Download 1 Model with Time-Consistent Agents

The first order conditions for the persistent allocations are
C
− ωm u0 cPm,0 = (1 − φ) πm µ,
(1 − φ) πm + λm − λTm+1
C
(1 − φ) πm + λm − λTm+1
− βωm u0 cPm,t = (1 − φ) πm µ, ∀t > 0,
1
[(1 − φ) πm + λm − ωm ] h0
θm
P
ym,0
θm
!
C
− λTm+1
1
θm+1
h0
P
ym,0
θm+1
!
= (1 − φ) πm µ + γm,0 − γm+1,0 ,
1
[(1 − φ) πm + λm − βωm ] h0
θm
P
ym,1
θm
!
C
− λTm+1
1
θm+1
h0
P
ym,1
θm+1
!
= (1 − φ) πm µ + γm,1 − γm+1,1 .
The first order condition on xm gives the following relationship for Lagrange
multipliers
X
λTmI0 |m .
λm = λTmC +
θm0
By the Kuhn-Tucker Theorem, the solution is characterized by the first order
conditions above and the complementary slackness conditions. Notice that if
ωm > 0, then there will be intertemporal distortions for type θm . The proof
will proceed to show how ωm > 0 for all θm > θ1 , there by proving part (i).
Suppose there exists θm > θ1 such that ωm = 0, then by the first order
R
P
P
R
R
P
P
conditions: cR
m,t = cm , cm,t = cm , ym,t = ym and ym,t = ym . If (11) is satisfied,
P
R
then cPm ≥ cR
m and ym ≥ ym for (10) to be satisfied. There are three cases: if
P
R
cPm > cR
m and cm = cm .
If cPm > cR
m , then by the first order condition,
C
λm − λTm+1
>1−
1+
(1 − φ) πm
22
P
θm0
I
λTm|m
0
φπm
.