Labor Supp

LongRunLaborSupply- Migration.- Endogenous population adjustment in closed economy

(Malthus, Becker, Willis). ShortRunLaborSupply Even if population is fixed in the short run, workers

still need to decide(i) whether to participate in the labor market;(ii) how many hours to supply.

Assumptions:- A worker maximizes own utility of the simple form

U = U(C,L)where C = consumption, L = leisure.

- The worker is endowed with 1 unit of time per period.- The worker earns a wage of w per unit time and faces a

price of p for the consumption good.- There is an unearned income of V even if the worker

does not work.

TheModelThe maximizing problem for the worker:

subject to V + w(1 - L) = pC

Lagrangian for the constrained max problem:

= U(C,L) + (V + w(1 L) pC)

where 8 = Lagrange multiplier for the budget constraint.

Max U C LC L,

( , )

InteriorSolutionAssuming 0 < L < 1, the FOCs for utility maximization:

C:

L:8:

where Ui = .

Y UL /UC = w/p

U C L pU C L wV w L pC

C

L

,

,

1

U i/

Holding utility constant

dU(C,L) =

Y

where MRS = marginal rate of substitution. Indifferent curves downward sloping if

(1) both C and L are goods;(2) non-satiation.

UC

dCUL

dL 0

CL

UU

MRSU

L

C

CLU1

U2

U3

Direction of preference

pC = w(1 L) + V

Y

w/p = opportunity cost of leisure

C V wp

wp

L

V+w = full income = value of endowment

CL1

V/p -w/p

(w +V )/p

CLL* 1

C*

H*=1-L*

V/p

U(C*,L*)

D

-w/p

(w +V )/p

LC

1

V/p A U1

B

(w +V )/p

LC

1

V/p A U1U2

B

(w +V )/p

LC

1

V/p A U1U (C*,L*)

D

L*

C*

(w +V )/p

Tangency condition:

Slope of indifference curve = slope of budget line

or

MRS UU

wp

L

C

Uw

Up

L C

Suppose, in absolute terms, slope of indifference curve < absolute slope of budget line:

or

Y 8 spending on C by $1 and 9 spending on L by $1:

)U = - UL /w + UC /p > 0

Y Utility can be increased by 9L and 8C

UU

wp

L

C

UwU

pL C

Taking total derivatives of the FOCs:

where Uij = MUi /Mj .

U dC U dL pd dp

U dC U dL wd dw

pdC wdL Cdp L dw dV

CC CL

LC LL

1

Effect of a change in unearned income on leisure (or labour supply):

where = (-UC2ULL + 2UCULULC - UL2 UCC)/2

> 0 by the SOC for maximization.

LV

U w U p U U U UCC LC LC C CC L

CLL0* 1

C0*

V0/p

U(C0*,L0*)

D

U

U

L1

V1/p

L1

c1

C1 F

F

Effect of a change in wage on leisure (or labor supply):

Slutsky Equation:Total effect = income effect + substitution effect

Lw

U w L p U L pCC CL 1 1

U U U U L U

L LV

U

CL C CC L C

C

1

1

2

2

CLL0 1

C0

H0 =1-L0

V/pU(C0,L0)

D

L1

C1

U(C1,L1)

-w0/p

-w1/p

F

CLL0 1

C0

V/pU(C0,L0)

D

L1

C1

U(C1,L1)

-w0/p

-w1/p

F

L

C G

-w1/p

CLL0 1

C0

V/pU(C0,L0)

D

L1

C1

U(C1,L1)

-w0/p

-w1/p

F

L

C G

Sub Effect: D 6 GIncome Effect: G 6 FTotal Effect: D 6 F

(Sub effect dominates)-w1/p

CLL0 1

C0

V/pU(C0,L0)

D

L1

C1

U(C1,L1)

-w0/p

-w1/p

F

-w2/p

U(C2,L2)

C2

L2

H

w increases from w1to w2,L increases from L1to L2Y Income effect dominates

CLL0 1

C0

V/pU(C0,L0)

D

L1

C1

U(C1,L1)

-w0/p

-w1/p

F

-w2/p

U(C2,L2)

C2

L2

H Price expansion path/offer curve

wH = 1-L

w0

w1

w2

H0H1H2

CL1

V/p

U(V/p,1)

-w/p E

CL1

V/p

U(V/p,1)

-w/p E-wR /p

wR = reservation wage= p uL(V/p,1)/uC (V/p,1)= p (MRS at E )

wH = 1-L

w0

w1

w2

H0H1H2

wR

wR

f(wR)

Different people have different preferences and incomeY wR differs across workers.

wR

f(wR)


w

wR

f(wR)


w

In the labor force

Out of the labor force

Assuming no interaction among labor supply of different workers,

Market labor supply = horizontal sum of individual workers labor supply

wRi

w

Hi

w

3HiH0iH1i

w0

w1

H ii

n

11

1

n1> n0Workers LS Market LS

H ii

n

01

0



As w increases, 2 margins of adjustment:



As w increases, 2 margins of adjustment:- intensive margin adjustment:

workers already in the market adjust working hours (8 or 9)

wH = 1-L

w0

wR

w1

H0H1

wH = 1-L

w0

wR

w1

H0H1H1H0wR



As w increases, 2 margins of adjustment:- intensive margin adjustment:

workers already in the market adjust working hours (8 or 9)

- extensive margin adjustment:number of workers increases.

wR

f(wR)

w0

Out of the labor force

In the labor force

w1

New entrants

Suppose V = 0, but the government guarantees GC

L11-H*

G/p

H* = G/w


LL1=11-H*

G/p

H* = G/w

U1

U0L0

-w/p


L1-H*

G/p

H* = G/w

U1U0

L0-w/p

L1=1

Suppose V = 0, but the government guarantees G and tax earnings at rate t

C

L11-H*

G/p

H* = G/w

U1U0

L0-w/p

1-H**

H** = G/w (1-t )-w (1-t )/p

L3

Time and purchased goods are inputs into the production of household commodities

Utility is a function of quantities of these household commodities

Suppose- there are n household commodities (Z1 Zn)- each commodity i is produced with purchased input

xi and time Ti according to production function:Zi = f (xi,Ti)

- each worker is endowed with 1 unit of time and unearned income V.

Workers maximization problem:

subject to

(1) Budget constraint:

(2) Time constraint:

(3) Technology constraint: for i = 1,,m,

p x V wTi ii

n

w

1

T Tii

n

w

1

1

Z f x Ti i i ,

MaxU U Z ZZ Z nn1

1...,...

U C L e dtt tT t,0

U C Lt t

tt

T ,

1 11

p C e dt w L e dt Vt tT rt t tT rt0 0 01

p C

r

w L

rVt tt

t

Tt t

tt

T

1

1

111 110

Worker maximizes dynamic/lifecycle utility:

or

Subject to intertemporal or lifetime budget constraint

or

wt

t

$

wt

t

$ Rt

wt

t

$ Rt

t T

Ht

t

$

t T

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