Extensions of the simple model of labor supply
§ Each reformulation modifies the simple model by introducing new assumptions
about the nature of the utility function or about the constraints subject to which the
individual maximizes utility.
1. Labor Supply of Family Members
§ Relation between family membership and labor supply.
 Labor supply decisions are interdependent. Eg. My labor supply can be affected by
the wage you earn, my notion of what reference group earns, or the wage settlement
that some reference group gets.
(1) Male chauvinist model
The wife views her husband’s earnings as a kind of property income when she
makes labor supply decisions, whereas the husband decisions on his labor supply
without reference to his wife’s labor supply decisions, solely on the basis of is
own wage and the family’s actual property income.
 For the labor supply point of views, the I0 relevant to the wife is assumed to
include the husband’s earned income as well as property income.
(2) Family Utility-Family Budget Constraint Model
U  u(C1 , C2 ,... , Cn , L1 , L2 , ..., Lm )
Max.
s.t.
n
m
i 1
j 1
 Pi Ci  W j H j  I 0
Ci , i  1,....n

L j , j  1,....m 
Consumption by the family of the ith consumer good.
Leisure time of the jth family member.
H j , j  1,....m 
Hours of work of the jth family member.
Pi , i  1,....n
Price of the ith consumer good.

The family is assumed to pool the total earnings of its different members, so that
the total family utility is maximized subject to a family budget constraint.
 All the familiar comparative static’s results derived from the consumer behavior
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model without labor supply and from the model of an individual person who
chooses between a single consumption good C and leisure time L, carry over to
this more general treatment with little or no modification.
§
Effect of wage increase of all family members
Since the prices of all members’ leisure times have remained in the same relation
to each other and similarly for all consumer goods, one may invoke Hick’s
composite commodity theorem.
 Simple static model where L and C represent two composite commodities
∴ An “income=compensated”(substitute) equal-proportionate rise in all members’
rates would always reduce composite L and increase composite C. Total family
earnings must increase due to the substitution effect of the rise in wage rates. If L
is a normal good then L will increase as total family earnings increase.
 In this model, there are two substitution effects that are relevant to labor supply of
any given family member.
a. Own-wage substitution: the substitution effect on the family members’ labor
supply of an increase in the family member’s own wage.
b. Cross-substitution effect: the effect on the family member’s labor supply of an
income-compensated rise in the wage of some other family member. The income
compensation involved is a change in property income that, when combined with
the rise in the wage of the other family member, keeps the family at the same
utility it initially enjoyed.
The cross-substitution effect of a rise in family member i’s wage on family
member j’s labor supply is positive or negative depending on whether the leisure
times of i and j are complements or substitutes. If the cross-substitution effect
entails a rise in j’s leisure when i’s wage rises, then j’s leisure and i’s leisure are
said to be substitutes; and if the cross-substitution effect of a rise in i’s wage is a
fall in j’s leisure, then the two leisure times are said to be complements.
*Note:
 2U
 2U

 the cross-substitution effect will be equal.
Li L j L j Li
But the gross or total effect of a rise in i’s wage on j’s labor supply need not equal
the total effect of a rise in j’s wage on i’s labor supply; this is because the income
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effects on the two family members need not be equal.
缺點： The family utility-family budget constraint model assumes that the family
as a whole derives utility from consumption as a whole. i.e., The distribution of
the family’s total consumption to its different members cannot affect the total
level of family utility. This assumption makes sense for public goods but not for
private goods such as food.
(3) Individual Utility-Family Budget Constraint Model.
Each individual family member maximize his or her own individual utility subject
to family budget constraint.
m
i.e., Max Ui(C, Li)
s.t.
PC  (W j H j  I 0 )
j 1
 Family resources and family consumption are pooled, but individuals maximize
their own individual utility.
Q: When everyone “dose his own thing”, is there any guarantee that the household
will be stable ?

