MATCHING MARKET CLEARING
JANUARY 25, 2017
Introduction
TIMING OF EVENTS



Matching market clearing …
… then wage determination
Focus just on extensive margin
January 25, 2017
2
Employment Margin
LABOR SUPPLY (LFP)

 lfpt
Representative household



t 
h
s 

max
E

u
(
c
)
h
1
k
s
n





0
t
t
t
 t

ct , nts , st
t 0
ct  (1   tn ) wt nts  (1  kth ) st b
nts  (1   x )nts1  st  kth

lfpt  (1  kth ) st  nts

Consumption-LFP optimality condition

 h '(lfpt 1 )  u '(ct 1 )b   1  kth1  
h '(lfpt )
h
n
h
h
 kt (1   t ) wt  (1  kt )  b  kt (1   x ) Et  t 1 
   h 
u '(ct )
u '(ct 1 )


  kt 1  
January 25, 2017
3
Employment Margin
LABOR SUPPLY (LFP)

Definition: Household optimality is a set of state-contingent
functions for ct , st , nts that satisfy



Consumption-LFP optimality condition

 h '(lfpt 1 )  u '(ct 1 )b   1  kth1  
h '(lfpt )
h
n
h
h
 kt (1   t ) wt  (1  kt )  b  kt (1   x ) Et  t 1 
   h 
u '(ct )
u
'
(
c
)

t 1

  kt 1  

Household budget constraint
ct  (1   tn ) wt nts  (1  kth ) st b

Perceived law of motion of employment (aka job-finding constraint)
nts  (1   x )nts1  st  kth
taking as given exogenous processes
January 25, 2017
k
h
t
, wt , tn  and n-1
4
Employment Margin
LABOR DEMAND (JC)

Representative firm


D
f



max
E
z
f
(
n
)
w
n

v
t
t t
t 
0   t |0  t
vt , ntD
 t 0

s.t. ntD  (1   x )ntD1  vt kt f

Job-creation condition

 
 zt f '(n )  w n  (1   x ) Et  t 1|t f 
f
kt
kt 1 


January 25, 2017
D
t
D
t t
5
Employment Margin
LABOR DEMAND (JC)

Definition: Firm optimality is a set of state-contingent functions
vt , ntD that satisfy



Job-creation condition

 
D
D





z
f
'(
n
)
w
n
(1

)
E
t
t
t t
x
t  t 1|t
f 
kt f
k
t 1 



Perceived law of motion of employment (aka job hiring constraint)
ntD  (1   x )ntD1  vt  kt f
taking as given exogenous processes
January 25, 2017
k
f
t
, wt 
and n-1
6
Employment Margin
MATCHING MARKET CLEARING

Definition: Equilibrium optimality is a set of state-contingent
functions  t , vt , st , nt , ct  that satisfies aggregate RC,

Consumption-LFP optimality condition
h '  (1  k h (t )) st  nts 
u '(ct )
 k h (t )(1   tn ) wt  (1  k h (t ))  b

 h '(lfpt 1 )  u '(ct 1 )b   1  k h (t 1 )  
 k (t )(1   x ) Et  t 1 
 h

u
c
k

'
(
)
(
)

t 1
t 1

 
 
h

Job-creation condition


 
D
D
z
f
n
w
n
E





'(
)
(1

)

t
t t
t  t 1|t
t
x
f
k f (t )
k
(

)
t

1


January 25, 2017
7
Employment Margin
MATCHING MARKET CLEARING

Definition: Equilibrium optimality is a set of state-contingent
functions  t , vt , st , nt , ct  that satisfies aggregate RC,

Consumption-LFP optimality condition
h '  (1  k h (t )) st  nts 
u '(ct )
 k h (t )(1   tn ) wt  (1  k h (t ))  b

 h '(lfpt 1 )  u '(ct 1 )b   1  k h (t 1 )  
 k (t )(1   x ) Et  t 1 
 h

u
c
k

'
(
)
(
)

t 1
t 1

 
 
h

Job-creation condition


 
D
D
z
f
n
w
n
E





'(
)
(1

)

t
t t
t  t 1|t
t
x
f
k f (t )
k
(

)
t

1



Aggregate LOM for employment
nt  (1   x )nt 1  m( st , vt )

AND…
January 25, 2017
8
Employment Margin
MATCHING MARKET CLEARING

Definition: Equilibrium optimality is a set of state-contingent
functions  t , vt , st , nt , ct  that satisfies aggregate RC,

