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 9

Labor demand – Topics

János Köllő

• Two factors: capital and labor

• More than two factors

Appendix 1: Demand for labor in the short run Appendix 2: Scale effect with homogeneous

production function

Demand for labor – a formal model

• The slides draw from P. Cahuc–A.

Zylberberg „Labor Economics” (MIT Press 2004), pp. 171–193

• We restrict ourselves to the case of homogeneous production functions.

Results for the general case are mentioned without proof.

• We study the two factors case in detail.

For the single factor case see Appendix 1.

The multifactor case is discussed briefly.

Two factors: capital and labor

Preparations: the production function

Bivariate, homogeneous of degree :

Returns to scale diminish if <1, constant if =1, increasing if >1.

Returns to factors of production are diminishing:

F

>0, F

>0, F

<0, F

<0 ) , ( ,

0 )

, ( )

,

( K L F K L K L

F

Preparations: cost function

The minimum cost at given output Y is written as K

r L w

The cost function describes minimum cost as a function of w, r and Y:

) , , ( w r Y C

C

a) C is homogeneous of degree 1 in prices.

b) C is concave: C_ww<0, C_rr<0.

c) Satisfies Shepard’s lemma, i.e. optimal factor demands are given by the partial derivatives of the cost function:

) , , ( )

, ,

( w r Y és K C w r Y

C

L

Perfect competition is an extreme case. Market power is measured with the non-positive elasticity of sales price wrt sales [drawn from the inverse

demand function P=P(Y)]

Y Y P

Y

p P

Y ( )

) (

Under perfect competition the firm is price-taker

Under imperfect competition the price varies with Y

p 0

Y p 0

Y

Market power is measured with defined as:

1 1

1

p Y

Under perfect competition the firm has no market power Under imperfect competition the firm has market power

1 1

Preparations: market power

Preparations: the profit function

When prices change, the optimal size of the firm changes, too.

The profit function is written as:

) , , ( )

( )

, ,

( w r Y P Y Y C w r Y

The FOC from differentiation by Y is given by:

formula above

the get to y

immediatel you

is function demand

inverse the

of elasticity output

that the recalling

and P(Y) out

g multiplyin By

0 ) , , ( )

( )

( )

, , (

) P'(Y)Y/P(Y η

Y r w C Y

P Y Y P Y

r w

p Y

Y Y

) 1

/(

1 ahol

) , , ( )

(

Y C

w r Y

P

Note that in the optimum : price=marginal cost*markup

After these preparation we look at the following questions

Conditional demand

(Optimal choice at given level of output in response to changing factor prices)

• Conditional demand for labor and capital

• Cross effects

Scale effects

(Optimal output in response to changing factor prices)

Unconditional demand

• The substitution and scale effects together

Conditional (compensated) demand for labor diminishes with the wage and increases with the user cost of capital

0 )

, ,

( _ww

w

C

w Y L

r w C

L

Similarly, the conditional demand for capital decreases with the price of capital and increases with the wage.

From Shepard’s lemma:

wr 0

r C

L

Since compensated demand depends only on relative factor prices, the demand for labor increases if the user cost of

capital goes up:

Cross effects

Cross elasticity: positive, non-symmetric

K w L

r K

w L

r

w

K K

w r

L L

r

Elasticity of substitution: positive and symmetric*

) 0 / (

) / (

r w

L K L

K r w

*) This is not necessarily true for more than two factors of production. See later.

Cross effects derived from the cost function

The elasticity of substitution can be derived from the cost function (Uzawa 1962, Cahuc-Zylberberg 2004, 237–238)

r w

wr

C C

C C

r w

L K L

K r w

) / (

) / (

r w wr wr

wr K

Y wr Y Y

Y L

r

C C

C C K

L C C K

r C L C r s

L C r L

r r

L r

L

1

: that see to rward straightfo

is it formula above

the g Considerin

log log

L r

0 0

s_K is the share of capital in total cost (s_K = rK/C = 1–s_L = 1–wL/C)

It follows that

The compensated price effects thus can be written as:

0 )

