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We have too little upward mobility, but it has not declined.

President Obama speaks in Osawatomie, Kan., in December.

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In this case, the goal of estimating the joint distribution of incomes — the shape of the soup-can display — is to determine the chance that someone growing up in poverty will make it to the middle class in adulthood. In terms of the display, the idea is to take all the cans that lie to the left of a line drawn down the display base from front to back (the line that defines “poverty” in childhood) and then within this group of cans, determine the fraction that also lie behind a line drawn across the base from left to right (the one that defines “middle class” in adulthood). The fraction of cans to the left that are also in the smaller group to the back gives the likelihood of upward mobility.

Studies typically do not have to estimate the joint distribution of incomes in this way to estimate upward-mobility rates. They have real-world data on a representative subset of families and simply do the math directly to determine how many adults were raised in poverty, and how many of them ended up in the middle class. In contrast, each of the president’s mobility estimates is based on just five numbers that describe the broad contours of the joint distribution of incomes and a sixth that defines “the middle class.”

The first four of these numbers describe inequality within the parent and child generations and average family incomes in each generation. These estimates — and a number of technical assumptions — are the basis for determining where from left to right the soup cans tend to fall, where they tend to lie from front to back, and how spread out they are in each direction. The fifth number that the modeling requires indicates the “correlation” between parent and child incomes — the extent to which the soup cans line up in a diagonal ridge or are evenly dispersed. Finally, the “median” income among adult children — the one that is right in the middle — is used to define “middle class.” (“Poverty” in childhood is defined as being below some percentile, such as being in the bottom 10 percent. Given the assumptions of the model, no real-world figure is necessary to estimate this cut point.)

One reason to doubt the reliability of the Soup Cans in Six Numbers model is the considerable work the technical assumptions do in producing its estimates. But an equally important shortcoming is the fact that the numbers on which the model is based are potentially bad proxies for the real-world numbers that are needed. Most strikingly, the level of the income correlation is simply assumed, and it is assumed not to change over time. The other figures are based on all families, young or old, with children or not, which is too broad a group. They should be confined to parents with children born within a narrow range of years and to adult children (whether they are in families or not) born around the same time. The model essentially determines how two other soup-can displays look from the front and uses that information and an assumed form for the new display to build it up from scratch. Whether the jerry-rigged joint distribution of incomes resembles the real one is an open question.

Determining how problematic this model is required that I first replicate the president’s estimates to make sure I understood the details sufficiently. When I used all the information provided by David Card, filling in a couple of gaps as best I could, I estimated that poor children born in the late 1940s had a 47 percent chance of ending up in the middle class as adults. I found that upward mobility fell, to 39 percent for poor children born in 1980 and to a projected 35 percent for poor children born in 2009. In other words, I was able to broadly replicate the figures President Obama had cited, though mine showed a somewhat smaller decline in mobility. Importantly, I discovered that the president’s estimates appear to define “upward mobility” in such a way that ending up better off than “middle class” does not count. That is, “middle class” in the president’s account means having at least a certain amount of income but no more than some maximum. My results suggest that the fraction starting out poor and ending up middle class or better is a bit higher than the president’s figures suggest, though they still show a decline from 51 to 41 percent.

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