LABOR ECONOMICS

LABOR ECONOMICS

LABOR ECONOMICS

Sponsored by a Grant TÁMOP-4.1.2-08/2/A/KMR-2009-0041 Course Material Developed by Department of Economics,

Faculty of Social Sciences, Eötvös Loránd University Budapest (ELTE) Department of Economics, Eötvös Loránd University Budapest

Institute of Economics, Hungarian Academy of Sciences Balassi Kiadó, Budapest

LABOR ECONOMICS

Author: János Köllő

Supervised by: János Köllő January 2011

ELTE Faculty of Social Sciences, Department of Economics

LABOR ECONOMICS

Week 5

Supply of skills – Topics

János Köllő

1. The Mincer equation

2. The Roy model

The Mincer equation

• As was discussed in Week 4, the supply of skills depends on the internal rate of return to education (r) which, in turn, depends on the supply of skills through its impact on skills-specific wage

differentials. At any given point in time, r has a decisive role in shaping educational decisions.

• The benchmark model of estimating r (under some simplifying

assumptions) is known as the Mincer equation developed by Jacob Mincer (1922–2006).

• There are two types of Mincer equations. Here we deal with the one known as the ‘accounting identity model’.

• The Mincer equation is one of the most important tools for labor market analysis. For a laudation and critique see: James J.

Heckman & Lance J. Lochner & Petra E. Todd, 2003. "Fifty Years of Mincer Earnings Regressions," NBER Working Papers 9732

Assumptions:

• Education incurs no direct costs.

• There is a unique rate of return (r) to all kinds of education and training.

Constraints:

• The accumulation of work-related experience is

difficult to observe. We can only observe time since leaving school.

• We evidently observe only the returns to general training (since firm-specific training has no effect on wages in the benchmark case)*.

*) This will be discussed in Week 8 Labor demand – Basics 2

0 0

0

1

Y r Y ( 1 r ) Y

Y

The cost of school based education in year t is Y

, the wage paid to workers with t–1 years of education

The rate of return to this investment is r. The

potential wage (marginal product) after completing the first school-year* therefore can be written as:

Step 1: School based education

*) Note that we deal with education after the completion of compulsory school-years

Foregone earnings in the second year amount to Y

and the return is rY

. Therefore, potential

earnings in years 1,2,…S are:

0 0

0

1

Y r Y ( 1 r ) Y

Y

0 2 1

1

2

Y r Y ( 1 r ) Y

Y

...

0 1

1

r Y ( 1 r ) Y Y

Y

S r S

S

r Y Y e

Y ( 1 )

Taking into consideration that (1+r)

e

if r is small enough:

(Why? Because in general x ln(1+x) if x is a small number)

rS Y

Y

ln

ln

Potential log wage after completing S years in

school:

S S

S S

S

S

Y r h Y r h Y

Y

( 1

)

After S years in school, the graduates go to work. They actually work less than their contracted working time as they spend fraction h_t of their time by accumulating work- related experience.

Their investment cost is: h_tY_t–1.

The rate of return to their investment is: r.

Potential wage in the first year after leaving school is:

Step 2: On the job training

Potential earnings in years 1, 2,…, x after leaving school are:

S S

S S

S

S

Y r h Y r h Y

Y

( 1

)

S S

S

S

r h r h Y

Y

( 1

)( 1

)

S S

x S x

S

r h r h Y

Y ( 1 )....( 1

)

…

S S

x S x

S

r h r h Y

Y ( 1 )....( 1

)

Taking logs and considering that ln(1+rh) rh we have:

x S

S t

t S

x s s

s S

x

S

Y rh rh rh Y r h

Y

1 2

1

... ln

ln ln

For year S+x we have a product of x+1

components:

In full detail:

x S

S t

t x

S

S t

t S

x

S

Y r h Y rS r h

Y

1 0

1

ln ln

ln

Simplifying the notation (by counting years from the date of school leaving) and turning to continuous

time yields:

x

t

t

x

Y r S r h dt

Y

1

ln

ln

T h h

T x x h

h dt

T t h h

dt h

x

t x

t

t

( )

2

2

1

0 0

1

We do not know. But we can assume that is starts at some value h₀>0 and falls to zero by the end on one’s career. Assuming linearity and a T years long career, we have:

T t h h

h

0

2 t

T t h

h

Substitution yields:

Step 3: How large is h _t ?

Substituting our approximation for the value of the integral yields:

Pro memo:

x

t

t

x

Y r S r h dt

Y

1

ln

ln

0 2 0

0 0

0

2 2

ln

ln x

T x rh

rh T rS

rh rh Y

Y

Actual wage is potential wage minus investment cost so we have:

0 2 0

0 0

0 0

0

] [ ] 2

[ln 2

ln x

T x rh

T rh h

S T r

h rh rh

w w

x x

x

h Y

w ( 1 )

ln w

ln( 1 h

) ln Y

h

ln Y

Substituting w for Y and considering that w₀=Y₀ and h_x=h₀–(h₀ /T)x, the equation for the log wage is:

This is an estimable equation with observable explanatory variables. For a pool of individuals (i=1,2,…,N) we can

estimate the regression:

2 3 2

1

-

ln w

= c + S

+ x

x

Step 4: Potential versus actual earnings

The Roy model

• Why some people study longer than others?

