LABOR ECONOMICS

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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

Author: János Köllő Supervised by: János Köllő

January 2011

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LABOR ECONOMICS

Week 10

Labor demand – Measurement

János Köllő

Cobb–Douglas

The production function is:

impying the first-order condition (MRTS=w/r):

In the FOC, the implicit restrictions of the model become immediately visible:

– The capital-labor ratio is linear function of the ratio of factor prices =1 – The share of labor in total cost is*: sL=

– The conditional elasticities are:

0 ,

1

0 a L

aK Y

r w L K F

F

K L

) 1 (

) 1 ( )

1 (

_L

K r L

w

s

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3 Note that in the empirical literature sL 0.7 and Lw -0.3 are regarded as consensus estimates. These parameters imply

=–(1–0.7)/–0,3=1 so the strong implicit assumptions of the C–D are not entirely groundless. The model is also fairly robust, and customarily used in explorative research.

Cross-section:

Panel with fixed firm effects:

Differentiated model:

Translog

i i

i

Y c u

r

L ln w ln

ln

₁ ₂

it i it it

it

Y c u

r

L ln w ln

ln

₁ ₂

i i i

i

Y u

r

L ln w ln

ln

₁ ₂

Let us write

) (ln )

(

ln C w f w

Taylor expansion at lnw=0 (w=1) yields:

1 1 1

(0) (0)

ln ( ) (ln ) (0) (ln ln1) (ln ln1)(ln ln1) other components

0! 1! 2!

K K K

i ij

i i j

f f

C w f w f w w w

After simplifications (ln(1)=0) and with simplified notation:

0

1 1 1

ln ln 1 ln ln other components

2

K K K

i i ij i j

i i i

C w w w

i i K

i

K

i iY j

i ij K

i i K

i i

Y Y w w w w Y

C ln ln ln ln

2 ln 1

ln ln

1 1

0

Let us write

) (ln )

(

ln C w f w

Taylor expansion at lnw=0 (w=1) yields:

1 1 1

(0) (0)

ln ( ) (ln ) (0) (ln ln1) (ln ln1)(ln ln1) other components

0! 1! 2!

K K K

i ij

i i j

f f

C w f w f w w w

After simplifications (ln(1)=0) and with simplified notation:

0

1 1 1

ln ln 1 ln ln other components

2

K K K

i i ij i j

i i i

C w w w

i i K

i

K

i iY j

i ij K

i i K

i i

Y Y w w w w Y

C ln ln ln ln

2 ln 1

ln ln

1 1

0

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4 We now have a set of equations, where factor shares in total costs are function of output and factor prices.

We have K equations and K+2 parameters, but the model is estimable thanks to the following restrictions:

The system to be estimated consists of K–1 equations since one cost share equation is redundant. Thus, for instance, a model with 1. unskilled labor, 2. skilled labor, 3. capital and 4. raw material can be written as*:

The excluded equation’s parameters are determined by the constraints.

The estimation method should allow for correlation between the residuals (GMM or seemingly unrelated regressions). Controlling for possibly non-neutral differences in productivity (by including Xn) is also recommended.

Repeating:

_i _i

K

i

K

i iY j

i ij K

i i K

i i

Y Y w w w w Y

C ln ln ln ln

2 ln 1

ln ln

1 1

0

Differentiate by lnw

_i

and apply Shepard’s lemma*.

i K

j

iY j ij i

i i

Y w

w s

C ln ln

ln

1

Repeating:

_i _i

K

i

K

i iY j

i ij K

i i K

i i

Y Y w w w w Y

C ln ln ln ln

2 ln 1

ln ln

1 1

Repeating:

0 _i _i

K

i

K

i iY j

i ij K

i i K

i i

Y Y w w w w Y

C ln ln ln ln

2 ln 1

ln ln

1 1

0

Differentiate by lnw

_i

and apply Shepard’s lemma*.

i K

j

iY j ij i

i i

Y w

w s

C ln ln

ln

1

K

i iY K

j ij K

i ij ji

ij

1 1

1

0 ,

4

1 1 1 1 1

1 4

2 2 2 2 1 2

1

3 3 3 3 1 3

1

Unskilled labor : ln ln

Skilled labor : ln ln

Capital : ln ln

n j j Y n n n in

j

n j j Y n n n n

j K

n j j Y n n n n

j

s w Y X u

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5 We are not done yet, for we are interested in the own-price and cross-price elasticities rather than the s/ w-s.

