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Fortin – Econ 561 Lecture 1C

I. Labour Supply 3. Self-Selection: Roy Model and Immigration 1. Overview: What kind of work to do? 2. Basic Roy Model 3. Who Immigrate to Where? a. Decision to Migrate b. Self-selection Properties


3.1 Overview Another important decision that individuals make with respect to their participation in the labour

market is their choice of line of work, occupation/industry, or career. More generally, self-selection models describe how workers choose a career, a location, a marital

status, etc..

First sketched in Roy’s (1951) “Thoughts on the Distribution of Earnings,” which discusses the optimizing choices of ‘workers’ selecting between fishing and hunting.

Today, salient choices might be between white vs. blue collar work, for example


Roy’s key observation is that there are three factors that affect this choice: 1a. Fundamental distribution of skills and abilities b. The correlations among these skills in the population 2. The technologies for applying these skills 3. Consumer tastes that impact demand for different types of outputs It helps understand the nature, determinants, and distribution of economic rents in labour markets

and the distribution of incomes.

At the time of Roy’s writing, the presumption was that the distribution of income that arises from economic processes is arbitrary. Roy’s article explains why this view is incorrect.

A motivation of the Roy model was to show that income inequality is a “natural” consequence of a

world where 1) people have different talents, 2) different jobs/sectors reward these talents differently and 3) people choose jobs to maximize their income.


For the economy as a whole, total output depends on how workers are assigned to jobs.

This occurs because jobs require many different tasks and human performance at those tasks are extremely diverse and because industrial sectors use different technologies that rely on different combinations of human skills.

Earnings are thus worker specific because workers productivity varies from job to job, and some

workers are better at some jobs than others.

The problem is then of how to assign workers to jobs. This assignment problem enters as an intermediate step in the connection between worker characteristics and earnings.


3.2 Basic Roy Model Two occupations in the village

o Fisher o Hunter

There is a competitive market for the price of fishes and rabbits

Let 푝 be the price of fish

푝 be the price of rabbits F number of fish caught (skill at fishing) R number of rabbits caught (skill at hunting)

Wages are thus : 푊 = 푝 F and 푊 = 푝 푅

Each individual chooses the occupation with the highest wage, that is o Choose 푅 if 푝 푅 > 푝 F , and choose F if 푝 F > 푝 푅,


To think of this graphically note that you are just indifferent between hunting and fishing when ln(푝 ) + ln(푅) = ln(푝 ) + ln(퐹)

which can be written as ln(푅) = ln(푝 ) − ln(푝 ) + ln(퐹)

If you are above this line you hunt, if you are below it you fish

Equal Income Line


Some special cases of skills/technology distribution

Case 1: No variance in Rabbits

Suppose everyone catches 푅∗ o If you hunt you receive 푊∗ = 푝 푅∗

o Fish if F > 푊∗/푝 ,

o Hunt if F < 푊∗/푝 ,

The best fishers fish,

All who fish make more than all who hunt


Case 2: Perfect positive correlation

Suppose that ln(푅) = 훼 + 훼 ln(퐹),

with 훼 > 0 var(ln(푅)) = 훼 var(ln(퐹)) Fish if

ln(푊 ) ≥ ln(푊 ) ln(푝 ) + ln(퐹) ≥ ln(푝 ) + ln(푅)

ln(푝 ) + ln(퐹) ≥ ln(푝 ) + 훼 + 훼 ln(퐹) (1 − 훼 ) ln(퐹) ≥ ln(푝 ) + 훼 + ln(푝 )

If 훼 < 1 then left hand side is increasing in ln(퐹) meaning that better fishers are more likely to fish o This also means that the best hunters fish o Absolute Advantage case

If 훼 > 1 pattern reverses itself


Case 3: Perfect negative correlation

Same as before with negative sign on ln(푝 ) (1 − 훼 ) ln(퐹) ≥ ln(푝 ) + 훼 − ln(푝 ) o Best fishers still fish (they are the worst

hunters),

o Best hunters hunt (they are the worst fishers)

o Comparative Advantage case


Case 4: Log Normal Random Variables Let’s assume that the distribution of skills in the village follows a joint log normal distribution

(ln(푅); ln(퐹)) ∼ 푁(흁,횺) where 흁 =휇휇 and 횺 =

휎 휎휎 휎

ln F

ln R

Figure 1


Covariance 휎 is related to the concept of absolute advantage and comparative advantage.