1

( _L

L r L

w s

0 )

1

( _L

L

r s

r  L w  L

similarly proceeds

of derivation the

symmetry, to

Thanks

) 1

( that

follows y

immediatel It

and :

lemma s

Shepard' to

According

/ 1

as capital of

share the

Define

) /

( :

yields on

Substituti

is that seen that

have We

L w

L K

L r

r w

L K

r w L

r

r w wr

r w

wr wr

L r

s C s

L K L r

K C

L C

C K r s

s

C L C rC

C C C C

C C

C and C

L C r r

L L

r

The effect of an exogeneous change of output

Under homogeneous production function

/ 1 /

1 ( , ) ( ,1)

) 1 , (

) , /

(

w r Y L w r Y és K w r Y K w r Y

L

If the production function is homogeneous, a rise in Y (without a change in relative factor prices) increases the demand for both capital and labor. From homogeneity of degree and Shepard’s lemma it follows that:

Remark: in the general case the demand for at least

one factor will rise.

Unconditional demand

When factor prices change, optimal output (Y^*) will change, too.

The firm’s problem is to solve:

) , , ( max )

,

(w r w r Y

Y

How Y^*, C and (as a consequence) L^* will change in response to a change in w?

) , , ( )]

, , ( )

1 )(

( [ ) ,

( C w r Y

w Y Y

r w C Y

P r

w _Y^p _{Y w}

w

Under optimality the term in brackets is zero (see Profit function). On the other hand, Shepard’s lemma states that C_w(w,r,Y^*)= L*. So we arrive at Hotelling’s lemma:

K r

w és

L r

w

w( , ) ( , )

Unconditional demands are decreasing in own prices*:

0

0 _rr

ww r

and K w

L

*) From the concavity of the cost function it follows that the profit function is convex, so the second derivatives are positive

Unconditional demand

– The effects of a change in the wage

The wage has a direct and an indirect effect:

w C Y

w C L

wY ww

Multiply throughout with w/L^*, and the second term with Y^*/Y^*

Y w wY

ww L

w wY

ww

L

C C Y

L w Y

Y w C Y

L C w

L w L

w w L

What is this? The first term is the conditional demand elasticity at output level Y=Y*

L w

ww

w w

L L w

L L

C w L

w

/ /

Unconditional demand

– The effects of a change in the wage

The wage has a direct and an indirect effect:

w C Y

w C L

wY ww

Multiply with w/L^*, and the second term with Y^*/Y^*

Y w wY

ww L

w wY

ww

L

C C Y

L w Y

Y w C Y

L C w

L w L

w w L

What is this? The first component of the second term is the output elasticity of demand at output level Y=Y* holding relative factor prices constant:

L Y wY

Y Y

L L Y

L L

Y L

C Y

/ /

Unconditional demand

– The effects of a change in the wage

The wage has a direct and an indirect effect:

w C Y

w C L

wY ww

Multiply with w/L

Y w wY

ww L

w wY

ww

L

C C Y

L w Y

Y w C Y

L C w

L w L

w w L

From step 1 and step 2 we finally have:

Y w L

Y L

w L

w

The total effect of a change in the wage is thus

Y w L

Y L

w L

w

Own-wage elasticity

Compensated elasticity of substitution

(–)

Scale effect (–)

The negativity of the scale effect is easy to prove if the production function is homogeneous. See two slides later!

Remark: in the general case it can be proven that the two components of the second term are differently signed (Cahuc–Zylberberg p. 184. and footnote 5 to Chapter 4)

Employment effect of a change in the user cost of capital

After similar steps we have:

Y r L Y L

r L

r

Cross price elasticity

(?)

Compensated elasticity of substitution

(+)

Scale effect (–)

s substitute gross

are L

and K

0

s complement gross

are L

and K

0

L r

L r

Own-wage elasticity under homogeneous production function

For the derivation see Appendix 2

θ Y d

L Y d

θ )

(w,r, L

(w,r,Y) L

Y r w L Y

r w

L 1

ln ln ln

1 1 ln

ln )

1 , , ( )

, ,

( ¹

) 1 (

: yields on

Substituti .

