43
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:
FL>0, FK>0, FLL<0, FKK<0
),(,0),(),( LKLKFLKF
Preparations: cost function
The minimum cost at given output Y is written as
KrLw
The cost function describes minimum cost as a function of w, r and Y:
),,( YrwCC
a) C is homogeneous of degree 1 in prices.
b) C is concave: Cww<0, Crr<0.
c) Satisfies Shepard’s lemma, i.e. optimal factor demands are
given by the partial derivatives of the cost function:
),,(),,( YrwCKésYrwCL rw
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)]
YYP
YPpY
)(
)(
Under perfect competition the firm is price-taker
Under imperfect competition the price varies with Y
0p
Y
0p
Y
Market power is measured with defined as:
11
1p
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:
),,()(),,( YrwCYYPYrw
The FOC from differentiation by Y is given by:
formula above theget toy immediatelyou isfunction demand
inverse theof elasticityoutput that therecalling and P(Y)out gmultiplyinBy
0),,()()(),,(
)P'(Y)Y/P(Yη
YrwCYPYYPYrw
p
Y
YY
)1/(1ahol),,()(p
YY YrwCYP
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),,( www Cw
LYrwCL
Similarly, the conditional demand for capital decreases with
the price of capital and increases with the wage.
From Shepard’s lemma:
0wrCr
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
Kw
Lr
Kw
Lr
w
K
K
w
r
L
L
r,,
Elasticity of substitution: positive and symmetric*
0)/(
)/(
rw
LK
LK
rw
*) 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)
rw
wr
CC
CC
rw
LK
LK
rw
)/(
)/(
rw
wrwrwr
K
wrYYYY
L
r
CC
CC
KL
CC
Kr
C
L
rC
s
L
rC
L
r
r
L
r
L
1
: thatsee torwardstraightfo isit formula above thegConsiderin
log
log
L
r
00
sK is the share of capital in total cost (sK = rK/C = 1–sL = 1–wL/C)
It follows that )1( LKL
r ss
The compensated price effects thus can be written as:
0)1( LL
rL
w s0)1( LL
r s
r L w L
similarly proceeds of derivation thesymmetry, toThanks
)1( thatfollowsy immediatelIt
and :lemma sShepard' toAccording
/1as capital of share theDefine
)/(:yieldson Substituti
isthat seen that have We
L
w
LK
L
r
rw
LK
rw
L
r
rwwr
rw
wrwr
L
r
ssCL
KLr
KCLC
CKrss
CLCrC
C
CCC
CC
CCandC
L
r
r
L
L
r
The effect of an exogeneous
change of output Under homogeneous production function
/1/1 )1,(),()1,(),/( YrwKYrwKésYrwLYrwL
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),( YrwrwY
How Y*, C and (as a consequence) L* will change in response to a change in w?
),,()],,()1)(([),( YrwCw
YYrwCYPrw wY
pYw
Under optimality the term in brackets is zero (see Profit function). On the other hand,
Shepard’s lemma states that Cw(w,r,Y*)= L*. So we arrive at Hotelling’s lemma:
KrwésLrw rw ),(),(
Unconditional demands are decreasing in own prices*:
00 rrwwr
Kand
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
YCC
w
LwYww
Multiply throughout with w/L*, and the second term with Y*/Y*
Yw
wYww
LwwYww
L
CYC
L
w
Y
Y
w
YC
L
wC
L
w
L
w
w
L
What is this? The first term is the conditional demand elasticity at output level
Y=Y*
Lwww
ww
LL
w
L
L
wC
L
w
/
/
Unconditional demand
– The effects of a change in the wage
The wage has a direct and an indirect effect:
w
YCC
w
LwYww
Multiply with w/L*, and the second term with Y*/Y*
Yw
wYww
LwwYww
L
CYC
L
w
Y
Y
w
YC
L
wC
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:
LY
wY
YY
LL
Y
L
L
Y
L
CY
/
/
Unconditional demand
– The effects of a change in the wage
The wage has a direct and an indirect effect:
w
YCC
w
LwYww
Multiply with w/L*, and the second term with Y*/Y*
Yw
wYww
LwwYww
L
CYC
L
w
Y
Y
w
YC
L
wC
L
w
L
w
w
L
From step 1 and step 2 we finally have:
Yw
LY
Lw
Lw
The total effect of a change in the wage is thus
Yw
LY
Lw
Lw
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:
Yr
LY
Lr
Lr
Cross price
elasticity
(?)
