Interpreting trends in intergenerational income mobility
August 31, 2012
Martin Nybom and Jan Stuhler
Draft - do not cite
Abstract
We examine how intergenerational income mobility responds to structural changes in a standard model
of intergenerational transmission, deviating from the existing literature by explicitly analyzing the transition
path between steady states. We note that mobility depends not only on current but also on past transmission
mechanisms, such that structural changes (e.g. in policies and institutions) generate long-lasting trends. Crosscountry variation in current mobility may thus be partly due to differences in past institutions; variation over
time may reflect institutional changes in the past. We further find that transitions between steady states are
often non-monotonic. For example, a shift towards a less plutocratic and more meritocratic economy (parental
status becomes less and own skill more important) will initially raise mobility but also cause a negative trend
over subsequent generations. Declining mobility today may then not reflect a recent decline in equality of
opportunity but rather major gains made in the past. Similarly, a reversal in the relative returns to different
skills (for example due to technological change) generates short-term gains in mobility but negative subsequent
trends. Times of changes thus tend to be times of high mobility, while mobility is bound to decrease when the
economic environment stabilizes.
Introduction
A question that has received much attention both in economic research and in public debate is to what degree economic inequality has been changing in recent decades. Two central dimensions of interest are the
cross-sectional inequality and intergenerational mobility in income. The first captures economic inequality in
a population, the latter to what degree this inequality persists across generations as parents’ economic status is
being transmitted to their offspring. Cross-sectional inequality is commonly regarded as more justifiable in a
mobile society since an individual’s relative economic position is then potentially to a larger extent related to
the individual’s own choices and actions, rather than inheritance from previous generations. In the sociological
literature, high mobility is also seen as contributing to the stability of ’liberal democracy’, by legitimating prevailing class and status inequalities and by reducing the potential for collective action of a class-based kind (see
Erikson and Goldthorpe, 1992).1
A large body of work has found that cross-sectional income inequality and skill differentials in earnings
have been increasing substantially since the late 1970s in the U.S. and U.K, and more recently in other OECD
countries.2 Many factors such as technological change, the role of unions, immigration and trade are being discussed as potential explanations. While the rise in cross-sectional inequality is thus well documented, we know
much less about trends in intergenerational mobility. We do observe however that mobility differs substantially
across countries, and that estimates of cross-sectional inequality and intergenerational mobility are negatively
correlated.3 A central theme in the current literature and public debate is thus if income inequality has not only
increased but also become more persistent across generations.
It is however not only the empirical evidence that is inconclusive. Most of the existing theoretical work
addresses the relationship between causal transmission mechanisms and steady-state outcomes of descriptive
measures of intergenerational mobility (e.g. Conlisk, 1974; Solon, 2004). The literature has however largely
overlooked how such descriptive measures respond dynamically to changes in transmission mechanisms, and
thus provided little guidance on how to interpret mobility trends. Our objective is to provide a first analysis of
that relationship.
Our interest in a dynamic view of mobility stems from the notion that transitions between steady states are of
particular importance in intergenerational research since a single step in the transmission process corresponds to
a whole generation. Transitions towards a new steady state level that were induced by changes in transmission
mechanisms will thus last many decades even if completed in a few generations. For example, structural reforms
in an education system that lead to a new steady state after only three or four generations would affect mobility
trends over approximately a full century. Transition paths between steady states are then important determinants
of mobility trends, especially if we expect countries to experience multiple structural changes over the course
of such long time frames. We will thus contribute to the existing literature by explicitly analyzing the transition
path towards a new steady state instead of solely focusing on steady state outcomes.
Moreover, we specify individual human capital as a vector of distinct skill types in our transmission framework. This is in contrast to the previous literature on intergenerational transmission, but in accordance with the
growing literature arguing that human capital (including informally produced ability or skills) is best regarded
as multidimensional (see e.g. Carneiro and Heckman, 2003; Heckman, Stixrud, and Urzua, 2006). A notion of
multidimensional skills is especially motivated by our long-run perspective, which may relate to such dramatic
changes as the shift from mainly physical labor to manufacturing and white-collar jobs.
1 This would, for example, explain the differences in unionization, taxation and ambitions of the welfare state that emerged in US and
Europe in the early 20th century. Several factors, maybe most notably the absence of a feudal tradition, are thought to have promoted
unusually high levels of mobility in the US at that time, which is also supported by recent evidence of long-run mobility (e.g., Long and
Ferrie, forthcoming).
2 Autor and Katz (1999) discuss trends in wage inequality across countries. Atkinson, Piketty, and Saez (2011) find a substantial raise in
top income shares in the U.S. and various other countries.
3 See the cross-country surveys in Björklund and Jäntti (2009), and Blanden (2011).
1
We first illustrate that the level of intergenerational mobility depends not only on contemporaneous transmission mechanisms, but also on the distribution of income and human capital in the parent generation – and
thus on past mechanisms. This has a number of implications. First, changes in transmission mechanisms can
generate long-lasting mobility trends. Likewise, current trends might be caused by major structural changes
in the past. Second, differences in mobility across countries might reflect differences not only in current but
also past transmission mechanisms. Relating current policies and institutions to current levels of mobility may
therefore lead to misleading conclusions about their long-run relationship to mobility. The interpretation of
mobility trends in actual data thus becomes a rather complex matter. On the other hand, its patterns may help
to identify shocks to transmission mechanisms, and a dynamic analysis can potentially provide complimentary
evidence on institutional impact on mobility levels.
Our most important finding is however that a fairly general class of changes in transmission mechanisms
cause non-monotonic transitions between steady states. First, a reversal in the relative heritability of, or returns
to, productive characteristics increases mobility initially, but is then followed by a decreasing trend in mobility
that lasts over multiple generations. Times of changes (e.g. as of technological change) thus tend to be times
of high mobility, while mobility is likely to decrease when the economic environment stabilizes. Second, shifts
from more plutocratic to more meritocratic societies (i.e., parental status becomes less and own skill more
important) will be characterized by an initial increase in mobility, but also by a negative trend for subsequent
generations. Even structural changes that are clearly mobility-enhancing in the long-run can thus cause negative
trends. Declining intergenerational mobility today might then not necessarily indicate a declining effectiveness
of current policies and institutions in promoting equality of opportunity, but might instead be an aftermath of
major improvements in the past.
In the next section we discuss the related literature. Following that we present our structural model of intergenerational transmission in Section 2. We derive current and steady-state levels of intergenerational mobility
in terms of structural parameters and analyze the dynamic content of the model. We examine some typical
extensions of the standard model in Section 3, thereby also addressing the robustness of our main results. Some
further implications and conclusions are found in Section 4.
1
The literature
While a large theoretical literature considers the relationship between causal transmission mechanisms and
steady state mobility levels, there is little work on transition paths between those steady states. In the standard
simultaneous equation approach as developed by Conlisk (e.g. in Conlisk, 1974) only Atkinson and Jenkins
(1984) consider a system that is not in steady state. While Atkinson and Jenkins show how failure of the
steady-state assumption impedes identification of invariable parameters of the structural model, we instead
consider how variations in structural parameters affect intergenerational mobility in subsequent generations.
