Intergenerational mobility measures in a bivariate normal model
aa r X i v : . [ q -f i n . E C ] J un Intergenerational mobility measures in a bivariate normal model
Yonatan Berman ∗ School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv, Israel (Dated: June 26, 2017)We model the joint log-income distribution of parents and children and derive analytic expressionsfor canonical relative and absolute intergenerational mobility measures. We find that both types ofmobility measures can be expressed as a function of the other.
For the past several decades, many scholars have beenstudying economic intergenerational mobility [1, 5, 7–10].The motivation for studying mobility stems from its rela-tionship to concepts like equality of opportunity [4, 13],the so-called “American Dream” [3, 6] and income in-equality [2, 7]. Typically measures of income intergenera-tional mobility are divided into two categories: relative –quantifying the propensity of individuals to change theirposition in the income distribution, and absolute – quan-tifying their propensity to change their income in moneyterms. The aim this note is to introduce a simple modelfor the joint income distribution of parents and childrenand use it for explicitly deriving canonical measures ofrelative and absolute mobility measures.Our starting point is a population of N parent-childpairs. We denote by Y ip and Y ic the incomes of the parentand the child (at the same age), respectively, for family i = 1 . . . N . We assume the incomes are all positive andmove to define the log-incomes X ip = log Y ip and X ic =log Y ic .The canonical measure of relative mobility is theelasticity of child income with respect to parent in-come, known as the intergenerational earnings elasticity(IGE) [4, 9, 11] and defined as the slope ( β ) of the linearregression X c = α + βX p + ǫ , (.1)where α is the regression intercept and ǫ is the error term.We note that IGE is a measure of immobility ratherthan of mobility and the larger it is, the stronger the rela-tionship between the parent and child income. Therefore,1 − β can be used as a measure of mobility.A standard approach to measure absolute intergener-ational mobility, recently used in [3] for studying thetrends in absolute mobility in the United States is tomeasure the fraction of children earning more than theirparents, denoted by A : A = P Nj =1 { i : Y ic >Y ip } (cid:0) Y jc (cid:1) N , (.2)where 1 S ( x ) is the indicator function for a set S and ar-gument x and (cid:8) i : Y ic > Y ip (cid:9) is the set of children earningmore than their parents. Since the logarithmic function preserves order we alsoget, A = P Nj =1 { i : X ic >X ip } (cid:0) X jc (cid:1) N . (.3)One hypothetical sample of such distribution is pre-sented in Fig. 1. It also depicts graphically how A and β are defined. The blue line is y = x , hence the rateof absolute mobility is defined as the fraction of parent-child pairs which are above it. The red line is the linearregression y = α + βx , for which β is the IGE. Parent log-income C h il d l og - i n c o m e y=xy= α + β · x FIG. 1: An illustration of the absolute and relative mobilitymeasures. The black circles are a randomly chosen sample of100 parent-child log-income pairs. The sample was createdassuming a bivariate normal distribution and the parametersused were µ p = 10 . σ p = 0 .
78 (for the parents marginaldistribution) and µ c = 10 . σ c = 1 .
15 (for the childrenmarginal distribution) with correlation of ρ = 0 .
57. The re-sulting α and β were 1 . .
84, respectively.
Since income distributions are known to be well ap-proximated by the log-normal distribution [12], a sim-ple plausible model for the joint distribution of parentand child log-incomes is the bivariate normal distribu-tion. Under this assumption, the marginal income distri-butions of both parents and children are log-normal andthe correlation between their log-incomes is defined by asingle parameter ρ . The marginal log-income distributionof the parents (children) follows N (cid:0) µ p , σ p (cid:1) ( N (cid:0) µ c , σ c (cid:1) ),hence the joint distribution is fully characterized by 5parameters: µ p , σ p , µ c , σ c and ρ .Assuming the bivariate normal approximation for thejoint distribution enables theoretically studying its prop-erties. In particular Both measures of mobility, A and1 − β , can be derived directly from the model and, no-tably, can both be expressed analytically as functions ofthe other. We first derive the IGE in terms of the distri-bution parameters: Proposition 1
For a bivariate normal distribution withparameters µ p , σ p (for the parents marginal distribution)and µ c , σ c (for the children marginal distribution) as-suming correlation ρ , the IGE is − β = 1 − σ c σ p ρ . (.4) Proof.
First, by definition, the correlation ρ , between X p and X c equals to their covariance, divided by σ p σ c ρ = Cov [ X p , X c ] σ p σ c . (.5) β can be directly calculated as follows, by the linearregression slope definition: β = P Ni =1 (cid:0) X ip − ¯ X p (cid:1) (cid:0) X ic − ¯ X c (cid:1)P Ni =1 (cid:0) X ip − ¯ X p (cid:1) , (.6)where ¯ X p and ¯ X c are the average parents and childrenlog-incomes, respectively.It follows that β = Cov [ X p , X c ] σ p . (.7)We immediately obtain β = σ c σ p ρ (.8)and therefore 1 − β = 1 − σ c σ p ρ (.9)Following Prop. 1 it is also possible to derive the rateof absolute mobility as a function of the distribution pa-rameters and the IGE: Proposition 2
For a bivariate normal distribution withparameters µ p , σ p (for the parents marginal distribution), µ c , σ c (for the children marginal distribution) and ρ = σ p β/σ c (where β is the IGE), the rate of absolute mobilityis A = Φ µ c − µ p q σ p (1 − β ) + σ c , (.10) where Φ is the cumulative distribution function of thestandard normal distribution. Proof.
We start by defining a new random variable Z = X c − X p . It follows that calculating A is equivalent tocalculating the probability P ( Z > Z ∼ N (cid:0) µ c − µ p , σ p + σ c − X p , X c ] (cid:1) , so ac-cording to Prop. 1 Z ∼ N (cid:0) µ c − µ p , σ p (1 − β ) + σ c (cid:1) . (.11)If follows that Z − ( µ c − µ p ) q σ p (1 − β ) + σ c ∼ N (0 , , (.12)so we can now write P ( Z >
0) = P Z − ( µ c − µ p ) q σ p (1 − β ) + σ c > − µ c − µ p q σ p (1 − β ) + σ c =Φ µ c − µ p q σ p (1 − β ) + σ c , (.13)where Φ is the cumulative distribution function of thestandard normal distribution.Proposition 2 shows that the rate of absolute mobilitycan be explicitly described as a function of the relativemobility. ∗ Electronic address: [email protected][1] Daniel Aaronson and Bhashkar Mazumder. Intergener-ational economic mobility in the united states, 1940 to2000.
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