• Our answer to the question, so far,

referred to differences in direct costs, discount rates and efficiency in learning.

• At one point we briefly touched upon the question of individual efficiency differences in production and (therefore) expected

wages.

• Pro memo: students, who did not want to continue their studies in college thought that they could achieve

relatively high earnings with a matura exam so their

individual gains from having a college diploma would be relatively low.

• This, taking into account the costs of further studies, made them profitable not to go to college …

• … even though they knew that the average college graduate earned more than the average secondary school graduate.

• The Roy model is concerned with these types of (typically unobserved) individual differences and

‘selection on unobservables’.

• Andrew Roy’s (1951) seminal paper (that was engaged with the question of why the wage

distribution is skewed on the left even more than the lognormal) is mostly verbal, quite difficult to read and was nearly forgotten for decades.

• Its rebirth started in the 1980s, when selection on unobservables became a central theme in

empirical social science. Heckman called it ‘one of the most important models in economics’.

• Most studies use a reformulation of the Roy model by George Borjas (1987). We, too.

George J. Borjas (1987): Self-selection and the earnings of immigrants, American Economic Review, Vol.

77, No.4, 531-553

• Roy: People have different abilities for hunting and fishing. How it affects specialization, productivity and the resulting income distribution?

• Borjas: People’s potential relative earnings differ by countries. Even the same person may expect

different relative earnings in countries A and B. How migration decisions are affected by the underlying

(mostly unobserved) productivity differences and how the composition of migrants is affected?

• The Borjas model can be applied to a wide range of decisions to specialize such as going/not going to college, choosing occupation, working in the private versus the public sector, and so on.

The frame of the Borjas model

• Let us have two worlds (country, occupation, level of education): 0 and 1

• The wages of the same person in the two worlds can be written as:*

The cost of mobility from 0 to 1, expressed in wage units: ⁱ = C/w₀ⁱ The person will move if:

An important detail: correlation between wages in 0 and 1:

1 0

01 1

0

1 0 ) cov(

0 )

( )

( _{1 0} ⁱ ₁ⁱ ₀ⁱ

*) The -s stand for mean earnings while the -s denote country-specific individual residuals

) ,

0

∼ ( ),

, 0

∼ ( ₀² _{1 1}²

0 1

1 1

0 0

0 w N N

wⁱ = + ⁱ and ⁱ = + ⁱ

A footnote: abilities versus

occupation-specific (country-specific, etc.) productivities

• Occupations rely on different mixtures of

innate abilities. They may require high IQ and less talent in manual tasks, and vice versa.

• Proposition: occupation-specific relative

productivities will be correlated even if innate

abilities are independent of each other.

Let us assume two occupations and two types of innate abilities ( ), which affect productivity (x) with different weights

) )

( )

( /

) _{1 2 1 2}

2 1 2

2 1

1 0

2 2 1

1 0

E(

= E(

= )

N(0,0,1,1,

~ ) , (

+ +

=

+ +

=

2 1 2i 1i

Var Var

b b

b x

a a

a x

i i

i i

The covariance of the x–s:

12 1 2 2

1 2

2 1

0 2

2 1

1 0 0 2

2 1

1

0 )( ) ( )

(a + a ⁱ+ a ⁱ a b +b ⁱ +b ⁱ b = a b + a b + a b + a b

= ₁

12x

The x–s will be correlated if there is at least one kind of ability affecting productivity in both occupations (a₁b₁>0 or a₂b₂ >0).

This remains true even if ₁₂ ^{= 0}

Back to the Roy model: the decision to move

N(0,1)

~ /

) N(0,

~

Define the random variable = ₁– ₀:

The probability of moving from 0 to 1 (if / is standard normal):

) ( 1

1 Pr

Pr _{0 1} ^{0 1 0 1} z

P

v v

The probability of moving depends on average wages, the costs of mobility and individual residual wages (marginal products) through .

Expected wages in 0 and 1

ratio) Mills

invers (

)]

( - 1 /[

) ( /

) ( where

- )

move

- )

move

0 1 1

0 1

1

1 0 1

0 0

0

z z

P z E(w E(w

See the next two slides for the proof!

Expected wages can be lower or higher than the -s depending on and the -s, that is, the variance of residual wages and their correlation.

Roy model: deriving the formulas on observed wages

Our starting point is an equation on the expected earnings of movers in 0:

z E

z E

w E

0 0 0 0 0

0 0mover) (

) 1 (

We shall use the following regression capturing the link between the conditional expectation of 0 and :

2 0 0 2 1

0 1

0 1 0 0

1

0 ( )

)]

[var(

) ,

) cov(

( ) ( ) 2

( E

Substituting (2) to the second term of (1) we have:

0 0

0 0 2 0 0

) 0

3

( E z

Using (3), equation (1) can be written as:

) ) (

( 1

) (

) mover (

) 4 (

0 0 0 0

0 0

0 0 0 0

0 0 0

0 0 0 0

z z z

z E

z E

z E

w E

The direction of selection bias is measured by the sigh of the parameter for the inverse Mills ratio ( ).

