ï»¿ WPS6254
Policy Research Working Paper 6254
Income Risk, Income Mobility and Welfare
Tom Krebs
Pravin Krishna
William F. Maloney
The World Bank
Development Research Group
Macroeconomics and Growth Team
October 2012
Policy Research Working Paper 6254
Abstract
This paper develops a framework for the quantitative the framework using data on individual incomes from
analysis of individual income dynamics, mobility and Mexico provides striking results. Much of measured
welfare. Individual income is assumed to follow a income mobility is driven by measurement error
stochastic process with two (unobserved) components, or transitory income shocks and therefore (almost)
component representing measurement error or welfare-neutral. A smaller part of measured income
transitory income shocks and an Autoregressive (AR(1)) mobility is due to either welfare-reducing income
component representing persistent changes in income. risk or welfare-enhancing catching-up of low-income
The analysis uses a tractable consumption-saving model individuals with high-income individuals, both of which
with labor income risk and incomplete markets to relate have economically significant effects on social welfare.
income dynamics to consumption and welfare, and Decomposing mobility into its fundamental components
derive analytical expressions for income mobility and is thus seen to be crucial from the standpoint of welfare
welfare as a function of the various parameters of the evaluation.
underlying income process. The empirical application of
This paper is a product of the Macroeconomics and Growth Team, Development Research Group. It is part of a larger
effort by the World Bank to provide open access to its research and make a contribution to development policy discussions
around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author
may be contacted at Wmaloney@worldbank.org .
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
WPS6254
Policy Research Working Paper 6254
Income Risk, Income Mobility and Welfare
Tom Krebs
Pravin Krishna
William F. Maloney
The World Bank
Development Research Group
Macroeconomics and Growth Team
October 2012
Policy Research Working Paper 0
Abstract
This paper develops a framework for the quantitative the framework using data on individual incomes from
analysis of individual income dynamics, mobility and Mexico provides striking results. Much of measured
welfare. Individual income is assumed to follow a income mobility is driven by measurement error
stochastic process with two (unobserved) components, or transitory income shocks and therefore (almost)
component representing measurement error or welfare-neutral. A smaller part of measured income
transitory income shocks and an Autoregressive (AR(1)) mobility is due to either welfare-reducing income
component representing persistent changes in income. risk or welfare-enhancing catching-up of low-income
The analysis uses a tractable consumption-saving model individuals with high-income individuals, both of which
with labor income risk and incomplete markets to relate have economically significant effects on social welfare.
income dynamics to consumption and welfare, and Decomposing mobility into its fundamental components
derive analytical expressions for income mobility and is thus seen to be crucial from the standpoint of welfare
welfare as a function of the various parameters of the evaluation.
underlying income process. The empirical application of
This paper is a product of the Macroeconomics and Growth Team, Development Research Group. It is part of a larger
effort by the World Bank to provide open access to its research and make a contribution to development policy discussions
around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author
may be contacted at Wmaloney@worldbank.org .
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
Income Risk, Income Mobility and Welfareâˆ—
Tom Krebs
University of Mannheim
Pravin Krishna
Johns Hopkins University and NBER
William F. Maloney
World Bank
âˆ—
We are grateful to seminar participants at Columbia, Harvard and Pennsylvania State University for
many useful comments. Particular thanks to Francois Bourguignon, Francisco Ferreira, Gary Fields, and the
other participants at the World Bank/Universitat Autonoma de Barcelona Workshop on Mobility for very
helpful discussions. Our appreciation to Edwin Goni and Mauricio Sarrias for inspired research assistance.
This work was partially supported by the Regional Studies Program of the Oï¬ƒce of the Chief Economist for
Latin America and the World Bank Research Support Budget.
I. Introduction
Individual income dynamics characterize society in important ways. The degree to which
individuals move across diï¬€erent sections of the income distribution is often summarized
by one parameter, income mobility. Indeed, income mobility is probably the single most
important indicator of individual income dynamics used in public policy discussions.1 Income
mobility is important as it informs us about the opportunities aï¬€orded by society to escape
oneâ€™s origins.2 At the same time, mobility may also be driven by variability in incomes that
reï¬‚ect the risk to which individuals are exposed in the economy.3 In this paper, we develop an
analytical framework for the estimation and welfare-theoretic evaluation of individual income
dynamics that takes into account these diï¬€erent drivers of income mobility. In addition, we
provide an application of our framework using individual income data from Mexico that
yields striking results: Much of measured income mobility is driven by measurement error or
transitory income shocks and therefore (almost) welfare-neutral. A smaller part of measured
income mobility is due to either welfare-reducing income risk or welfare-enhancing catching-
up of low-income individuals with high-income individuals, both of which have economically
signiï¬?cant eï¬€ects on social welfare. Decomposing mobility into its fundamental components
is thus crucial from the standpoint of welfare evaluation.
The literature on income mobility has often focused on two important questions: the
quantitative/empirical measurement of the extent and nature of the change in individual
incomes and, separately, the social-welfare-theoretic evaluations of such changes.4 Two
methodological issues have arisen in this area. First, the parametric formulations used in the
1
In a developing country context, see, for example, a ï¬‚agship publication of the World Bank in 2012,
â€œEconomic Mobility and the Rise of the Middle Class,â€? that focuses on the income mobility in Latin America.
For the US, the New York Times article â€œHarder for Americans to Rise From Lower Rungs,â€? by Jason deParle
(Jan 04,2012) describes the importance of economic mobility for the upcoming presidential election.
2
See, for instance, Shorrocks (1978), Lillard and Willis (1978a), Atkinson, Bourguignon and Morrison
(1992), Danziger and Gottschalk (1998) and Benabou and Ok (2001a, 2001b).
3
See again Atkinson, Bourguignon and Morrison (1992).
4
For the former, see Lillard and Willis (1978), Shorrocks (1978b), Geweke, Marshal and Zarkin (1986),
Conlisk (1990) and Fields and Ok (1996). For the latter, see, Atkinson (1983), Markandya (1982, 1984),
Atkinson, Bourguignon and Morrison (1992), Dardononi (1993), and Gottschalk and Spolaore (2002). Ad-
ditionally, the discussion over suitable social (income) mobility measures (indices), which may be used to
evaluate mobility given the pattern of individual income changes in society, constitutes a very well researched
area that has generated a number of important contributions in recent years. See Fields and Ok (1999) for
a survey discussion.
1
measurement of income changes are not easily used as inputs to the quantitative welfare-
theoretic analysis, thereby constituting a problematic gap between these two literatures.
