ECN 741: Public Economics
Fall 2008
A BNE is a strategy : D A (t is t measurable) such that
T
c
t1 (T ) u(gt (t (T ), ) v
c
gt (t (T ), )
t
c
t1 (T ) u(gt (t (T ), ) v
c
gt (t (T ), )
t
t=1
T
t=1
for all : D A (t is t measurable) and (a) = cfw_T |(T )=a
ECN 741: Public Economics
Fall 2008
FOC implies
Uct
= (1 lt )wt
Ult
pt vt = pt+1 vt+1 + pt+1 (1 dt+1 )dt+1
And therefore implementability constraint is
t [Uct ct + Ult lt ] = Uc0 [v0 + (1 d0 )d0 ] s0
(27)
t=0
There is a corporation that maximizes the pre
ECN 741: Public Economics
Fall 2008
Exercise: Show that a feasible allocation is implementable if and only if it satisfy (38).
Ramsey problem
Ramsey problem is the following
t [U (ct,0 , lt,0 ) + U (ct,1 , lt,1 )]
max
t=0
subject to
Uct,0 ct,0 + Ult,0 lt
ECN 741: Public Economics
sub. to.
Fall 2008
T
t1
t=1
(t ) u(ct (t )
T D
v (yt (t )
w
t
PF I (w) is the value to the planner from delivering utility w to individual, if we ignore
incentive constraint. Note that PF I (w) > P (w). Also, PF I (w) is stri
ECN 741: Public Economics
Fall 2008
Consumers problem is the following (for generation t > 0)
max U (ct,0 , lt,0 ) + U (ct,1 , lt,1 )
subject to
c
l
(1 t,0 )ct,0 + at,1 (1 t,0 )wt z0 lt,0
c
l
k
(1 t,1 )ct,1 (1 t,1 )wt z1 lt,1 + (1 + (1 t,1 )(rt )at,1
Ther
ECN 741: Public Economics
Fall 2008
z
z
1 = E (t , zt+1 )(1 + FK (Kt+1 (t ), Yt (t , zt+1 ), z t , zt+1 )|z t
z
Note that (using Jensens inequality)
1
( , zt+1 ) =
z
E
t+1 , z t , zt+1
tt
t+1 , z t , z
t+1 )
u (ct ( , z )
u (ct+1 (
E u (c+1 (t+1 , z t ,
ECN 741: Public Economics
Fall 2008
like a lump-sum tax. But since lump-tax is allowed here, it is not necessary. However, when
individuals are heterogeneous in their initial wealth, then taxing wealth for redistribution is
desirable.
Example: Now conside
ECN 741: Public Economics
() u(c(, w)
P (w) = max
c,y,w
Fall 2008
v (y (, w)
c(, w) + y (, w) + P (w (, w)
(51)
sub. to
u(c(, w)
v (y (, w)
v (y ( , w)
+ w (, w) u(c( , w)
+ w ( , w) ,
() u(c(, w)
v (y (, w)
+ w (, w) w
We want to show that in th
ECN 741: Public Economics
Fall 2008
Note that Pm (w) is strictly concave, Pm (w) P (w). Also, note that only the rst term
depends on w. Therefore
Pm (w) = 1 E
1
v (v (y (w, 0 ) + w0 w)
and limw Pm (w) = 1.
Therefore, limw P (w) limw Pm (w) = 1. Hence, we
ECN 741: Public Economics
Fall 2008
Uc0 + (Uc0 + Ucc0 c0 + Ulc0 l0 )
= (1 + fkt+1 )
Uc1 + (Uc1 + Ucc1 c1 + Ulc1 l1 )
Note that this in general does not imply zero tax on capital. When prole of labor productivity, zj , is not at over lifetime, in general c
ECN 741: Public Economics
Fall 2008
also from envelope condition we have
K (w) =
therefore
K (w) =
()K (w (, w)
Start from a given w0 , construct a stochastic process wt as
wt+1 = w (t , wt )
then
K (wt ) = Et [K (wt+1 )]
hence wt is a martingale. By ma
ECN 741: Public Economics
Fall 2008
Proposition 9 (Revelation Principle)
A allocation (cn , yn )N=1 is implementable if and only if it is truthfully implementable in a
n
direct mechanism.
Proof.
