why do some groups experience greater mobility than others?

an IGE-based conditional-distribution framework
linking absolute and relative intergenerational mobility,
with origin-specific uncertainty

mauricio bucca

puc chile

the problem

context

social mobility is a central theme in public discussions about fairness, opportunity, and the “rules of the game.”
framed—alongside wealth, poverty, and inequality—as a defining marker of a society’s character (e.g., “the land of opportunity”).
commonly invoked as a public good: metaphors like “the social elevator” or a “level playing field” signal that rising should be attainable for all.

the problem

social mobility means different things in public debate and academic research
two core concepts are often conflated:
- relative mobility: how tightly children’s positions track their parents’
- absolute mobility: where children end up compared with their parents (in levels)

so what counts as “good” or “bad” mobility? seems straightforward enough…
- weaker parent-child associations → more “equality of opportunity”
- more people surpassing their parents → more upward mobility

but not so fast!: these two mobility types can move in opposite directions and have different consequences for individuals within the same society.

two ways this story could be true:

the bus driver’s son ends up owning the bus company, while the doctor’s son winds up driving the bus (rank reversal)
the bus driver’s son owns the bus company, and the doctor’s son owns the clinic (upward mobility for both)

this paper

core claim: properly understanding mobility requires examining both types simultaneously. this paper introduces a simple unified framework that allows for that

analytical framework

definitions

Relative mobility captures how tightly children’s economic standing tracks their parents’.

Represented by the conditional distribution
\(f(y \mid x)\), where \(x\) is parental income and \(y\) is child income.

the standard model is the IGE workhorse: \(E[y \mid x] = \alpha + \beta x\)

where \(x\) is parents’ permanent log income and \(y\) is children’s log income.
the slope \(\beta\) is the intergenerational elasticity (IGE), typically in the [0.2–0.6] range.

population-level concept: non-directional and essentially a zero-sum measure of positional movement.

definitions

absolute mobility compares children’s income to their parents on a level scale, not a rank or positional scale.

with a meaningful threshold \(\delta\), classify mobility using the gap \(y - x\):
- upward mobility: \(y > x + \delta\)
- downward mobility: \(y < x - \delta\)
- immobility: \(x - \delta \le y \le x + \delta\)

individual-level outcome: directional — everyone can, in principle, move up, move down, or stay roughly the same.

linking relative and absolute mobility

standard IGE specification:

\[y_i \mid x_i \sim \text{Normal}\!\big(\mu_i = \alpha + \beta x_i,\; \sigma \big)\]

four key components:

parameter	interpretation
\(\alpha\)	income floor — expected income at a low reference origin \(c\) (not literally the bottom)
\(\beta\)	persistence (IGE) — degree of relative immobility
\(\sigma\)	uncertainty — unpredictability in intergenerational transmission
\(x\)	parental income — the baseline for absolute comparison

extension: the present framework allow both expected income \(\mu_i(x)\) and income uncertainty \(\sigma(x)\) vary with parental income

linking relative and absolute mobility

absolute mobility probabilities:

\[ \begin{aligned} \Pr(M^{\uparrow}) &= \Pr(y - x > \delta \mid x) && \text{(upward mobility)} \\ \Pr(M^{\downarrow}) &= \Pr(y - x < -\delta \mid x) && \text{(downward mobility)} \\ \Pr(M^{0}) &= 1 - \big[ \Pr(M^{\uparrow}) + \Pr(M^{\downarrow}) \big] && \text{(immobility)} \end{aligned} \]

linking relative and absolute mobility

we can express absolute mobility probabilities as a function of the IGE components:

step 1: start with the income generation model: \(y \mid x \sim \text{Normal}\big(\mu(x) = \alpha + \beta \widetilde{x},\; \sigma(x)\big)\), with \(\widetilde{x}=x-c\) centered at a low reference point \(c\)

step 2: standardized mobility thresholds (comparison is still against raw \(x\))

\[ \begin{aligned} z^{\uparrow} &= \frac{(x + \delta) - \mu(x)}{\sigma(x)} \\ z^{\downarrow} &= \frac{(x - \delta) - \mu(x)}{\sigma(x)} \end{aligned} \]

step 3: express absolute mobility probabilities using standard normal CDF \(\Phi(\cdot)\)

\[ \begin{aligned} \Pr(M^{\uparrow}) &= 1 - \Phi(z^{\uparrow}) \\ \Pr(M^{\downarrow}) &= \Phi(z^{\downarrow}) \\ \Pr(M^{0}) &= \Phi(z^{\uparrow}) - \Phi(z^{\downarrow}) \end{aligned} \]

drivers of absolute mobility

driver 1: income floor (\(\alpha\))

question: how do intercept shifts (uniform income growth) affect absolute mobility?