Furthermore, as the literature has often pointed out, the measurement of dynamic income
changes is itself confronted by (at least) the following two problems. First, income data are
subject to measurement error and, second, a signiï¬?cant proportion of the observed income
changes may be simply temporary in nature - resulting, typically, in an overestimation of the
relevant mobility in income (Lillard and Willis (1978) Solon (2001), Luttmer (2002), Fields
et al (2003), Glewwe (2004), Antman and McKenzie (2007)). This is also important from
the perspective of welfare analysis, as measurement error has no eï¬€ect on workersâ€™ welfare
and transitory shocks to income are perhaps easily smoothed out, resulting in very small
welfare eï¬€ects. In addition, welfare analysis is confronted by an additional challenge. Since
individual utility is postulated as taking consumption rather than income as its argument,
its direct valuation requires reliable data on individual consumption levels, which are often
unavailable for developing countries. To use the more easily available data on incomes, a
theoretical framework is required that translates the estimated income dynamics into con-
sumption changes taking into account the institutional constraints individual agents face.
In this paper, we develop a tractable analytical framework to study income mobility
that provides a close link between the welfare theory and the empirical methodology used
in the measurement of the income dynamics, thereby helping to bridge the gap between
these literatures. At the same time, this framework overcomes many of the methodological
problems that we have just discussed. We note, at the outset, that our focus is on income
mobility within individual lifetimes (intra-generational mobility). We postulate (Section II)
that individuals face a stochastic income process that is highly parameterized, but, following
much of the literature, is suï¬ƒciently elaborate to distinguish between changes in income
resulting from trend growth and other predictable factors and changes in income that are
unpredictable. The unpredictable part of income change, in turn, has two components, one
ï¬?rst degree autoregressive (AR(1)) component reï¬‚ecting persistent shocks to income and
another component that is i.i.d and captures transitory shocks and measurement error in
the income data. We show how income mobility, measured in relation to the correlation
of incomes over time,5 relates to the various parameters of the underlying income process.
5
Speciï¬?cally, we use a quite basic and familiar measure, the Hart Index, which is the complement of
the correlation between the logarithm of incomes over times (see Hart (1981) and Shorrocks (1993)). As
2
We also discuss how these parameters can be estimated using individual income data and
econometric techniques that exploit both the longitudinal and repeated cross-section features
of our data set on individual incomes from Mexico (Sections III-V).6
Finally, we use a tractable consumption-saving model with labor income risk and incom-
plete markets (Section VI) that yields closed-form solutions for equilibrium consumption and
welfare as a function of the preference and income parameters. This theoretical framework,
based on the work of Constantinides and Duï¬ƒe (1996) and Krebs (2007), focuses on the per-
sistent component of labor income and abstracts from the i.i.d component, an abstraction
motivated by results in the literature demonstrating that workers can eï¬€ectively self-insure
against transitory income shocks, as we have already mentioned.7 One of the main insights
of this literature is that in equilibrium consumption responds one-for-one to permanent in-
come shocks.8 In this paper, we exploit this property of equilibrium consumption to derive
an explicit formula for social welfare as a function of the underlying income parameters.
The analytical framework we develop in this paper has the merit of linking income dy-
namics, income mobility and social welfare in a simple and transparent manner âˆ’ allowing
for a clearer analytical and quantitative discussion of these interrelated concepts, and specif-
ically the role of income variability, than has generally been possible in the past. We discuss
in detail how diï¬€erent determinants of measured income mobility may have quite diï¬€erent
Fields and Ok (1996) discuss, however, the literature has recently made important advances in studying
the â€œmulti-faceted conceptâ€? of mobility and a number of diï¬€erent theoretical measures, each capturing a
diï¬€erent aspect of mobility have been introduced. We have no contribution to make to this discussion and
simply use the Hart Index as our basic measure of mobility.
6
For an interesting exercise which compares results on poverty vulnerability (the propensity to move
into poverty) obtained using panel data on incomes with those obtained from repeated cross-sections instead
and ï¬?nds that model parameters recovered from pseudo-panels approximate reasonably well those estimated
directly from a true panel, see Bourguignon, Goh and Kim (2006)
7
See, for example, Aiyagari (1994), Heaton and Lucas (1996), and Levine and Zame (2002) and Section
III for further discussion of the issue of transitory income shocks. Clearly, in the case of measurement error
there is even more compelling reason to neglect the i.i.d. component in the welfare analysis. We should also
note that while we consider self-insurance against persistent income shocks, we do not model any alternative
schemes that may provide insurance against variations in persistent income. We believe this characterization
to be closer to that of developing economies, but this analysis would be relevant in any contexts where such
(social) insurance schemes are absent.
8
Moreover, Krebs (2007) shows that even in the case of persistent, but not necessarily permanent, income
shocks (AR(1) process with auto-correlation coeï¬ƒcient less than one) consumption still responds one-for-
one to income shocks if there are costs of ï¬?nancial intermediation that generate a suï¬ƒciently large spread
between borrowing rate and lending rate.
3
implications for welfare. Speciï¬?cally, we show that the auto-correlation coeï¬ƒcient of the
AR(1) process (the catching-up parameter) measures â€œgood mobilityâ€? in the sense that a
reduction in this parameter increases both mobility and welfare. In contrast, social welfare is
(almost) unaï¬€ected by measurement error or transitory income shocks even though mobility
increases with the variance of the i.i.d. component of labor income. Finally, the variance of
persistent income shocks (income risk) increases mobility, but decreases social welfare. This
implies that two societies with the same initial distribution of income and the same level of
measured income mobility and aggregate growth may experience quite diï¬€erent social welfare
changes depending upon the diï¬€erent combinations of the underlying income parameters.
We present a quantitative implementation of our framework that underscores the impor-
tance of decomposing income dynamics into its components, as we have discussed. Specif-
ically, an application using data on individual incomes from Mexico yields striking results.
Most of measured income mobility is driven by measurement error or transitory income
shocks and therefore (almost) welfare-neutral, and only a small part of measured income
mobility is due to either welfare-reducing income risk or welfare-enhancing catching-up of
low-income individuals with high-income individuals. However, despite the small mobility
eï¬€ects, (idiosyncratic) persistent income risk has signiï¬?cant negative eï¬€ects on social welfare
â€“ eliminating or insuring it would generate welfare gains that are equivalent to an increase
in lifetime consumption by about 10 percent even if workers are only moderately risk-averse
(log-utility).9 Eliminating the catch-up of low income individuals with high income indi-
viduals yields a loss in social welfare of similar magnitude. Decomposing mobility into its
fundamental components is thus seen to be crucial from the standpoint of welfare evaluation.