Consumption-LFP optimality condition
h '  (1  k h (t )) st  nts 
u '(ct )
 k h (t )(1   tn ) wt  (1  k h (t ))  b

 h '(lfpt 1 )  u '(ct 1 )b   1  k h (t 1 )  
 k (t )(1   x ) Et  t 1 
 h

u
c
k

'
(
)
(
)

t 1
t 1

 
 
h

Job-creation condition


 
D
D
z
f
n
w
n
E





'(
)
(1

)

t
t t
t  t 1|t
t
x
f
k f (t )
k
(

)
t

1



Aggregate LOM for employment
nt  (1   x )nt 1  m( st , vt )

Matching-market clearing
January 25, 2017
k h (t )  st  k f (t )  vt  m( st , vt )
9
Employment Margin
LABOR SUPPLY (LFP)


Consider ρx = 1, b =0
Definition: Household optimality is a set of state-contingent
s
functions for ct , st , nt that satisfy



Consumption-LFP optimality condition
h '  (1  kth ) st  nts 
u '(ct )

 kth (1   tn ) wt
Household budget constraint
ct  (1   tn ) wt nts

Perceived “law of motion” of employment (aka job-finding constraint)
nts  st  kth
taking as given exogenous processes
January 25, 2017
k
h
t
, wt , tn  and n-1
10
Employment Margin
LABOR SUPPLY (LFP)


Consider ρx = 1, b =0
Definition: Household optimality is a set of state-contingent
s
functions for ct , st , nt that satisfy



Consumption-LFP optimality condition
h '  (1  kth ) st  nts 
u '(ct )

 kth (1   tn ) wt
Household budget constraint
ct  (1   tn ) wt nts

Perceived “law of motion” of employment (aka job-finding constraint)
n  st  k
s
t
h
t
nts
 st  h
kt
taking as given exogenous processes
January 25, 2017
k
h
t
, wt , tn  and n-1
11
Employment Margin
LABOR SUPPLY (LFP)


Consider ρx = 1, b =0
Definition: Household optimality is a set of state-contingent
s
functions for ct , nt
that satisfy



Consumption-LFP optimality condition
 nts

h '  h  nts  nts 
 kt
  k h (1   n ) w
t
t
t
u '(ct )

Household budget constraint
ct  (1   tn ) wt nts
taking as given exogenous processes
January 25, 2017
k
h
t
, wt , tn  and n-1
12
Employment Margin
LABOR SUPPLY (LFP)


Consider ρx = 1, b =0
Definition: Household optimality is a set of state-contingent
s
functions for ct , nt
that satisfy



Consumption-LFP optimality condition
 nts 
h ' h

 k ( nt )   k ( )(1   n ) w
t
t
nt
u '(ct )

Household budget constraint
ct  (1   tn ) wt nts
taking as given exogenous processes
January 25, 2017
k
h
( nt ), wt , tn  and n-1
13
Employment Margin
LABOR DEMAND (JC)

Definition: Firm optimality is a set of state-contingent functions
vt , ntD that satisfy



Job-creation condition

kt f

 zt f '(ntD )  wt
Perceived law of motion of employment (aka job hiring constraint)
ntD  vt  kt f
taking as given exogenous processes
January 25, 2017
k
f
t
, wt 
and n-1
14
Employment Margin
LABOR DEMAND (JC)

Definition: Firm optimality is a set of state-contingent functions
vt , ntD that satisfy



Job-creation condition

D
'
(
)  wt

z
f
n
t
t
f
k ( nt )

Perceived law of motion of employment (aka job hiring constraint)
ntD  vt  k f ( nt )
taking as given exogenous processes
January 25, 2017
k
f
( nt ), wt 
and n-1
15
Employment Margin
MATCHING MARKET CLEARING

Definition: Equilibrium optimality is a set of state-contingent
functions  t , vt , st , nt , ct  that satisfies aggregate RC,
ct    vt  zt f (nt )

Aggregate RC

Consumption-LFP optimality condition
 n 
h ' h t 
 k ( nt )   k h ( )(1   n ) w
t
t
nt
u '(ct )

Job-creation condition

 zt f '(nt )  wt
k f ( nt )

Solve for θ using hh job
finding constraint
Aggregate LOM for employment
Solve for θ using firm job
hiring constraint
nt  m( st , vt )

Matching-market clearing
January 25, 2017
k h ( nt )  st  k f ( nt )  vt  m( st , vt )
16
Employment Margin
MATCHING MARKET CLEARING