e.g., Two-person model: 類似 reaction curves in models of duopolists.
Q is the only place where the two family members’ actions are consistent with each
other. At other points, supplies will be inconsistent.
This process of reaction will be stable iff the slope of the husband’s reaction curve in
the HmHf plane exceeds that of the wife. A sufficient condition for this to be true is
that consumer goods are normal goods for both spouses.
*Note: In this model, there are no intrafamily cross-substitution effects. (of a change
in i’s wage on j’s labor supply) arise in the family utility model due to the
common utility function; in the individual utility model, there are instead what
may be called “indirect income effect.” 類似 cross-substitution effect in family
utility model.
*Note: Whereas the family utility model entails equal cross-substitution effects of
indeterminate sign, the analogous indirect income effects on labor supply in the
individual utility model are necessarily negative(provided leisure is a normal good
for each spouse), but not necessarily equal.
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2. Female Labor Supply
Two issues:
(1) Multiple Time Uses : There are multiple choice of allocation of time.
i.e. choice among work in the market, work at home and leisure.
(2) Measurement of Wage Rate.
Problem: no wage measure at all if work at home.
(1) Approach to handle “multiple time uses”
§
Introducing the idea of home production.(household production)
What we consume is home commodity produced through home production
function, with inputs time and market foods, e.g, dinner.
I. General Model of Household Production.
Z: a commodity vector with Zr, r=1, …, m as elements.
X: a market good vector with Xi, r=1, …, n as elements.
(one of the X could be leisure)
Z= F(X): a multidimensional production surface.
 f 1(X ) 


  
In general Z=  f r ( X ) 


  
 f m ( X )


If the production processes for the commodities are independent (i.e. no joint
product) then
 f 1( X ) 


  
Z   f r (X )


  
 f m ( X )


m
with X i   y ri
r 1
where yri ≡quantity of ith market good used as an input in rth commodity.
 U=U(Z)
If fr are linear homogeneous, then we can define implicit prices and incomes for
commodities that would operate in a manner completely analogous to standard
demand theory.
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 Define the cost function for commodities as:
C(P,Z)  min
n
P X
k
k 1
s.t. produce Z
k
then the implicit commodity prices will be marginal cost of production i.e.
C ( P, Z )
 r(P,Z) 
where  r ≡ implicit price
Z r
no joint product impies:
C(P,Z) 
m
 C (P Z
r
r 1
r
total cost is the sum of the cost of
each commodity Zr.
)
Constant returns implies: (Cost function is linear homo. in Z)
(fr are linear homogeneous)
C r (P,Z r )  C r ( P,1)  Z r
m
 C ( P, Z )   C r ( P,1)  Z r
r 1


  Cr ( P,1)  Z r 
C ( P, Z r )
  C ( P,1) independent of activity level
r 
  r 1
r
Z r
Z r
m
The implicit budget constraint is:
m

r 1
r
( P)Z r   ;   implicit income .
 The max of U(Z1,….., Zm) subject to the implicit budget constraint will
generate commodity demand functions with usual properties.
 The demand functions for goods (Xi) are “derived” or factor demand.
II. An Explicit Model of Leisure/ Labor Supply with Home Production
U=U(Zr)
r=1,…..,m.
Assume one market good X used to produce m commodities,
 X 
m
X
r 1
r
Time t can be used in work or in the r commodity activities.
 t  t w  tc
m
tc   tr
r 1
 Zr = fr (Xr, tr)
r= 1,….., m.
Budget constraint:
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m
P X
r 1
r
 wt w  I 0 ; I0 ≡property income.
Max problem:
Max U  U  f r ( X r , t r )
m
s.t.
P X
r 1
r
 wt w  I 0
m
m