II. Income and Mobility
II.1. Income Process
Consider a large number of workers indexed by i. For notational ease, we focus on one cohort
of workers who enter the labor market for the ï¬?rst time in period t = 0 so that t = 0, 1, . . .
stands for both calendar time and age (experience) of the worker. Let yit stand for the labor
income of worker i in period t. Following a longstanding tradition in micro-econometrics,
9
In comparison, for the same preference parameters, Lucas (2003) computes welfare cost of aggregate
consumption ï¬‚uctuations in the US that are two orders of magnitude smaller. Thus, even though our
estimates of persistent income risk seem small when measured mobility is the yardstick, their welfare eï¬€ect
is large indeed.
4
we postulate that the log of yit is a random variable that is the sum of two components, a
persistent component, Ï‰it , and a transitory component, Î·it .10 In addition, we set the mean
of lnyit to Âµ. In short, we have:
log yit = Ï‰it + Î·it + Âµ . (1)
The persistent component, Ï‰it , follows an AR(1) process
Ï‰i,t+1 = Ï?Ï‰it + Ç«i,t+1 , (2)
where Ï? is a parameter measuring the persistence of shocks. The term Ç« denotes a stochastic
innovation to labor income, which we assume to be i.i.d. over time and across individuals.
We further assume that the transitory component of labor income, Î·it , is i.i.d. over time
and across individuals. Moreover, Î·it and Ç«i,t+n are uncorrelated for all t and n. All random
variables are normally distributed so that labor income is log-normally distributed. More
2 2 2
speciï¬?cally, we assume that Ç«it âˆ¼ N (0, ÏƒÇ« ), Î·it âˆ¼ N (0, ÏƒÎ· ), and Ï‰i0 âˆ¼ N (0, ÏƒÏ‰ 0
).
Equations (1) and (2) together imply that:
tâˆ’1
t
ln yit = Ï? Ï‰i0 + Ï?tâˆ’nâˆ’1 Ç«i,n+1 + Î·it + Âµ . (3)
n=0
Thus, labor income in period t is determined by initial condition, Ï‰0 , and stochastic changes,
the latter being represented by the transitory shocks, Î· , and permanent shocks, Ç«. From (3)
and our assumptions about Ç«, Î· , and Ï‰0 it follows that expected labor income is E [lnyit ] = Âµ
and labor income uncertainty before Ï‰i0 is known is given by
âˆ’Ï? 2t
Ï?2t ÏƒÏ‰
2 2
+ ÏƒÎ· +1 Ïƒ 2 if Ï? = 1
1âˆ’Ï?2 Ç«
var[lnyit ] = 2
0
2 2
. (4)
ÏƒÏ‰0 + ÏƒÎ· + tÏƒÇ« if Ï? = 1
As we have mentioned earlier, our study examines income mobility within individual
lifetimes, i.e., intra-generational income mobility.11 From (2), the parameter Ï? measures
persistency of income and thus (1âˆ’Ï?) measures the extent to which individuals with low levels
10
See Gottschalk and Moï¬ƒtt (1994) and Carroll and Samwick (1997) for similar speciï¬?cations and Baker
and Solon (2003) for a detailed discussion.
11
For recent work on intra-generational mobility, see Antman and McKenzie (2007), Cuesta and Pizzolitto
(2010), Dang et. al. (2011), and Cruces et. al (2011).
5
of income â€œinitiallyâ€? will catch up with individuals with high income. In our context, the
â€œinitialâ€? period corresponds to the time of entry into the work force after the completion of
formal education. Since labor income may vary initially for equivalent individuals, catching-
up in this context measures the extent to which individuals with initially low incomes catch
up to those with initially high incomes.12 In the terminology of the growth literature, it
measures convergence.13
II.2. Mobility
As noted in the introduction, our empirical measure of income mobility between 0 and t,
which we denote by mt , is the Hart index, deï¬?ned as the complement of the correlation in
(log) incomes at 0 and t (see Shorrocks (1993)):
mt = 1 âˆ’ corr(lnyi0 , lnyit ) (5)
cov (lnyi0 , lnyit )
= 1âˆ’ ,
Ïƒlnyi0 Ïƒlnyit
where we have used the notation Ïƒlnyi0 = var[lnyi0 ] and Ïƒlnyit = var[lnyit ]. Using
our income speciï¬?cation from the previous section, we ï¬?nd the following expression for the
co-variance:
tâˆ’1
t
cov (lnyi0 , lnyit ) = cov (Ï‰i0 + Î·i0 , Ï? Ï‰i0 + Ï?tâˆ’nâˆ’1 Ç«i,n+1 + Î·it + Âµ) (6)
n=0
2
= Ï?t ÏƒÏ‰ 0
12
In this theoretical section, our discussion relates to initial income diï¬€erences and subsequent mobility
between ex-ante identical individuals. In our discussion of empirical methodology and in our empirical
application to Mexican data, we will study mobility between observationally equivalent individuals. That is
to say, we examine income diï¬€erences and mobility in residual income after conditioning for the standard
determinants of income such as education and experience.
13
To see this, suppose Ï? < 1. In this case, we have convergence towards the â€œsteady stateâ€?: E [lnyit |Ï‰i0 ] â†’
Âµ. Let âˆ†0 = lnyi0 âˆ’ d Â¯ be the initial distance from the steady state and âˆ†t = lnyit âˆ’ d Â¯ be the distance in
period t. We can then deï¬?ne the time, T , it takes to get halfway towards the steady state, which is simply
the solution to âˆ†T /âˆ†0 = 1/2. Using the expression for âˆ†T and âˆ†0 , it is straightforward to see that T is
increasing in Ï? for Ï? < 1, that is, an increase in Ï? reduces the speed of convergence.
6
Using (3) and (6), we ï¬?nd the following expression for income mobility:14
ï£± 2
Ï?t Ïƒ Ï‰
ï£´ 1 âˆ’ âˆš
ï£´ 0
2t
if Ï? = 1
2t Ïƒ 2 +Ïƒ 2 + 1âˆ’Ï? Ïƒ 2
ï£² 2 +Ïƒ 2
ÏƒÏ‰ Ï?
mt = 0 Î·
2
Ï‰0 Î· 1 âˆ’ Ï?2 Ç«
. (7)
ÏƒÏ‰
âˆš âˆš
ï£³ 1 âˆ’ Ïƒ2 +Ïƒ2 Ïƒ2 +Ïƒ2 +tÏƒ2 if Ï? = 1
ï£´
ï£´ 0
Ï‰0 Î· Ï‰0 Î· Ç«
2 2
Equation (7) deï¬?nes income mobility as a function of the parameters of interest, ÏƒÇ« , ÏƒÎ· ,
and Ï?. It is straightforward to show that mobility is increasing in the volatility parameters
2 2
ÏƒÇ« and ÏƒÎ· . This is intuitive as an increase in the variance of income shocks increases the
variability of individual incomes, lowering the correlation between incomes across time, thus
increasing mobility.
