þ
u uþ1 2 log u 1u u
ðð2u 2 ðu þ 1Þ log uÞ þ ð1 uÞðu 1 u log uÞÞ ¼
1 ð1 uÞ2
ðu2 þ 4u 3 þ ðu2 2u 1Þ log uÞ:
We are left to verify that S(u):¼ u2 + 4u 3 + (u2 2u 1)log u is positive. In fact, its derivative is 2u + 4 + (2u 2)log u + (u2 2u 1)/u = u + 2 1/u 2(1 u)logu = ((1 u)/u)(1 u + 2ulogu), which is negative for u 6 0.28 (incidentally, this is shown in Appendix B.3, which refers to the Burdett–Mortensen model). So, in the whole (0,0.28] interval, S decreases with u. Unfortunately, at u = 0.28, S is already negative (approximately 7.2 102), which makes the Iu > 0 result necessarily weaker than in the original Burdett–Mortensen model. Nevertheless, S(0.15) 1.07 103 (and S(0.16) 1.35 102), whence S (and Qu) is positive for u 6 0.15, making N an increasing function of u in the required interval. References Acemoglu, D., 1999. Changes in unemployment and wage inequality: an alternative theory and some evidence. American Economic Review 89, 1259–1278. Albrecht, J., Vroman, S., 2002. A matching model with endogenous skill requirements. International Economic Review 43, 283–305. Björklund, A., 1991. Unemployment and income distribution: time-series evidence from Sweden. Scandinavian Journal of Economics 93, 457–465. Blau, D.M., 1992. An empirical analysis of employed and unemployed job search behavior. Industrial and Labor Relations Review 45, 738–752. Blázquez, M., Jansen, M., 2008. Search, mismatch and unemployment. European Economic Review 52, 498–526. Blejer, M.I., Guerrero, I., 1990. The impact of macroeconomic policies on income distribution: an empirical study of the Philippines. Review of Economics and Statistics 72, 414–423. Blinder, A.S., Esaki, H.Y., 1978. Macroeconomic activity and income distribution in the postwar United States. Review of Economics and Statistics 60, 604– 609. Bontemps, C., Robin, J.-M., van den Berg, G.J., 2000. Equilibrium search with continuous productivity dispersion: theory and nonparametric estimation. International Economic Review 41, 305–358. Burdett, K., Mortensen, D.T., 1998. Wage differentials, employer size, and unemployment. International Economic Review 39, 257–273. Burdett, K., Lagos, R., Wright, R., 2004. An on-the-job search model of crime, inequality, and unemployment. International Economic Review 45, 681–706. Cahuc, P., Postel-Vinay, F., Robin, J.-M., 2006. Wage bargaining with on-the-job search: theory and evidence. Econometrica 74, 323–364. Cardoso, E., Paes de Barros, R., Urani, A., 1995. Inflation and unemployment as determinants of inequality in Brazil: the 1980s. In: Dornbusch, R., Edwards, S. (Eds.), Reform, Recovery, and Growth: Latin America and the Middle East. University of Chicago Press, Chicago, pp. 37–64. Castañeda, A., Dı´az-Giménez, J., Ríos-Rull, J.-V., 1998. Exploring the income distribution business cycle dynamics. Journal of Monetary Economics 42, 93– 130. Cysne, R.P., 2009. On the positive correlation between income inequality and unemployment. Review of Economics and Statistics 91, 218–226. Gaumont, D., Schindler, M., Wright, R., 2006. Alternative theories of wage dispersion. European Economic Review 50, 831–848.
R.P. Cysne, D. Turchick / Journal of Macroeconomics 34 (2012) 454–469
469
Greiner, A., Rubart, J., Semmler, W., 2004. Economic growth, skill-biased technical change and wage inequality: a model and estimations for the US and Europe. Journal of Macroeconomics 26, 597–621. Jäntti, M., 1994. A more efficient estimate of the effects of macroeconomic activity on the distribution of income. Review of Economics and Statistics 76, 372–378. Jaumotte, F., Lall, S., Papageorgiou, C., 2008. Rising income inequality: technology, or trade and financial globalization? IMF Working paper 08/185. McCall, J.J., 1970. Economics of information and job search. Quarterly Journal of Economics 84, 113–126. Meghir, C., Narita, R., Robin, J.-M., 2010. Wages and Informality in Developing Countries. University College London, Mimeo. Metcalf, C.E., 1969. The size distribution of personal income during the business cycle. American Economic Review 59, 657–668. Mocan, H.N., 1999. Structural unemployment, cyclical unemployment, and income inequality. Review of Economics and Statistics 81, 122–134. Mortensen, D.T., 2000. Equilibrium unemployment with wage posting: Burdett–Mortensen meet Pissarides. In: Bunzel, H., Christianesen, B.J., Keifer, N.M., Mortensen, D.T. (Eds.), Panel Data and Structural Labour Market Models. Elsevier, Amsterdam, pp. 281–292. Mortensen, D.T., 2003. How monopsonistic is the (Danish) labor market? In: Aghion, P., Frydman, R., Stiglitz, J.E., Woodford, M. (Eds.), Knowledge, Information, and Expectations in Modern Macroeconomics: In Honor of Edmund S. Phelps. Princeton University Press, Princeton, pp. 285–308. Mortensen, D.T., Pissarides, C.A., 1999. Unemployment responses to ‘skill-biased’ technology shocks: the role of labour market policy. Economic Journal 109, 242–265. Nickell, S., Nunziata, L., Ochel, W., Quintini, G., 2002. The Beveridge curve, unemployment and wages in the OECD from the 1960s to the 1990s. Centre for Economic Performance, London School of Economics and Political Science