Intergenerational mobility

Intergenerational mobility is the extent to which children’s economic position is independent of their parents’. It is measured by the intergenerational elasticity $\beta$, the slope of a regression of children’s log earnings on their parents’ log earnings. A family’s earnings gap from the average shrinks by the factor $\beta$ each generation. Estimates are about 0.2 for Canada and between 0.4 and 0.5 for the United States, so family advantages fade much faster in Canada.

Why parents’ and children’s earnings are linked

Parents pass on earning power in three ways:

• Investment. Higher-income parents spend more on their children’s schooling, health and development. Here parents invest in their children’s human capital, rather than individuals in their own as in the basic schooling model.
• Inherited traits. Some abilities are passed on genetically or through upbringing.
• Social capital. The neighbourhood, the school and its students, and the community and religious organizations a child grows up with shape the human capital the child acquires. In the United States, areas with more upward mobility have less residential segregation, less income inequality, better primary schools, more social capital and more stable families.¹

These channels make the skills of parents and children positively correlated. Three forces pull children back toward the average:

• Diminishing returns to investment. Each extra dollar a rich family invests raises the child’s human capital by less.
• Regression in ability. The children of unusually able parents are, on average, less unusual than their parents.
• Luck. Part of any parent’s high or low earnings is luck, which the child does not inherit.

This process is regression toward the mean: children of parents far from the average end up, on average, closer to it.

The intergenerational regression

For family $i$,

$$\ln w_{\text{child},i} = a + \beta \ln w_{\text{parent},i} + \varepsilon_i ,$$

where $a$ captures growth in average earnings between generations and $\varepsilon_i$ is everything about the child’s earnings unrelated to the parent’s. Measured as deviations from each generation’s average, the expected gap of the child is $\beta$ times the gap of the parent:

• $\beta = 1$: children keep their parents’ whole gap; there is no regression toward the mean.
• $\beta = 0$: children’s earnings are independent of their parents’; regression is complete.
• $0 \lt \beta \lt 1$: children regress part of the way toward the mean.

Intergenerational elasticity β

0.40

Parent's earnings gap

50%

Child's expected gap

20.0%

Grandchild's expected gap

8.0%

Share of the gap passed on each generation

40%

Open in Graph Lab

Figure 1. The intergenerational regression line has slope β; the 45-degree line (β = 1) is full persistence and the horizontal line (β = 0) is complete regression. With β = 0.4, the child of a parent 50 percent above the average is expected to be 20 percent above it.

The slope is closely related to the intergenerational correlation $\rho$: $\rho = \beta \, \sigma_{\text{parent}} / \sigma_{\text{child}}$, where $\sigma$ is the standard deviation of log earnings in each generation. When earnings are equally dispersed in both generations the two coincide, which is why textbooks often use the terms interchangeably. When inequality rises from one generation to the next, the elasticity exceeds the correlation.

Applied repeatedly, the regression implies that a gap $g_0$ shrinks to $\beta^n g_0$ after $n$ generations. Gaps are differences in log earnings; for gaps of a few tenths they are close to percentage differences, and the course states the arithmetic in percentages.

Parents' gap above the average

50%

Children's gap, β = 0.2

10.0%

Children's gap, β = 0.4

20.0%

Grandchildren's gap, β = 0.2

2.0%

Grandchildren's gap, β = 0.4

8.0%

Generations until 5% or less, β = 0.2

2

Generations until 5% or less, β = 0.4

3

Open in Graph Lab

Figure 2. A family starts 50 percent above the average. With β = 0.2 the gap falls to 10 percent for the children and 2 percent for the grandchildren; with β = 0.4 it falls to 20 and 8 percent, and takes a third generation to drop below 5 percent.

Key idea

Each generation keeps a fraction $\beta$ of the previous generation’s gap from the average. Doubling $\beta$ from 0.2 to 0.4 doubles the children’s gap, quadruples the grandchildren’s and adds about a generation to the time an advantage takes to fade.

Worked example

Two fathers: one earns 50 percent more than the other.

1. Low persistence, $\beta = 0.2$. The sons’ expected gap is $0.2 \times 50 = 10$ percent; the grandsons’, $0.2 \times 10 = 2$ percent.

2. High persistence, $\beta = 0.4$. The sons’ gap is $0.4 \times 50 = 20$ percent, the grandsons’ $0.4 \times 20 = 8$ percent and the great-grandsons’ $3.2$ percent.

3. Generations until the gap is 5 percent or less. Solve $50\beta^n \le 5$, so $\beta^n \le 0.1$ and $n \ge \ln 0.1 / \ln \beta$. With $\beta = 0.2$: $n \ge -2.303 / -1.609 = 1.43$, so 2 generations. With $\beta = 0.4$: $n \ge -2.303 / -0.916 = 2.51$, so 3 generations.

4. The exact log version. A 50 percent gap is $\ln 1.5 = 0.405$ in logs. With $\beta = 0.2$ the sons’ gap is $0.2 \times 0.405 = 0.081$, or $e^{0.081} - 1 = 8.4$ percent, close to the 10 percent approximation.

Exercise: a larger starting gap

A family earns 80 percent above the average and $\beta = 0.5$. The children’s expected gap is 40 percent, the grandchildren’s 20 percent, then 10, 5 and 2.5 percent. The gap first reaches 5 percent or less in the fourth generation: $n \ge \ln(5/80)/\ln 0.5 = -2.773 / -0.693 = 4$.