Solon (2004), noting that the interpretation of mobility trends would benefit from a theoretical perspective,
makes a first step towards such dynamic analysis by considering how variation in the return to human capital or
in the progressivity of public investment affect intergenerational mobility in the first generation after the change.
Among the few papers that discuss transitions in other types of structural frameworks, Davies et al. (2005)
compare mobility and inequality under private and public education, noting that the pattern of mobility trends
may help to identify structural changes in the underlying transmission model. Deviations from steady-state
mobility rates arise in their model from changes in cross-sectional inequality over time. We instead explore
the interaction of multiple transmission channels, such as the inheritance of and the return to different types of
human capital, as well as the direct impact of parental income.
There has been much recent empirical work on mobility trends in the U.S. and other countries. A vast lit-
2
erature is concerned with trends in occupational and class mobility, with the recent contribution by Long and
Ferrie (forthcoming) being particularly noteworthy. Using new data, they compare intergenerational occupational mobility in Britain and the US since 1850, and document that nineteenth century mobility was indeed
higher in the US than in Britain. The US mobility lead was however erased by the 1950s, as US mobility fell
from its nineteenth century levels.4
The emerging evidence on trends in income mobility is however highly contrasting.5 The most reliable
US study is by Lee and Solon (2009), who find no evidence of major changes in mobility across the cohorts
born 1952-1975. Measuring intergenerational income mobility ideally requires income data that fully span two
generations, but often only sparse data are available or exploited (see Lee and Solon, 2009).6 A central concern
in many of these papers, especially in more policy-related outlets and in the public press, is if mobility has
declined in conjunction with the recent rise in income inequality.7 Many potential causal factors for observed
mobility trends – such as educational expansion, rising returns to education, or changes in welfare policies –
have been discussed in the literature. Common to all explanations is that they relate trends to recent events
that may have directly affected the respective cohorts. We aim to illustrate why the key to understand recent
mobility trends might instead lie in the more distant past.
2
A simple model
Measuring intergenerational mobility.
A popular descriptive measure of persistence in economic status is the
intergenerational income elasticity. It is defined as the coefficient from a linear OLS regression of the statistical
relationship
yi,t = βt yi,t−1 + i,t .
(1)
We consider a simplified one-parent one-offspring family structure, with yi,t as log lifetime income of the offspring in generation t of family i and yi,t−1 as log lifetime income of his parent. The error i,t is uncorrelated
with the regressor by construction. Under stationarity of yi,t the elasticity βt equals the intergenerational correlation parameter, which adjusts the elasticity for changes in cross-sectional inequality. A low elasticity or
correlation indicates high mobility.
A model of intergenerational transmission. We consider a stochastic linear difference-equation model of intergenerational transmission in the tradition of the simultaneous equation approach developed by Conlisk (see
Conlisk, 1969, 1974), as for example also considered in Atkinson and Jenkins (1984). Although we will not explicitly address optimizing behavior as in Becker and Tomes (1979) or Solon (2004), the mechanistic pathways
represented by the structural equations can be derived from an underlying utility-maximization framework (see
Goldberger, 1989).8
4 The findings of Long and Ferrie have been questioned in comments by Hout and Guest (forthcoming) and Xie and Killewald (forthcoming).
5 See Levine and Mazumder (2002), Fertig (2002), Mayer and Lopoo (2005), Aaronson and Mazumder (2008), Levine and Mazumder
(2007), Hertz (2007), and Lee and Solon (2009) for trends in intergenerational mobility and (related) sibling correlations in the United
States. Blanden, Goodman, Gregg, and Machin (2004), Pekkala and Lucas (2007), Levine and Jellema (2007), Bratberg, Nilsen, and Vaage
(2007), Blanden and Machin (2007), and Nicoletti and Ermisch (2007) analyze trends in other countries.
6 See Solon (1999) and Haider and Solon (2006) for a discussion of the early empirical literature and the currently preferred method
to estimate mobility parameters based on incomplete income data. Nybom and Stuhler (2011) discuss the latter and argue that the recent
literature still suffers from significant measurement problems. Estimates of mobility trends can be subject to life-cycle bias even when
incomes are measured at the same age for all individuals, since income profiles over age are unlikely to have a constant shape across
cohorts.
7 Exemplary for the U.S. are Bernstein (2003); Wooldridge (2005); Wessel (2005); Scott and Leonhardt (2005); Trusts (2008),Noah
(February 8, 2012), and a recent speech (January 12th, 2012) by Alan Krueger, Chairman of Council of Economic Advisors, who warned
that intergenerational mobility should be expected to decline further as of the recent rise in income inequality in the U.S..
8 We briefly discuss how explicit modeling of optimizing behavior of parents affects our findings in section 3.
3
Assume a simple causal model of income determination and intergenerational transmission,
yi,t
=
γt yi,t−1 + ρ0t ei,t + ui,t
(2)
ei,t
=
Λt ei,t−1 + vi,t .
(3)
The parameter γt captures a direct effect of parental income on the income of offspring, which may arise as
of parental information and networks, nepotism, credit constraints, statistical discrimination under imperfect
information on individual productivity, or (if total market income is considered) returns to bequests. The exact
mechanism and the distinction between earnings and income are not central for our purposes. We denote the
human capital of the offspring of family i in generation t by ei,t , a Jx1 vector with elements e1,it , e2,it , ..., eJ,it
that reflect distinct productive characteristics such as health, physical attributes, cognitive and non-cognitive
abilities. These characteristics are valued on the labour market according to a Jx1 price vector ρt with elements
ρ1,t , ρ2,t , ...ρJ,t . The random shock term ui,t captures factors that do not relate to parental background. For
our analysis it makes no difference if these are interpreted as (labour market) luck or as the impact of other
characteristics that are not transmitted within families. Income is thus assumed to be potentially determined by
parental income, individual characteristics, and chance. The elements of the JxJ matrix Λt govern to what
degree parental characteristics determine the characteristics of the offspring generation. We consider Λt to
represent a broad concept of heredity potentially working through both nature and nurture. For simplicity we
assume no cross-correlations between characteristics, so that Λt is diagonal with elements λ1,t , λ2,t , ..., λJ,t .9
Both ui,t and the elements of the Jx1 shock vector vi,t are assumed to be uncorrelated with each other and to
past values of {yi,t , ei,t , ui,t , vit }. For some of our examples it will suffice to consider a single characteristic
ei,t and thus scalar versions of equations (2) and (3).
For convenience we drop the individual subscript i in the subsequent analysis and make a few simplifying
assumptions. Assume that yt and all ej,t are measured as trendless indices with constant mean zero (as in
Conlisk, 1974) such that we do not need to include constants. To avoid the need for case distinctions assume
further that the indices measure positive characteristics (so that the elements of ρt are non-negative) and that
parent and offspring characteristics are not negatively correlated (so that the elements of Λt are non-negative
for all t). For stability we assume that all slope parameters are less than one. The reduced form of equations (3)
and (4) is
yt
et
!
=
γt
ρ0t Λt
0Jx1
Λt
!
yt−1
!
et−1
+
ut + ρ0t vt
vt
!
,
which we may shorten to
xt
= At xt−1 + wt .