: lemma s

Shepard' .

/

. ) / (

seen that have

We

L L

r r

w L

r w L

r r

w wr wr

L r

s K

C és L C C

L w s

C L C C rC

C C and C

L C r r L L (a) r

(b)

(c)

Y w L

Y L

w L

w

(a)

(b)

(c)

0 )

1

( - <

-

= _L s

L

w s

q

1 _w ^Y s

L L

w L

L L

w

( 1 s ) s or s

In these formulas it is easy to observe that*:

a) The demand for labor decreases with the wage.

b) The substitution and scale effects are additive.

c) Demand is more elastic if capital and labor are ‘easy to substitute’ ( is large) HM–2.

d) The stronger is market power, the weaker the scale effect. If competition is strong ( ) the scale effect is large and the demand for labor is highly elastic HM–1.

e) The elasticity of demand for labor increases with labor’s share in total cost provided that < /( - ). The validity of HM–4 depends on how the scope for capital-labor substitution relates to the elasticity of product demand.

*) „The formulas in large measure confirm the laws of demand put forward by Marshall (1920) and Hicks (1932)” (C–Z 186).

More than two factors of production

(different types of capital and labor, land, raw materials, etc.)

The firm’s problem is to solve:

Y X

X F f

k X

w ^{i n}

n

i

i X

X ⁿ

) ,...,

( .

.

min ¹

... 1

1

The FOC is essentially identical to that discussed in the two-factors case:

n j

w i w X

X F

X X

és F Y

X X

F

j i n

j

n n i

,..., 1 ) ,

,..., (

) ,...,

) ( ,...,

( 1

1 1

The cost function is first order homogeneous in w and

homogeneous of degree 1/ in Y if F(.) is homogeneous of degree . It is concave and satisfies Shepard’s lemma:

) , ,..., (

) , ,..., (

1 1

Y w w

C X

Y w w

C

n i

i

n

The demand for factors of production diminishes with own prices:

n i

C w

Y X w w

C

X _ii

i i n

i

i ( ¹,..., , ) 0 1,...,

But the demand for a given factor does not necessarily increase if the price of another factor goes up. If the price of factor j goes up, its employment will fall and the employment of at least one other factor will rise. However, we cannot predict how the demand for a particular factor i will change (without knowing the technology).

n j

i w C

X w

X

i ij j j

i

,..., 1 ,

j 0

i

w

X  i and j are p-subsitutes (Hicks–Allen substitutes)

j 0

i

w

X  i és j are p-complements (Hicks–Allen complements)

Cross effects

Cross elasticity. Ambigously signed, non-symmetric

j i i

j i ij

j i

j j i i

j C

X w X

w w X

Direct elasticity of substitution. Defined as in the two-factors case:

) /

(

) / ( ) / (

) /

(

j i

i j i

j

j i

i

j X X

w w

w w

X d X

Difficult to interpret: a change in the price of j starts a chain of substitutions so the demand for i will change for several reasons.

Allen’s partial elasticity of substitution (derivable from the cost function) tells more:

j i

j i

j j

i ij j

j i j i

j s

C C

C C X

w C

The formula known from the two-factors case

j i

j i

j

s

continues to hold but is ambigously signed (unlike in the two- factors case).

Corollary: changes in the price of capital, materials and land may affect the demand for different types of labor in

different ways. Unskilled labor and capital are usually found

to be substitutes, for instance, while skilled labor and capital

are complements according to several estimates (capital-

skill complementarity).

It also continues to hold that

n j

Y

i

i i Y i

j i

j

, 1 ,...,

Therefore the sign of the uncompensated elasticity remains an empirical question :

s complement gross

are and

0

s substitute gross

are and

0

j i

j i

i j i j

If the production function is homogeneous and the market is not fully competitive then:

j i s_j ⁱ_j

i

j ,

If the term in the bracket is positive, a rise in the price of j will increase the demand for i. If it is negative, the demand for both factors will fall.

