labor supp
TRANSCRIPT
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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.
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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.
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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,
( , )
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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/
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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
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CLU1
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CLU1
U2
U3
Direction of preference
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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
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CLL* 1
C*
H*=1-L*
V/p
U(C*,L*)
D
-w/p
(w +V )/p
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LC
1
V/p A U1
B
(w +V )/p
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LC
1
V/p A U1U2
B
(w +V )/p
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LC
1
V/p A U1U (C*,L*)
D
L*
C*
(w +V )/p
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Tangency condition:
Slope of indifference curve = slope of budget line
or
MRS UU
wp
L
C
Uw
Up
L C
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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
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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
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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
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CLL0* 1
C0*
V0/p
U(C0*,L0*)
D
U
U
L1
V1/p
L1
c1
C1 F
F
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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
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CLL0 1
C0
H0 =1-L0
V/pU(C0,L0)
D
L1
C1
U(C1,L1)
-w0/p
-w1/p
F
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CLL0 1
C0
V/pU(C0,L0)
D
L1
C1
U(C1,L1)
-w0/p
-w1/p
F
L
C G
-w1/p
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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
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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
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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
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wH = 1-L
w0
w1
w2
H0H1H2
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wH = 1-L
w0
w1
w2
H0H1H2
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CL1
V/p
U(V/p,1)
-w/p E
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CL1
V/p
U(V/p,1)
-w/p E
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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 )
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wH = 1-L
w0
w1
w2
H0H1H2
wR
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wR
f(wR)
Different people have different preferences and incomeY wR differs across workers.
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wR
f(wR)
Different people have different preferences and incomeY wR differs across workers.
w
-
wR
f(wR)
Different people have different preferences and incomeY wR differs across workers.
w
In the labor force
Out of the labor force
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Assuming no interaction among labor supply of different workers,
Market labor supply = horizontal sum of individual workers labor supply
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wRi
w
Hi
w
3HiH0iH1i
w0
w1
H ii
n
11
1
n1> n0Workers LS Market LS
H ii
n
01
0
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Assuming no interaction among labor supply of different workers,
Market labor supply = horizontal sum of individual workers labor supply
As w increases, 2 margins of adjustment:
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Assuming no interaction among labor supply of different workers,
Market labor supply = horizontal sum of individual workers labor supply
As w increases, 2 margins of adjustment:- intensive margin adjustment:
workers already in the market adjust working hours (8 or 9)
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wH = 1-L
w0
wR
w1
H0H1
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wH = 1-L
w0
wR
w1
H0H1H1H0wR
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Assuming no interaction among labor supply of different workers,
Market labor supply = horizontal sum of individual workers labor supply
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
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Suppose V = 0, but the government guarantees GC
L11-H*
G/p
H* = G/w
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Suppose V = 0, but the government guarantees GC
LL1=11-H*
G/p
H* = G/w
U1
U0L0
-w/p
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Suppose V = 0, but the government guarantees GC
L1-H*
G/p
H* = G/w
U1U0
L0-w/p
L1=1
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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
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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.
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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...,...
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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
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wt
t
$
-
wt
t
$ Rt
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wt
t
$ Rt
t T
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Ht
t
$
t T