(4)
The stability condition lims→∞ Ast = 0 is satisfied as the parameter constraints assure that all eigenvalues of
At are below one in modulus. Finally, since our focus is on intergenerational transmission we will initially only
consider changes in the relative strength of transmission mechanisms by holding the cross-sectional inequality
in income and characteristics constant. We further normalize the variance of all variables to one by choosing
appropriate values for the variances of ut and elements of vt .10 These restrictions simplify the discussion and
9 It might be argued that parental income should be included as explanatory variable in eq. (3). To keep the model simple we choose
instead to capture the causal impact of parental income in only one parameter. All effects from parental income (also those indirectly via
offspring human capital) are thus reflected by γt in eq. (2), and Λt reflects effects from parents human capital that are orthogonal to parental
income. Our qualitative results arise also without this simplification, as we will discuss in the next section.
10 Taking the covariance of (4) and denoting the covariance matrices of x and w by S and W , respectively, gives
t
t
t
t
St = At St−1 A0t + Wt .
4
lead to more illustrative derivations by abstracting from the additional source of dynamics that stems from the
transition path of cross-sectional inequality. We discuss implications from the latter instead in Section 3.
In comparison to previous models cited above, we assume that income depends on human capital through a
vector of distinct productive characteristics instead of only a single factor. While this generalization will prove
to be central for some of our findings it should not constitute a significant change of the character of the model.
Our second deviation from previous work is simply the addition of explicit t subscripts on all parameters, since
we want to consider the effects of changes in the transmission framework over time. Similarity to the existing
literature is advantageous for our purposes since it should support our claim that our findings are general and
do not arise due to any specific modeling choices.
The importance of past transmission mechanisms
The intergenerational income elasticity, which coincides with the intergenerational correlation by virtue of the
assumption of constant cross-sectional inequality, is derived by plugging equations (2) and (3) from our model
of intergenerational transmission into equation (1),
βt =
Cov(yi,t , yi,t−1 )
V ar(yi,t )
= γt + ρ0t Λt Cov(ei,t−1 , yi,t−1 ).
(5)
Thus, βt depends on current transmission mechanisms (parameters γt , ρt and Λt ) and on the cross-covariance
between income and productive characteristics in the parent generation. Expression (5) illustrates that two
populations that are subject to similar transmission mechanisms (e.g. exposed to similar institutions and policies) can nevertheless differ in their levels of intergenerational mobility, since current mobility depends on the
distribution of productive characteristics in the parent generation.
The cross-covariance between income and productive characteristics in the parent generation is in turn
determined by past transmission mechanisms, and thus depends on past values of {γt , ρt , Λt }. We can iterate
backwards to express βt in terms of parameter values,
βt
= γt + ρ0t Λt Cov(ei,t−1 , yi,t−1 )
= γt + ρ0t Λt [Λt−1 Cov(ei,t−2 , yi,t−2 )γt−1 + ρt−1 ]
= ...
= γt + ρ0t Λt ρt−1 + ρ0t Λt
∞
X
r
Y
r=1
s=1
!
γt−s Λt−s
!
ρt−r−1
,
(6)
where for simplicity we assume that the process is infinite.11 The level of intergenerational mobility today thus
depends on current and past transmission mechanisms. If parameters have been constant for past generations,
{γs = γ, ρs = ρ, Λs = Λ}∀s≤t , then equation (7) simplifies to the steady state intergenerational elasticity
β = γ + ρ0 Λ
∞
X
s
(γΛ) ρ
s=0
0
= γ + ρ Λ (IJxJ − γΛ)
−1
ρ,
(7)
Expansion yields that V ar(ej,t ) = 1 ∀ j, t iff V ar(ej,0 ) = 1 ∀ j and V ar(vt ) = IJxJ − Λt Λt ; and that V ar(yt ) = 1 for all j, t
iff V ar(y0 ) = 1 and V ar(ut ) = 1 − γt2 − 2γt Cov(yt−1 , et−1 )Λt ρt − ρ0t ρt . The requirement for variances to be non-negative is
satisfied iff λj,t ≤ 1, ρ0t ρt ≤ 1, and γt ≤ 1/2(−2Cov(yt−1 , et−1 )Λt ρt + ((2Cov(yt−1 , et−1 )Λt ρt )2 + 4(1 − ρ0t ρt ))1/2 ) for
all j, t.
11 For a finite process, β will depend on past parameter values and the initial condition Cov(e
t
i,0 , yi,0 ).
5
where the second line follows since the geometric series
∞
P
s
(γΛ) converges (the absolute value of each eigen-
s=0
value of γΛ is below one). Most of the literature has focused on how changes in structural parameters affect
intergenerational mobility in steady state mobility given by (7). We will now instead analyse the path of transition towards the new steady state as determined by eq. (6).
Examples of transition dynamics
Unfortunately, intergenerational data that would allow for empirical estimation of our model (i.e., spanning
many generations) are not available. Nevertheless, rough orders of magnitude of the parameters discussed
above can be inferred from the empirical literature. The evidence in the literature, and our cross-validations
within the model,12 suggest the following parameter values for the U.S. case:
0.45 ≤ β ≤ 0.55, 0.15 ≤ γ ≤ 0.25, 0.60 ≤ ρ ≤ 0.70, 0.50 ≤ λ ≤ 0.65
(8)
Although our qualitative results do not rely on specific choices for the parameters, the relevance of the
quantitative implications of our numerical examples increase from using such values that are consistent with the
empirical evidence. We provide a detailed motivation of our choices of parameter values in Appendix A1.
From Simple Examples to Non-Monotonic Trends
For our first examples it will be sufficient to consider a single characteristic ei,t, and thus scalar versions of
equations (2) and (3):
yi,t = γt yi,t−1 + ρi,t ei,t + ui,t ,
(9)
ei,t = λt ei,t−1 + vi,t .
(10)
and
Unless indicated to the contrary we will assume that the slope parameters are positive to avoid case distinctions.
It will however be useful to first look at an even simpler case in which parental income has no effect on offspring
income.
EXAMPLE 1: A simple meritocratic economy. Assume that parental income has no direct effect on child
income (γt = 0 ∀t). Changes in the heredity of characteristics then shift mobility within one generation.
In contrast, changes in the returns to characteristics affect mobility over the course of two generations.
From equation (6) it follows that a change in the heredity of characteristics from λt<T = λ1 to λt≥T = λ2 leads
to a one-time shift in βt equal to ∆βT = βT − βT −1 = ρ(λ2 − λ1 )ρ. Mobility remains constant afterwards.
A shift in returns from ρ1 to ρ2 at time T instead leads to a shift in βt lasting over two generations. The first
shift is equal to
∆βT = βT − βT −1 = (ρ2 − ρ1 )λCov(eT −1 , yT −1 )
= (ρ2 − ρ1 )λρ1 ,
12 After having specified our preferred values of the parameters and the intergenerational elasticity – which we set to match recent US
evidence – we can cross-validate each of the values given the other ones. The set of values we choose based on the literature is surprisingly
consistent with our model.