2 2 2
Importantly, income mobility is decreasing in Ï? if either t is small and ÏƒÏ‰ 0
< ÏƒÎ· + ÏƒÇ« or t
2 2 2
is large and ÏƒÏ‰0 < ÏƒÇ« /(1 âˆ’ Ï? ):
âˆ‚mt âˆ‚mt âˆ‚mt
2
> 0 , 2
> 0 , < 0. (8)
âˆ‚ÏƒÇ« âˆ‚ÏƒÎ· âˆ‚Ï?
Intuitively, any increase in Ï? increases income persistence, reducing the catching-up eï¬€ect and
2 2 2 2 2
therefore reducing mobility. Note that both conditions ÏƒÏ‰ 0
< ÏƒÎ· + ÏƒÇ« and ÏƒÏ‰ 0
< ÏƒÇ« /(1 âˆ’ Ï?2 )
are satisï¬?ed in our empirical application (see section V).
III. Econometric Implementation
The discussion in the preceding sections has described how the diï¬€erent parameters of the
2 2 2
income process (ÏƒÏ‰ 0
, ÏƒÇ« , ÏƒÎ· and Ï?) aï¬€ect mobility. To get to a quantitative assessment of
these linkages, we turn next to the methodology and data used to estimate these parameters.
III.1. Estimation
We continue to assume that log labor income, ln yit , is speciï¬?ed as in (1). We further
assume that the deterministic mean component, Âµ, depends on xit = (xâ€²it , zit ), where zit
14
For Ï? < 1, the Ï‰ -process has a stationary distribution. If we choose as initial distribution this stationary
2 2 2
distribution, the Ï‰ -process becomes stationary with ÏƒÏ‰ t
= ÏƒÏ‰ 0
= ÏƒÇ« /(1 âˆ’ Ï?2 ). In this case the mobility
t 2 2
expression (7) reduces to mt = 1 âˆ’ Ï? / 1 + ÏƒÎ· /ÏƒÇ« .
7
denotes the age of worker i in year t and xâ€²it is vector of observable individual characteristics
beyond age (education, education2 , gender). We also make the functional form assumption
Âµt (xâ€²it , zit ) = Î»t + Î»(xâ€² ) Â· xâ€²it + z Î»(z )Î´ (zit ), where Î»t is a constant that varies by calendar time
period (thus absorbing the eï¬€ects of macroeconomic factors such as aggregate productivity
growth and aggregate economic ï¬‚uctuations on income), Î»(xâ€² ) is a vector of coeï¬ƒcients for
the vector of worker characteristics xâ€² , and Î´ (zit ) are age-dummies. Thus, log labor income
can be written as:
lnyit = Î»t + Î»(xâ€² ) Â· xit + Î»(z )Î´ (zit ) + vit (1â€² )
z
vit = Ï‰it + Î·it .
Equation (1â€™) resembles a typical Mincer speciï¬?cation for labor income for which the
residual, vit , is the sum of two unobserved stochastic components, Ï‰it and Î·it . As in Carroll
and Samwick (1997), we ï¬?rst use equation (1â€™) to estimate the residuals vit and then use
these estimated residuals to estimate, in a second step, the parameters of interest. As noted
above, this implies, importantly, that our mobility measure relates to residual income v
rather than unconditional income lny .
For notational simplicity, assume that all individuals i â€œare bornâ€? in period t = 0, so that
t and z simultaneously stand for age of the individual and calendar time. Equations (1) and
(2) which describe our labor income process imply that the the change in residual income
variance with age is given by:
V ar[viz ] = var[(Ï‰iz + Î·iz )]
1 âˆ’ Ï?2z 2
2
= ÏƒÎ· 2
+ Ï?2z ÏƒÏ‰ + Ïƒ (4â€² )
0
1 âˆ’ Ï?2 Ç«
(4â€™) links the changes in cross sectional residual income variances over for any age cohort z
with our parameters of interest. Unfortunately, however, (4â€™) is not suï¬ƒcient to separately
2 2
identify ÏƒÏ‰ 0
and ÏƒÇ« since, as can be seen from the expression on the right hand side, both
evolve at the same rate with z . We therefore also use the covariance restriction,
cov (viz , vi,z+1 ) = cov ((Ï‰iz + Î·iz ), (Ï‰i,z+1 + Î·i,z+1 ))
1 âˆ’ Ï?2z 2
= Ï?2z+1 ÏƒÏ‰
2
+ Ï?Ïƒ (6â€² )
0
1 âˆ’ Ï?2 Ç«
8
to achieve identiï¬?cation of all four parameters. Notice that (4â€™) requires, on the left hand
side, estimates of the cross-sectional variance of residual income for each age group z , while
(6â€™) requires that we use the panel dimension of our data set to estimate the covariances
in individualsâ€™ residual incomes viz over time. Thus, our estimation strategy exploits both
the panel dimension and the repeated cross sections available in the data set. As in Caroll
and Samwick (1997), we use residual income data at the individual level to obtain unbiased
2
estimators of the terms on the left hand side of (4â€™) and (6â€™). Speciï¬?cally, viz and viz vi,z+1
serve as as individual level â€?observationsâ€? of the variance and covariance terms on the left
hand sides of (4â€™) and (6â€™) respectively. We estimate our system of two equations ((4â€™)
and (6â€™)) using a simultaneous, non-linear, seemingly unrelated regressions model (NLSUR)
(as described in Gallant, 1975 and Amemiya, 1983). This permits the estimation of the two
non-linear equations, with the cross-equation restrictions implied by the common parameters,
simultaneously and achieves additional estimation eï¬ƒciency by combining information from
both equations (Davidson & MacKinnon, 2004).15
IV. Data
Using the estimation methodology described in the preceding section, we estimate income
mobility parameters using individual income data from Mexico. Speciï¬?cally, the individual
income data are taken from the Encuesta Nacional de Empleo Urbano (ENEU, Mexican
National Urban Employment Survey) which was conducted by the Instituto Nacional de
Estadistica, Geograï¬?a e Informatica (INEGI, National Institute of Statistics, Geography
and Information), the primary statistical agency in Mexico, and the Secretaria del Trabajo y
Prevision Social (STPS, Secretariat of Labor and Social Security), Mexicoâ€™s Labor Ministry.