For making inspection easier we can write equation (4) in the form used in (5a) and (5b):

) ( )

( )

átlépő (

) 5 (

1 0 1

0 0 0

0 0

0 z z

w E a

) ( )

( )

átlépő (

) 5 (

0 1 1 0 0 0

0 1

1 z z

w E b

Why? Let us move backwards from (5a) to (4) and consider that:

2 0 01 0 0 1

0) [( ) ]

,

cov( E

) ( )

( )

( )

(

) ( )

( )

( )

mover (

0 0 0 0

0 0 0 0

0 1

0 2 0 01 1 0 0

1 0

0 0 1 0

01 1 0 0 1

0 1 0

01 1 0 0 1

0 1

0 0 0

z z

z z

z z

z w

E



Roy model: deriving the formulas on observed wages

Our starting point is an equation on the expected earnings of movers in 0:

z E

z E

w E

0 0 0 0 0

0 0mover) (

) 1 (

We shall use the following regression capturing the link between the conditional expectation of ₀ and :

2 0 0 2 1

0 1

0 1 0 0

1

0 ( )

)]

[var(

) ,

) cov(

( ) ( ) 2

( E

Substituting (2) to the second term of (1) we have:

0 0

0 0

2 0 0

) 0

3

( E z

Using (3), equation (1) can be written as:

) ) (

( 1

) (

) mover (

) 4 (

0 0 0 0

0 0

0 0 0 0

0 0 0

0 0 0 0

z z z

z E

z E

z E

w E

The direction of selection bias is measured by the sigh of the parameter for the inverse Mills ratio ( ).

For making inspection easier we can write equation (4) in the form used in (5a) and (5b):

) ( )

( )

átlépő (

) 5 (

1 0 1

0 0 0

0 0

0 z z

w E a

) ( )

( )

átlépő (

) 5 (

0 1 1 0 0 0

0 1

1 z z

w E b

Why? Let us move backwards from (5a) to (4) and consider that:

2 0 01 0 0 1

0) [( ) ]

,

cov( E

) ( )

( )

( )

(

) ( )

( )

( )

mover (

0 0 0 0

0 0 0 0

0 1

0 2 0 01 1 0 0

1 0

0 0 1 0

01 1 0 0 1

0 1 0

01 1 0 0 1

0 1

0 0 0

z z

z z

z z

z w

E



The quality of movers

• Let us measure the quality of movers (Q) with the excess of their earnings [E(w|mover)] above expected earnings ( )

• Let us examine how self-selection affects the quality of movers at the source and the destination. Three arrangements are

possible 

• Let us examine these arrangements at the equality of expected wages ( ₁ = ₀)!

0 1 0

1 1

1 1

1 0 0

1 0

0 0

- átlépő

| (

- ) átlépő

| (

) w

E Q

w E Q

Positive hierarchical sorting

• Movers are selected from the high performers and they perform better than average at the destination, too: Q₀>0 and Q₁>0.

• True if > ₀/ ₁ és ₁/ ₀>1. ( ₀/ ₁<1 would imply > ₀/ ₁ and >1 that is impossible in the case of a correlation coefficient)

• What does it mean? If the range of residual earnings is wider in 1 and the two worlds reward the same skills ( is sufficiently high)  brain drain.

• Low ₀ punishes potential high performers and insures

less productive workers  mobility will improve the quality of the labor force at the destination.

Negative hierarchical sorting

• Movers are selected from the low performers and they perform worse than average in 1, too: Q₀<0 and Q₁<0.

• True if > ₁/ ₀ és ₀/ ₁>1.

• What does it mean? If the range of residual earnings is wider in 0 and the two worlds reward the same skills ( is sufficiently high)  the less skilled and the less able will move.

• Why? Low ₁ „insures” less productive workers  mobility will deteriorate the quality of the labor force at the

destination.

„Refugee sorting”

• Movers are selected from among the low-wage earners but they perform better than average in 1:

Q₀<0 and Q₁>0.

• True if <min( ₁/ ₀ , ₀/ ₁).

• What does it mean? is low, 0 and 1 rewards

different skills  movers are selected from among people paid under their productivity in 0.

• Examples: East European scientists under socialism.

Members of discriminated minorities if there is less (or no) discrimination in 1.

The non-existing fourth case

• Movers are selected from people with

high residual earnings in 0, who perform below average in 1: Q

>0 and Q

<0.

• Would be true if >max(

/

,

/

),

which implies >1 – impossible in the

case of a correlation coefficient.

Empirical applications

• Migration (Borjas 1987, Dahl 2002, Brücker and Trübswetter 2004)

• Higher education (Rosen and Willis 1979)

• Choice of occupation (Hoxby and Leigh 2003)

• Public versus private sector (Borjas

2002)