6
and is induced by the change in returns for the offspring generation in T . The second shift equals
∆βT +1 = βT +1 − βT = ρ2 λ (Cov(eT , yT ) − Cov(eT −1 , yT −1 ))
= ρ2 λ(ρ2 − ρ1 ),
and is induced by the change in the correlation between income and characteristics among the parents of the
offspring generation T + 1, in turn caused by changing returns to those characteristics in the previous period.
If returns increase (ρ2 − ρ1 > 0) then the second shift will be larger than the first.13 Mobility remains constant
afterwards. Figure 1 gives a numerical example.
Figure 1: A shift in the heredity of, or returns to, characteristics
Ÿ
0.38
Ÿ
Ÿ
increase in Ρ
0.36
0.34
Ÿ
0.32
0.3
Ÿ
Ÿ
0.28
decrease in Λ
0.26
0.24
Ÿ
Ÿ
Ÿ
Ÿ
T
T+1
T+2
T+3
0.22
T-2
T-1
t
Numerical example with ρ = ρ1 = 0.7, λ = λ1 = 0.6 and shifts to λ2 = 0.5 and ρ2 = 0.8,
respectively.
Hence, even in this simple example we find that the effect on the intergenerational income elasticity from
changes in the returns to characteristics cannot be fully evaluated until after both the parent and child generations
have experienced the new price regime. Relating this finding to the rising skill differential in wages from the
late 1970s in the U.S. and U.K. (and more recently in other OECD countries) might serve to illustrate its practical implications. Many authors have argued that widening wage differentials could decrease intergenerational
mobility (see for example Blanden et al., 2004, Solon, 2004, and Aaronson and Mazumder, 2008), a hypothesis
that is also one of the main motivations for the recent interest in mobility trends. But recent trend studies do not
yet observe offspring cohorts whose parents have fully experienced the changing wage regime.14 The impact of
rising wage differentials on mobility levels may thus become fully evident only in future empirical studies that
observe more recent cohorts.
13 Co-movements in the cross-sectional variance of income (see section 3) can affect the relative size of the first and second elasticity shifts
further. Allowing for variable cross-sectional inequality and for a direct effect of parental income (γ 6= 0) does not affect the implication
that current mobility levels do not fully reflect a change in prices before at least two generations.
14 For example, the youngest offspring cohort observed in Lee and Solon (2009) was born in 1975. The early careers of many of their
parents will not have been subject to the widening skill differential. Income measures for such younger offspring cohorts are further based
on income observations at early age, and are thus subject to potentially strong life-cycle biases (see Nybom and Stuhler, 2011).
7
We proceed with two simple examples on equalizing opportunities, in which the outcome of offspring
becomes less dependent upon parental income.15
EXAMPLE 2: Equalizing opportunities (type I). Assume that the relative importance of parental status
diminishes and that the importance of factors that do not relate to parental background increases. Mobility
increases monotonically in subsequent generations, at a decreasing rate.
Assume that the direct effect of parental income declines from γt<T = γ1 to γt≥T = γ2 , such that γ1 > γ2 .16
From (6) income mobility of subsequent generations changes according to,
∆βT
∆βT +1
= γ2 − γ1
=
..
.
(γ2 − γ1 )ρλ2 Cov(eT −1 , yT −1 )
Mobility will thus be increasing monotonically in subsequent generations, at a decreasing rate. Figure 2 plots
a numerical example. Mobility adapts over the course of multiple generations and not instantly since the correlation of productive characteristics and income depends on the direct impact of income in the past. A decline
in the strength of the direct income mechanism diminishes that correlation in subsequent generations, thereby
increasing the intergenerational income mobility.
Figure 2: A decline in the importance of parental income
0.6
Ÿ
Ÿ
0.55
0.5
Ÿ
0.45
Ÿ
Ÿ
Ÿ
T+1
T+2
T+3
0.4
T-2
T-1
T
t
Numerical example with ρ = 0.6, λ = 0.6 and a decline in γ from γ1 = 0.3 to γ2 = 0.2 at
generation T .
The example illustrates that changes in structural mechanisms should be expected to affect mobility trends
over multiple generations. Mobility trends today are thus not necessarily indicative of a changing effectiveness
of current policies and institutions in the promotion of equality of opportunity, but could be the residual effect
of institutional changes in the past.
15 Conlisk (1974) notes that “opportunity equalization” is an ambiguous term that may relate to different types of structural changes in
intergenerational transmission models.
16 Constant cross-sectional inequality then requires σ
u,t<T < σu,t≥T .
8
From equation (6) it follows that other types of structural changes have similar dynamic implications. Mobility will increase monotonically if the returns to partially inherited characteristics decrease (a fall in ρ) or
if the heredity of these characteristics decreases (a fall in λ). Convergence occurs through two channels. A
lower degree of heredity, for example, decreases the intergenerational correlation of individual characteristics
and thus the intergenerational persistence in income. The direct effect of parental income (γ) then becomes less
detrimental to income mobility since the offspring from rich families are less likely to inherit productive characteristics. The correlation between income and characteristics within a given generation declines. This latter
indirect effect follows a transition path that can last over multiple generations, thus causing the convergence to
the new steady state.
Mobility levels converged fast to the new steady state in this example, with the initial drop in βt in the first
generation dominating the subsequent trend. As we will see in the next example the latter depends however on
the type of structural change.
EXAMPLE 3: Equalizing opportunities (type II). Assume that the importance of parental status diminishes
and that productive characteristics that are partially inherited within families are instead more strongly
rewarded.
In other words, assume that the economy becomes less plutocratic and more meritocratic, which in our model
corresponds to the assumption that γ1 > γ2 and ρ1 < ρ2 .
From (5) it follows
∆βT
=
(γ2 − γ1 ) + (ρ2 − ρ1 )λCov(eT −1 , yT −1 )
∆βT +1
=
..
.
ρ2 λ(ρ2 − ρ1 ) + ρ2 λ2 (γ2 − γ1 )Cov(eT −1 , yT −1 )
β∞ − βT −1
=
(γ2 − γ1 ) +
ρ22 λ
ρ21 λ
−
.
1 − γ2 λ 1 − γ1 λ
The first line illustrates that a shift to a more meritocratic society tends to increase mobility initially, specifically
iff
γ1 − γ2
> λCov(eT −1 , yT −1 ).
ρ2 − ρ1
(11)
However, mobility then decreases in subsequent generations iff
ρ2 − ρ1
> λCov(eT −1 , yT −1 ).
γ1 − γ2
(12)
Conditions (11) and (12) will be satisfied for any shifts γ1 − γ2 and ρ2 − ρ1 that are sufficiently symmetric,
i.e., relatively equal in absolute terms. Intuitively, when the economy becomes more meritocratic, individuals
with high productivity become more likely to rise to the top of the income distribution. This process initially
increases income mobility since individuals with high productivity are not necessarily from previously rich
families (if λ<1). Individuals from families with high productive characteristics will subsequently however tend
to keep their higher ranks in the income distribution, thus decreasing mobility. Conditions (11) and (12) are
more likely to be satisfied when heredity λ is low since high-productivity individuals are then less likely to
come from previously rich families.
Figure 3 plots a numerical example.