Evidence

Measurement matters. Early studies found weak links between fathers’ and sons’ earnings; reviewing them, Becker and Tomes concluded that in rich countries almost all of an ancestor’s earnings advantage or disadvantage disappears within three generations.² Many of those studies measured fathers’ earnings in a single year, which includes temporary ups and downs that act like noise in the explanatory variable and bias the slope toward zero (attenuation bias). Using several years of fathers’ earnings, Solon found the intergenerational correlation in long-run income was at least 0.4, implying much less mobility than the earlier research.³ Higher estimates of this kind reflect better measurement, not a change in mobility.

Canada versus the United States. Using income tax records on about 400,000 Canadian father-son pairs, Corak and Heisz estimated an intergenerational earnings elasticity of about 0.2, with more mobility at the bottom of the distribution than at the top.⁴ Corak’s cross-country comparison puts the US elasticity at about 0.5, twice the Canadian figure; less than a fifth of a father’s advantage passes to his son in Finland, Norway and Denmark, against about half in Italy, the United Kingdom and the United States. The two countries differ mainly at the extremes: in the United States, more than half of sons of top-decile fathers stay in the top three deciles, and about half of sons of bottom-decile fathers stay in the bottom three.⁵ Mobility also varies within each country, and the Canada-US border separates high- and low-mobility regions only in part.⁶

The Great Gatsby curve. Countries with more income inequality a generation ago have higher intergenerational elasticities, that is, less mobility. This relationship is called the Great Gatsby curve.⁵ It is a correlation, not proof that inequality causes low mobility.

Is persistence rising? The evidence is mixed. One set of US estimates shows the elasticity rising from 0.38 to about 0.55 after 1980, as the return to college rose, but other studies find no significant change.⁵ Using tax records, Chetty and coauthors find that rank-based mobility, the link between a child’s percentile and the parents’ percentile, was stable for Americans born between 1971 and 1993; in the United States as a whole, a 10-percentile higher parental income rank is associated with a 3.4-percentile higher child rank.⁷¹ Because inequality grew, the same ranks now mean larger differences in income. Absolute mobility, the share of children who earn more than their parents, fell from about 90 percent for Americans born in 1940 to about 50 percent for those born in the 1980s.⁸

Common mistakes

Traps on exams

• Reading $\beta$ backwards. $\beta = 1$ means no mobility (full persistence); $\beta = 0$ means children’s earnings are independent of their parents’. A higher $\beta$ means less mobility.
• Thinking regression toward the mean shrinks inequality. Each generation draws new luck and abilities, so the distribution stays spread out even as each family drifts toward the average.
• Reading a higher estimate as falling mobility. Single-year measures of parents’ earnings bias $\beta$ toward zero, so studies that average several years find larger values for the same population.

On the exam

For “explain inequality across generations”, write the regression, interpret $\beta = 0$ and $\beta = 1$, work the 50 percent example for $\beta = 0.2$ and $\beta = 0.4$ through two generations, give two reasons for regression toward the mean, and state the evidence: about 0.2 in Canada, 0.4 to 0.5 in the United States, with no consensus on whether US persistence is rising.

Check yourself

1.

In the regression $\ln w_{\text{child}} = a + \beta \ln w_{\text{parent}}$, what does $\beta = 1$ mean?

1Children keep their parents' entire earnings gap from the average, so there is no regression toward the mean2Children's earnings are independent of their parents' earnings3Every child earns exactly the average4Children's earnings gaps are, on average, twice their parents'

Check answer

2.

Why do children's earnings regress toward the mean instead of fully reflecting their parents' earnings?

1Diminishing returns to investment in children's human capital, regression toward the mean in ability, and luck in parents' earnings that is not inherited2Children of high-earning parents invest less in their own human capital, so they earn less than average3Governments tax the children of high earners at higher rates4Earnings are measured in logs, which mechanically shrinks every gap

Check answer

3.

In explaining differences in children's human capital, what does social capital refer to?

1The resources of the community a child grows up in, such as neighbourhood quality, schools and classmates, and community or religious organizations2The physical capital, such as housing and savings, that a family owns3Government spending on social programs such as Employment Insurance4The schooling a child completes before entering the labour market

Check answer

Practise more questions on this topic

Sources

1. Chetty, R., Hendren, N., Kline, P. and Saez, E. (2014). Where is the land of opportunity? The geography of intergenerational mobility in the United States. Quarterly Journal of Economics, 129(4), 1553 to 1623. NBER working paper 19843. ↩ ↩²

2. Becker, G. S. and Tomes, N. (1986). Human capital and the rise and fall of families. Journal of Labor Economics, 4(3, part 2), S1 to S39. ↩

3. Solon, G. (1992). Intergenerational income mobility in the United States. American Economic Review, 82(3), 393 to 408. ↩

4. Corak, M. and Heisz, A. (1999). The intergenerational earnings and income mobility of Canadian men: Evidence from longitudinal income tax data. Journal of Human Resources, 34(3), 504 to 533. ↩

5. Corak, M. (2013). Income inequality, equality of opportunity, and intergenerational mobility. Journal of Economic Perspectives, 27(3), 79 to 102. The rising US elasticity is from Aaronson and Mazumder’s estimates as reported there; Corak notes that other studies find no significant change. ↩ ↩² ↩³

6. Connolly, M., Corak, M. and Haeck, C. (2019). Intergenerational mobility between and within Canada and the United States. Journal of Labor Economics, 37(S2), S595 to S641. ↩

7. Chetty, R., Hendren, N., Kline, P., Saez, E. and Turner, N. (2014). Is the United States still a land of opportunity? Recent trends in intergenerational mobility. American Economic Review, 104(5), 141 to 147. NBER working paper 19844. ↩

8. Chetty, R., Grusky, D., Hell, M., Hendren, N., Manduca, R. and Narang, J. (2017). The fading American dream: Trends in absolute income mobility since 1940. Science, 356(6336), 398 to 406. NBER working paper 22910. ↩