analytical result:

\[ \frac{\partial \Pr(M^{\uparrow})}{\partial \alpha} = \frac{\phi(z^{\uparrow})}{\sigma}, \qquad \frac{\partial \Pr(M^{\downarrow})}{\partial \alpha} = -\frac{\phi(z^{\downarrow})}{\sigma} \]

where \(\phi(\cdot)\) is the standard normal PDF

key insights:

raising \(\alpha\) shifts the entire income distribution upward
increases upward mobility, decreases downward mobility
effect is uniform across the parental income distribution

driver 2: relative (im)mobility (\(\beta\))

question: how do changes in relative mobility affect absolute mobility?

analytical result:

\[ \frac{\partial \Pr(M^{\uparrow})}{\partial \beta} = \frac{\widetilde{x} \cdot \phi(z^{\uparrow})}{\sigma}, \qquad \frac{\partial \Pr(M^{\downarrow})}{\partial \beta} = -\frac{\widetilde{x} \cdot \phi(z^{\downarrow})}{\sigma} \]

key insights:

\(\beta\) tilts the distribution around the reference origin, rather than shifting it uniformly like \(\alpha\)
since \(c\) is a low floor (\(\widetilde{x}=x-c>0\) throughout), increasing \(\beta\) (more persistence) increases upward mobility and decreases downward mobility, with effect proportional to parental income

driver 3: income uncertainty (\(\sigma\))

question: how does the “predictability” of children’s income affect absolute mobility?

analytical result:

\[ \frac{\partial \Pr(M^{\uparrow})}{\partial \sigma} = \frac{z^{\uparrow} \phi(z^{\uparrow})}{\sigma}, \qquad \frac{\partial \Pr(M^{\downarrow})}{\partial \sigma} = -\frac{z^{\downarrow} \phi(z^{\downarrow})}{\sigma} \]

key insights:

effect depends on distance to thresholds (\(z^{\uparrow}\), \(z^{\downarrow}\))
at the bottom: increases upward mobility (lottery-ticket effect)
at the top: increases downward mobility (greater vulnerability)

driver 4: parental income (\(x\))

question: how do social origins (parental income) affect absolute mobility?

analytical result (fixed dispersion):

\[ \frac{\partial \Pr(M^{\uparrow})}{\partial x} = \frac{\beta - 1}{\sigma} \phi(z^{\uparrow}), \qquad \frac{\partial \Pr(M^{\downarrow})}{\partial x} = \frac{1 - \beta}{\sigma} \phi(z^{\downarrow}) \]

key insights:

since empirically \(0 < \beta < 1\) :
- upward mobility decreases with parental income \(x\)
- downward mobility increases with parental income \(x\)
even under fixed persistence, mobility varies substantially by origin
origin effects are structural, not just compositional

driver 4 under heteroskedasticity

question: once uncertainty \(\sigma(x)\) also varies with origin, what else moves with parental income?

total derivative (both channels at once):

\[ \frac{d \Pr(M^{\uparrow})}{dx} = \phi(z^{\uparrow})\left[\frac{-(1-\beta)}{\sigma(x)} + \lambda\, z^{\uparrow}\right], \qquad \frac{d \Pr(M^{\downarrow})}{dx} = \phi(z^{\downarrow})\left[\frac{1-\beta}{\sigma(x)} - \lambda\, z^{\downarrow}\right] \]

key insights:

bracket #1 is mean reversion (driver 4, as before): pushes upward mobility down, downward mobility up at higher origins
bracket #2 is a new dispersion-gradient channel, \(\lambda z^{\uparrow}\) — absent when \(\sigma\) is constant
sign of \(\lambda\) relative to \(z\) decides whether the two channels reinforce or offset:
- \(\lambda<0\) (narrowing uncertainty at the top) reinforces mean reversion
- \(\lambda>0\) (widening uncertainty at the top) offsets it
this generalizes the mean-reversion result (Bloome 2025) by adding a dispersion-driven channel

counterfactual decomposition

the parameter-swap counterfactual

goal: isolate the marginal contribution of each structural parameter to group differences

challenge: the effect of any single parameter depends on the levels of all others

solution: replace one parameter with its counterfactual value while holding others fixed. For groups \(g\) and \(\bar{g}\), I do three swaps:

\(\alpha\)-swap: plug in \(\alpha_{\bar{g}}\) in place of \(\alpha_g\)
\(\beta\)-swap: plug in \(\beta_{\bar{g}}\) in place of \(\beta_g\)
\(\sigma\)-swap: plug in \(\sigma_{\bar{g}}(x)\) in place of \(\sigma_g(x)\)

example question: “how much upward mobility would group A experience if, holding everything else constant, it had group B’s level of relative mobility?”

pointwise effect at income \(x\): \(\Delta^{(g,\theta)}_k(x) = \widetilde{\Pr}^{(g,\theta)}(M^k \mid x) - \Pr^{(g)}(M^k \mid x)\)

aggregation and composition

population-level consequences depend on the distribution of parental origins, which may differ across groups.

aggregation — average change in probability of mobility type \(k\) when swapping parameter \(\theta\) for group \(g\), holding distribution of \(x\) fixed:

\[ \overline{\Delta}^{(g,\theta)}_k = \int \Delta^{(g,\theta)}_k(x)\, f^{(g)}(x)\, dx \]

composition — change in probability of mobility type \(k\) induced solely by swapping the parental income distribution:

\[ \overline{\Delta}^{(g,x)}_k = \int p^{(g)}_k(x)\, \big[f^{(\bar{g})}(x) - f^{(g)}(x)\big]\, dx \]

distinguishes structural transmission (via parameters) from distributional positioning (via origins)

empirical application

data: PSID 1968–2019

sample: 4,342 Black and White sons and daughters

birth cohorts 1944–1989
at least 2 positive earnings observations, ages 25–60
valid parental earnings anchor + positive survey weight
models estimated separately by gender

earnings measures:

labor earnings: wages, salaries, self-employment, bonuses/tips, inflation-adjusted
permanent child earnings: individual mean of log earnings
permanent parental earnings: max(mother, father) age-42 anchor, from a centered 7-wave rolling average

mobility profiles: women

floors, persistence, and dispersion are nearly identical across race — conditional distributions essentially overlap.
dispersion narrows slightly with parental earnings (\(\lambda<0\)): parental resources behave like insurance against shocks.

pointwise counterfactual effects: women

aggregation and composition: women

factual upward mobility: 33.2% Black vs. 20.2% White
\(\alpha\)-, \(\beta\)-, \(\sigma\)-swaps are all small (consistent with near-identical fitted parameters).
composition — swapping the parental income distribution alone moves upward mobility by +13.2 to +13.7 points — it accounts for essentially the whole gap.

mobility profiles: men

persistence is similar (\(\beta\approx0.28\) vs. \(0.26\)), but White men’s earnings floor is ~0.29 log points higher (\(\approx34\%\) more earnings at every origin).
again, dispersion narrows slightly with parental earnings (\(\lambda<0\))

pointwise counterfactual effects: men

aggregation and composition: men

factual upward mobility: 44.7% Black vs. 39.1% White.
floor swap is the dominant effect (\(\beta\), \(\sigma\) swaps stay near zero).
composition swap is comparably large but points the other way.
floor disadvantage and compositional advantage are of similar size and opposite sign — they largely cancel, so the small raw gap conceals a real conditional disadvantage.

key findings

same aggregate summary — “composition dominates” — describes two different situations:

women: virtually no conditional gap → composition explains almost all of the observed (pro-Black) upward-mobility gap

men: a real earnings-floor disadvantage for Black men is masked by an offsetting compositional advantage from poorer origins

disentangling conditional (floor/persistence/dispersion) from compositional (origins) margins — separately by gender — is what makes this visible.