Appendix 1:

Demand for labor in the short run

Demand for labor in the short run

Labor is the only factor of production. The production function can be written as:

Y=F(L), Y^’>0, Y’’<0

The firm may have market power. Market power is measured by the elasticity, where P(Y) is an isoelastic inverse demand function:

Y Y P

Y

p P

Y ( )

) (

Under perfect competition the firm is price-taker 

Under imperfect competition the price is affected by Y 

p 0

Y

p 0

Y

The profit function is:

wL L

F L F P wL Y

Y P

L) ( ) ( ) ( ) (

The FOC from differentiation by L is 0 )

( )

( ) ( )

(L F L P Y P Y Y w

Making use of the fact that: ^Y _Y^P

Y P

Y P

) (

) (

0 )

1 )(

( ) ( )

(L F L P Y _Y^P w

So the optimum is at:

1 1 where 1

)

( _p

P Y

L w F

Under perfect competition the marginal product is equal to the wage.

Under imperfect competition the optimal marginal product is higher (at given w/P) and employment is lower.

The cost function is

) ( )

(Y wL wF ¹ Y C

Marginal cost is equal to*:

) ( / )

(Y w F L

C

*) Note that the derivative of F^-1(L) is 1/F’(L)

Recalling that F’(L)= w/P, that is, P= w/F’(L), it follows that:

) ) (

( C Y

L F P w

Under perfect competition ( =1) the product price is equal to marginal cost. The price-setting firm applies a price markup . Product price is higher and output end employment are lower.

Prices

The FOC was:

Let us rearrange the FOC to arrive at the formula below (taking into account that L is affected by the wage) and look at how the optimal L varies with w!

P L w

F ( )

0 }

) ( {

)

(w P F L w w L

F

Differentiation by w and re-arranging terms yields:

2

0

F P P

w F L

(+)(-) (+)(-)

The demand for labor decreases with the wage.

The size of the effect varies with technology (F), elasticity of product demand (P’) and market power ( ).

Effect of the wage on the demand for labor

Appendix 2:

The scale effect under homogeneous

production function (proof)

The scale effect under homogeneous production function

Prove that if the production function is homogeneous of degree then:

s CZ21) _w^Y

(

The proof can be found on pp. 185-186 of Cahuc-Zylberberg (CZ 2004) but an intermediate formula above their equation (21) is wrong. We give a more detailed proof by correcting the formula in question.

Our starting point is optimum condition (CZ 15) which has the form (CZ 15a) if the production function is homogeneous:

) , , ( )

( ) 15 (

) , , ( )

( )

15 (

Y r Y w Y C

P a CZ

Y r w C Y

P

CZ _Y

We shall take into consideration the following definitions:

The indicator of market power ( > , see footnote 4 on CZ, p 814), derived from the inverse demand function P=P(Y):

(a) 0

) (

) ( 1

1

Y P

Y Y where _Y^p P

p Y

Shepard’s lemma:

(b) C_w L

The share of labor in total cost:

(c) C s wL

The output elasticity of demand for labor:

1 log

logY log 1

r,1) (w, L log Y) r, (w, L log )

1 , , ( ) , , ( )

( ¹

Y d

L Y d

r w L Y r w L d

After these preparations let us start with:

P(Y) = vCY(W, R, Y), where v = 1/(1 + ) and = Y

Important: R is independent of w, v is independent of w, Y varies with w!

LHS: Log differentiating P(Y) by w yields:

RHS:

Following the rules of differentiating multivariate composite functions we have:

(vectorial)

That is:

This is close to the formula above equation (21) on CZ, p, 185 but the in the book the last term within the bracket (+1) is missing.

Using definitions (a), (b) and (c) and the corrected formula above it is easy to get to CZ’s equation (21) :

C L Y

dw dY C

Y C C

C Y

P Y Y P Y

P Y w

w

Y / 1 1

) 1 (

) ( 1

Multiply throughout with w:

C s wL w

Y dw

dY _P

Y Y w P

Y 1 1

1 1 /

/

Rearrange terms and multiply the right hand side with / :

P s

Y Y

w ( 1) 1

and then with (taking into account definition (a)). We have:

Y s

w

