Economics of Migration and Job Search: Problem Set 1
Dr. Suphanit Piyapromdee
January 23, 2023
Consider the Roy model where log earnings for individual i in the immigration and emigration country
are given by Y ji = µ j + b j1 S1i + b j2 S2i , where j = E, I denote the emigration and immigration country
respectively, i is an index for individual i and S1 and S2 are cognitive and manual skills, with prices b j1 , b j2 in
country j. Assume that S1 and S2 are standard normally distributed with zero covariance. Finally, µ j = lnR j
where R j is the rental rate of human capital in country j.
1. Suppose the moving cost k is zero, and the skill prices for cognitive and manual skills are the same,
but differ across country, i.e. bE1 = bE2 = c and bI1 = bI2 = b where E = Mexico and I = the US.
(a) Suppose c > b, what does the Roy model predict about the selection of immigrants from Mexico
to the US? Provide the economic intuition.
Solution: As c > b, σMX2 = 2c2 > σ 2 2
MX,US = 2bc > σUS = 2b . Given that the variance of income
distribution in Mexico is relatively high, in the context of the Roy model, immigrants from Mexico
will be negatively selected. The lower variance of the US income distribution implies that the US
insures low-income workers against poor labor market outcomes while taxing high-income workers.
This generates greater incentives for low-income Mexicans to migrate than high-income Mexicans,
and leads to immigrants being negatively selected from the population.
(b) Assume c > b and the mean income in the US is higher than the mean income in Mexico. If the
mean income in Mexico increases, what would happen to the average quality of Mexican immigrants
in the US and why? Draw a diagram. Hint: YE = µE − bc µI + bc YI . Also, explain what would happen
if the mean income in Mexico increases a lot so that µE > bc µI .
Solution: Recall that we derive in class YE = µE − bc µI + bc YI .1 So we know that the mapping from YI to
YE has a slope greater than one and a negative intercept on the YE axis when µE < bc µI . The increase in
the mean income of home country’s distribution reduces the incentive of marginal migrants to migrate.
Given that immigrants are negatively selected, the marginal immigrants are more productive than the
average immigrants. So only the immigrants from the lower part of the distribution will move from
Mexico to the US; therefore the average quality of immigrants decreases.
1 Start with YE = µE + c(S1 + S2 ) then add the sum of zero to RHS: ± bc µI and rearrange the terms.
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Dr. Suphanit Piyapromdee
January 23, 2023
Consider the Roy model where log earnings for individual i in the immigration and emigration country
are given by Y ji = µ j + b j1 S1i + b j2 S2i , where j = E, I denote the emigration and immigration country
respectively, i is an index for individual i and S1 and S2 are cognitive and manual skills, with prices b j1 , b j2 in
country j. Assume that S1 and S2 are standard normally distributed with zero covariance. Finally, µ j = lnR j
where R j is the rental rate of human capital in country j.
1. Suppose the moving cost k is zero, and the skill prices for cognitive and manual skills are the same,
but differ across country, i.e. bE1 = bE2 = c and bI1 = bI2 = b where E = Mexico and I = the US.
(a) Suppose c > b, what does the Roy model predict about the selection of immigrants from Mexico
to the US? Provide the economic intuition.
Solution: As c > b, σMX2 = 2c2 > σ 2 2
MX,US = 2bc > σUS = 2b . Given that the variance of income
distribution in Mexico is relatively high, in the context of the Roy model, immigrants from Mexico
will be negatively selected. The lower variance of the US income distribution implies that the US
insures low-income workers against poor labor market outcomes while taxing high-income workers.
This generates greater incentives for low-income Mexicans to migrate than high-income Mexicans,
and leads to immigrants being negatively selected from the population.
(b) Assume c > b and the mean income in the US is higher than the mean income in Mexico. If the
mean income in Mexico increases, what would happen to the average quality of Mexican immigrants
in the US and why? Draw a diagram. Hint: YE = µE − bc µI + bc YI . Also, explain what would happen
if the mean income in Mexico increases a lot so that µE > bc µI .
Solution: Recall that we derive in class YE = µE − bc µI + bc YI .1 So we know that the mapping from YI to
YE has a slope greater than one and a negative intercept on the YE axis when µE < bc µI . The increase in
the mean income of home country’s distribution reduces the incentive of marginal migrants to migrate.
Given that immigrants are negatively selected, the marginal immigrants are more productive than the
average immigrants. So only the immigrants from the lower part of the distribution will move from
Mexico to the US; therefore the average quality of immigrants decreases.
1 Start with YE = µE + c(S1 + S2 ) then add the sum of zero to RHS: ± bc µI and rearrange the terms.
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