Appendix
Clemens, Montenegro, and Pritchett (2009)—referred to as CMP—use data on 41 developing source countries j and the USA, and examine various income ratios (denoted here by lowercase letters), where the denominator is the average income of source country residents, Y
0 = α
0
S = α
0(A + H), where A is unobservable ability and H is observable education. CMP’s objective is to obtain the average income ratio, \( {y}_d^{\prime \prime }={Y}_d^{\prime \prime }/{Y}_0 \), of migrants living in the USA who were educated in their home country, relative to the income of home country residents with the same A and H levels, i.e., \( {y}_d^{\prime \prime }={Y}_d^{\prime \prime }/{Y}_0={\alpha}_d\left( A+ H\right)/{\alpha}_0\left( A+ H\right)={\alpha}_d/{\alpha}_0 \). The problem with such comparisons is of course that migrants self-select on both ability and education, whose levels are denoted by A
M and H
M, respectively, and that their observed income is Y
d
= α
d
S
M = α
d
(A
M + H
M) rather than \( {Y}_d^{\prime \prime } \).
CMP find that for the 42 source countries, migrants’ average income ratio \( {y}_d=\frac{\alpha_d{S}^M}{\alpha_0 S}=\frac{\alpha_d\left({A}^M+{H}^M\right)}{\alpha_0\left( A+ H\right)}=7.99 \), i.e., migrants’ average income is 7.99 times that of source country residents. They first correct y
d
for migrants’ self-selection with respect to observable H in order to obtain \( {y}_d^{\prime }={\alpha_d{S}^{M\prime }/{\alpha}_0 S=\alpha}_d\left({A}^M+ H\right)/{\alpha}_0\left( A+ H\right) \) where, from Eq. (14), \( \Delta H\equiv {H}^M- H=\left[2\pi \lambda \phi P\left(1- P\right)\right] Vai \). They find that \( {y}_d^{\prime }=5.11=0.64{y}_d \) for the 42 countries. Thus, correcting for selection on observables (i.e., education) reduces migrants’ income by\( \frac{\alpha_d\Delta H}{Y_0}=2.88=0.36{y}_d \) or a reduction in migrants’ relative income of 36%.
US immigrants and home country residents may also differ in terms of non-observable characteristics associated with migrants’ self-selection on ability. CMP correct for migrants’ self-selection on ability, replacing \( {y}_d^{\prime }={\alpha}_d\left({A}^M+ H\right)/{Y}_0 \) by\( {y}_d^{\prime \prime }={\alpha}_d\left( A+ H\right)/{Y}_0 \), where ΔA ≡ A
M − A= \( \frac{\pi}{\phi P\left(1- P\right)} V\left({a}_i\right) \), and\( {\ y}_d^{\prime }-{y}_d^{\prime \prime }={\alpha}_d\Delta A/{Y}_0 \).
Two conditions make it possible to obtain the value of the ability drain, AD; brain drain, BD; and their relative size, AD/BD, from the relationship between \( {y}_d-{y}_d^{\prime } \) and \( {\ y}_d^{\prime }-{y}_d^{\prime \prime } \):
-
(i)
The relationship between AD and BD is identical to that between ΔA and ΔH. From Eq. (14), \( {\Delta A\equiv A}^M- A=\frac{\pi}{\phi \left(1- P\right)} V\left({a}_i\right)+\frac{\pi}{\phi P} V\left({a}_i\right)=\frac{\pi}{\phi P\left(1- P\right)} V\left({a}_i\right) \), so that \( \mathrm{AD}=\frac{\pi}{\phi \left(1- P\right)} V\left({a}_i\right)= P\ast \Delta A \).Footnote 15 Similarly, BD = P*ΔH. Thus, AD/BD = ΔA/ΔH.
-
(ii)
ΔA and ΔH are multiplied by the same parameter, α
d
, to obtain the income changes associated with the vetting system, so that AD/BD can be obtained from the difference between relative incomes \( {\ y}_d^{\prime } \) and \( {y}_d^{\prime \prime } \).
CMP use various methods, based on both macroeconomic and microeconomic evidence, to obtain an estimate of the impact on migrants’ average income of selection on (non-observable) ability, η, in \( {y}_d^{\prime \prime }= \)
\( {y}_d^{\prime }/\eta \), where η ≥ 1. They conclude that the degree of positive selection on unobserved wage determinants results in a bias, η, between 1.0 (no bias) and 1.45 in the case of Peru, i.e., η ϵ [1.0, 1.45]. The average ability drain obtained over these η values is AD = 1.0742BD, as shown below.
CMP obtained the range of η values for workers with 9 years of education and state that selection on ability for less-educated workers is likely to be attenuated by the fact that they tend to work in occupations without plausibly high returns to unobserved skill, a result confirmed by the model.