If 휎 = 휌휎 휎 > 0, then people who are good at fishing, tend to be good at hunting too, 휌 the

coefficient of correlation between fishing and hunting will be positive.

Do the best hunters hunt? Do the best fisherman fish?

It turns out that the answer to this question depends on the relative variance of the skills o Whichever occupation (휎 ≶ 휎 ) happens to have the largest variance in logs will tend to have

more sorting.


Will the earnings of fishers (or hunters) be higher or lower than they would be if the assignment to

occupations were random?

The question is: on average do people choose to fish (퐷 = 1) earn more than the average fisher (without self-selection)?

E[ln(푊 ) |퐷 = 1] > E(ln(푊 ))

E[ln(푊 ) |ln(푊 ) ≥ ln(푊 )] > E(ln(푊 ))

퐸[ln(푝 ) + ln(퐹)|ln(푝 ) + ln(퐹) > ln(푝 ) + ln(푅)]>ln(푝 ) + 휇 (1)

Notice that this is the difference between the conditional and unconditional mean which is called the selection bias.


To simplify (1), let first consider the probability of choosing to fish: Pr(퐷 = 1)

For 푗 ∈ {푅,퐹}, let the mean u = ln 푝 + 휇 include the price of the catch and let ε = ln(퐽 ) − 휇 , be the catch in deviation from the mean Pr(퐷 = 1) = Pr(ln(푝 ) + ln(퐹) > ln(푝 ) + ln(푅)) = Pr (푢 + ε > 푢 + ε ) = Pr (푢 − 푢 > −(ε − ε )) = Pr > − where σ is the standard deviation of ξ = ε − ε Note that the variance of ξ E((ε − ε ) ) = E ε − E ε − 2E(ε ε ) = 휎 + 휎 − 2휎 = 휎 + 휎 − 2휌휎 휎


E(ln(푊 ) |퐷 = 1) = 푢 + 퐸(ε |퐷 = 1)

= 푢 + 퐸 ε ≥

= 푢 + 휎 퐸 ≥

Using the fact that if (휀 , 휀 ) are two normal random variables, we can write 휀 = 훽 + 훽 휀 + 휈 as a regression with 휐 ⊥ 휀 and E(휈) = 0, where 훽 = ( , )

( )

Here, we have ( , )

( )=

[ ( )]

/=


Then 퐸 ≥ =,

퐸 ≥

=

휎 − 휎σ 휎

1 퐸 휉 휉 ≥ 푢 − 푢

σ

= (휎 − 휌σ σ )퐸 휉 휉 ≥

So E(ln(푊 ) |퐷 = 1) = 푢 + (휎 − 휌σ σ )퐸 휉 휉 ≥

= 푢 + ( − 휌)퐸 휉 휉 ≥

where 퐸 휉 휉 ≥ = 휆 the inverse Mills ratio!

Thus if σ > σ , i.e. fish is more difficult to catch than rabbit − 휌 > 0, and E(ln(푊 ) |퐷 = 1) > 퐸(ln(푊 ))


The average catch for those who choose to become fishers is greater than the average catch for the whole population

A skilled occupation will yield greater earnings for those who have the skills to do it.

How does change in price affect average ability levels?


Absolute Advantage (best fishers are best hunters)

F

H

H=(PF /PH)F

Skills (H, F) on this line

Here, decrease in P H or increase in P F induces higher ability hunter to enter the fishing occupation, which raises skill level in both

Become fishers


Comparative Advantage Case (best fishers are worst hunters)

F

H

H=(PF/PH)F

Skills (H,F) on this line

Here, decrease in P H or increase in P F induces lower ability hunter to enter thefishing occupation, which raises avg skill level in hunting, but lowers it in fishing.


Summary

Economic rents enter in that model both in the sense of relative talents (comparative advantage) and in the sense of absolute talents (absolute advantage). Workers locate closest to the log price line receive the least surplus for their talents while workers

furthest for the line receive the largest surplus. If 0 , this means that a person whose earnings prospects are larger in job R is likely to have better

than average prospects in job F. Better rabbit hunters are also better trout fishers. In Figure 1, people with the highest earnings prospects in both jobs tend to do job F; those with

lesser overall prospects tend to work in job R. Selection would be hierarchical.

The sign of the correlation coefficient is crucial to whether or not selection tends to be hierarchical.