Compensated elasticity
of substitution
(+)
Scale effect
(–)
ssubstitute gross are L andK 0
scomplement gross are L andK 0
L
r
L
r
Own-wage elasticity under homogeneous production function
For the derivation see Appendix 2
θYd
LdYθ)(w,r,L(w,r,Y)LYrwLYrwL
1
ln
lnln11lnln )1,,(),,( 1
)1( :yieldson Substituti. :lemma sShepard' ./
.)/( seen that have We
L
L
rrwL
rw
L
r
rw
wrwr
L
r
sKCésLCCLws
CLCrCCC
CCandC
L
r
r
L
L
r(a)
(b)
(c)
Yw
LY
Lw
Lw
(a)
(b)
(c)
0 ) 1 ( < - - = s L
L w s
q
1 s Y
w
L
L
wLL
L
w sss or)1(
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:
YXXFfkXw nin
i
i
XX n),...,(..min 1
1...1
The FOC is essentially identical to that discussed in the two-factors case:
njiw
w
XXF
XXFésYXXF
j
i
nj
nin ,...,1,
),...,(
),...,(),...,(
1
11
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
YwwCX
YwwC
ni
i
n
The demand for factors of production diminishes with own prices:
niCw
XYwwCX iii
in
ii ,...,10),,...,( 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).
njiCw
X
w
Xiji
j
j
i
,...,1,
0j
i
w
X i and j are p-subsitutes (Hicks–Allen substitutes)
0j
i
w
X i és j are p-complements (Hicks–Allen complements)
Cross effects Cross elasticity. Ambigously signed, non-symmetric
ji
ijiji
j
i
j
j
iij C
X
w
X
w
w
X
Direct elasticity of substitution. Defined as in the two-factors case:
)/(
)/(
)/(
)/(ji
ij
ij
jiij
XX
ww
ww
XXd
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:
jij
ij
ji
ij
jj
ij
ij s
CC
CC
Xw
C
The formula known from the two-factors case
jij
ij 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
njiYi
iY
ij
ij ,...,1,
Therefore the sign of the uncompensated elasticity remains an
empirical question :
scomplement gross are and 0
ssubstitute gross are and 0
ji
ji
i
j
i
j
If the production function is homogeneous and the market is not fully
competitive then:
jis ijj
ij ,
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:
YYP
YPpY
)(
)(
Under perfect competition the firm is price-taker
Under imperfect competition the price is affected by Y
0p
Y
0p
Y
The profit function is:
wLLFLFPwLYYPL )()()()(
The FOC from differentiation by L is
0)()()()( wYYPYPLFL
Making use of the fact that: PYY
YP
YP
)(
)(
0)1)(()()( wYPLFL PY
So the optimum is at:
11
1where)(
p
YP
wLF
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
)()( 1 YwFwLYC
Marginal cost is equal to*:
)(/)( LFwYC
*) 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:
)()(
YCLF
wP
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
wLF )(
0})({)( wwLFPwLF
Differentiation by w and re-arranging terms yields:
02 FPPFw
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:
sCZ Yw)21(
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(
YrwY
CYPaCZ
YrwCYPCZ 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
YP
YYPwhere p
Yp
Y
Shepard’s lemma:
(b) LCw
The share of labor in total cost:
(c) C
wLs
The output elasticity of demand for labor:
1
log
loglogY1r,1)(w,LlogY)r,(w,Llog )1,,(),,()( 1
Yd
LdYrwLYrwLd
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
dwdY
C
CY
C
C
YP
YYP
Y
PY
ww
Y 11/
1)(
)(1
Multiply throughout with w:
sC
wL
wY
dwdY PY
Yw
PY 1
11
1
/
/
Rearrange terms and multiply the right hand side with / :
sPY
Yw
1)1(
and then with (taking into account definition (a)). We have:
sYw