Until recently, the ENEU was the primary survey instrument for collecting earnings and
employment data in Mexico. The survey is sampled to be representative geographically and
by social strata (see INEGI 2000). The basic sampling unit is the dwelling. Demographic in-
formation is collected on the household (households) occupying each dwelling. Subsequently,
an employment questionnaire is administered for each individual aged 12 and above in the
household on position in the household, level of education (years of schooling), age and sex
15
See also Davidson and MacKinnon (2004) for a through discussion of the asymptotic equivalence between
estimates obtained using a non-linear-least-squares methodology and the generalized method of moments.
9
as well as standard measures related to participation in the labor market: occupation, hours
worked, employment conditions, search and earnings. Importantly, the ENEU is constructed
as a rotating panel, where individuals are surveyed every quarter for a total of ï¬?ve quarters.16
Worker earnings include overall earnings in the individualâ€™s principal occupation from ï¬?xed
salary payments, hourly or daily wages, piece-meal work, commissions, tips and self employ-
ment earnings. The ENEU, in its modern form, has employed a consistent survey instrument
from 1987 to 2004; it is thus one of very few long-running surveys with a panel dimension in
the developing world. In our study, we are able to use this 18 year span comprising a total
of 72 quarters of data.17
We note that while the ENEU survey records employment information on all members
of the household above 12 years old, for younger workers employment is generally transient
and time is often divided among schooling, unpaid support to the household and paid work.
Similarly, much later in life, work again becomes more transient. In our analysis, we focus
on individuals between the ages 20 and 65.
V. Results
As discussed in the previous section, our estimation methodology proceeds in two steps.
As in Carroll and Samwick (1997), we ï¬?rst use individual data to estimate a Mincer earnings
regression. In a second step, the residuals from the Mincer regression are used to estimate
income mobility parameters using (4â€™) and (6â€™). Table 2 reports the estimates from the
ï¬?rst stage earnings regression using the ENEU data described in the preceding section. Our
estimates are consistent with earlier ï¬?ndings in the literature. Speciï¬?cally, earnings increase,
but at a decreasing rate, with education. Further, earnings increase with potential experience
(age) up until the age of 44 after which they decrease again. Males appear to earn 31 percent
more than women, conditional on the other covariates.18
16
In each round of the rotating panel, the questionnaire records absent members, adds any new members
who have joined the household, and records any changes in schooling that have taken place. If none of the
original group of household members is found to be living in the dwelling unit in the follow-up survey, the
household is recorded as a new household. The interviewers do not track households that move, so they
leave the panel. Rates of attrition are comparable to other developing countries (See Antman and McKenzie,
2007).
17
Since 2004, the ENEU has been replaced by the Encuesta Nacional de Ocupacion y Empleo (ENOE,
Survey of Occupation and Employment) in 2005. Unfortunately, however, the ENOE instrument diï¬€ers from
ENEU in important ways that make it impossible to match the surveys with conï¬?dence.
18
For robustness we have also run alternate earnings speciï¬?cations, allowing for both more and less
10
We use next the residuals from the earnings regression, vit , to construct individual level
2
â€œobservationsâ€? of income variances vit and covariances vit vi,t+1 ,19 that are to be used on the
left hand side of equations (4â€™) and (6â€™) to estimate the income mobility parameters. The
age proï¬?le of the constructed variance and covariance measures are indicated in Figures 1
and 2, which are generated by regressing the two variables respectively on a complete set of
age and time dummies and then plotting the former against age (see Deaton and Paxson,
1994, for a similar exercise). Consistent with equations (4â€™) and (6â€™), the accumulation of
2
persistent shocks, ÏƒÇ« , as age increases, gives both relationships an upward slope, albeit at
rates diï¬€ering by a factor of Ï?.
Estimation results from the joint estimation of (4â€™) and (6â€™), as described in the previous
section, yield the parameter estimates listed in Table 3. The ï¬?rst column presents the results
using the full sample, while the second column provides results obtained using data from just
those households that enter the sample in the ï¬?rst quarter of each year. Our estimates of the
income mobility parameters are also in line with those obtained previously in the literature.
The autoregressive component, Ï?, is estimated to be 0.977, which suggests that persistent
shocks to income experienced by any individual i will indeed last a long time. The estimated
2
variance of transitory shocks to income ÏƒÎ· = 0.202, is signiï¬?cantly larger than the variance
2
of persistent shocks to income ÏƒÇ« = 0.015. This is not surprising given that the transitory
shocks in our speciï¬?cation subsume measurement error in income, which we expect to be
2
quite large in our data set.20 Finally, the estimated variance in initial incomes, ÏƒÏ‰ 0
= 0.104.
As the results in the second column indicate, the estimates are not appreciably diï¬€erent with
the restricted sample of households who enter the survey in just the ï¬?rst quarter of each
year.
Given our estimates of the income parameters, we can use expressions (7) to analyze
mobility patterns. In particular, we can compute how much the individual parameters
contribute to overall mobility. Table 4 shows that mobility in residual income across 1 year
is 0.67 and it increases as the span of measurement increases to 10 years (0.76), and 25
years (0.84). The reasons behind the surprisingly high 1 year mobility level, and relatively
temporal variation, by allowing all parameters to vary in each time period, and separately by constraining
even the constant to be invariant across periods (unlike in the speciï¬?cation reported on in Table 2, which
includes year ï¬?xed eï¬€ects). The results do not change appreciably.
19
Note that vt+1 denotes individual iâ€™s residual one year (four quarters) after t
20
See Antman and McKenzie (2007) for a discussion of measurement error and mobility using this data.
11
modest increases thereafter become clearer in the next rows which set to zero each of the key
parameters and calculate the resulting change in mobility. Notice, ï¬?rst, that 1-year mobility
2
falls by a full 90 percent if we set ÏƒÎ· , which represents transitory shocks and measurement
error to zero. As we have noted earlier, our analysis proceeds under the understanding that
individuals can largely smooth such transitory shocks through own savings and these shocks
2
are therefore limited welfare impact. By contrast, â€œbad mobilityâ€? ÏƒÇ« due to risk and â€œgoodâ€?
mobility due to convergence, Ï?) account for roughly 1 percent each across 1 year.21
The relative impact of these parameters clearly changes as we increase the span over which
we are measuring mobility. At 25 years, setting transitory shocks to zero reduces mobility
by a still large, but much reduced by 23 percent (as transitory shocks are, by deï¬?nition,
transitory and mobility over this duration is driven to a greater extent by the cumulative
eï¬€ect of persistent shocks experienced by individuals over this period). By contrast, mobility
due to persistent risk accounts for 7.4 percent and due to convergence, to 8.6 percent. Having
identiï¬?ed which parameters have the largest inï¬‚uence on measured mobility, we now turn to
their relative contribution to welfare.