When 0 , a person who is good at one job is likely below average in the other. The ellipse in Figure 1, would be negatively sloped. Selection is not hierarchical.


The assignment of workers with different earnings prospects also depends on the variance of earnings

in each sector ( 21 and 2

2 ).

Sattinger (1993) also shows that the unequal variances between the sectors will play as least as strong a role as the correlation between sector performances in shaping the distribution of income. Increasing a sector’s variance tends to increase positive selection into that sector. Better example is the immigration example below.

In addition, a shift in demand (e.g. an increase in the price of trout in terms of rabbits) can influence

the distribution of earnings by increasing the relative earnings of trout fishers and leading some workers to move from catching rabbits to fishing for trout.


3.3. Who decides to immigrate, and how do they decide where to go?

a. The Decision to Migrate Who chooses to immigrate to the United States? Who chooses to immigrate to Canada? Why does Canada attract less immigrants from Europe and more from Asia than it did in the 1960s?

One ready-made answer to the first question is that workers from low wage countries will immigrate to

the U.S. This may be true on average, but it’s probably too simple. o The workers immigrating to the United States are probably not a random subset of the Mexican

workforce. Rather, we should expect that potential migrants make some rough comparison of their wages in the

home country and their expected wages in the U.S. On average, we should expect those who immigrate to have higher expected earnings in the U.S. than Mexico and vice versa for those who stay.


Borjas (1987) applies the Roy model to the decision to migrate.

He analyses the way in which the earnings of the immigrant population may be expected to differ from

the earnings of the native population because of the endogeneity of the decision to migrate. Borjas revisits the assumption of first generation studies that

1) the age-earnings profile of immigrants is steeper than the age-earnings profile of native population with the same measured skills

2) the age-earnings profile of immigrants crosses the age-earnings profile of natives about 10 to 15 years after immigration, a crossing point called the over-taking age.

He raises two issues concerning these findings

1) the analysis of a single cross-section cannot separate the aging effect (and assimilation effect) from the cohort effect

2) changes in cohort quality arise from secular shifts in the skill mix of immigrants and the nonrandom return migration propensities


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20 25 30 35 40 45 50 55 60 65

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40 6020

1960 Wave

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P*

Q*

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Early studies (Chiswick, 1978) using single cross-sectional analysis had misleadingly argued that immigrants’s earnings overtook those of natives 10 to 15 years after arrival


Borjas tries to link the issue of changes in cohort quality with the immigrant self-selection.

Are immigrants selected from the upper or lower tail of ability (or income) distribution of the sending countries?

Consider two countries 0 and 1, where 0 denote the home country (e.g. Mexico) and 1 the host country (e.g. United States).

Log earnings in the source country are given by

푤 = 휇 + 휀 where 휀 ~ N(0, 휎 ). It’s useful to think of 휀 as the de-meaned value of worker’s ‘skill’ in Mexico

(the source country).


If everyone from Mexico were to migrate to the U.S. (host country), their earnings would be (ignoring any general equilibrium effects!):

푤 = 휇 + 휀 where 휀 ~ N(0, 휎 ).

Assume that the cost of migrating is C, which Borjas puts into ‘time equivalent’ terms as π = C/푤 . Borjas further assumes that π is constant, meaning that C is directly proportional to 푤 .

Assume further that each worker knows C, 휇 , 휇 and his individual epsilons: 휀 , 휀 .

You, the econometrician, only observe a worker in one country or the other, and hence you only know 휀 or 휀 for any individual.

What can you infer about what wages for immigrants in the United States would have been had they stayed in their source countries? What would wages in the United States be for non-migrants had they come to the United States?

The Roy Model answers these questions.


The correlation between source and host country earnings is 휌 =

휎휎 휎

where 휎 is the covariance 푐표푣(휀 , 휀 ).

To implement this model, we need to know ρ, although we do not need to know both 휀 , 휀 for any worker.

A worker will choose to migrate if 휇 − 휇 –휋 + (휀 − 휀 ) > 0 (1)

Borjas defines the indicator variable I, equal to 1 if this selection condition is satisfied, 0 otherwise).

Now, let 휈 = 휀 − 휀 . The probability that a randomly chosen worker from the source country chooses

to migrate is equal to

Pr[I = 1] = Pr[휈 > −(휇 − 휇 –휋)] = Pr휈휎

>휇 − 휇 + 휋

휎


= 1 −Φ휇 − 휇 + 휋

휎= 1 −Φ(z)

where 푧 = the “normalized” difference in average earnings minus migration cost, and Φ(∙) is the CDF of the standard normal.