9
Figure 3: A decline in the importance of parental income and increasing returns to inherited characteristics
0.65
0.6
Ÿ
Ÿ
0.55
Ÿ
Ÿ
Ÿ
T+2
T+3
0.5
Ÿ
0.45
T-2
T-1
T
T+1
t
Numerical example with λ = 0.6 and a decline in γ from γ1 = 0.4 to γ2 = 0.2 as well as a rise
in ρ from ρ1 = 0.5 to ρ1 = 0.7 at generation T .
The example illustrates that the response in mobility trends can be non-monotonic and long-lasting. A
decline in the importance of parental status increases mobility initially, but mobility declines in subsequent
generations. Mobility trends become insignificant only in the third generation, which may be more than half
a century after the structural shock. These are important implications for the interpretation of mobility trends.
First, structural changes that are mobility-enhancing in the long-run may nevertheless cause decreasing mobility
trends. Second, these negative mobility trends can last over many decades.
In the numerical example, mobility changes much more strongly in the first and second generation than in
the subsequent ones. One might thus expect that the lagged impact of more distant structural changes plays
only a negligible role for current mobility trends. That depends however on the relative magnitude of those
past changes. For example, in the late 19th and early 20th century the U.S. experienced rapid industrialization,
massive immigration, internal migration, and urbanization, a decline in the share of agriculture and self employment, and an expansion of public secondary schooling that affected a large part of the population. The country
participated in two world wars and went through a highly turbulent interwar period. These events may have
affected intergenerational transmission to a greater degree than more recent changes that have been considered
as potential determinants of current mobility trends (such as an increase in private schooling or the increase in
higher education).17 The recent empirical literature measures mobility trends for offspring cohorts born from
around 1960, cohorts that are separated by only a couple of generations from events in the early 20th century.
We thus suspect that mobility trends observed over these cohorts may not primarily reflect contemporaneous
changes in policies or institutions, but instead the lagged impact of major changes in past generations.
17 The hypothesis that past structural changes had large consequences is consistent with the sociological literature on long-lasting trends
in intergenerational occupational mobility (an alternative measure of economic mobility, see Hauser 2010 for a discussion), which implies
that mobility has increased substantially between the late 19th and early 20th century and declined thereafter in the U.S. (see Grusky, 1986;
and Ferrie, 2004). The hypothesis that industrialisation tends to increase mobility is also consistent with evidence on mobility trends in
Finland, where mobility increased strongly for cohorts born around 1950 compared to cohorts born in the early 1930s (see Pekkala and
Lucas, 2007).
10
Times of Change and Mobility Trends
Our finding that a shift from plutocracy to meritocracy can lead to non-monotonic mobility trends could be
relevant for the interpretation of recent mobility trends. However, such shifts are rather specific examples
of structural changes, non-monotonic adjustments in mobility levels could thus be expected to occur only in
exceptional cases. In our next examples we aim to show that non-monotonic responses are instead quite general
phenomena, which tend to occur whenever the relative heredity of characteristics or their relative returns change.
These examples will also illustrate that a transmission framework with multiple characteristics as in (2) and (3)
may provide additional implications that cannot be captured by models that are based on a single inheritable
characteristic.
EXAMPLE 4: Changing labour market conditions. Assume that the returns to individual characteristics
change on the labour market (ρ1 6= ρ2 ).
Changes in the returns to characteristics could stem from changes in relative labour demand as of industrialization, technological change, or changes in the relative supply of these characteristics as of immigration or the
production function of these skills. A specific example is the shift in the demand for physical ability (strength,
endurance, etc) relative to cognitive ability as a labor market shifts from agricultural to white-collar jobs. It has
been documented in the literature that returns to characteristics can indeed change rapidly even in periods that
are much shorter than the time scale underlying our intergenerational analysis.18
To grasp the intuition consider first a simple symmetric case in which two characteristics k and l are equally
transmitted within families (λk = λl = λ), but their prices on the labour market are swapping at time T
(p2,k = ρ1,l 6= p1,k = ρ2,l ). From (5) it follows that
∆βT
=
(ρk,2 − ρk,1 )λCov(ek,T −1 , yT −1 ) − (ρl,1 − ρl,2 )λCov(el,T −1 , yT −1 )
=
− (ρk,2 − ρk,1 ) λ
< 0
1 − γλ
2
and
∆βT +1
= ρk,2 λ(ρk,2 − ρk,1 ) − ρl,2 λ(ρl,1 − ρl,2 )
=
2
(ρk,2 − ρk,1 ) λ > 0
Figure (4) provides a numerical example.
18 A
typical example is the job-polarization literature which highlights how the IT revolution has implied a dramatic shift in demand from
substitutable manual skills to complementary abstract skills (see e.g.Levy, Murnane, and Autor, 2003).
11
Figure 4: A swap in prices
0.55
0.5
Ÿ
Ÿ
Ÿ
Ÿ
T+2
T+3
Ÿ
0.45
Ÿ
0.4
T-2
T-1
T
T+1
t
Numerical example with γ = 0.2, λ = 0.6 and a shift in prices from ρk,1 = 0.3 to ρk,2 = 0.6
and ρl,1 = 0.6 to ρl,2 = 0.3 at generation T .
Intuitively, those partially inherited characteristics that have been more strongly rewarded are also more
strongly correlated with parental income. As a consequence, mobility tends to initially increase if prices change,
since characteristics for which prices increase from low levels are less prevalent among the rich than characteristics for which prices decrease from high levels. In subsequent generations, the characteristics for which prices
increased become increasingly correlated with parental income, leading to decreasing mobility trends.
We can derive that such v-shaped response in mobility trends is typical for the general case in which the
prices of any number of characteristics change. From (5) and (7) we have
2βT = ρ01 Λ (I − γΛ)
−1
= ρ01 Λ (I − γΛ)
|
{z
−1
=βT −1
−1
ρ2 + ρ01 Λ (I − γΛ)
−1
ρ1 + ρ02 λ (I − γΛ)
} |
{z
=β∞
ρ2
−1
ρ2 − (ρ02 − ρ01 )Λ (I − γΛ)
}
(ρ2 − ρ1 ).
The quadratic form in the last term is greater than zero (for ρ2 6= ρ1 ) since Λ (I − γΛ)
−1
(13)
is positive-
semidefinite, indicating that price changes tend to increase intergenerational mobility initially. From (13) it
follows that βT is below both the previous steady state βT −1 and the new steady state β∞ if the shift in the
steady-state elasticity (∆β = β∞ − βT −1 ) is not too large„ specifically if
|∆β| < (ρ02 − ρ01 )Λ (I − γΛ)
−1
(ρ2 − ρ1 )
or, plugging in for steady state values,
ρ02 Λ (I − γΛ)
−1
(ρ2 − ρ1 )
>
0
>
−1
ρ01 Λ (I − γΛ)
(ρ2 − ρ1 ).
Any symmetric changes (as in the numerical example) fulfill these conditions and will thus lead to nonmonotonic, v-shaped adjustments as in Figure 3. More generally, changes in the renumeration of individual
12
characteristics that do not affect long-run mobility much (e.g. some prices go down while others go up) will
increase mobility in the short-run but cause a decreasing trend in mobility in subsequent generations. We believe
that these are new results. While some authors have shown that specific events can lead to non-monotonic
mobility trends through repeated changes in structural parameters (such as returns to characteristics),19 we find
that even a one-time reversal in returns generates non-monotonic trends.