contributions and implications

provides a unified, IGE-based conditional-distribution framework linking absolute and relative mobility, extended to allow origin-dependent uncertainty.
shows uncertainty can reinforce or offset mean reversion — a second channel beyond persistence through which origin shapes absolute mobility.
offers non-additive, one-factor-swap counterfactuals to disentangle floor, persistence, dispersion, and compositional (origin-distribution) contributions to group gaps.
applied to Black–White earnings mobility:
- the same aggregate gap (or the lack thereof) can hide complex, offsetting effects.
- “good news” (upward mobility) due to “bad reasons” (poor origins)

thank you

mauricio bucca — puc chile · mebucca@uc.cl

all replication materials available

appendix

monte carlo validation

it works! trust me.

model and estimation

statistical model

heteroskedastic IGE model:

\[ \begin{aligned} y_i \mid g, x_i &\sim \text{Normal}\Big(\mu_g(x_i),\ \sqrt{\eta_{ig}^2/T_{ig} + \sigma_g^2(x_i)}\Big) \\ \mu_g(x_i) &= \alpha_g + \beta_g \widetilde{x}_i, \qquad \widetilde{x}_i = x_i - c\\ \log\sigma_g(x_i) &= \gamma_g + \lambda_g \widetilde{x}_i \end{aligned} \]

bayesian estimation approach:

\(\eta_{ig}^2/T_{ig}\): sampling noise from averaging \(T_{ig}\) annual earnings into “permanent” earnings — enters estimation, not the mobility thresholds
hierarchical structure with group-specific parameters, non-centered parameterization
weakly informative priors; informative prior for \(\beta\): centered on empirically plausible range (0.3–0.6)
likelihood weighted by longitudinal survey weights; full posterior uncertainty quantification

estimated parameters, by gender

parameter	black women	white women	black men	white men
\(\alpha\)	9.47 [9.28, 9.68]	9.45 [9.33, 9.57]	9.93 [9.66, 10.18]	10.22 [10.12, 10.32]
\(\beta\)	0.34 [0.23, 0.43]	0.34 [0.29, 0.39]	0.28 [0.16, 0.41]	0.26 [0.22, 0.29]
\(\gamma\)	-0.26 [-0.51, -0.07]	-0.16 [-0.27, -0.04]	-0.36 [-0.56, -0.07]	-0.40 [-0.52, -0.28]
\(\lambda\)	-0.04 [-0.13, 0.08]	-0.05 [-0.10, -0.00]	-0.12 [-0.27, -0.01]	-0.06 [-0.10, -0.01]

key patterns:

women: floors, persistence, and dispersion are nearly identical across race — conditional distributions essentially overlap.
men: persistence is similar (\(\beta\approx0.28\) vs. \(0.26\)), but White men’s earnings floor is ~0.29 log points higher (\(\approx34\%\) more earnings at every origin).
for both genders, dispersion narrows with parental earnings (\(\lambda<0\)): parental resources behave like insurance against large shocks.

robustness: selection into positive earnings

estimates condition on positive earnings — selection into that state is not origin-neutral.
men: retention roughly flat by origin for both races.
women: diverges by origin, in opposite directions — Black women’s earnings-retention rises with origin (\(\approx0.85\to0.94\)); White women’s falls (\(\approx0.95\to0.89\), plausibly household specialization).
imposing a hypothetical zero-earnings share (\(q\) up to 15%) mainly shifts levels at lower origins; counterfactual signs/ordering unchanged, just scaled by \((1-q)\).

robustness: equivalized family income

replacing own earnings with equivalized family income (less sensitive to labor-force attachment) preserves the compositional story overall…
…but changes the picture for women: a 0.42 log-point White–Black floor gap emerges that is absent in the earnings analysis.
under family income, women now show the same opposing-forces pattern as men: floor disadvantage vs. compositional advantage.
conclusion: the compositional finding is robust across outcomes; which conditional parameter matters depends on whether mobility is measured in own earnings or pooled family resources.