Notice that he higher is z, the lower is the probability of migration (from Mexico to the U.S.). This is

because z is rising in the mean earnings of Mexico and the cost of migration. Also ∂P/∂휇 < 0, ∂P/∂휇 > 0, ∂P/∂π < 0.

But these are mean effects. In our example, high mean wages in the U.S. relative to those in Mexico,

create a net incentive for workers to migrate from Mexico to the U.S. Variation in the ”other” costs of migration also create such incentives.

It’s useful to assume from here forward that 휇 ≈ 휇 , so we can focus on self-selection rather than mean differences.


b. Self-selection Properties Let’s begin by finding the expected earnings of workers in each case

From what we have seen previously, the expected earnings of workers who choose to immigrate to the

host country is

E[w |I = 1] = 휇 + 휎 휌φ(z)Φ(−z)

= 휇 +휎 휎휎

휎휎− 휌 휆(푧)

Similarly, we would find that the expected earnings of the immigrants in the home country are

E[w |I = 1] = 휇 + 휎 휌φ(z)Φ(−z)

= 휇 +휎 휎휎

휌 −휎휎

휆(푧)

Define 푄 ≡ E[w |I = 1] − E[w ] = E[ε |I = 1] and

푄 ≡ E[w |I = 1] − E[w ] = E[ε |I = 1] where 푄 and 푄 are the truncated means of the unobserved components of earnings in Mexico and earnings in the U.S., given migration from Mexico to the U.S.


Whether we will observe positive or negative selection will depend on the signs of 푄 , the income differential between the average immigrant and the average person in the home country and 푄 the income differential between the average immigrant and the average person in the host country.

Now we could get four different cases easily.

Case 1: 푄 > 0, 푄 > 0 Case 2: 푄 < 0, 푄 < 0 Case 3: 푄 < 0, 푄 > 0 Case 4: 푄 > 0, 푄 < 0

But the fourth case, 푄 > 0, 푄 < 0, would suggest irrational migration, where people leave the upper tail of the source country income distribution to join the lower tail of the host country distribution. This is inconsistent with income maximization.

So, there are really are 3 cases:

1) Positive selection or positive hierarchical sorting 푄 > 0, 푄 > 0 the earnings of immigrants are larger than average in both countries and 휎 > 휎 , the earnings dispersion is larger in the host country and 휌 > the skill correlation is sufficiently high.


Because the skills valued in the host and home country is sufficiently high. If you were a skilled

worker in the home country, you would not want to migrate to a host country with a very high return to skills if the skills valued in the host country were uncorrelated (or negatively correlated) with skills value in the home country.

2) Negative selection or negative hierarchical sorting 푄 < 0, 푄 < 0 the earnings of immigrants are smaller than average in both countries and 휎 < 휎 , 휌 > , but the earnings dispersion is larger in the home country, so that when they migrate immigrants end up with higher earnings than in their home country.

Here, the home country is unattractive to low earnings workers because of high wage dispersion. Again assuming that wages are sufficiently correlated between the home and host country, low skill workers will want to migrate.


3) Refugee sorting: 푄 < 0, 푄 > 0 the immigrants does worse than average in their home country, but better than average in the host country, this happens when the correlation is very low 휌 < 푚푖푛 , .

This might occur, for example, for a minority group whose opportunities in the home country are depressed by prejudice. Or for the case of migration from a non-market economy where the set of skills rewarded is quite different from the economy in the receiving country (e.g., European Jews in the first case, intellectuals from the former Communist block in the second).

How relevant is the Borjas/Roy selection model to the problem he studies in the 1987 paper, self-selection of immigrants? The evidence is not overwhelming.

There are probably more relevant applications of this model. The growing focus of empirical economists on applying instrumental variables and other methods

for causal estimation is in large part a response to the realization that self-selection (i.e., optimizing behavior) plagues interpretation of ecological relationships.


Basic readings: Roy, A. “Some Thoughts on the Distribution of Earnings,” Oxford Economic Papers, Vol 3. (June 1951): 135-

146.

Borjas, G.J. “Self-Selection and the Earnings of Immigrants,” American Economic Review, Vol. 77 (Sep. 1987) 531-553.

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