VI. Welfare Analysis
The voluminous literature on consumption and saving with individual income risk and
incomplete insurance markets has generated a number of insights.22 One important insight is
that workers can eï¬€ectively self-insure against transitory income shocks through borrowing
or own saving, and that the eï¬€ect of these shocks on equilibrium prices and quantities are
relatively small.23 A second important insight of this literature is that very persistent or
fully permanent income shocks have substantial eï¬€ects on consumption and welfare even if
individual households have own savings, but no or only limited access to insurance markets.
Indeed, when labor income is the main source of income and labor income shocks are highly
persistent, we would expect that consumption responds (almost) one-for-one to labor income
shocks. This point has been made more formally by Constantinides and Duï¬ƒe (1996) and
21
Note that since mobility is highly non-linear in its underlying parameters, measured mobility does not
decompose additively into its component parts.
22
See, for example, Heathcote, Storesletten, and Yaron (2009) for a recent survey.
23
See, for example Aiyagari (1994) and Heaton and Lucas (1996) for quantitative work and Levine and
Zame (2002) for a theoretical argument. Kubler and Schmedders (2002) show that welfare cost of â€œtransitoryâ€?
labor income shocks are non-negligible, but the labor income process they consider has Ï? = 0.5.
12
Krebs (2007) using dynamic general equilibrium exchange models with incomplete markets.
Constantinides and Duï¬ƒe (1996) only consider the case in which income follows a random
walk (Ï? = 1), but Krebs (2007) also analyzes an extension with Ï? < 1 and costs of ï¬?nancial
intermediation that introduce a spread between the borrowing rate and the lending rate. In
this section, we discuss the main ideas and results of the model analyzed in Krebs (2007).
VI.1. Consumption
The model features long-lived, risk-averse workers with homothetic preferences who make
consumption/saving choices in the face of uninsurable income shocks. Workersâ€™ preferences
over consumption plans, {cit }, allow for a time-additive expected utility representation with
one-period utility function of the CRRA-type, where in this paper we conï¬?ne attention to
the log-utility case (degree of relative risk aversion of 1):
âˆž
U ({cit }|Ï‰i0 ) = E Î² t lncit |Ï‰i0 . (9)
t=0
Workers maximize expected lifetime utility subject to a sequential budget constraint that
allows them to transfer wealth across periods through saving (or borrowing). The model is
an exchange economy with endogenous interest rate (general equilibrium).
In order to apply the equilibrium characterization result of Krebs (2007), we need to
introduce three modiï¬?cation of the labor income process (1). First, we abstract from ex-
ante heterogeneity and time-eï¬€ects: Âµt (xit ) = Âµ. For simplicity, we set Âµ = 0 so that the
mean of labor income (aggregate labor income) is normalized to one (see below). Second,
measurement error should not enter into the workerâ€™s budget constraint, and the part of Î·
that represents measurement error should therefore be omitted. Further, as we have argued
before, the part of Î· that is due to true income shocks is expected to have only small eï¬€ects
on equilibrium consumption and welfare. To simplify the analysis, we neglect these small
eï¬€ects of transitory income shocks and set lnyit = Ï‰it , where {Ï‰it } is an AR(1) process as in
the previous section. Third, the distribution of the innovation term, Ç«, and the distribution of
2 2 2 2
initial income, Ï‰0 , include a mean-adjustment: Ç« âˆ¼ N (âˆ’ÏƒÇ« /2, ÏƒÇ« ) and Ï‰0 âˆ¼ N (âˆ’ÏƒÏ‰ 0
/2, ÏƒÏ‰ 0
).
2 2
This adjustment is necessary to ensure that ÏƒÇ« and ÏƒÏ‰0 can be interpreted as uncertainty
parameters (see below).24
24
The main part of the analysis in Krebs (2007) deals with the random walk case, but the Appendix
13
Our speciï¬?cation of the labor income process implies that
Ï?
E [yi,t+1 |It ] = yit (10)
2
ÏƒÇ«
var[yi,t+1 |It ] = e âˆ’1
E [yi0 ] = 1
2
var[y0 ] = eÏƒÏ‰0
where It denote the information available at time t. Thus, increases in either ÏƒÇ« or ÏƒÏ‰0 in-
crease the variance of labor income without any change in the (conditional) mean â€“ they lead
to a mean-preserving spread. In other words, the two parameters measure risk/uncertainty.25
If Ï? = 1 and labor income follows a random walk, then the equilibrium interest rate will
adjust so that individual workers will optimally decide to set consumption equal to labor
income (see Constantinides and Duï¬ƒe (1996) and Krebs (2007) for details). If Ï? is not equal
to one, but not too far away from one, then a suï¬ƒciently large diï¬€erence in the borrowing
and lending rate (cost of ï¬?nancial intermediation) will ensure that in equilibrium households
still choose to set consumption equals labor income (see the Appendix of Krebs (2007) for
details). In short, in equilibrium we have cit = yit , that is, consumption and labor income
move one-for-one.
VI.2. Mobility and Welfare
Using cit = yit = Ï‰it and the income speciï¬?cation discussed above, we can evaluate the
expected lifetime utility (9) of an individual with initial income Ï‰i0 . Taking the expectation
over Ï‰i0 yields social welfare, W , where we assume that each individual household is assigned
equal weight in the social welfare function. In other words, social welfare is the expected
lifetime utility from an ex ante point of view when the initial condition, Ï‰0 , is not yet known
discusses the extension to labor income shocks that are not fully permanent. The labor income process
speciï¬?ed in the Appendix of Krebs (2007) is equivalent to an AR(1) process with an innovation term that
has ï¬?nite support, which rules out the case of a normal distribution. One way to apply the results of Krebs
(2007) to the present analysis is to truncate all normal distributions at an arbitrarily large point, and to think
of all equilibrium results as approximate results for which the approximation error can be made arbitrarily
small.
25 2
The n-period ahead variances, var[yi,t+n |It ], in general depend on ÏƒÇ« for n â‰¥ 2 if Ï? < 1. We can correct
for these â€œhigher-orderâ€? eï¬€ects without essentially changing the main results of the paper. More precisely, a
modiï¬?ed version of the welfare formula (.), which adjusts for the change in mean income, yields quantitative
results that are very close to the results reported here. Details are available on request.
14
(veil of ignorance). More formally, we have
âˆž
W = E Î² t lncit (11)
t=0
âˆž
= E E Î² t lncit |Ï‰0
t=0
2
Î² ÏƒÇ« 1
= E âˆ’ + Ï‰0
(1 âˆ’ Î² )(1 âˆ’ Î²Ï?) 2 1 âˆ’ Î²Ï?