£  U  f r ( X r , t r )    P  X r  w(t   t r )  I 0 
r 1
 r 1

Implications from F.O.C.
f r
X r
P
(i)

f r
W
t r
in production of any Z
time-intensive commodity or goods-intensive commodity.
 try to economize more expensive inputs.
 Production efficiency
(ii) across Zr
m
C
 m
 U
£  U ( Z r )    P X r  w t r  ( wt  V )
 
r  1,......, m
Z r
r 1
 r 1
 Z r
U
Z i  i

 substitute less expensive commodity for more expensive
U  j
Z i
commodity.
Where implicit price of commodity is
i  P
X i
t
w i
Z i
Z i
C ( Z , P, w)  P  X  w  t c
m
m
r 1
r 1
 P  ( X r )  w( t r )
r 
C ( Z r )
X
t
 P  r  w r
Z r
Z r
Z r
*Note: The change in behavior may be caused by change in production not change in
preference. e.g., ,microwave oven, washer and dryer make house-keeping less
time-intensive => Female LFP ↑
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(2) Measurement of Wage Rate – An Empirical Issue
The empirical studies are of interest for four reasons:
(i) Test the predictions and implications of theoretical models.
e.g., Is the own-substitution effect of a wage increase on labor supply positive?
(ii) Provide information on the signs and magnitude of effects about which
theoretical models makes no a priori predictions.
e.g., Is leisure a normal good?
(iii)Shed light on a variety of important labor market developments.
e.g., ↑ in LFP of married women.
(iv) An important tool for evaluation of proposed government policies.
e.g. tax cut, childcare subsidy
Linear Labor Supply Function—First Generation of the Labor
Supply Function
Idea: F.O.C => Leisure Demand Function
L
L
L
dT 
dW 
dI 0
Differential: dL 
T
W
I 0
(Recall: Differential System in discussing Income Effect & Substitution Effect)
H T L
Labor Supply
dH  (1 
L
L
L
dT ) 
dW 
dI 0
T
W
I 0
 In empirical studies, researchers assume:
H   0  1 w   2 I 0  
ε: differences in taste for work or measurement error; i.e. unmeasured factors,
factors that are known to the individual but are not known or observed by
researchers.
Then giving the data of I0, W and H, we can perform the OLS to estimate α1, α2.
Specification of linear labor supply function consistent with the theory.
Test of theory: Most of the studies focus on the sign of the substitution effect.
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From theory of demand
or
L
W
H
W
subst
subst
0
0
We know from differential system of F.O.C.
L P 2
L

 (T  L)
W
D
I 0

L
L

W W

L
W
subst
 (T  L)
since  
L
I 0
L
L
 (T  L)
W
I 0
 -ˆ1  (T  L)(-ˆ 2 )

subst
 -ˆ1  (T  L)ˆ 2
or
H
W
subst
Uc
0
P
 ˆ1  (T  L)ˆ 2
0
0
From Table 3.2 in Killingworth (1983) labor supply elasticity estimates for men and
women, we can see most of the compensated wage elasticity is positive as suggested
from the theory of demand.
Can this function (linear labor supply) be generated from some utility function?
The direct utility function:
U (C , L)  (
T LB
2
   (C  A)  
) exp  1  2

  B  (T  L)  
0

 12
 2 ( 2 )

B 1
2
where A 
This utility plus the linear budget constraint can give us the linear demand function
H   0  1 w   2 I 0
Problem with linear supply model:
Not a complete model of labor supply: neglect “reservation wage”.
*Note: the individual will work and have M= W/P only if his real wage W/P exceeds
his reservation wage ML=T, where ML=T is the value of
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M (
U / L
) when L  T and H  0.
U / C
Linear labor supply function ignores values of W/P that are below the reservation
wage, i.e., says nothing about what happens if the demand for leisure (desired leisure)
exceeds total available time.
 Let actual hours of leisure be L and desired hours of leisure be L*.
The complete leisure demand model becomes:
L*=T-α0 -α1 W-α2I0
L= L*
L=T
if and only if
if and only if
W/P > ML=T (reservation wage)
W/P ≦ ML=T
i.e. the complete model of labor supply becomes:
H*= α0 +α1 W+α2I0
H= H* if and only if
H= 0 if and only if
W/P > ML=T
W/P ≦ ML=T
In practice, there is a problem. We do not observe W for those who has H=0.
 Econometric issues in the estimation of labor supply function.
1. Discrete Choice Model: Probit, Logit, Tobit
2. Sample Selection Model: Heckman’s Two Stage Model
3. Non-Parametric Model: Distribution-Free
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