EXAMPLE 5: Changes in the intergenerational transmission of characteristics. Assume that the heredity
of multiple characteristics change (Λ1 6= Λ2 ). Mobility will initially increase but decrease in subsequent
generations if gains and losses in heredity of individual characteristics are of comparable magnitude.
For example, changes in the school system may affect the heredity of formal education, or reforms of the
health care system may affect intergenerational transmission of health. As with changes in prices, when some
characteristics become more and some less transmitted within families mobility tends to increase initially but
then follows a negative trend in subsequent generations. The intuition is similar as well.20
From (7) it follows that the mobility trend will follow such v-shape (∆βT < 0 < ∆βT +1 ) iff
ρ0 (Λ2 − Λ1 ) (I − γΛ1 )
−1
ρ
<
0
<
−1
ρ0 Λ2 (Λ2 − Λ1 ) (I − γΛ1 )
ρ
Again, any symmetric parameter changes, as considered in the previous numerical example, fulfill these conditions.
These last two examples have quite general implications that do not depend much on the nature of changes
in transmission mechanisms. Relative changes in the returns to, or heredity of, characteristics will tend to raise
intergenerational mobility in the short run. Times of changes thus tend to be times of high mobility. Second,
such mobility gains will be succeeded by longer-lasting negative trends in intergenerational income mobility if
no further structural changes occur. Countries experiencing a period of stable economic conditions will thus
tend to be characterized by negative mobility trends if they were preceded by more turbulent times.
3
Extensions and sensitivity analysis
Although our model is broadly in line with the previous literature it is still based on various simplifying assumptions, some of which deserve further discussion. To match the empirical literature on mobility trends we
introduce a cohort dimension into our model, from which we derive some additional implications.We then discuss the sensitivity of our results to the way we model the influence of parental income or status. In the final
subsection we relax the assumption of constant cross-sectional inequality and consider how the transition path
of income dispersion adds an additional source of dynamics. For simplicity we consider scalar cases throughout
the section.
From Generations to Cohorts
While the theoretical literature considers how intergenerational mobility evolves over generations, the empirical
literature instead typically estimates mobility trends over cohorts.21 These two dimensions, which do not match
19 For example, Galor and Tsiddon (1997) consider how the life-cycle of technological progress might lead to repeated changes in the relative returns to ability and parent-related human capital, and thus to non-monotonic trends in cross-sectional inequality and intergenerational
mobility over time.
20 Characteristics that are more strongly inherited are also more strongly correlated with parental income. Characteristics for which
heredity increase from low levels are thus less prevalent among the rich than characteristics for which heredity decrease from high levels,
and changes in the heredity of characteristics will tend to increase mobility initially. The characteristics for which heredity increased then
become increasingly correlated with parental income in subsequent generations, leading to a decreasing mobility trend.
21 Mobility measures are usually indexed to offspring cohorts, a convention that we will follow here.
13
if parental age at birth varies across families or time, have to our knowledge not previously been linked in the
literature.
To link theoretical implications to empirical trends we thus introduce a cohort or year dimension into our
model of intergenerational transmission. We adopt the following notation to distinguish between cohorts and
generations. Let the random variable Ct denote the cohort into which a member of generation t of a family is
born. Let At−1 (Ct ) be a random variable that denotes the age of the parent at birth of the offspring born in
cohort Ct . For simplicity we assume At−1 (Ct ) to be independent of parental income and characteristics, but
we allow for dependence on Ct so that the distribution of parental age at birth can change over time. Member
t − j of a family is then born in cohort
Ct−j = Ct − At−1 (Ct ) − ... − At−j (Ct−j+1 ) = Ct−j (Ct , At−1 , . . . , At−j ).
(14)
Member t − j becomes parent at age
At−j = At−j (Ct−j+1 ) = At−j (Ct , At−1 , . . . , At−j+1 ).
(15)
Denote realizations of these random variables by lower case letters. Our model for intergenerational transmission between offspring born into cohort Ct = c and a parent born in cohort Ct−1 = ct−1 is then given
by
yt (c) = γc yt−1 (ct−1 ) + ρc et (c) + ut (c)
(16)
et (c) = λc et−1 (ct−1 ) + vt (c),
(17)
and
where we assume a single productive characteristic and keep the same simplifying assumptions on parameters
and variables as in our generations-only model in equations (2) and (3).
By considering a single set of equations per cohort we abstract from life-cycle effects within a given cohort.
The transmission parameters in (16) and (17) can thus be interpreted as representing an average of effective
transmission mechanisms over the life-cycle. For example, the price parameter ρc does not reflect returns to
characteristics in year c, but average returns throughout the working life of an individual born in year c.
Using (16) and (17), the intergenerational income elasticity for offspring of generation t born in cohort
Ct = c is then
βc
=
Cov yt (c), yt−1 (Ct−1 )
V ar yt (c)
= γc + ρc λc Cov et−1 (Ct−1 ), yt−1 (Ct−1 .
where we do not explicitly note that all random variables are conditional on Ct = c in order to keep the notation
short. Income mobility for a given cohort thus depends on cohort-specific transmission mechanisms (γc , ρc
and λc ) and the cross-covariance of income and characteristics in the parent generation. This cross-covariance
may vary with parental age, since different cohorts of parents might have been subject to different policies and
institutions. Using (14) and the law of iterated expectations we can rewrite
βc
h
i
= γc + ρc λc EAt−1 Cov et−1 (c − At−1 ), yt−1 (c − At−1 )|At−1
X
= γc + ρc λc
fc at−1 Cov et−1 (c − at−1 ), yt−1 (c − at−1 ) ,
at−1
where fc is the probability mass function for parental age at birth of cohort c. Income mobility thus depends
14
on current transmission mechanisms and a weighted average of the cross-covariance of income and characteristics in previous cohorts, where the weights are given by the cohort-specific distribution of parental age in the
population.
As in our simple model we can iterate backwards to express βc in terms of parameter values,
βc
h
h
i
i
= γc + ρc λc EAt−1 λCt−1 EAt−2 Cov et−2 (Ct−2 ), yt−2 (Ct−2 )|At−2 γCt−1 + ρCt−1 |At−1
=
...
=
γc + ρc λ c
X
fc (at−1 )ρc−at−1
at−1
+ρc λc
∞ X
X
r=1 at−1
fc (at−1 )
X
X fct−1 (at−2 ) . . .
at−2
fct−r (at−r−1 )
at−r−1
r Y
γct−s λct−s
ρct−r−1 (18).
s=1
Given assumptions or data on the time series of parental age at birth we can thus use eq. (18) to analyse
the dynamic response of mobility trends over cohorts to parameter changes. The insights from the more simple
generations-only model still hold (e.g. past transmission mechanisms affect mobility of cohorts today), but
comparison of equations (18) and (6) leads to a number of additional implications.