2 2
Î² ÏƒÇ« 1 ÏƒÏ‰
= âˆ’ âˆ’ 0
(1 âˆ’ Î² )(1 âˆ’ Î²Ï?) 2 1 âˆ’ Î²Ï? 2
The formula (11) shows how social welfare depends on the various income parameters
and the preference parameter Î² . In particular, (11) shows that an increase in uncertainty,
either about initial conditions or about future labor market conditions, will reduce social
welfare. Further, an increase in Ï? increases uncertainty about lifetime income, and therefore
reduces welfare:
W W W
2
<0 , 2
<0 , <0 (12)
âˆ‚ÏƒÏ‰0 âˆ‚ÏƒÇ« âˆ‚Ï?
In order to express welfare changes in economically meaningful units, we calculate the cor-
responding change in consumption in each period and possible future state that is necessary
to compensate the worker for the change in uncertainty. For example, suppose we compare
2 2
two economies, one with income parameters (ÏƒÏ‰ 0
, ÏƒÇ« , Ï?) and one with income parameters
2 2
ÏƒÏ‰
(Ë† 0
Ë†Ç«
,Ïƒ Ë†). We then deï¬?ne the consumption-equivalent welfare change, âˆ†, of moving from
,Ï?
2 2 2 2
ÏƒÏ‰
(ÏƒÏ‰0 , ÏƒÇ« , Ï?) to (Ë† 0
Ë†Ç«
,Ïƒ Ë†) as
,Ï?
âˆž âˆž
E Î² t ln (cit (1 + âˆ†)) = E Î² t lnc
Ë†it , (13)
t=0 t=0
where c is consumption in the ï¬?rst economy and c Ë† is consumption in the second economy.
Using the deï¬?nition (13) and the welfare formula (11), we ï¬?nd:
2 2
Î² ÏƒË†Ç« Ë†Ï‰
(1 âˆ’ Î² ) Ïƒ
ln(1 + âˆ†) = + 0
(14)
Ë†) 2
(1 âˆ’ Î² Ï? Ë†) 2
(1 âˆ’ Î² Ï?
2 2
Î² ÏƒÇ« (1 âˆ’ Î² ) ÏƒÏ‰
âˆ’ âˆ’ 0
(1 âˆ’ Î²Ï?) 2 1 âˆ’ Î²Ï? 2
15
As mentioned before, measurement error and transitory shocks have (almost) no eï¬€ect
on welfare. In contrast, the eï¬€ect of the other two mobility parameters, ÏƒÇ« and Ï?, turn
out be quite substantial. For example, based on the welfare formula (14) and an annual
discount factor of Î² = 0.96, a value that is standard in the macro-economic literature (for
2
example, Cooley and Prescott, 1995), we ï¬?nd that removing all â€œbad mobilityâ€?, ÏƒÇ« = 0,
leads to a welfare gain of about 12 percent of lifetime consumption. Using the same discount
factor, the welfare cost of removing all â€œgood mobilityâ€?, Ï? = 1, is equal to 8 percent of
lifetime consumption, again a signiï¬?cant welfare eï¬€ect. Finally, removing both â€œgoodâ€? and
2
â€œ badâ€? mobility at the same time, ÏƒÇ« = 0 and Ï? = 1, leads to a net welfare gain of about
10 percent of lifetime consumption. The last result shows that the welfare formula (14) is
highly non-linear and that the positive welfare eï¬€ect of catching-up, Ï? < 1, is closely linked
to the presence or absence of persistent income shocks, Ç«. Calculations with other values of
Î² yield similar results as indicated in Table 5.
In sum, the application of our general framework to Mexico provides striking results. The
parameter that accounts for the largest part of measured mobility, ÏƒÎ· , has (almost) no eï¬€ect
on welfare, and the two parameters that have large eï¬€ects on welfare, ÏƒÇ« and Ï?, have only
a modest contribution to measured mobility, and least over small time durations. Clearly,
our welfare results depend on the choice of preference parameters, namely the degree of risk
aversion and the degree of impatience (discounting). However, by using a logarithmic utility
function we have already chosen a relatively low degree of (relative) risk aversion, namely
one, and any increase in the degree of risk aversion would only increase the welfare eï¬€ects.
Further, lowering the discount factor Î² will lower the welfare eï¬€ects, but for a wide range of
values of Î² the welfare eï¬€ects remain substantial and the ranking of the diï¬€erent parameters
remains the same (see Table 5). In short, our welfare results are valid for a wide range of
preference parameters.
16
VII. Conclusions
This paper develops an analytically tractable framework linking individual income dy-
namics, social mobility and welfare. This analytical framework that we develop has the
merit that the links between diï¬€erent determinants of income mobility and social welfare
are drawn out in a simple and transparent manner âˆ’ allowing for a clearer analytical and
quantitative discussion of these interrelated concepts than has generally been possible in
the past. In particular, we discuss in detail how diï¬€erent determinants of measured income
mobility (shocks to income, and convergence forces, for instance) may have quite diï¬€erent
implications for welfare. This implies that two societies with the same initial distribution
of income and the same level of measured income mobility may be characterized by quite
diï¬€erent levels of social welfare. Decomposing the determinants of mobility is thus shown to
be crucial from the standpoint of welfare evaluation.
An important strength of the proposed framework is its empirical implementability. The
quantitative evaluation of mobility and welfare in our context entails the estimation of income
process parameters may be achieved using combined cross sectional and longitudinal data
on individual incomes and relatively straightforward econometric techniques. The results
from Mexico are striking. Most of measured mobility is estimated to be driven by transitory
shocks to income and is therefore (almost) welfare neutral. Only a small part of mobility
(i.e., mobility in permanent income) is driven by either social-welfare-reducing persistent
income shocks or welfare-enhancing catching-up of low-income individuals with high-income
individuals. Despite their small contributions to measured mobility, the implications for
welfare are large. Decomposing mobility into its fundamental components is thus crucial
from the standpoint of welfare evaluation.
17
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21
Figure 1: Variance of Unpredicted Part of Earnings vs. Age (1987-2003)
.8
Age effects in variance of Mincerâ€™s residuals
.7
.6
.5
.4
.3
.2
20 30 40 50 60 70
age
Variance Upper bound Lower bound
Note: Variance is the coeï¬ƒcient on age from a regression of the Mincer residual squared on age and year dummies. Estimates
from Mexican Urban Employment Survey using individuals age 20-65. 5% conï¬?dence intervals
22
Figure 2: Covariance of Unpredicted Part of Earnings across 5 Quarters vs. Age (1987-2003)
.4
Age effects in covariance of Mincerâ€™s residuals
.35
.3
.25
.2
.15
.1
20 30 40 50 60 70
age
Covariance Upper bound Lower bound
Note: Covariance is the coeï¬ƒcient on age from a regression of the covariance of the Mincer residual in quarter 1 vs. quarter
5 on age and year dummies. Estimates from Mexican Urban Employment Survey using individuals age 20-65. 5% conï¬?dence
intervals.