Functional forms
m( st , vt )  st n vt1 n  ms ( st , vt )   n st n 1vt1 n   n nt1 n
k h ( nt )   nt1 n
f (nt )  nt
h(lfpt ) 
n
1  1/ n
lfpt11/n 
h '(lfpt )   nlfpt1/n
 n 
h ' h t 
 k ( nt )   k h ( ) w
nt
t
u '(ct )
mv ( st , vt )  (1   n ) st n vt n  (1   n ) nt n
k f ( nt )   nt n

f (nt )   nt 1
u (ct )  ct

u '(ct )  1

 zt f '(nt )  wt
k f ( nt )
Substitute…(and set tax rate = 0)
January 25, 2017
17
Employment Margin
MATCHING MARKET CLEARING

In (θ, n) space
 n   nt  nt
n 1

1/ n
  nt1 n wt
   nt  zt nt 1  wt 
n
…and solve for θLFP and θJC


w
t
 ntLFP  
1/ 
 n  nt 
n
January 25, 2017

1
(1 n )(11/n )
 zt nt
 ntJC  

 1
 wt 



1
n
18
Employment Margin
MATCHING MARKET CLEARING

In (θ, n) space
 n   nt  nt
n 1

1/ n
   nt  zt nt 1  wt 
  nt1 n wt
n
…and solve for θLFP and θJC


w
t
 ntLFP  
1/ 
 n  nt 

1
(1 n )(11/n )
 zt nt
 ntJC  

n

 wt 



1
n
Arseneau and Chugh (2012 JPE, Proposition 1 (p. 950))

Efficient allocations characterized by
 MRS
ct ,lfpt

 hu'('(lfpc ))   m m(s(s, v, v) )   MRT 
t
s
t
t
ct ,lfpt
t

 1
v
t
t
Given functional forms, stated in (θ, n) space?...
January 25, 2017
19
Employment Margin
MATCHING MARKET CLEARING

In (θ, n) space
 n   nt  nt
n 1

1/ n
   nt  zt nt 1  wt 
  nt1 n wt
n
…and solve for θLFP and θJC


w
t
 ntLFP  
1/ 
 n  nt 

1
(1 n )(11/n )
 zt nt
 ntJC  

n

 wt 



1
n
Arseneau and Chugh (2012 JPE, Proposition 1 (p. 950))

Efficient allocations characterized by
 MRS
ct ,lfpt

 hu'('(lfpc ))   m m(s(s, v, v) )   MRT 
t
s
t
t
ct ,lfpt
t

 1
v
t
t
Given functional forms, stated in (θ, n) space
 ntLFP   ntJC
January 25, 2017
Obvious in (decentralized equilibrium that
supports…) efficient allocations….
20
Employment Margin
MATCHING MARKET CLEARING

In (θ, n) space
 n   nt  nt
n 1

1/ n
   nt  zt nt 1  wt 
  nt1 n wt
n
…and solve for θLFP and θJC


w
t
 ntLFP  
1/ 
 n  nt 

1
(1 n )(11/n )
 zt nt
 ntJC  

n

 wt 



1
n
Arseneau and Chugh (2012 JPE, Proposition 1 (p. 950))

Efficient allocations characterized by
 MRS
ct ,lfpt

 hu'('(lfpc ))   m m(s(s, v, v) )   MRT 
t
s
t
t
ct ,lfpt
t

 1
v
t
t
Given functional forms, stated in (θ, n) space
 ntLFP , EFF   ntJC , EFF
January 25, 2017
Obvious in (decentralized equilibrium that
supports…) efficient allocations….
21
Employment Margin
MATCHING MARKET CLEARING

Aggregate matching-market clearing
Aggregate matching
market clearing
kth  st  kt f  vt  m( st , vt )
 new employeest
January 25, 2017
22
Employment Margin
MATCHING MARKET CLEARING


Aggregate matching-market clearing
Experiment 1: labor income tax rate τn > 0
WEDGEn , n 

n*
n
LOW
 nLFP , EFF (ni ; )   nVC , EFF (ni ; )  dni
Aggregate matching
market clearing
kth  st  kt f  vt  m( st , vt )
 new employeest
January 25, 2017
23
Employment Margin
MATCHING MARKET CLEARING


Aggregate matching-market clearing
Experiment 2: unemployment benefit b > 0
WEDGEn ,b 

n HIGH
n*
 nLFP , EFF (ni ; )   nVC , EFF (ni ; )  dni
Aggregate matching
market clearing
kth  st  kt f  vt  m( st , vt )
 new employeest
January 25, 2017
24