First, note that both expressions simplify to the same steady state elasticity given in eq. (7). The explicit
consideration of cohorts in intergenerational transmission models has thus only consequences for transitions between steady states, which presumably explains why the theoretical literature, with its focus on steady states, has
not yet been linked to measures of cohort-specific mobility. Second, from (18) it follows that the importance of
past transmission mechanisms (and thus of past institutions and policies) on current mobility rises with parental
age at birth.22 Likewise, the impact of structural shifts on mobility trends will die out faster in populations in
which individuals tend to become parents at younger ages.
This finding might be of interest for cross-country comparisons, especially between developed and developing countries. Cross-country mobility differentials are not only driven by differences in current and past
transmission mechanisms, but according to eq. (18) also by different weights on those past mechanisms. The
literature has proposed various reasons to why developing countries show a tendency of
lower levels of intergenerational mobility than developed countries in the available evidence (see Levine
and Jellema, 2007). Our results imply that differences in the current transmission framework might for example
be partially offset if countries share a partly common trend towards more meritocratic societies, since mobility
levels in developing countries are less dependent on past institutions given that people on average become
parents at younger ages.
Parental investments in offspring human capital
The primary effect of parental income may be related to human capital investments in their children, such as in
Becker and Tomes (1979, 1986), rather than direct on offspring income as implied by equation (2). We therefore
briefly examine the sensitivity of our results by introducing a new parameter that governs the impact of parental
income yt−1 on offspring human capital et , such that equation (3) becomes
et = λt et−1 + φt yt−1 + ut .
(19)
Examples of what can cause φt to change are college tuition or health care policies. The intergenerational
22 While consideration of life-cycle effects would give a more detailed insight, this implication would hold for intergenerational transmission mechanisms that tend to be effective in early life (e.g. genetic transmission and parental education).
15
elasticity at time t is now given by
βt = γt + ρt φt + ρt λt Cov(et−1 , yt−1 ),
(20)
and the corresponding steady-state elasticity becomes
β = γ + φρ +
ρλ(ρ + φγ)
.
1 − λγ
(21)
In this extended model parental income may thus affect offspring income directly (γt ) or indirectly by
affecting offspring human capital (φt ). To explore how the implications of these channels differ we revisit
example 3, in which we considered an increase in the return to human capital and a decrease in the relevance
of parental income. For illustration we consider parametrizations that lead to the same pre-shock and long-run
elasticities but that give different weights on each of the two income channels. Figure 5 plots the transition paths
for various initial levels of γt .23
Figure 5: A decline in the importance of parental income and increasing returns to inherited characteristics,
various cases
Β
0.58
0.56
ì
ò
æ
à
ì
ò
æ
à
0.54
ì
ò
à
æ
ì
ò
à
æ
ì
ò
à
æ
ì
0.52
à
æ
Γ=0.3
æ
à
Γ=0.2
ì
Γ=0.1
ò
Γ=0
0.5
ò
0.48
T-2
T-1
T
T+1
T+2
T+3
t
Numerical example with λ = 0.6 and a rise in ρ from ρ1 = 0.6 to ρ1 = 0.7 at generation T .
The four cases considered are for a decline in γ from γ1 = 0.3 to γ2 = 0.2 (parental income has
only a direct effect); from γ1 = 0.2 to γ2 = 0.1; from γ1 = 0.1 to γ2 = 0; and for γ1 = γ2 = 0
(parental income has no direct effect).
Two findings are noteworthy. First, mobility trends are similar for the baseline case (φ = 0) and the extreme
alternative case (γ = 0). Mobility does not trend much after two generations and the initial increase in mobility
is larger if parental income works exclusively through offspring human capital, but mobility trends follow
similar non-monotonic patterns. Second, the two intermediate parametrizations (φ > 0, γ > 0), illustrate that
the initial jump may be smaller if both income channels are important initially but only the direct impact of
income becomes less important after generation T . To sum up, inclusion of parental income in equation (3)
yields results that are qualitatively similar to our more simple baseline model.
23 By restricting the steady-state elasticities the value of φ implicitly follows from our choice of γ such that φ = (β + γ 2 λ − γ − λρ2 −
βγλ)/ρ.
16
Moreover, it is worth noting that the above extension with a mechanistic effect of parental income on offspring human capital is consistent with standard modeling of utility-maximizing investment behavior of parents
such as in Solon (2004). Following Solon (2004), we can rewrite equation 19 replacing φt yi,t−1 with θt ii,t−1 ,
where ii,t−1 denotes log investment in offspring human capital, and θt the efficiency of those investments, and
assume log-linear preferences. Abstracting from public investments, optimal private investments in offspring is
then given by
∗
Ii,t−1
= W (αt , θt , ρt , γt ) ∗ Yi,t−1 ,
(22)
where αt is the familiar “altruism” parameter that governs the preference for offspring income relative to
own consumption, and W a function of parameters only. Since optimal investments in log form implies separability between parental income and the parameters, there are no interactions between the structural parameters
and parental income affecting βt . A change in ρt , for example, will induce differential investment adjustments
depending upon absolute level of parental income, but the adjustments are proportionally equivalent and thus
irrelevant in the log-linear case.24
Co-movements in cross-sectional inequality
In the previous section we considered examples in which the marginal distributions of income and characteristics remained constant. Various researchers have however illustrated that cross-sectional inequality and intergenerational mobility are likely interdependent.25 Interest in this interdependence is further warranted because
intergenerational persistence is more consequential when cross-sectional inequality is high, and empirically because countries with large cross-sectional inequality tend to be characterized by low mobility (see Björklund
and Jäntti, 2009; and Blanden, 2009). Indeed, in view of this observation some economists have warned that
mobility might trend downwards in countries that have recently been characterized by rising cross-sectional
inequality, as the United States.26 We thus discuss additional implications that arise from co-movements in
cross-sectional inequality.
As in example 1, we consider a shift in the heredity parameter (λ2 6= λ1 ) while assuming that γ is zero, but
now we keep the variance of the error terms in equations (2) and (3) constant such that cross-sectional inequality
is affected. The intergenerational elasticity then shifts according to
∆βT = βT − βT −1 =
ρλ2 Cov(eT −1 , yT −1 ) ρλ1 Cov(eT −2 , yT −2 )
−
V ar(yT −1 )
V ar(yT −2 )
= ρ(λ2 − λ1 )ρ.
in the first generation affected. In contrast to the case considered in the previous section, mobility however
trends also in subsequent generations, e.g.
∆βT +1 = βT +1 − βT =
ρλ2 V ar(eT )ρ
ρλ2 V ar(eT −1 )ρ
−
ρV ar(eT )ρ + V ar(uT ) ρV ar(eT −1 )ρ + V ar(uT −1 )
The trend is positive if λ2 > λ1 and V ar(uT ) 6= 0 since V ar(et ) increases in λt . The convergence of crosssectional inequality to its new steady state thus constitutes another source of dynamics.
For empirical analysis of mobility trends it might thus be more informative to consider intergenerational
24 This
implies that choosing a parametrization such that θt ii,t−1 /yi,t−1 = φt would give the same results as the above case with
purely mechanistic transmission.