23
Table 1: Summary Statistics: 1987-2003
Mean Sd Min Max
Age 36.271 10.626 20 65
Schooling 10.624 5.460 0 22
Sex 0.737 0.440 0 1
Note: Based on the Mexican Monthly Urban Employment Survey, 1987-2003 using individuals between 20 and 65 years of age.
Age and schooling in years.
24
Table 2: Mincer Regression
Coef Sd t p > |t|
Cons 3.699 0.009 422.450 0.000
Sex 0.310 0.002 191.140 0.000
Sch 0.077 0.001 143.160 0.000
Sch2 -0.001 0.000 -45.380 0.000
Age
21 0.044 0.005 8.730 0.000
22 0.088 0.005 17.540 0.000
23 0.127 0.005 25.740 0.000
24 0.173 0.005 34.570 0.000
25 0.208 0.005 41.530 0.000
26 0.242 0.005 47.780 0.000
27 0.268 0.005 52.450 0.000
28 0.288 0.005 56.510 0.000
29 0.309 0.005 60.240 0.000
30 0.328 0.005 64.350 0.000
31 0.348 0.005 66.740 0.000
32 0.360 0.005 68.540 0.000
33 0.370 0.005 71.270 0.000
34 0.382 0.005 71.890 0.000
35 0.389 0.005 73.360 0.000
36 0.391 0.005 73.160 0.000
37 0.407 0.005 75.150 0.000
38 0.422 0.005 77.940 0.000
39 0.421 0.005 76.910 0.000
40 0.426 0.006 77.270 0.000
41 0.442 0.006 76.630 0.000
42 0.451 0.006 77.750 0.000
43 0.448 0.006 76.700 0.000
44 0.459 0.006 74.430 0.000
45 0.455 0.006 74.150 0.000
46 0.450 0.006 70.690 0.000
47 0.452 0.007 66.900 0.000
48 0.441 0.007 64.660 0.000
49 0.430 0.007 61.210 0.000
50 0.434 0.007 60.680 0.000
51 0.431 0.008 56.920 0.000
52 0.430 0.008 54.200 0.000
53 0.423 0.008 52.360 0.000
54 0.420 0.009 48.510 0.000
55 0.398 0.009 44.750 0.000
56 0.400 0.009 42.730 0.000
57 0.393 0.010 39.600 0.000
58 0.367 0.011 34.870 0.000
59 0.356 0.011 32.000 0.000
60 0.322 0.011 28.640 0.000
61 0.307 0.012 24.860 0.000
62 0.302 0.014 22.000 0.000
63 0.286 0.015 19.640 0.000
64 0.309 0.016 19.460 0.000
65 0.247 0.016 15.360 0.000
Year and wave dummies Yes
N 782179
R2 Adj 0.595
Note: Regression of log income on sex, age as a dummy variable, schooling, schooling square and a year time speciï¬?c dummy
and a dummy for whether the data correspond to the ï¬?rst period or the ï¬?fth. Data are pooled across all years. Based on the
Mexican Monthly Urban Employment Survey, 1987-2003, using individuals between 20 and 65 years of age.
25
Table 3: Estimation of Mobility Parameters
Full Sample Restricted
Ï? 0.977*** 0.976***
(0.0019) (0.0037)
2
ÏƒÏ‰ 0.104*** 0.104***
(0.0038) (0.0068)
2
ÏƒÇ« 0.015*** 0.016***
(0.0008) (0.0016)
2
ÏƒÎ· 0.203*** 0.217***
(0.0039) (0.0073)
Time Dummies Yes Yes
N 387460 99570
Note: Estimation using Non-linear SUR estimation. Dependent variables: Eq 1 variance, Eq 2 covariance. Variance calculated
as the square of the residual of the mincer regression. Covariance as the covariance of the residual in the ï¬?rst quarter observed
with that of the ï¬?fth quarter. Ï? represents the autoregressive coeï¬ƒcient or convergence parameter. ÏƒÏ‰ 2 represents the variance of
2 2
the initial distribution of income. ÏƒÇ« represents the variance of permanent shocks. ÏƒÎ· represents the variance of the transitory or
measurement error component of income. A complete and separate set of time dummies is included in each equation. Estimates
using the Mexican Monthly Urban Employment Survey, 1987-2003, using individuals between 20 and 65 years of age. Column
1 uses all observations. Column 2 just those beginning Q1 of each year. Robust standard errors in parentheses. âˆ— p < 0.1, âˆ—âˆ—
p < 0.05, âˆ—âˆ—âˆ— p < 0.01.
Table 4: Mobility Analysis
Span of measurement (t, in years)
1 10 25
Actual mobility 0.674 0.763 0.846
% âˆ† if:
Ï?=0 -0.77 -5.2 -8.6
2
ÏƒÇ« =0 -1.2 -6.6 -7.4
2
ÏƒÎ· =0 -89.7 -45.7 -23.3
Note: Table shows the percentage decline in mobility as component parameters are individually set to zero relative to actual
mobility calculated from equation (7) using parameters estimated in Table 3 based on the Mexican Monthly Urban Employment
Survey, 1987-2003. Ï? represents the autoregressive coeï¬ƒcient or convergence parameter. ÏƒÇ«2 represents the variance of permanent
shocks. ÏƒÎ·2 represents the variance of the transitory or measurement error component of income. Mobility is calculated across
a span, t, of 1, 10 and 25 years.
26
Table 5: Welfare Analysis
2 2
ÏƒÇ« = 0 Ï? = 1 ÏƒÇ« = 0 and Ï? = 1
% âˆ† if:
Î² = 0.96 12.56 -8.04 10.5
Î² = 0.95 10.64 -5.82 8.91
Î² = 0.94 9.21 -4.45 7.72
Î² = 0.90 5.87 -2.05 4.93
2 , the variance
Note: Table shows the percentage change in welfare calculated measured as a percent of lifetime consumption as ÏƒÇ«
of permanent shocks, is set to 0 (no income risk) and Ï?, the convergence parameter, is set to one (no convergence). Î² is the
annual discount factor. Welfare is calculated using equations (11) and (14) and the estimated values in Table 3 using the
Mexican Monthly Urban Employment Survey 1987-2003.
27