Extensions of the Simple Labor Supply Model § 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 Max. U = u (C1 , C 2 ,... , C n , L1 , L2 , ..., Lm ) s.t.

n

m

i =1

j =1

∑ Pi Ci = ∑W j H j + I 0

Ci , i = 1,....n

 Consumption by the family of the ith consumer good.

L j , j = 1,....m 

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 -1-

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 -2-

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. 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 maximizes 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 decision 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. -3-

*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.

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

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 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 f rare linear homogeneous, then we can define implicit prices and incomes for commodities that would operate in a manner completely analogous to standard demand theory.  Define the cost function for commodities as: n

C(P,Z) = min ∑ Pk X k

s.t. produce Z

k =1

then the implicit commodity prices will be marginal cost of production i.e. ∂C ( P, Z ) where π r ≡ implicit price π r(P,Z) = ∂Z r no joint product implies: C(P,Z) =

total cost is the sum of the cost of each commodity Zr.

m

∑ C (P Z r

r =1

r

)

Constant returns implies: (Cost function is linear homo. in Z) (f r 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

  ∂ ∑ C r ( P,1) ⋅ Z r  ∂C ( P, Z r )  = C ( P,1) independent of activity level =  r =1 πr = r ∂Z r ∂Z r m

The implicit budget constraint is: m

∑π r =1

r

( P)Z r = µ ; µ ≡ 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.

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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: 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 )] + λ ∑ 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 ) + λ  P ∑ X r + w∑ t r − ( wt + V )⇒ ∀r = 1,......, m = −λ ∂Z r r =1  r =1  ∂Z r ∂U ∂Z i π i = ⇒ substitute less expensive commodity for more expensive ∂U π j ∂Z i

commodity. -6-

Where implicit price of commodity is

πi = 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 ) ∂X ∂t ∂C ( Z r ) πr = = 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 ↑ (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 Differential: dL = dI 0 dW + dT + ∂I 0 ∂W ∂T (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 = α 0 + α1w + α 2 I 0 + ε -7-

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.

From theory of demand or

∂L ∂W

∂H ∂W

subst

0

subst

We know from differential system of F.O.C. ∂L ∂L ∂P 2 = + (T − L) ∂I 0 D ∂W ⇒

∂L ∂L = ∂W ∂W

⇒

∂L ∂W

subst

+ (T − L)

since λ = ∂L ∂I 0

∂L ∂L − (T − L) ∂I 0 ∂W = -αˆ1 − (T − L)(-αˆ 2 )

=

subst

= -αˆ1 + (T − L)αˆ 2 or

∂H ∂W

subst

−Uc 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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