25 See for example Solon (2004), Davies, Zhang, and Zeng (2005) and Hassler, Mora, and Zeira (2007).
26 E.g. Alan Krueger, Chairman of the Council of Economic Advisors, in a speech given on January 12th, 2012.
17
correlations, defined as
rt = Corr(yt , yt−1 ) = βt
σyt−1
,
σyt
to abstract from changes in the marginal distributions across generations.27 The implied long-run shifts in
(steady state) elasticities and correlations are equal since
∆r∞ = β∞
σy∞−1
σy∞
σy
− βT −1 σyT −2
T −1
= ∆β∞ ,
but the transition paths differ. For example,
σyT −1
σy
− βT −1 T −2
σyT
σyT −1
σyT −1
− λ1 )ρ,
= ρ(λ2
σyT
∆rT = rT − rT −1 = βT
is smaller than ∆βt when heredity and thus cross-sectional inequality increases (λ2 > λ1 ).
In the case of variable cross-sectional inequality we thus need to consider an additional source of dynamic
transitions, which depends on the speed of convergence in V ar(et ) and V ar(yt ). This strengthens our motivating argument that transitions between steady states should be considered important, but the interpretation of
mobility statistics becomes more complicated. Elasticity trends can be partially due to changes in the marginal
distributions across cohorts; the initial response in elasticities is amplified when structural parameters shift that
are positively related to both intergenerational persistence and cross-sectional inequality.
4
Conclusions
We hope that our analysis illustrates why dynamic transitions between steady states are an interesting dimension
in intergenerational mobility studies. We focused on general properties, noting for example that structural
changes in the past can have a lagged impact on mobility levels several generations later and often cause nonmonotonic trends. Yet we also referred to specific cases to exemplify the implications of our results. It may
be useful to summarize these more specific implications as a first step to interpret recent empirical estimates of
trends in intergenerational mobility.
On the one hand, some of our results are reason for concern. For example, the finding that the effect of an
increase in the skill-premium will not be fully reflected in mobility levels before a parent generation is affected
implies that the full impact of the recent rise in cross-sectional inequality on intergenerational mobility levels
are yet to be uncovered by empirical research. Likewise, our result that changes in the relative returns to characteristics lead to a short-lived rise in mobility might be worrying for mobility proponents – if recent technological
change led to such changes in relative returns then its impact on long-run mobility might currently be covered
by such short-term gains. On the other hand, our results suggest that any recent decline in mobility may have a
rather harmless explanation. A negative mobility trend is exactly what we would expect when an economy has
undergone major transformation from plutocracy to meritocracy some generations ago. Paradoxically, negative
mobility trends can thus be indicative of major improvements in equality of opportunity in the past.
Our model is however highly stylized and several of the topics touched upon in this paper deserve more
careful analysis. We nevertheless see our simple approach as useful guidance for future research. It underlined
that the covariance between human capital and income in the parental generation should be expected to be an
important predictor of both levels and trends in current income mobility. This is relevant for future empirical
27 Björklund and Jäntti (2009) note that intergenerational correlations might for the same reasons be preferred in cross-country comparisons.
18
research agendas. Moreover, our findings highlight the difficulty of interpreting and causally explain recent
estimates of mobility trends. Given the restrictive data requirements within the field, an alternative route ahead
could be to look at trends in sibling correlations to learn more about the role of parental background across time
(see Björklund, Jäntti, and Lindquist, 2009, and Levine and Mazumder, 2007).
19
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Appendix
A.1 Choices of parameter values
The parameters of our model reflect total effects of broad concepts of parental economic status (γ), parental
human capital (λ), and offspring human capital (ρ). All these are imperfectly captured by actual data, existing
evidence can thus only provide indications of what these parameters may be. Lefgren, Lindquist, and Sims
(2012) examine the relative importance of different mechanisms in a transmission framework that is similar
to ours. Using imperfect instruments that are differentially correlated with parental human capital and income
they estimate that in Sweden the effect from parental income explains about a third of the elasticity, while the
effect from parental human capital explains the remaining two thirds. In any case, a reasonable lower bound,
given that we are considering mobility in total income, is the contribution of wealth to the intergenerational
correlation that is reported to be about 25 percent in Bowles and Gintis (2002).
The literature provides more guidance on the mechanistic transmission of human capital (λ). All since the
classic study of Galton (1886) on the intergenerational correlation of height to more recent evidence concerning
transmission of genes, schooling, and measures of ability, the results imply father-son correlations of about 0.30.4, and almost the double when considering both parents.28 While estimates may capture to various degrees
direct effects of parental income they nevertheless provide an upper bound for λ if we assume that the effects
of parental income are non-negative. Values of λ in the range 0.5-0.8 seem thus reasonable.
Finally, a reasonable lower-bound estimate of ρ can be approximated by evidence on the explanatory power
of earnings equations. Studies that observe richer sets of covariates, including measures of cognitive and noncognitive ability, typically yield estimates of R2 in the neighborhood of 0.40.29 On the one hand, all the reported
estimates are likely to underestimate the explanatory power of human capital (broadly defined) as of imperfect
measurement and omitted variables. On the other hand, the inclusion of parental income in our structural
equation would imply that the reported estimates are overestimates for our purposes if parental income and
human capital are positively correlated. Values of ρ in the range of 0.6-0.8 should then be roughly consistent
with the empirical evidence.30
In steady state these implied parameter values are roughly consistent with recent US evidence on the intergenerational income elasticity, which typically report estimates of 0.45-0.55 (see Mazumder, 2005; Lee and
Solon, 2009). Conversely, given reliable estimates of β we can cross-validate the chosen values for the structural parameters of the model. This both enables us to evaluate the consistency of our parameter choices and
the model, and potentially narrow down the implied ranges for our parameters. We can write each parameter as
a function of the others in steady state:
β=γ+
λ=
r
ρ=
ρ2 λ
1 − γλ
β−γ
βγ + ρ2 − γ 2
(β − γ) (1 − γλ)
λ
28 For estimates of correlations in measures of cognitive ability see Bowles and Gintis (2002) and the studies they cite, for measures of
both cognitive ability and non-cognitive ability see Grönqvist, Öckert, and Vlachos (2010).
29 Examples include Zax and Rees (2002), Mueller and Plug (2006) for the US; Groves (2005) for US and UK, Lindqvist and Vestman
(2011) for Sweden, Heineck and Anger (2010) for Germany. Fixed-effects models (e.g., individual or job level) as employed by Mueller
and Plug (2006) yield even higher estimates, although some of the difference may be capturing persistent luck rather than unobserved
characteristics. Moreover, the explanatory power for earnings in general seem to be somewhat lower than wages.
30 If we maintain the simplifying assumption that V ar(y) = V ar(e) = 1, then R2 = 0.4 translates into ρ ≈ 0.63 given steady state.
24
γ=
βλ + 1 ±
p
β 2 λ2 − 2βλ + 4λ2 ρ2 + 1
2λ
Plugging in the values discussed above on the right-hand sides of the above equations results in imputations
of the parameters that are surprisingly consistent with our reading of the empirical literature. We can also
narrow down some of the implied ranges (most notably we rule out too high values of λ and ρ, as they cause γ
to approach zero. As a result we arrive at the following implied ranges:
0.45 ≤ β ≤ 0.55, 0.15 ≤ γ ≤ 0.25, 0.60 ≤ ρ ≤ 0.70, 0.50 ≤ λ ≤ 0.65
25
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