Recalling that h
i
= 1 represents 20 years of education, it follows that h
i
= 0.45 for 9 years of education. From Eq. (12), we have \( {h}_i=\frac{\alpha_0}{\phi}+2\lambda {a}_i \), or \( {a}_i=\frac{1}{2\lambda \phi}\left(\phi {h}_i-{\alpha}_0\right) \). For individuals with 9 years of education, we have \( {a}_i=\frac{1}{2\lambda \phi}\left(0.45\phi -{\alpha}_0\right) \). Migrants who enter the USA via the H1-B visa program must have at least a bachelor’s degree or a minimum of 16 years of education, i.e., a level of h
i
equal to 0.8 or higher, and the equation for a
i
and the correction for selection on ability must take the difference in education levels into account. Rothwell and Ruiz (2013) report that 90% of US companies’ H-1B applications are for occupations that require high-level STEM (i.e., high-level science, technology, engineering, and math) knowledge. These typically require a graduate degree or equivalent, which takes at least 1 year and often 2 years to complete.Footnote 16 Thus, it seems reasonable to assume that H-1B immigrants average one to two more years of education. Assuming conservatively that they have one more year of education implies that h
i
= 0.85. Then, \( {a}_i=\frac{1}{2\lambda \phi}\left(0.85\phi -{\alpha}_0\right) \), and the correction for selection on ability, η, becomes \( {\eta}^{\prime }=\eta \left(\frac{0.85\phi - {\alpha}_0}{0.45\phi -{\alpha}_0}\right) \). Thus, for these individuals, \( {y}_d^{\prime \prime }={y}_d^{\prime }/{\eta}^{\prime } \) and ability drain’s impact is \( {y}_d^{\prime }-{y}_d^{\prime \prime }={y}_d^{\prime}\left[\left({\eta}^{\prime }-1\right)/{\eta}^{\prime}\right]=0.64{y}_d\left[\left({\eta}^{\prime }-1\right)/{\eta}^{\prime}\right] \).
With the correction for self-selection on education equal to 0.36y
d
, we have
$$ \frac{\mathrm{AD}}{\mathrm{BD}}=\frac{0.64}{0.36}\left[\left({\eta}^{\prime }-1\right)/{\eta}^{\prime}\right]=1.778\left[\left({\eta}^{\prime }-1\right)/{\eta}^{\prime}\right],{\eta}^{\prime }=\eta \left(\frac{0.85\phi -{\alpha}_0}{0.45\phi -{\alpha}_0}\right) $$
(A.1)
Probability p
i
= π(a
i
+ h
i
) ≤ 1, with \( \pi \le \frac{1}{a_i+{h}_i} \), ∀a
i
ϵ [0, a
M
] , h
i
ϵ [0, 1], implying that \( \pi \le \frac{1}{1+{a}_M}. \) Individual education is \( {h}_i=\frac{\alpha_0}{\phi}+2\lambda {a}_i=\frac{\alpha_0+2\pi \left({\alpha}_d-{\alpha}_0\right){a}_i\ }{1-2\pi \left({\alpha}_d-{\alpha}_0\right)}\le 1 \). Define x ≡ α
d
− α
0 > 0, so that \( {h}_i=\frac{\alpha_0+2\pi x{a}_i\ }{1-2\pi x}\le 1 \), or α
0 ≤ 1 − 2πx(1 + a
i
), \( \forall \pi \left(0,\frac{1}{1+{a}_M}\right] \), i.e., \( {\alpha}_0\le 1-2 x\frac{\left(1+{a}_i\right)}{\left(1+{a}_M\right)} \). With h
i
(a
M
) = 1, we have α
0 = 1 − 2x = 1 − 2(α
d
− α
0). Thus, \( {\alpha}_d=\frac{1-{\alpha}_0}{2} \).
I proceed now to “guess” a solution for α
d
/α
0, namely α
d
/α
0 = 2 (this is verified below). Then, from \( {\alpha}_d=\frac{1-{\alpha}_0}{2} \), it follows that α
0 = 0.2 and α
d
= 0.4. Thus, α
d
− α
0 = 0.2 and ϕ = 1 − 2π(α
d
− α
0) = 1 − 0.4π. From \( \lambda \le \frac{1}{2} \) and h
i
(a
M
) = 1, we have π ≤ 0.8 or π ϵ (0, 0.8].
I verify now whether the guess that α
d
/α
0 = 2 is correct. The average ability drain relative to the brain drain, AD/BD, is obtained by averaging the AD/BD values obtained for η ϵ [1, 1.45] and π ϵ (0 , 0.8]. For instance, take π = 0.2 and η = 1.25. Then, ϕ = 1 − 0.2∗0.4 = 0.92. Substituting the values for ϕ and η into Eq.(A.1), the ratio AD/BD = 1.0301, which means that the ability drain is 3% larger than the average brain drain.
The average value is AD/BD = 1.0742. Thus, AD = 1.0742BD = 0.3867y
d
= 3.09, and the impact of selection on unobservable traits is to reduce \( {y}_d^{\prime }=5.11 \) by 3.09, so that \( {y}_d^{\prime \prime }=2.02 \) (or 1% above 2). Thus, developing country natives with a bachelor’s degree (or more) who migrate to the USA would be expected to earn on average about twice the real income they earned in their country of origin.Footnote 17