Suppose allocation (cn , yn )N=1 is implementable as outcome
ECN 741: Public Economics
Fall 2008
Denition 11 An allocation is incentive compatible if
T
t1
T
z (z )
t=1
z T Z T
(T |z T ) u(ct (T , z T ) v
T T
T
T
z (z )
t1
t=1
z T Z T
T
T
T
T
( |z ) u(ct (t ( , z ) v
T T
yt (T , z T )
t
(54)
yt (t (T , z T )
t
ECN 741: Public Economics
Fall 2008
Example : one period problem
Suppose T = 1 and there are only two types, H and L with H > L . Consider the
utilitarian planers problem
max (H ) u(c (H ) v
y (H )
H
+ (L ) u(c (L ) v
y (L )
L
sub. to.
y (H )
H
y (L )
u(c
ECN 741: Public Economics
Fall 2008
C (st ) + K (st ) + g (st ) = F (K (st1 ), L(st ), st , t) + (1 )K (st1 )
Make our usual change of variable
i i U i (hi (C, L; )
W (C, L; , , )
i
m
m
+i UC (C, L; )hi (C, L; ) + UL (C, L; )hi (C, L; )
c
y
and rewrite
ECN 741: Public Economics
1
Fall 2008
Ramsey Taxation - Primal Approach
Consider an economy with n types of consumption good that are produced using labor input:
F (c1 + g1 , . . . , cn + gn , l) = 0
(1)
ci is private and gi is public consumption of good
ECN 741: Public Economics
Fall 2008
the ones that satisfy the following incentive compatibility constraints
(n ) u(cn (n , n ) + v
n
yn (n , n )
n
(n ) u(cn ( , n ) + v
n
yn ( , n )
n
n, n , .
We are going to primarily focus on environment with unit mea
ECN 741: Public Economics
Fall 2008
The outcome is determined according to outcome function
Denition 5 Let (A, g c , g y ) be a Mechanism. A Bayesian Nash Equilibrium (BNE) is a
collection of strategies cfw_n N=1 , n : An such that
n
c
(n ) u(gn (, n (n
ECN 741: Public Economics
Fall 2008
Finally, feasibility and market clearing
c1 + g1 = x1
c2 + g 2 + z = x 2
f (x1 , z, l1 ) = 0
h(x2 , l2 ) = 0
The Ramsey problem is
max U (c1 , c2 , l1 + l2 )
subject to
U1 c1 + U2 c2 + Ul (l1 + l2 ) = 0
f (c1 + g1 , z,
ECN 741: Public Economics
Fall 2008
lnt and nt are both private information but ynt is observable. In what follows we make a
nt
change of variable lnt = ynt .
Denition 2 An allocation is a sequence of functions (cn , yn )N=1
n
cn : T
N
RT
+
yn : T
N
RT
ECN 741: Public Economics
Fall 2008
Note that any feasible allocation that satises (18) can be implemented by two of the four
taxes (that is we only need two of the c , l ,x and k to implement the same allocations).
This in turn implies that
kt = 0
1 + ct
ECN 741: Public Economics
Fall 2008
discussion above implies that
T (y (L ) > 0, T (y (H ) = 0
3.1
The New Dynamic Public Finance
So far we have characterized the set of achievable allocations by any mechanism. The goal
of the planner is to nd the best ac
ECN 741: Public Economics
Fall 2008
t W (c1t , c2t , l1t , l2t , 1 , 2 )
max
t=0
subject to
c1t + c2t + kt+1 = F (kt , l1t , l2t ) + (1 )kt
; t
where W i is dened the obvious way.
First order conditions imply
Wcit = Wcit+1 (1 + Fkt+1 )
and in the steady
ECN 741: Public Economics
Fall 2008
Consider a relaxed planning problem with only type H s I.C. constraint
max (H ) u(c (H ) v
y (H )
H
+ (L ) u(c (L ) v
y (L )
L
sub. to.
u(c (H ) v
y (H )
H
u(c (L ) v
y (L )
H
; (H )
(H ) [c(H ) y (H )] + (H ) [c(H )
ECN 741: Public Economics
Fall 2008
Dene
i
W (c1 , c2 , l1 , l2 , 1 , 2 ) = U 1 (c1 , l1 ) + i Uc ci + Uli li
First order conditions
2
2
t Ucct c2t + Uct + t = 0
t = t+1 (1 + Fkt+1 )
in steady state t+1 = t and therefore
1 = (1 + Fkt+1 )
and again, tax o
ECN 741: Public Economics
Fall 2008
Next we check that under these allocations, the I.C. constraints for type L is slack.
v
y (H )
L
v
y (H )
y (L )
L
y
L
v
=
y (L )
y (H )
v
dy >
y (L )
y
H
dy
y (H )
y (L )
v
H
H
= u(c(H ) u(c(L )
=v
rearrange terms
u(c(
ECN 741: Public Economics
Fall 2008
Note that this is the FOC with respect to a ct (T ) at a particular draw T . But we know
that ct (T ) is t measurable, therefore we dont need to sum over all T D, but only
those that contain the particular history t
(T
ECN 741: Public Economics
1.2.1
Fall 2008
Additive separable utility functions
Suppose U is of the form
U (c1 , c2 , l) = u1 (c1 ) + u2 (c2 ) v (l)
then
Hi =
Uii ci
Ui
Our goal to relate Hi to income elasticity of demand for good i. In order to do that,