Own-price elasticities: Cross-price elasticities: Allen partial elasticity of substitution

Why?

You can calculate the cross-price elasticities in a similar fashion.

* The index n=1,…,N relates to the observed firms or sectors.

Empirical estimates

Published estimates on compensated own-wage elasticities (from time series, industry, or firm level data) typically fall between –0.15 and –1.0. The consensus is –0.3.

There are fewer estimates of uncompensated own-wage elasticities. The typical result (–

1.0) points to a strong scale effect when compared to compensated own-wage elasticity.

i i i ii i i i i

ii

s s s

x w w

x (

²

) /

_j

i ij i

j j i

ij

s

s x w w

x

j ij

ij

s

i i i ii i

ii i

i i i

i i

i i ii i

i i

i i i

i i

ii i

i i

i i i

i

i i i

s s s s

s

x w w

s C x

w w

C x

w w

s x w

s C w

C w

s x

w s C w w

C s w

w s C w

x

w s C C x

x s w

/ ) (

1

²

2 2

1 2 1

1

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6 Substitutes and complements*:

• Labor, capital, material and energy are typically found to be substitutes.

• Unskilled labor – capital: substitutes and gross substitutes.

• Skilled labor – capital: often found to be gross complements.

• For skilled and unskilled labor: 1.0-3.0.

* See : D. Hamermesh: Labor demand, 1993, 110-111

Empirical C–D and translog estimates from Hungary

Compensated own-wage elasticities from various models

Model Period Type of firm, type of labor Estimates Source Homogeneous labor

Differenced single equation C-D 1992 Large exporting firms -0.80 Kőrösi (2000)

estimated with OLS 1993-95 -0.56

1996-99 -0.23

Differenced single equation C-D 1992-95 Medium and large firms⁵ -0.61 Kőrösi (2002)

estimated with OLS 1996-97 -0.32

Differenced single equation C-D 2000-2001 Firms 5-20 employees -0.39 Kertesi-Köllő (2004)

estimated with IV⁴ Firms 20-50 employees -0.41

Firms 51-300 employess -0.43 Firms over 300 employees -0.04 Heterogeneous labor

Translog cost function, estimated 1996-99 Large firms Halpern et al. (2004)

in repeated cross-sections with Unskilled labor¹ -0.47

SURE⁶ Young-skilled labor² -0.14

Old-skilled labor³ -0.19

1) Lower than secondary education. 2) Secondary or higher education, lower than median experience 3) Secondary or higher education, higher than median experience 4) Exposure to the minimum wage increase used as an instrument 5) From the model with the variables in first differences, p 12. Mean of the annual estimates. 6) Mean of the annual estimates

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7 Translog estimates of cross-price elasticities from a model with 1. unskilled labor 2.

skilled-old labor 3. skilled-young labor 4. capital*

*) Köllő 2002, translog, large firms. Skilled = at least high school attainment. Old: older than 35

Estimates for large Hungarian firms, 1996

Own-price elasticities:

Unskilled labor –0,485

Skilled, old labor –0,175

Skilled, young labor –0,110

Capital –0,894

Cross-price elasticities:

Unskilled – old skilled –0,057

Unskilled–young skilled –0,001

Unskilled– capital 0,543

Old skilled – unskilled –0,098

Old skilled – young skilled –0,054

Old skilled – capital 0,326

Young skilled – unskilled –0,002

Young skilled – old skilled –0,049

Young skilled – capital 0,160

Capital–unskilled 0,582

Capital – old skilled 0,203

Capital – young skilled 0,109

Source: Köllő (2008)